Physical Sciences and Mathematics Commons^™

The Future Of Ios Development: Evaluating The Swift Programming Language, Garrett Wells

CMC Senior Theses

Swift is a new programming language developed by Apple for creating iOS and Mac OS X applications. Intended to eventually replace Objective-C as Apple’s language of choice, Swift needs to convince developers to switch over to the new language. Apple has promised that Swift will be faster than Objective-C, as well as offer more modern language features, be very safe, and be easy to learn and use. In this thesis I test these claims by creating an iOS application entirely in Swift as well as benchmarking two different algorithms. I find that while Swift is faster than Objective-C, it does …

On The Characterization Of Prime Sets Of Polynomials By Congruence Conditions, Arvind Suresh

CMC Senior Theses

This project is concerned with the set of primes modulo which some monic, irreducible polynomial over the integers has a root, called the Prime Set of the polynomial. We completely characterise these sets for degree 2 polynomials, and develop sufficient machinery from algebraic number theory to show that if the Galois group of a monic, irreducible polynomial over the integers is abelian, then its Prime Set can be written as the union of primes in some congruence classes modulo some integer.

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 …

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 …

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 …

Stability Of Ideal Lattices From Quadratic Number Fields, Lenny Fukshansky

CMC Faculty Publications and Research

We study semi-stable ideal lattices coming from real quadratic number fields. Specifically, we demonstrate infinite families of semi-stable and unstable ideal lattices of trace type, establishing explicit conditions on the canonical basis of an ideal that ensure stability; in particular, our result implies that an ideal lattice of trace type coming from a real quadratic field is semi-stable with positive probability. We also briefly discuss the connection between stability and well-roundedness of Euclidean lattices.

Height Bounds On Zeros Of Quadratic Forms Over Q-Bar, Lenny Fukshansky

CMC Faculty Publications and Research

In this paper we establish three results on small-height zeros of quadratic polynomials over Q. For a single quadratic form in N ≥ 2 variables on a subspace of Q N , we prove an upper bound on the height of a smallest nontrivial zero outside of an algebraic set under the assumption that such a zero exists. For a system of k quadratic forms on an L-dimensional subspace of Q N , N ≥ L ≥ k(k+1) 2 + 1, we prove existence of a nontrivial simultaneous small-height zero. For a system of one or two inhomogeneous quadratic and …

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 …

Spherical 2-Designs And Lattices From Abelian Groups, Albrecht Böttcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj

CMC Faculty Publications and Research

We consider lattices generated by finite Abelian groups. The main result says that such a lattice is strongly eutactic, which means the normalized minimal vectors of the lattice form a spherical 2-design, if and only if the group is of odd order or if it is a power of the group of order 2. This result also yields a criterion for the appropriately normalized minimal vectors to constitute a uniform normalized tight frame.

Enhancement On Counting Invariant On Symmetric Virtual Biracks, Melinda Ho

Scripps Senior Theses

This thesis introduces a new enhancement for virtual birack counting invariants. We first introduce knots and other general types of knots (oriented knots, framed knots, racks, and biracks). Then we’ll discuss the methods, knot invariants, mathematicians use to identify whether two knots are different. Next we’ll look at knots with virtual crossings and knots with a good involution. Finally, we introduce a new symmetric enhancement for virtual birack counting invariants and provide an example.

San Diego’S Options For Alternate Sources Of Water: A Comparative Analysis Of Water Recycling And Desalination As Alternative Methods To Importing Water, Alana O. Pokorny

Scripps Senior Theses

This paper describes the processes, methods, backgrounds, and economic challenges, of Desalination and Water Recycling and provide current examples of both. To create a baseline with which to compare the two methods, I will also delve into the history of California water policy. This complicated past is the reason water importation into Southern California remains the main method of obtaining water. Yet, as the current drought continues and technology advances, the need for imported water will become obsolete as the methods for recycling and desalinating water become less expensive, more convenient and more equitable. In the conclusion, all the methods …

Utilizing Hydrology And Geomorphology Relationships To Estimate Streamflow Conditions On Maui And O‘Ahu, Hawai‘I, Brytne Okuhata

Scripps Senior Theses

As the population on the island of Maui drastically increases, water resource demands continue to rise. In order to match water demands and to manage water resources, it is important to understand streamflow and drainage basin interactions. If relationships between a drainage basin’s hydrologic and geomorphologic characteristics can be quantified, then streamflow conditions of ungaged streams can potentially be estimated. The baseflow recession constant is an important variable to analyze for water management, yet until this study, recession constants were not calculated for the island of Maui, or Hawai‘i as a whole. Recession constants of currently gaged streams on Maui …

Price, Perceived Value And Customer Satisfaction: A Text-Based Econometric Analysis Of Yelp! Reviews, Eleanor A. Dwyer

Scripps Senior Theses

We examine the antecedents of customer satisfaction in the restaurant sector, paying particular attention to perceived value and price level. Using Latent Dirichlet Allocation, we extract latent topics from the text of Yelp! reviews, then analyze the relationship between these topics and satisfaction, measured as the difference between review rating and user average review rating.

