Mathematics Commons^™

An Exposition Of Kasteleyn's Solution Of The Dimer Model, Eric Stucky

HMC Senior Theses

In 1961, P. W. Kasteleyn provided a baffling-looking solution to an apparently simple tiling problem: how many ways are there to tile a rectangular region with dominos? We examine his proof, simplifying and clarifying it into this nearly self-contained work.

Toeplitz Determinants With Perturbations In The Corners, Albrecht Böttcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj

Pomona Faculty Publications and Research

This paper is devoted to exact and asymptotic formulas for the determinants of Toeplitz matrices with perturbations by blocks of fixed size in the four corners. If the norms of the inverses of the unperturbed matrices remain bounded as the matrix dimension goes to infinity, then standard perturbation theory yields asymptotic expressions for the perturbed determinants. This premise is not satisfied for matrices generated by so-called Fisher-Hartwig symbols. In that case we establish formulas for pure single Fisher-Hartwig singularities and for the Hermitian matrices induced by general Fisher-Hartwig symbols.

One-Bit Compressive Sensing With Partial Support, Phillip North, Deanna Needell

CMC Faculty Publications and Research

The Compressive Sensing framework maintains relevance even when the available measurements are subject to extreme quantization, as is exemplified by the so-called one-bit compressed sensing framework which aims to recover a signal from measurements reduced to only their sign-bit. In applications, it is often the case that we have some knowledge of the structure of the signal beforehand, and thus would like to leverage it to attain more accurate and efficient recovery. This work explores avenues for incorporating such partial support information into the one-bit setting. Experimental results demonstrate that newly proposed methods of this work yield improved signal recovery …

Permutation Invariant Lattices, Lenny Fukshansky, Stephan Ramon Garcia, Xun Sun

CMC Faculty Publications and Research

We say that a Euclidean lattice in Rn is permutation invariant if its automorphism group has non-trivial intersection with the symmetric group Sn, i.e., if the lattice is closed under the action of some non-identity elements of Sn. Given a fixed element τ ∈ Sn, we study properties of the set of all lattices closed under the action of τ: we call such lattices τ-invariant. These lattices naturally generalize cyclic lattices introduced by Micciancio in [8, 9], which we previously studied in [1]. Continuing our investigation, we discuss some basic properties of permutation invariant lattices, in particular proving that the …

An Exhibition Of Exponential Sums: Visualizing Supercharacters, Paula Burkhardt '16, Gabriel Currier '16, Stephan Ramon Garcia, Mathieu De Langis '15, Bob Lutz '13, Hong Suh '16

Pomona Faculty Publications and Research

We discuss a simple mathematical mechanism that produces a variety of striking images of great complexity and subtlety. We briefly explain this approach and present a selection of attractive images obtained using this technique.

Compressive Sensing With Redundant Dictionaries And Structured Measurements, Felix Krahmer, Deanna Needell, Rachel Ward

CMC Faculty Publications and Research

Consider the problem of recovering an unknown signal from undersampled measurements, given the knowledge that the signal has a sparse representation in a specified dictionary D. This problem is now understood to be well-posed and efficiently solvable under suitable assumptions on the measurements and dictionary, if the number of measurements scales roughly with the sparsity level. One sufficient condition for such is the D-restricted isometry property (D-RIP), which asks that the sampling matrix approximately preserve the norm of all signals which are sufficiently sparse in D. While many classes of random matrices are known to satisfy such conditions, such matrices …

On Lattices Generated By Finite Abelian Groups, Albrecht Böttcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj

CMC Faculty Publications and Research

This paper is devoted to the study of lattices generated by finite Abelian groups. Special species of such lattices arise in the exploration of elliptic curves over finite fields. In the case where the generating group is cyclic, they are also known as the Barnes lattices. It is shown that for every finite Abelian group with the exception of the cyclic group of order four these lattices have a basis of minimal vectors. Another result provides an improvement of a recent upper bound by M. Sha for the covering radius in the case of the Barnes lattices. Also discussed are …

A Combinatorial Exploration Of Elliptic Curves, Matthew Lam

HMC Senior Theses

At the intersection of algebraic geometry, number theory, and combinatorics, an interesting problem is counting points on an algebraic curve over a finite field. When specialized to the case of elliptic curves, this question leads to a surprising connection with a particular family of graphs. In this document, we present some of the underlying theory and then summarize recent results concerning the aforementioned relationship between elliptic curves and graphs. A few results are additionally further elucidated by theory that was omitted in their original presentation.

