Physical Sciences and Mathematics Commons^™

A New Subgroup Chain For The Finite Affine Group, David Alan Lingenbrink Jr.

HMC Senior Theses

The finite affine group is a matrix group whose entries come from a finite field. A natural subgroup consists of those matrices whose entries all come from a subfield instead. In this paper, I will introduce intermediate sub- groups with entries from both the field and a subfield. I will also examine the representations of these intermediate subgroups as well as the branch- ing diagram for the resulting subgroup chain. This will allow us to create a fast Fourier transform for the group that uses asymptotically fewer opera- tions than the brute force algorithm.

Experimental Realization Of Slowly Rotating Modes Of Light, Fangzhao A. An

HMC Senior Theses

Beams of light can carry spin and orbital angular momentum. Spin angular momentum describes how the direction of the electric field rotates about the propagation axis, while orbital angular momentum describes the rotation of the field amplitude pattern. These concepts are well understood for monochromatic beams, but previous theoretical studies have constructed polychromatic superpositions where the connection between angular momentum and rotation of the electric field becomes much less clear. These states are superpositions of two states of light carrying opposite signs of angular momentum and slightly detuned frequencies. They rotate at the typically small detuning frequency and thus we …

Arithmetical Graphs, Riemann-Roch Structure For Lattices, And The Frobenius Number Problem, Jeremy Usatine

HMC Senior Theses

If R is a list of positive integers with greatest common denominator equal to 1, calculating the Frobenius number of R is in general NP-hard. Dino Lorenzini defines the arithmetical graph, which naturally arises in arithmetic geometry, and a notion of genus, the g-number, that in specific cases coincides with the Frobenius number of R. A result of Dino Lorenzini's gives a method for quickly calculating upper bounds for the g-number of arithmetical graphs. We discuss the arithmetic geometry related to arithmetical graphs and present an example of an arithmetical graph that arises in this context. We also discuss the …

Characterizing Forced Communication In Networks, Samuel C. Gutekunst

HMC Senior Theses

This thesis studies a problem that has been proposed as a novel way to disrupt communication networks: the load maximization problem. The load on a member of a network represents the amount of communication that the member is forced to be involved in. By maximizing the load on an important member of the network, we hope to increase that member's visibility and susceptibility to capture. In this thesis we characterize load as a combinatorial property of graphs and expose possible connections between load and spectral graph theory. We specifically describe the load and how it changes in several canonical classes …

Reed's Conjecture And Cycle-Power Graphs, Alexa Serrato

HMC Senior Theses

Reed's conjecture is a proposed upper bound for the chromatic number of a graph. Reed's conjecture has already been proven for several families of graphs. In this paper, I show how one of those families of graphs can be extended to include additional graphs and also show that Reed's conjecture holds for a family of graphs known as cycle-power graphs, and also for their complements.

There And Back Again: Elliptic Curves, Modular Forms, And L-Functions, Allison F. Arnold-Roksandich

HMC Senior Theses

L-functions form a connection between elliptic curves and modular forms. The goals of this thesis will be to discuss this connection, and to see similar connections for arithmetic functions.

Infinitely Many Rotationally Symmetric Solutions To A Class Of Semilinear Laplace-Beltrami Equations On The Unit Sphere, Emily M. Fischer

HMC Senior Theses

I show that a class of semilinear Laplace-Beltrami equations has infinitely many solutions on the unit sphere which are symmetric with respect to rotations around some axis. This equation corresponds to a singular ordinary differential equation, which we solve using energy analysis. We obtain a Pohozaev-type identity to prove that the energy is continuously increasing with the initial condition and then use phase plane analysis to prove the existence of infinitely many solutions.

A Mathematical Framework For Unmanned Aerial Vehicle Obstacle Avoidance, Sorathan Chaturapruek

HMC Senior Theses

The obstacle avoidance navigation problem for Unmanned Aerial Vehicles (UAVs) is a very challenging problem. It lies at the intersection of many fields such as probability, differential geometry, optimal control, and robotics. We build a mathematical framework to solve this problem for quadrotors using both a theoretical approach through a Hamiltonian system and a machine learning approach that learns from human sub-experts' multiple demonstrations in obstacle avoidance. Prior research on the machine learning approach uses an algorithm that does not incorporate geometry. We have developed tools to solve and test the obstacle avoidance problem through mathematics.