An Examination Of El Niño's And Agricultural Runoff's Effect On Harmful Algal Blooms And California Sea Lion (Zalophus Californianus) Health In Monterey Bay, Nadine Lafeber

Scripps Senior Theses

An increase in marine mammal stranding and die-off events has been observed along the California coast. The exact cause to explain for these recent events is unknown, but El Niño and harmful algal blooms are established sources for temporary decreases in marine mammal health. To determine whether El Niño could be causing and amplifying harmful algal blooms, particularly in Monterey Bay where they occur frequently, data was analyzed from the Marine Mammal Center in Sausalito, California. Data analysis focused on California sea lions (Zalophus californianus), because they have the largest data set and are directly affected by harmful …

Reforestation, Renewal, And The Cost Of Coal: Opposing A Manichean Worldview In Central Appalachia, Elizabeth R. Hansen

Pomona Senior Theses

Surface coal mining is a major form of land change and environmental degradation in Central Appalachia. Traditional mine reclamation iresults in unmanaged, unproductive grasslands that fail to mitigate many of the environmental costs of coal mining and are of minimal use to communities. Forestry reclamation is an alternative reclamation tactic that has the potential to address both environmental and socioeconomic concerns in Central Appalachia. A case study of Laurel Fork Mine in Eastern Kentucky is included.

Innovation Or Inundation: The Political Economy Of Sea Level Rise In The San Francisco Bay Area, Nicole Quilliam

Pomona Senior Theses

Sea levels are rising around the world due to climate change and the rise in greenhouse gas emissions causing an environmental climate crisis. This thesis paper analyzes how sea level rise is affecting the San Francisco Bay Area. It walks through the geographical factors contribute to more severe impacts of sea level rise, potential adaptation strategies, and discusses the economic impacts and political challenges of protecting communities from sea level rise.

Geographic Relevance For Travel Search: The 2014-2015 Harvey Mudd College Clinic Project For Expedia, Inc., Hannah Long

Scripps Senior Theses

The purpose of this Clinic project is to help Expedia, Inc. expand the search capabilities it offers to its users. In particular, the goal is to help the company respond to unconstrained search queries by generating a method to associate hotels and regions around the world with the higher-level attributes that describe them, such as “family- friendly” or “culturally-rich.” Our team utilized machine-learning algorithms to extract metadata from textual data about hotels and cities. We focused on two machine-learning models: decision trees and Latent Dirichlet Allocation (LDA). The first appeared to be a promising approach, but would require more resources …

A Mathematical Model Of The Effect Of Aspirin On Blood Clotting, Breeana J. Johng

Scripps Senior Theses

In this paper, we provide a mathematical model of the effect of aspirin on blood clotting. The model tracks the enzyme prostaglandin H synthase and an important blood clotting factor, thromboxane A₂, in the form of thromboxane B₂. Through model analysis, we determine conditions under which the reactions of prostaglandin H synthase are self-sustaining. Lastly, through numerical simulations, we demonstrate that the model accurately captures the steady-state chemical concentrations of interest in blood, both with and without aspirin treatment.

A Cryptographic Attack: Finding The Discrete Logarithm On Elliptic Curves Of Trace One, Tatiana Bradley

Scripps Senior Theses

The crux of elliptic curve cryptography, a popular mechanism for securing data, is an asymmetric problem. The elliptic curve discrete logarithm problem, as it is called, is hoped to be generally hard in one direction but not the other, and it is this asymmetry that makes it secure.

This paper describes the mathematics (and some of the computer science) necessary to understand and compute an attack on the elliptic curve discrete logarithm problem that works in a special case. The algorithm, proposed by Nigel Smart, renders the elliptic curve discrete logarithm problem easy in both directions for elliptic curves of …

Moving Towards A Greener Future: An Investigation Of How Transit-Oriented Development Has The Potential To Redefine Cities Around Sustainability, Margaret E. Smith

Scripps Senior Theses

How does transportation shape the cities we live in? This paper takes a close look at the practice of transit-oriented development to assess its implications for the future of urban areas. Through the design of a hypothetical light rail station in the suburb of Redmond, WA, this paper demonstrates how targeting sustainable development around transit has the potential to influence entire towns to “go green,” and proposes that, moving forward, cities be designed to maximize mobility, livability, and sustainability.

