Physical Sciences and Mathematics Commons^™

Line-Of-Sight Pursuit And Evasion Games On Polytopes In R^N, John Phillpot

HMC Senior Theses

We study single-pursuer, line-of-sight Pursuit and Evasion games in polytopes in $\mathbb{R}^n$. We develop winning Pursuer strategies for simple classes of polytopes (monotone prisms) in Rⁿ, using proven algorithms for polygons as inspiration and as subroutines. More generally, we show that any Pursuer-win polytope can be extended to a new Pursuer-win polytope in more dimensions. We also show that some more general classes of polytopes (monotone products) do not admit a deterministic winning Pursuer strategy. Though we provide bounds on which polytopes are Pursuer-win, these bounds are not tight. Closing the gap between those polytopes known to be …

Graph Cohomology, Matthew Lin

HMC Senior Theses

What is the cohomology of a graph? Cohomology is a topological invariant and encodes such information as genus and euler characteristic. Graphs are combinatorial objects which may not a priori admit a natural and isomorphism invariant cohomology ring. In this project, given any finite graph G, we constructively define a cohomology ring H*(G) of G. Our method uses graph associahedra and toric varieties. Given a graph, there is a canonically associated convex polytope, called the graph associahedron, constructed from G. In turn, a convex polytope uniquely determines a toric variety. We synthesize these results, and describe the …

Interval Graphs, Joyce C. Yang

HMC Senior Theses

We examine the problem of counting interval graphs. We answer the question posed by Hanlon, of whether the formal power series generating function of the number of interval graphs on n vertices has a positive radius of convergence. We have found that it is zero. We have obtained a lower bound and an upper bound on the number of interval graphs on n vertices. We also study the application of interval graphs to the dynamic storage allocation problem. Dynamic storage allocation has been shown to be NP-complete by Stockmeyer. Coloring interval graphs on-line has applications to dynamic storage allocation. The …

Fibonomial Tilings And Other Up-Down Tilings, Robert Bennett

HMC Senior Theses

The Fibonomial coefficients are a generalization of the binomial coefficients with a rather nice combinatorial interpretation. While the ordinary binomial coefficients count lattice paths in a grid, the Fibonomial coefficients count the number of ways to draw a lattice path in a grid and then Fibonacci-tile the regions above and below the path in a particular way. We may forgo a literal tiling interpretation and, instead of the Fibonacci numbers, use an arbitrary function to count the number of ways to "tile" the regions of the grid delineated by the lattice path. When the function is a combinatorial sequence such …

Adinkras And Arithmetical Graphs, Madeleine Weinstein

HMC Senior Theses

Adinkras and arithmetical graphs have divergent origins. In the spirit of Feynman diagrams, adinkras encode representations of supersymmetry algebras as graphs with additional structures. Arithmetical graphs, on the other hand, arise in algebraic geometry, and give an arithmetical structure to a graph. In this thesis, we will interpret adinkras as arithmetical graphs and see what can be learned.

Our work consists of three main strands. First, we investigate arithmetical structures on the underlying graph of an adinkra in the specific case where the underlying graph is a hypercube. We classify all such arithmetical structures and compute some of the corresponding …

Convexity Of Neural Codes, Robert Amzi Jeffs

HMC Senior Theses

An important task in neuroscience is stimulus reconstruction: given activity in the brain, what stimulus could have caused it? We build on previous literature which uses neural codes to approach this problem mathematically. A neural code is a collection of binary vectors that record concurrent firing of neurons in the brain. We consider neural codes arising from place cells, which are neurons that track an animal's position in space. We examine algebraic objects associated to neural codes, and completely characterize a certain class of maps between these objects. Furthermore, we show that such maps have natural geometric implications related to …

Hopper Bands: Locust Aggregation, Ryan C. Jones

HMC Senior Theses

Locust swarms cause famine and hunger in parts of Sub-Saharan Africa as they travel across croplands and eat vegetation. Current models start with biological properties of locusts and analyze the macroscopic behavior of the system. These models exhibit the desired migratory behavior, but do so with too many parameters. To account for this, a new model, the Alignment and Intermittent Motion (AIM) model, is derived with minimal assumptions. AIM is constructed with regards to locust biology, allowing it to elicit biologically correct locust behavior: the most noteworthy being the fingering of hopper bands. A Particle-in-Cell method is used to optimize …

Visual Properties Of Generalized Kloosterman Sums, Paula Burkhardt '16, Alice Zhuo-Yu Chan '14, Gabriel Currier '16, Stephan Ramon Garcia, Florian Luca, Hong Suh '16

Pomona Faculty Publications and Research

For a positive integer m and a subgroup A of the unit group (Z/mZ)^x, the corresponding generalized Kloosterman sum is the function K(a, b, m, A) = Σ_uEA e(au+bu^-1/m). Unlike classical Kloosterman sums, which are real valued, generalized Kloosterman sums display a surprising array of visual features when their values are plotted in the complex plane. In a variety of instances, we identify the precise number-theoretic conditions that give rise to particular phenomena.