Experiments On Surfactants And Thin Fluid Films, Peter Megson

HMC Senior Theses

We investigate the spatiotemporal dynamics of a surfactant monolayer on a thin fluid film spreading inward into a region devoid of surfactant, a system motivated by the alveolus of the human lung. We perform experiments that simultaneously measure the fluid height profile and the fluorescence intensity due to our fluorescent surfactant, NBD-PC. We perform experiments on both a Newtonian layer of glycerol and a shear-thinning fluid layer consisting of xanthan gum mixed with glycerol. We can very successfully extract height profiles on the xanthan gum fluid, although the simultaneous measurement of fluorescent intensity profiles proved problematic, as the laser tended …

Cycle Lengths Of Θ-Biased Random Permutations, Tongjia Shi

HMC Senior Theses

Consider a probability distribution on the permutations of n elements. If the probability of each permutation is proportional to θ^K, where K is the number of cycles in the permutation, then we say that the distribution generates a θ-biased random permutation. A random permutation is a special θ-biased random permutation with θ = 1. The m^th moment of the r^th longest cycle of a random permutation is Θ(n^m), regardless of r and θ. The joint moments are derived, and it is shown that the longest cycles of a permutation can either be positively or …

Energy-Driven Pattern Formation In Planar Dipole-Dipole Systems, Jaron P. Kent-Dobias

HMC Senior Theses

A variety of two-dimensional fluid systems, known as dipole-mediated systems, exhibit a dipole-dipole interaction between their fluid constituents. The com- petition of this repulsive dipolar force with the cohesive fluid forces cause these systems to form intricate and patterned structures in their boundaries. In this thesis, we show that the microscopic details of any such system are irrelevant in the macroscopic limit and contribute only to a constant offset in the system’s energy. A numeric model is developed, and some important stable domain morphologies are characterized. Previously unresolved bifurcating branches are explored. Finally, by applying a random energy background to …

Knotted And Linked Products Of Recombination On T(2,N)#T(2,M) Substrates, Erica Flapan, Jeremy Grevet, Qi Li, Chen Daisy Sun, Helen Wong

Pomona Faculty Publications and Research

We develop a topological model of site-specific recombination that applies to substrates which are the connected sum of two torus links of the form T(2,n)#T(2,m). Then we use our model to prove that all knots and links that can be produced by site-specific recombination on such substrates are contained in one of two families, which we illustrate.

An Extremal Problem For Characteristic Functions, Stephan Ramon Garcia, Isabelle Chalendar, Williams T. Ross, Dan Timotin

Pomona Faculty Publications and Research

Suppose E is a subset of the unit circle T and H^infinity C L^infinity is the Hardy subalgebra. We examine the problem of finding the distance from the characteristic function of E to zⁿH^infinity. This admits an alternate description as a dual extremal problem. Precise solutions are given in several important cases. The techniques used involve the theory of Toeplitz and Hankel operators as well as the construction of certain conformal mappings.

A Look Into The Industry Of Video Games Past, Present, And Yet To Come, Chad Hadzinsky

CMC Senior Theses

Since its inception, the video game industry has been both a new medium for art and innovation as well as a major driving force in the advancements of many technologies. The often overlooked video game industry has turned from a hobby to a multi-billion dollar industry in its short, forty year life. People of all ages and genders across the world are playing video games at a higher clip than ever before. With so many new gamers and emerging technologies, it is an exciting time for the industry. The landscape is constantly changing and successful business models of the past …

Finding Zeros Of Rational Quadratic Forms, John F. Shaughnessy

CMC Senior Theses

In this thesis, we introduce the notion of quadratic forms and provide motivation for their study. We begin by discussing Diophantine equations, the field of p-adic numbers, and the Hasse-Minkowski Theorem that allows us to use p-adic analysis determine whether a quadratic form has a rational root. We then discuss search bounds and state Cassels' Theorem for small-height zeros of rational quadratic forms. We end with a proof of Cassels' Theorem and suggestions for further reading.

