Physical Sciences and Mathematics Commons^™

Kaleidoscopes, Chessboards, And Symmetry, Tricia M. Brown

Journal of Humanistic Mathematics

This paper describes the n-queens problem on an n by n chessboard. We discuss the possible symmetries of n-queens solutions and show how solutions to this classical chess question can be used to create examples of colorful artwork.

A Short Walk Can Be Beautiful, Jonathan M. Borwein

Journal of Humanistic Mathematics

The story I tell is of research undertaken, with students and colleagues, in the last six or so years on short random walks. As the research progressed, my criteria for `beauty' changed. Things seemingly remarkable became `simple' and other seemingly simple things became more remarkable as our analytic and computational tools were refined, and understanding improved. I intentionally display some rather advanced mathematics as it is my contention -- as with classical music -- that one can learn to appreciate and enjoy complex formulas without needing to understand them deeply.

A Beautiful Proof By Induction, Lars-Daniel Öhman

Journal of Humanistic Mathematics

The purpose of this note is to present an example of a proof by induction that in the opinion of the present author has great aesthetic value. The proof in question is Thomassen's proof that planar graphs are 5-choosable. I give a self-contained presentation of this result and its proof, and a personal account of why I think this proof is beautiful.

A secondary purpose is to more widely publicize this gem, and hopefully make it part of a standard set of examples for examining characteristics of proofs by induction.

Malheur Occupation In Oregon: Whose Land Is It Really?, Char Miller

Pomona Faculty Publications and Research

The Malheur National Wildlife Refuge, a 187,757-acre haven for greater sandhill cranes and other native birds in eastern Oregon, is usually a pretty peaceful place. But its calm was shattered on Saturday, January 2 when Ammon Bundy and a group of armed men broke into and occupied a number of federal buildings on the refuge, vowing to fight should the government try to arrest them. Their insurrectionary goal appears to be, simply put, to destroy the national system of public lands – our forests, parks and refuges – that was developed in the late 19th century to conserve these …

Review: Transitivity And Bundle Shifts, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

A New Radiometric Date And Assessment Of The Last Glacial Megafauna Of Dream Cave, Derbyshire, Uk, Donald A. Mcfarlane, Joyce Lundberg, Guy Van Rentergem, Eliza Howlett, Chris Stimpson

KGI Faculty Publications and Research

The extinct fauna of Dream Cave, Derbyshire, has played a significant role in the history of British cave paleontology, a near-complete woolly rhinoceros from the cave having been famously illustrated in 1823. The fauna was not subsequently re-studied until 2000, with the publication of an indirect radiometric date by uranium-series disequilibrium dating of a presumed-overlying flowstone. Here we present a direct radiocarbon date of 43,330 +/- 1800 rcyBP, 45083 - 48613 calBP (1 σ) on a representative Bos/Bison bone, with additional comments on the fauna and the taphonomy of the site.

Using A Data Warehouse As Part Of A General Business Process Data Analysis System, Amit Maor

CMC Senior Theses

Data analytics queries often involve aggregating over massive amounts of data, in order to detect trends in the data, make predictions about future data, and make business decisions as a result. As such, it is important that a database management system (DBMS) handling data analytics queries perform well when those queries involve massive amounts of data. A data warehouse is a DBMS which is designed specifically to handle data analytics queries.

This thesis describes the data warehouse Amazon Redshift, and how it was used to design a data analysis system for Laserfiche. Laserfiche is a software company that provides each …

Existence Of Positive Solutions For A Semipositone P-Laplacian Problem, Alfonso Castro, Djairo G. De Figueredo, Emer Lopera

All HMC Faculty Publications and Research

We prove the existence of positive solutions to a semipositone p-Laplacian problem combining mountain pass arguments, comparison principles, regularity principles and a priori estimates.

Bernstein’S Lethargy Theorem In Fréchet Spaces, Asuman Güven Aksoy, Grzegorz Lewicki

CMC Faculty Publications and Research

In this paper we consider Bernstein’s Lethargy Theorem (BLT) in the context of Fréchet spaces. Let X be an infinite-dimensional Fréchet space and let V = {V_n} be a nested sequence of subspaces of X such that V_n ⊆ V_n+1 for any n ∈ N and X = S∞ _n=1 V_n. Let e_n be a decreasing sequence of positive numbers tending to 0. Under an additional natural condition on sup{dist(x, V_n)}, we prove that there exists x ∈ X and no ∈ N such that

e_n/3 ≤ dist(x, V …

Constrained Adaptive Sensing, Mark A. Davenport, Andrew K. Massimino, Deanna Needell, Tina Woolf

