Physical Sciences and Mathematics Commons^™

Direct, Biomimetic Synthesis Of (+)-Artemone Via A Stereoselective, Organocatalytic Cyclization, Eric D. Nacsa, Brian C. Fielder, Shannon P. Wetzler, Veerasak Srisuknimit, Jonathan P. Litz, Mary J. Van Vleet, Kim Quach, David A. Vosburg

All HMC Faculty Publications and Research

We present a four-step synthesis of (+)-artemone from (–)- linalool, featuring iminium organocatalysis of a doubly diastereoselective conjugate addition reaction. The strategy follows a proposed biosynthetic pathway, rapidly generates stereochemical complexity, uses no protecting groups, and minimizes redox manipulations.

Counting On R-Fibonacci Numbers, Arthur Benjamin, Curtis Heberle

All HMC Faculty Publications and Research

We prove the r-Fibonacci identities of Howard and Cooper using a combinatorial tiling approach.

Review: On Pairs Of Generalized And Hypergeneralized Projections In A Hilbert Space, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

A Mathematician's Villanelle, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

The Cantor Trilogy, Harun Šiljak

Journal of Humanistic Mathematics

The Cantor trilogy is a mathematical dystopia featuring JHM as an important part of that world... at least to humans.

My Finite Field, Matthew Schroeder

Journal of Humanistic Mathematics

A love poem written in the language of mathematics.

Prisoner's Dilemma, Raymond N. Greenwell

Journal of Humanistic Mathematics

No abstract provided.

Book Review: Love And Math: The Heart Of Hidden Reality By Edward Frenkel, Emily R. Grosholz

Journal of Humanistic Mathematics

This review traces Edward Frenkel’s attempt to convey the excitement of mathematical research to a popular audience. In his expositions and explanations of his own research program, he shows how processes of mathematical discovery depend on the juxtaposition of various iconic and symbolic modes of representation as disparate fields of research are brought together in the service of problem solving. And he shows how crucial the encouragement of various older mathematicians was to his own development, as they guided his choice of problems, and served as inspiration.

Abscissas And Ordinates, David Pierce

Journal of Humanistic Mathematics

In the manner of Apollonius of Perga, but hardly any modern book, we investigate conic sections as such. We thus discover why Apollonius calls a conic section a parabola, an hyperbola, or an ellipse; and we discover the meanings of the terms abscissa and ordinate. In an education that is liberating and not simply indoctrinating, the student of mathematics will learn these things.

The Symbolic And Mathematical Influence Of Diophantus's Arithmetica, Cyrus Hettle

Journal of Humanistic Mathematics

Though it was written in Greek in a center of ancient Greek learning, Diophantus's Arithmetica is a curious synthesis of Greek, Egyptian, and Mesopotamian mathematics. It was not only one of the first purely number-theoretic and algebraic texts, but the first to use the blend of rhetorical and symbolic exposition known as syncopated mathematics. The text was influential in the development of Arabic algebra and European number theory and notation, and its development of the theory of indeterminate, or Diophantine, equations inspired modern work in both abstract algebra and computer science. We present, in this article, a selection of problems …

Love Games: A Game-Theory Approach To Compatibility, Kerstin Bever, Julie Rowlett

Journal of Humanistic Mathematics

In this note, we present a compatibility test with a rigorous mathematical foundation in game theory. The test must be taken separately by both partners, making it difficult for either partner alone to control the outcome. To introduce basic notions of game theory we investigate a scene from the film "A Beautiful Mind" based on John Nash's life and Nobel-prize-winning theorem. We recall this result and reveal the mathematics behind our test. Readers may customize and modify the test for more accurate results or to evaluate interpersonal relationships in other settings, not only romantic. Finally, we apply Dyson's and Press's …

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 …

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 …

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.

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 …

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.

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.

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, …

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 …

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 …

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.

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.