Recovery Techniques For Finite Element Methods And Their Applications, Hailong Guo

Wayne State University Dissertations

Recovery techniques are important post-processing methods to obtain improved approximate solutions from primary data with reasonable cost. The practical us- age of recovery techniques is not only to improve the quality of approximation, but also to provide an asymptotically exact posteriori error estimators for adaptive meth- ods. This dissertation presents recovery techniques for nonconforming finite element methods and high order derivative as well as applications of gradient recovery.

Our first target is to develop a systematic gradient recovery technique for Crouzeix- Raviart element. The proposed method uses finite element solution to build a better approximation of the exact gradient based …

Dynamics Of Climate Change: Explaining Glacier Retreat Mathematically, Robert Guillette

Undergraduate Review

Climate change is an important topic that has become extremely relevant this day and age. The world’s climate is undergoing monumental shifts with over two-thirds of the estimated 150 glaciers existing in 1850 disappearing by 1980. The melting of glaciers offers tangible evidence of broader environmental changes as they respond directly to long-term trends in temperature, precipitation, and solar radiation. Since the study of glacier retreat provides a barometer of climate change, it is important to better understand the effects of climatic factors on glaciers. In my project I created a mathematical model for the melting of glaciers and used …

The Relaxed Edge-Coloring Game And K-Degenerate Graphs, Charles Dunn, David Morawski, Jennifer Firkins Nordstrom

Faculty Publications

The (r, d)-relaxed edge-coloring game is a two-player game using r colors played on the edge set of a graph G. We consider this game on forests and more generally, on k-degenerate graphs. If F is a forest with ∆(F) = ∆, then the first player, Alice, has a winning strategy for this game with r = ∆ − j and d ≥ 2j + 2 for 0 ≤ j ≤ ∆ − 1. This both improves and generalizes the result for trees in [10]. More broadly, we generalize the main result in [10] …

Combinatorial Potpourri: Permutations, Products, Posets, And Pfaffians, Norman B. Fox

Theses and Dissertations--Mathematics

In this dissertation we first examine the descent set polynomial, which is defined in terms of the descent set statistics of the symmetric group. Algebraic and topological tools are used to explain why large classes of cyclotomic polynomials are factors of the descent set polynomial. Next the diamond product of two Eulerian posets is studied, particularly by examining the effect this product has on their cd-indices. A combinatorial interpretation involving weighted lattice paths is introduced to describe the outcome of applying the diamond product operator to two cd-monomials. Then the cd-index is defined for infinite posets, with …

New Characterizations Of Sobolev Spaces On Heisenberg And Carnot Groups And High Order Sobolev Spaces On Eucliean Spaces, Xiaoyue Cui

Wayne State University Dissertations

This dissertation focuses on new characterizations of Sobolev spaces .

It encompasses an in-depth study of Sobolev spaces on Heisenberg groups, as well as Carnot groups, second order and high order Sobolev spaces on Euclidean spaces.