CMC Faculty Publications and Research

Suppose that we wish to estimate a vector x∈Cn from a small number of noisy linear measurements of the form y=Ax+z, where z represents measurement noise. When the vector x is sparse, meaning that it has only s nonzeros with s≪n, one can obtain a significantly more accurate estimate of x by adaptively selecting the rows of A based on the previous measurements provided that the signal-to-noise ratio (SNR) is sufficiently large. In this paper we consider the case where we wish to realize the potential of adaptivity but where the rows of A are subject to physical constraints. In …

On Arithmetic Lattices In The Plane, Lenny Fukshansky, Pavel Guerzhoy, Florian Luca

CMC Faculty Publications and Research

We investigate similarity classes of arithmetic lattices in the plane. We introduce a natural height function on the set of such similarity classes, and give asymptotic estimates on the number of all arithmetic similarity classes, semi-stable arithmetic similarity classes, and well-rounded arithmetic similarity classes of bounded height as the bound tends to infinity. We also briefly discuss some properties of the j-invariant corresponding to similarity classes of planar lattices.

Lattice Theory And Toeplitz Determinants, Albrecht Böttcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj

CMC Faculty Publications and Research

This is a survey of our recent joint investigations of lattices that are generated by finite Abelian groups. In the case of cyclic groups, the volume of a fundamental domain of such a lattice is a perturbed Toeplitz determinant with a simple Fisher-Hartwig symbol. For general groups, the situation is more complicated, but it can still be tackled by pure matrix theory. Our main result on the lattices under consideration states that they always have a basis of minimal vectors, while our results in the other direction concern exact and asymptotic formulas for perturbed Toeplitz determinants. The survey is a …

Methods For Quantized Compressed Sensing, Hao-Jun Michael Shi, Mindy Case, Xiaoyi Gu, Shenyinying Tu, Deanna Needell

CMC Faculty Publications and Research

In this paper, we compare and catalog the performance of various greedy quantized compressed sensing algorithms that reconstruct sparse signals from quantized compressed measurements. We also introduce two new greedy approaches for reconstruction: Quantized Compressed Sampling Matching Pursuit (QCoSaMP) and Adaptive Outlier Pursuit for Quantized Iterative Hard Thresholding (AOP-QIHT). We compare the performance of greedy quantized compressed sensing algorithms for a given bit-depth, sparsity, and noise level.

Lattices From Tight Equiangular Frames, Albrecht Böttcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj, Deanna Needell

CMC Faculty Publications and Research

We consider the set of all linear combinations with integer coefficients of the vectors of a unit tight equiangular (k,n) frame and are interested in the question whether this set is a lattice, that is, a discrete additive subgroup of the k-dimensional Euclidean space. We show that this is not the case if the cosine of the angle of the frame is irrational. We also prove that the set is a lattice for n=k+1 and that there are infinitely many k such that a lattice emerges for n=2k. We dispose of all cases in dimensions k at most 9. In …

Totally Isotropic Subspaces Of Small Height In Quadratic Spaces, Wai Kiu Chan, Lenny Fukshansky, Glenn Henshaw

CMC Faculty Publications and Research

Let K be a global field or Q, F a nonzero quadratic form on KN , N ≥ 2, and V a subspace of KN . We prove the existence of an infinite collection of finite families of small-height maximal totally isotropic subspaces of (V, F) such that each such family spans V as a K-vector space. This result generalizes and extends a well known theorem of J. Vaaler [16] and further contributes to the effective study of quadratic forms via height in the general spirit of Cassels’ theorem on small zeros of quadratic forms. All bounds on height are …

On An Effective Variation Of Kronecker's Approximation Theorem, Lenny Fukshansky

CMC Faculty Publications and Research

Let Λ ⊂ Rn be an algebraic lattice, coming from a projective module over the ring of integers of a number field K. Let Z ⊂ Rn be the zero locus of a finite collection of polynomials such that Λ |⊂ Z or a finite union of proper full-rank sublattices of Λ. Let K1 be the number field generated over K by coordinates of vectors in Λ, and let L1, . . . , Lt be linear forms in n variables with algebraic coefficients satisfying an appropriate linear independence condition over K1. For each ε > 0 and a ∈ Rn, …

Turning Waste Into Compost In Napa, California, Liana D. Solis

Pomona Senior Theses

Two significant pieces of legislation in California have mandated that cities and counties must reduce their waste streams. Assembly Bill 341 establishes that California must divert 75% of its waste from landfills by the year 2020. The first bill that included composting, Assembly Bill 1826, was passed in 2014 and requires that commercial users enact composting beginning in 2016. These initiatives have led cities and counties to seek ways of implementing composting programs. Using the City of Napa as a case study, this thesis argues that a composting program can be integrated into any existing waste hauling service. Although there …