Lattices From Hermitian Function Fields, Albrecht Böttcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj

Pomona Faculty Publications and Research

We consider the well-known Rosenbloom-Tsfasman function field lattices in the special case of Hermitian function fields. We show that in this case the resulting lattices are generated by their minimal vectors, provide an estimate on the total number of minimal vectors, and derive properties of the automorphism groups of these lattices. Our study continues previous investigations of lattices coming from elliptic curves and finite Abelian groups. The lattices we are faced with here are more subtle than those considered previously, and the proofs of the main results require the replacement of the existing linear algebra approaches by deep results of …

Best Approximations, Lethargy Theorems And Smoothness, Caleb Case

CMC Senior Theses

In this paper we consider sequences of best approximation. We first examine the rho best approximation function and its applications, through an example in approximation theory and two new examples in calculating n-widths. We then further discuss approximation theory by examining a modern proof of Weierstrass's Theorem using Dirac sequences, and providing a new proof of Chebyshev's Equioscillation Theorem, inspired by the de La Vallee Poussin Theorem. Finally, we examine the limits of approximation theorem by looking at Bernstein Lethargy theorem, and a modern generalization to infinite-dimensional subspaces. We all note that smooth functions are bounded by Jackson's Inequalities, but …

Bikei Cohomology And Counting Invariants, Jake L. Rosenfield

CMC Senior Theses

This paper gives a brief introduction into the fundaments of knot theory: introducing knot diagrams, knot invariants, and two techniques to determine whether or not two knots are ambient isotopic. After discussing the basics of knot theory an algebraic coloring of knots knows as a bikei is introduced. The algebraic structure as well as the various axioms that define a bikei are defined. Furthermore, an extension between the Alexander polynomial of a knot and the Alexander Bikei is made. The remainder of the paper is devoted to reintroducing a modified homology and cohomology theory for involutory biquandles known as bikei, …

Topic Analysis Of Tweets On The European Refugee Crisis Using Non-Negative Matrix Factorization, Chong Shen

CMC Senior Theses

The ongoing European Refugee Crisis has been one of the most popular trending topics on Twitter for the past 8 months. This paper applies topic modeling on bulks of tweets to discover the hidden patterns within these social media discussions. In particular, we perform topic analysis through solving Non-negative Matrix Factorization (NMF) as an Inexact Alternating Least Squares problem. We accelerate the computation using techniques including tweet sampling and augmented NMF, compare NMF results with different ranks and visualize the outputs through topic representation and frequency plots. We observe that supportive sentiments maintained a strong presence while negative sentiments such …

Monte Carlo Approx. Methods For Stochastic Optimization, John Fowler

Pomona Senior Theses

This thesis provides an overview of stochastic optimization (SP) problems and looks at how the Sample Average Approximation (SAA) method is used to solve them. We review several applications of this problem-solving technique that have been published in papers over the last few years. The number and variety of the examples should give an indication of the usefulness of this technique. The examples also provide opportunities to discuss important aspects of SPs and the SAA method including model assumptions, optimality gaps, the use of deterministic methods for finite sample sizes, and the accelerated Benders decomposition algorithm. We also give a …

How Can We Have A Better Public Transportation System? –An Exploratory Agent Based Model, Boyu Liu

Pomona Senior Theses

Public transportation plays an integral part in a city's development, but transportation professionals disagree about whether it is feasible to increase the capacity of public transportation systems at a reasonable cost; and if it is, how. This study develops an agent based model that aims to answer this question and provide a framework to compare the effects of improvements in different aspects of the public transportation service. The results of this study show that it is possible to increase ridership enough to compensate for the increased operational cost, but only in certain circumstances. Interesting phenomenon that might have showed up …

Towards A New Gis Maturity Model: An Organizational Usage Perspective, Omer Abdulaziz Alrwais

CGU Theses & Dissertations

The first condition required for an Information Technology (IT) system to produce value is that it be used by its designated target group of users. Despite the prevalence of “system use” in IS literature, it has been often limited to the individual level. The organizational perspective is rarely considered. This dissertation focuses on system usage in the GIS domain through an organizational lens. GIS is a technology with the potential to transform government by enhancing business processes and providing a platform to manage spatial and non-spatial data, which is expected to result in better decision-making. However, little is known about …

Exploration Of Curvature Through Physical Materials, Lucinda-Joi Chu-Ketterer

Pitzer Senior Theses

Parametric equations are commonly used to describe surfaces. Looking at parametric equations does not provide tangible information about an object. Thus through the use of physical materials, an understanding of the limitations of the materials allows someone to gain a broader understanding of the surface. A M$\ddot{o}$bius strip and Figure 8 Klein bottle were created through knitting due to the precision and steady increase in curvature allowed through knitting. A more standard Klein bottle was created through crochet due to the ease in creating quick increases in curvature. Both methods demonstrate the change in curvature for both surfaces where the …

Review: A Short Introduction To De Branges-Rovnyak Spaces, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

A Note On Practical Approximate Projection Schemes In Signal Space Methods, Xiaoyi Gu, Deanna Needell, Shenyinying Tu

