Mathematics Commons^™

On The Persistence And Attrition Of Women In Mathematics, Katrina Piatek-Jimenez

Journal of Humanistic Mathematics

The purpose of this study was to investigate what motivates women to choose mathematics as an undergraduate major and to further explore what shapes their future career goals, paying particular attention to their undergraduate experiences and their perceptions of the role of gender in these decisions. A series of semi-structured, individual interviews were conducted with twelve undergraduate women mathematics majors who were attending either a large public university or a small liberal arts college. This study found that strong mathematical identities and enjoyment of mathematics heavily influenced their decisions to major in mathematics. At the career selection stage, these women …

Probing The Inverted Classroom: A Study Of Teaching And Learning Outcomes In Engineering And Mathematics, Nancy K. Lape, Rachel Levy, Darryl Yong

All HMC Faculty Publications and Research

Flipped classrooms have started to become commonplace on university campuses. Despite the growing number of flipped courses, however, quantitative information on their effectiveness remains sparse. Active learning is a mode of instruction that focuses the responsibility of learning on learners. Multiple studies have shown that active learning leads to better student outcomes. Given that instructors in flipped classrooms are generally able to create more opportunities for students to apply or practice course material, we hypothesized that students in a flipped classroom would exhibit more learning compared to students in an unflipped class. This case study describes our research comparing …

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 …

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 …

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.

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 …

Model Spaces: A Survey, Stephan Ramon Garcia, William T. Ross

Pomona Faculty Publications and Research

This is a brief and gentle introduction, aimed at graduate students, to the subject of model subspaces of the Hardy space.

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.

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.

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.

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 …

Elliptic Curves And The Congruent Number Problem, Jonathan Star

CMC Senior Theses

In this paper we explain the congruent number problem and its connection to elliptic curves. We begin with a brief history of the problem and some early attempts to understand congruent numbers. We then introduce elliptic curves and many of their basic properties, as well as explain a few key theorems in the study of elliptic curves. Following this, we prove that determining whether or not a number n is congruent is equivalent to determining whether or not the algebraic rank of a corresponding elliptic curve E_n is 0. We then introduce L-functions and explain the Birch and …

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 …

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.

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.

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.

Combinatorial Proofs Of Fibonomial Identities, Arthur Benjamin, Elizabeth Reiland

All HMC Faculty Publications and Research

We provide a list of simple looking identities that are still in need of combinatorial proof.

Review: Crystal Bases Of Q-Deformed Kac Modules Over The Quantum Superalgebras Uq(Gl(Mln)), Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

Op-Ed: Solve This Math Problem: The Gender Gap, Francis Su

All HMC Faculty Publications and Research

Women may not face such blatant impediments to doing math and science today. But Mirzakhani's achievement aside, we are still a long way from adequately recognizing the outstanding work of women.

Math Talk: Preparing Your Conference Presentation, Gizem Karaali

Pomona Faculty Publications and Research

If you are a typical reader of this blog, then you recently wrapped up your finals week and then dutifully made a summer plan. And then came the summer. Your plan may have involved working on a manuscript, preparing for a qualifying exam or a new course coming up in the fall, drafting a grant proposal, learning a new language (human or machine), eating kale in four different forms, and perhaps some fun times under the sun. Some, like me, also made plans to travel to conferences and give talks. Gearing up to get ready for my first conference of …

Four Quotient Set Gems, Stephan Ramon Garcia, Michael Someck '14, Bob Lutz '13, Bryan Brown '15, Michael Dairyko '13

Pomona Faculty Publications and Research

Our aim in this note is to present four remarkable facts about quotient sets. These observations seem to have been overlooked by the MONTHLY, despite its intense coverage of quotient sets over the years.

Exponential Decay Of Reconstruction Error From Binary Measurements Of Sparse Signals, Richard Baraniuk, Simon Foucart, Deanna Needell, Yaniv Plan, Mary Wootters

CMC Faculty Publications and Research

Binary measurements arise naturally in a variety of statistical and engineering applications. They may be inherent to the problem—e.g., in determining the relationship between genetics and the presence or absence of a disease—or they may be a result of extreme quantization. A recent influx of literature has suggested that using prior signal information can greatly improve the ability to reconstruct a signal from binary measurements. This is exemplified by onebit compressed sensing, which takes the compressed sensing model but assumes that only the sign of each measurement is retained. It has recently been shown that the number of one-bit measurements …

Near Oracle Performance And Block Analysis Of Signal Space Greedy Methods, Raja Giryes, Deanna Needell

CMC Faculty Publications and Research

Compressive sampling (CoSa) is a new methodology which demonstrates that sparse signals can be recovered from a small number of linear measurements. Greedy algorithms like CoSaMP have been designed for this recovery, and variants of these methods have been adapted to the case where sparsity is with respect to some arbitrary dictionary rather than an orthonormal basis. In this work we present an analysis of the so-called Signal Space CoSaMP method when the measurements are corrupted with mean-zero white Gaussian noise. We establish near-oracle performance for recovery of signals sparse in some arbitrary dictionary. In addition, we analyze the block …