Block Kaczmarz Method With Inequalities, Jonathan Briskman

CMC Senior Theses

The Kaczmarz method is an iterative algorithm that solves overdetermined systems of linear equalities. This paper studies a system of linear equalities and inequalities. We use the block version of the Kaczmarz method applied towards the equalities with the simple randomized Kaczmarz scheme for the inequalities. This primarily involves combining Needell and Tropp's work on the block Kaczmarz method with the application of a randomized Kaczmarz approach towards a system of equalities and inequalities performed by Leventhal and Lewis. We give an expected linear rate of convergence for this kind of system and find that using the block Kaczmarz scheme …

Scalable Collaborative Filtering Recommendation Algorithms On Apache Spark, Walker Evan Casey

CMC Senior Theses

Collaborative filtering based recommender systems use information about a user's preferences to make personalized predictions about content, such as topics, people, or products, that they might find relevant. As the volume of accessible information and active users on the Internet continues to grow, it becomes increasingly difficult to compute recommendations quickly and accurately over a large dataset. In this study, we will introduce an algorithmic framework built on top of Apache Spark for parallel computation of the neighborhood-based collaborative filtering problem, which allows the algorithm to scale linearly with a growing number of users. We also investigate several different variants …

Changing The Climate Narrative: How A Long-Term Climate Change Might Save Our Lives, Natalie P. Harreld

CMC Senior Theses

The goal of this paper is to offer new insights into the climate change debate by shifting away from the heated anthropologic arguments that dominate politics, media, and popular science. Instead, I choose to rely on the long-term impacts of a changing climate on our planet. The paper begins with a break down of key processes involved in short-term and long-term climate change, using the latest research. After a foundational understanding of climate sciences is established, we will discuss the failure of the climate change debate in educating the general public about the facts of a changing climate. Finally, the …

Colormoo: An Algorithmic Approach To Generating Color Palettes, Joshua Rael

CMC Senior Theses

Selecting one color can be done with relative ease, but this task becomes more difficult with each subsequent color. Colormoo is an online tool aimed at solving this problem. We implement three algorithms for generating color palettes based off of a starting color. Data is collected for each palette that is generated. Our analysis reveals two of the algorithms are preferred, but under different circumstances. Furthermore, we find that users prefer palettes containing colors that are compatible, but not too similar. With refined heuristics, we believe these techniques can be extended and applied beyond the field of graphic design alone.

Cultivating Human-Nature Relationships: The Role Of Parents And Primary Caregivers In Development Of Environmental Identity, Anne E. Bremer

Pitzer Senior Theses

Industrialized societies have been characterized by a trend of disconnecting humans from our natural environment, leading to environmental and psychological damage. Therefore, in order to work toward repairing such damage, reconnecting humans and the natural environment is critical. One way of conceptualizing human-nature relationships is through “environmental identity,” a term that describes self-identification as part of a larger ecosystem, aesthetic, spiritual, or recreational enjoyment of nature, environmentally positive behaviors, and a social, political, or moral identification with environmentalists. Despite the literature having emphasized childhood experience in nature as being essential to the development of an environmental identity, parental influence in …

A Discussion Of The Impact Of Political And Economic Forces On Equitable Access To Potable Water In Ecuador And Recommendations For Improvement Through Better Watershed Management, Eliza States

Pitzer Senior Theses

This thesis will address the impact of political and economic forces on the equitable access to fresh water in Ecuador. Demographic factors such as the rural-to-urban migration and the political and economic forces have strongly influenced the debate over the privatization of the provision of potable water and sanitation services. Within the context of Ecuador, two different approaches by the largest cities, Quito and Guayaquil, are analyzed; in Guayaquil, the services were privatized, while in Quito, the public utility was corporatized, remaining under public control. It concludes arguing that in the face of political instability and a lack of regulatory …

“It’S Like We Are Free”: An Analysis Of Soccer-Based Programming In A California Prison, Lillian S. Barrett-O'Keefe

Pitzer Senior Theses

The concept of space goes well beyond just buildings and infrastructure; it can represent feelings of attachment and belonging, it can interact with us and generate meaning. The built environment is not just the “backdrop” of our lives, but rather it plays a major role in them. In the state of California prisons have become a prominent element of our communal landscape, now housing 2.4 million Americans today. This paper explores prisons as a rich site of analysis in terms of how our built environment affects our daily lives. In order to delve into this analysis, I will explore sport-based …