On The Theory Of Zeta-Functions And L-Functions, Almuatazbellah Awan

Electronic Theses and Dissertations

In this thesis we provide a body of knowledge that concerns Riemann zeta-function and its generalizations in a cohesive manner. In particular, we have studied and mentioned some recent results regarding Hurwitz and Lerch functions, as well as Dirichlet's L-function. We have also investigated some fundamental concepts related to these functions and their universality properties. In addition, we also discuss different formulations and approaches to the proof of the Prime Number Theorem and the Riemann Hypothesis. These two topics constitute the main theme of this thesis. For the Prime Number Theorem, we provide a thorough discussion that compares and contrasts …

Finite-Difference Approximations And Optimal Control Of Differential Inclusions, Yuan Tian

Wayne State University Dissertations

This dissertation concerns the study of the generalized Bolza type problem for dynamic systems governed by constrained differential inclusions. We develop finite-discrete approximations of differential inclusions by using the implicit Euler scheme and the Runge-Kutta scheme for approximating time derivatives, while an appropriate well-posedness of such approximations is justified. Our principal result establishes the uniform approximation of strong local minimizers for the continuous-time Bolza problem by optimal solutions to the corresponding discretized finite-difference systems by the strengthen $W^{1,2}$-norm approximation of this type in the case ``intermediate" (between strong and weak minimizers) local minimizers under additional assumptions. Especially the implicitly discrete …

A Remark On Estimating The Mean Of A Normal Distribution With Known Coefficient Of Variation, Rasul A. Khan

Mathematics and Statistics Faculty Publications

Let X1, X2, …, Xn be iid N(μ, aμ2) (a>0) random variables with an unknown mean μ>0 and known coefficient of variation (CV) √a. The estimation of μ is revisited and it is shown that a modified version of an unbiased estimator of μ [cf. Khan RA. A note on estimating the mean of a normal distribution with known CV. J Am Stat Assoc. 1968;63:1039–1041] is more efficient. A certain linear minimum mean square estimator of Gleser and Healy [Estimating the mean of a normal distribution with known CV. J Am Stat Assoc. 1976;71:977–981] is also modified and …

Lattice Polytopes In Coding Theory, Ivan Soprunov

Mathematics and Statistics Faculty Publications

In this paper we discuss combinatorial questions about lattice polytopes motivated by recent results on minimum distance estimation for toric codes. We also prove a new inductive bound for the minimum distance of generalized toric codes. As an application, we give new formulas for the minimum distance of generalized toric codes for special lattice point configurations.

Generation And Validation Of Optimal Topologies For Solid Freeform Fabrication, Purnajyoti Bhaumik

Masters Theses

"The study of fabricating topologically optimized parts is presented hereafter. The mapping of topology optimization results for Standard Tessellation Language (STL) writing would enable the solid freeform fabrication of lightweight mechanisms. Aerospace leaders such as NASA, Boeing, Airbus, European Aeronautic Defense And Space Company (EADS), and GE Aero invest in topology optimization research for the production of lightweight materials. Certain concepts such as microstructural homogenization, discretization, and mapping are reviewed and presented in the context of topology optimization. Future biomedical applications of solid freeform fabrication such as organ printing stand to save millions of lives through the robust development of …

Day Of The Week Effect In Returns And Volatility Of The S&P 500 Sector Indices, Juan Liu

Masters Theses

"Previous studies have shown that returns associated with the stock market or foreign exchange's futures show variations across the day of the week. On such study, that employs a modified GARCH model for estimation, shows that returns associated with the S&P 500 stock index is highest on Wednesday and lowest returns on Monday. The same study shows that volatility is highest on Fridays and lowest on Wednesdays. In this study we investigate if this day-of-the-week effect on returns and volatility is present in the different sectors that constitute the S&P 500 index. The data set used provides daily returns from …

Vassiliev Invariants Of Virtual Legendrian Knots, Patricia Cahn, Asa Levi

Mathematics Sciences: Faculty Publications

We introduce a theory of virtual Legendrian knots. A virtual Legendrian knot is a cooriented wavefront on an oriented surface up to Legendrian isotopy of its lift to the unit cotangent bundle and stabilization and destabilization of the surface away from the wavefront. We show that the groups of Vassiliev invariants of virtual Legendrian knots and of virtual framed knots are isomorphic. In particular, Vassiliev invariants cannot be used to distinguish virtual Legendrian knots that are isotopic as virtual framed knots and have equal virtual Maslov numbers.