CMC Faculty Publications and Research

Compressive sensing (CS) is a new technology which allows the acquisition of signals directly in compressed form, using far fewer measurements than traditional theory dictates. Recently, many socalled signal space methods have been developed to extend this body of work to signals sparse in arbitrary dictionaries rather than orthonormal bases. In doing so, CS can be utilized in a much broader array of practical settings. Often, such approaches often rely on the ability to optimally project a signal onto a small number of dictionary atoms. Such optimal, or even approximate, projections have been difficult to derive theoretically. Nonetheless, it has …

Convergence Properties Of The Randomized Extended Gauss-Seidel And Kaczmarz Methods, Anna Ma, Deanna Needell, Aaditya Ramdas

CMC Faculty Publications and Research

The Kaczmarz and Gauss-Seidel methods both solve a linear system Xβ=y by iteratively refining the solution estimate. Recent interest in these methods has been sparked by a proof of Strohmer and Vershynin which shows the randomized Kaczmarz method converges linearly in expectation to the solution. Lewis and Leventhal then proved a similar result for the randomized Gauss-Seidel algorithm. However, the behavior of both methods depends heavily on whether the system is under or overdetermined, and whether it is consistent or not. Here we provide a unified theory of both methods, their variants for these different settings, and draw connections between …

Review: The Classical Hom-Yang-Baxter Equation And Hom-Lie Bialgebras, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

Small Molecule Inhibitor Design For Anaplastic Lymphoma Kinase Inhibition, Theodore D. Hansel, David J. Grabovsky

Interface Compendium of Student Work

The Anaplastic Lymphoma Kinase (ALK) gene has been linked to tumorigenesis in a number of human cancers, including anaplastic large cell lymphoma (ALCL) and neuroblastoma. While ALK mutations in ALCL and many other cancers occur as a result of gene fusions with wild type kinase domains, those in neuroblastoma stem from single nucleotide polymorphisms (SNPs) in the kinase domain. These lead to autophosphorylation and constitutive signaling by ALK for cell growth and division, ultimately causing cancer. Crizotinib, an ATP-competitive ALK inhibitor, has proven to be an effective inhibitor of both ALKWT and ALKMutant kinase domains, and is in the middle …

On Montel And Montel–Popoviciu Theorems In Several Variables, Asuman Güven Aksoy, Jose M. Almira

CMC Faculty Publications and Research

We present an elementary proof of a general version of Montel’s theorem in several variables which is based on the use of tensor product polynomial interpolation. We also prove a Montel-Popoviciu’s type theorem for functions f:R^d→Rf:Rd→R for d > 1. Furthermore, our proof of this result is also valid for the case d = 1, differing in several points from Popoviciu’s original proof. Finally, we demonstrate that our results are optimal.

L.A. River Project, Erin Payne

The STEAM Journal

A field note that reflected the artists' experience of the city and the making of art through an activity at the L.A. River.

Unseen Minutia, Sara Kapadia

The STEAM Journal

A brief thought on the unseen details.

Art, Math, And Physics; All About For, Chris Brownell, Steve Pauls

The STEAM Journal

Anish Kapoor’s public sculpture “Cloud Gate” and Frame of Reference.

Relationships, Michael Goodman

The STEAM Journal

No abstract provided.

The Threshold Of Hearing, Mike Wereski

The STEAM Journal

No abstract provided.

Ecoscience + Art Initiative: Designing A New Paradigm For College Education, Scholarship, And Service, Changwoo Ahn

The STEAM Journal

The paper presents a new initiative, EcoScience + Art, which blooms at George Mason University. The creator explains the background, history, and recent activities of the initiative, and also introduces an on-going special project called “The Rain Project”, a student participatory project to design, construct, and monitor a green infrastructure (i.e., floating wetland) for sustainable stormwater management on campus. The special project is geared to design and present a new paradigm to integrate college education, scholarship, and service. The relevance of the initiative and the special project to STEAM education is discussed.

Putting The Team In Steam: The Art Of Robot Making, Teresa Walker, David Thompson

The STEAM Journal

What began as a life partnership has evolved into an early learning STEAM team. Artist, David Thompson, uses science, technology, engineering and math, on a daily basis, to create robots and much more. Teresa Day Walker is an assistant professor of early childhood education. Through necessity they discovered their combined talents could be used to promote STEAM in early childhood. David and Teri teamed up to provide a nearly impromptu robot making demonstration for 100 kindergarten students. After reading their co-authored book, Robot Hide and Seek, both, clay based and 3-D modeled images were used to generate interest and …

Guidelines For Good Mathematical Writing, Francis Su

All HMC Faculty Publications and Research

Communicating mathematics well is an important part of doing mathematics. Many of us know from writing papers or giving talks that communicating effectively not only serves our audience but also clarifies and structures our own thinking. There is an art and elegance to good writing that every writer should strive for. And writing, as a work of art, can bring a person great personal satisfaction.

Within the MAA, we value exposition and mathematical communication. In this column, I’m sharing the advice I give my students to help them write well. There are more extensive treatments (e.g., see Paul Halmos’s How …