Missing Voices, Hidden Fields: The Gendered Struggles Of Female Farmworkers, Keiko A. Budech

Pitzer Senior Theses

Known for its fertile soil and ideal climate, California has been one of the most agriculturally productive areas in the world. Often left out of this picture are the farmworkers who make it possible. Within this farmworker community, females are a sub-class that has been even more marginalized. This thesis investigates the gendered aspects of fieldwork and exposes female leadership working towards changing these specific struggles, such as sexual harassment in the fields, domestic abuse, pesticide exposure, and the perpetuation of submissive gender roles in the household and workplace. An in-depth case study of Lideres Campesinas, a community- based grassroots …

Extended Book Review: Mathematics In Popular Culture: Essays On Appearances In Film, Fiction, Games, Television And Other Media, Edited By Jessica K. Sklar And Elizabeth S. Sklar; Loving+Hating Mathematics: Challenging The Myths Of Mathematical Life, By Reuben Hersh And Vera John-Steiner; Mathematicians: An Outer View Of The Inner World, By Mariana Cook, Gizem Karaali

Pomona Faculty Publications and Research

I was delighted to have the opportunity to review three books on a topic near and dear to my heart. In recent years it has become a passion of mine to think of and speak about the place of mathematics in the real world, in the world of those who are not doing mathematics for a living. I care about the applications and the implications of mathematics, but more than that, I care about the feelings and the impressions attached to it. Often math anxiety or skepticism comes up; the latter may be due to how frequently others (mis)use statistics, …

Quotients Of Gaussian Primes, Stephan Ramon Garcia

Pomona Faculty Publications and Research

It has been observed many times, both in the Monthly and elsewhere, that the set of all quotients of prime numbers is dense in the positive real numbers. In this short note we answer the related question: "Is the set of all quotients of Gaussian primes dense in the complex plane?"

Kaczmarz Algorithm With Soft Constraints For User Interface Layout, Noreen Jamil, Deanna Needell, Johannes Muller, Christof Lutteroth, Gerald Weber

CMC Faculty Publications and Research

The Kaczmarz method is an iterative method for solving large systems of equations that projects iterates orthogonally onto the solution space of each equation. In contrast to direct methods such as Gaussian elimination or QR-factorization, this algorithm is efficient for problems with sparse matrices, as they appear in constraint-based user interface (UI) layout specifications. However, the Kaczmarz method as described in the literature has its limitations: it considers only equality constraints and does not support soft constraints, which makes it inapplicable to the UI layout problem.

In this paper we extend the Kaczmarz method for solving specifications containing soft constraints, …

Two-Part Reconstruction In Compressed Sensing, Yanting Ma, Dror Baron, Deanna Needell

CMC Faculty Publications and Research

Two-part reconstruction is a framework for signal recovery in compressed sensing (CS), in which the advantages of two different algorithms are combined. Our framework allow s to accelerate the reconstruction procedure without compromising the reconstruction quality. To illustrate the efficacy of ou r two-part approach, we extend the author’s previous Sudocodes algorithm and make it robust to measurement noise. In a 1- bit CS setting, promising numerical results indicate that our algorithm offers both a reduction in run-time and improvement in reconstruction quality

Review: Unitary Equivalence To Truncated Toeplitz Operators, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

Book Review: Encyclopedia Of Mathematics And Society, Gizem Karaali

Pomona Faculty Publications and Research

The Encyclopedia of Mathematics and Society is an impressive achievement of collective effort and serious thought, an amazing collection of delightful, unexpected essays, a sourcebook for students and teachers alike. The experience of reading the EMS was, for this encyclopedia enthusiast, enlightening and enjoyable.

If any librarians out there are still wondering, let me be clear: I strongly recommend this encyclopedia. For individuals, the expense might well be an initial turn-off; keeping in mind that the hard copy books come with online access to the same content might alleviate some of the pain.

As for me? I expect to enjoy …

Equation, Nilanjan De

Journal of Humanistic Mathematics

No abstract provided.