Comparing Bulk Aerosol Profiles In The Mixed Layer In Coastal Los Angeles And The Inland Empire, Taia Sean Wu

Scripps Senior Theses

Characteristic westerly sea breeze carries air over the Los Angeles Basin in Southern California to the Inland Empire approximately 50 miles inland, directly impacting air quality in both of these two highly polluted regions. As particles play a critical role in air quality and human health, this study compares the bulk aerosol profiles of the Los Angeles pollution "source" and Inland Empire "receptor" regions during the 2013 and 2014 NASA Student Airborne Research Program (SARP) campaigns onboard the NASA DC-8 airborne laboratory. The source and receptor regions were characterized by a series of missed approaches at the Los Angeles International …

Exploration Of Materials Used In 3-Dimensional Printing For The Dental Industry, Holly Chang Hayden

Scripps Senior Theses

A limiting factor in the digitization of dental devices is the availability of materials suitable for use in both dentistry and the new digital technologies. As a rapidly growing industry, three-dimensional printing (3DP) has the potential to disrupt traditional manufacturing and prototyping methods. A review of both restorative materials and the current 3DP materials has lead to a focus on fiber- reinforced composites in the exploration for a new 3DP material. In addition, another area worth exploring and investing in would be 3D bioprinting as it opens up the possibility of regenerative dentistry.

Discovery Islands, Earth Islands: The Theory And Practice Of Island Imagery In Environmental Thought, Naomi A. Bosch

Pomona Senior Theses

Earth Island is a core metaphor of activist thought often applied in Environmental Analysis and related fields as a tool for thinking about the planet’s limited resources. It puts forth the claim that if only we thought of the earth as more like an island, we would better understand our connectivity to other living things and be drawn to develop better and more extensive practices of environmental stewardship. This thesis uses personal accounts of environmental life philosophies and political practices collected from residents of the Discovery Islands in British Columbia as a site for analytical comparison between the theory and …

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

Pomona 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 S_n, i.e., if the lattice is closed under the action of some non-identity elements of S_n. Given a fixed element T E S_n, we study properties of the set of all lattices closed under the action of T: we call such lattices T-invariant. These lattices naturally generalize cyclic lattices introduced by Micciancio in [7,8], which we previously studied in [1]. Continuing our investigation, we discuss some basic properties of …

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.

A Plausibly Deniable Encryption Scheme For Personal Data Storage, Andrew Brockmann

HMC Senior Theses

Even if an encryption algorithm is mathematically strong, humans inevitably make for a weak link in most security protocols. A sufficiently threatening adversary will typically be able to force people to reveal their encrypted data. Methods of deniable encryption seek to mend this vulnerability by allowing for decryption to alternate data which is plausible but not sensitive. Existing schemes which allow for deniable encryption are best suited for use by parties who wish to communicate with one another. They are not, however, ideal for personal data storage. This paper develops a plausibly-deniable encryption system for use with personal data storage, …

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.

Chromatic Polynomials And Orbital Chromatic Polynomials And Their Roots, Jazmin Ortiz

HMC Senior Theses

The chromatic polynomial of a graph, is a polynomial that when evaluated at a positive integer k, is the number of proper k colorings of the graph. We can then find the orbital chromatic polynomial of a graph and a group of automorphisms of the graph, which is a polynomial whose value at a positive integer k is the number of orbits of k-colorings of a graph when acted upon by the group. By considering the roots of the orbital chromatic and chromatic polynomials, the similarities and differences of these polynomials is studied. Specifically we work toward proving a conjecture …

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.

Topological Complexity In Protein Structures, Erica Flapan, Gabriella Heller '14

Pomona Faculty Publications and Research

For DNA molecules, topological complexity occurs exclusively as the result of knotting or linking of the polynucleotide backbone. By contrast, while a few knots and links have been found within the polypeptide backbones of some protein structures, non-planarity can also result from the connectivity between a polypeptide chain and inter- and intra-chain linking via cofactors and disulfide bonds. In this article, we survey the known types of knots, links, and non-planar graphs in protein structures with and without including such bonds and cofactors. Then we present new examples of protein structures containing Möbius ladders and other non-planar graphs as a …