Using Cleaved Amplified Polymorphic Sequence (Caps) Genetic Markers To Determine The Extent Of Hybridization Between Castilleja Affinis And Castilleja Mollis As A Mechanism For Adapting To Climate Change On Santa Rosa Island, Elizabeth Medford

Scripps Senior Theses

Hybridization, the process of interbreeding between individuals of different species, is one method by which plants and animals adapt to a changing environment. One example of such adaptation through hybridization may be occurring on the California Channel Islands with two species of Castilleja. While United State Geological Survey (USGS) researchers have been studying the populations of Castilleja affinis and Castilleja mollis to determine if hybridization is occurring on Santa Rosa Island since the early 1990s, up until this point primarily overt phenotypic characteristics have been used to differentiate between the two species. Genetic methods of differentiation were adopted to …

On The Similarity Of Ab And Ba For Normal And Other Matrices, Stephan Ramon Garcia, David Sherman, Gary Weiss

Pomona Faculty Publications and Research

It is known that AB and BA are similar when A and B are Hermitian matrices. In this note we answer a question of F. Zhang by demonstrating that similarity can fail if A is Hermitian and B is normal. Perhaps surprisingly, similarity does hold when A is positive semidefinite and B is normal.

Lattices From Tight Equiangular Frames, Albrecht Böttcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj, Deanna Needell

Pomona Faculty Publications and Research

We consider the set of all linear combinations with integer coefficients of the vectors of a unit tight equiangular (k,n) frame and are interested in the question whether this set is a lattice, that is, a discrete additive subgroup of the k-dimensional Euclidean space. We show that this is not the case if the cosine of the angle of the frame is irrational. We also prove that the set is a lattice for n = k + 1 and that there are infinitely many k such that a lattice emerges for n = 2k …

What's In A Name? A Critical Review Of Definitions Of Quantitative Literacy, Numeracy, And Quantitative Reasoning, Gizem Karaali, Edwin H Villafane Hernandez '18, Jeremy Alexander Taylor '18

Pomona Faculty Publications and Research

This article aims to bring together various threads in the eclectic literature that make up the scholarship around the theme of Quantitative Literacy. In investigating the meanings of terms like "quantitative literacy," "quantitative reasoning," and "numeracy," we seek common ground, common themes, common goals and aspirations of a community of practitioners. A decade ago, these terms were relatively new in the public sphere; today policy makers and accrediting agencies are routinely inserting them into general education conversations. Having good, representative, and perhaps even compact and easily digestible definitions of these terms might come in handy in public relations contexts as …

Computational Progress Towards Maximum Distinguishability Of Bell States By Linear Evolution And Local Measurement, Victor Shang

HMC Senior Theses

Many quantum information protocols rely on the ability to distinguish between entangled quantum states known as Bell states. However, theoretical limits exist on the maximal distinguishability of these entangled states using linear evolution and local measurement (LELM) devices. In the case of two particles entangled in multiple qubit variables, the maximum number of distinguishable Bell states is known. However, in the more general case of two particles entangled in multiple qudit variables, only an upper bound is known under additional assumptions. I have written software in Matlab and Mathematica to explore computationally the maximum number of Bell states that can …

An Interactive Tool For The Computational Exploration Of Integrodifference Population Models, Kennedy Agwamba

HMC Senior Theses

Mathematical modeling of population dynamics can provide novel insight to the growth and dispersal patterns for a variety of species populations, and has become vital to the preservation of biodiversity on a global-scale. These growth and dispersal stages can be modeled using integrodifference equations that are discrete in time and continuous in space. Previous studies have identified metrics that can determine whether a given species will persist or go extinct under certain model parameters. However, a need for computational tools to compute these metrics has limited the scope and analysis within many of these studies. We aim to create computational …

Mathematical Modeling Of Blood Coagulation, Joana L. Perdomo

HMC Senior Theses

Blood coagulation is a series of biochemical reactions that take place to form a blood clot. Abnormalities in coagulation, such as under-clotting or over- clotting, can lead to significant blood loss, cardiac arrest, damage to vital organs, or even death. Thus, understanding quantitatively how blood coagulation works is important in informing clinical decisions about treating deficiencies and disorders. Quantifying blood coagulation is possible through mathematical modeling. This review presents different mathematical models that have been developed in the past 30 years to describe the biochemistry, biophysics, and clinical applications of blood coagulation research. This review includes the strengths and limitations …

The Global Stability Of The Solution To The Morse Potential In A Catastrophic Regime, Weerapat Pittayakanchit

HMC Senior Theses

Swarms of animals exhibit aggregations whose behavior is a challenge for mathematicians to understand. We analyze this behavior numerically and analytically by using the pairwise interaction model known as the Morse potential. Our goal is to prove the global stability of the candidate local minimizer in 1D found in A Primer of Swarm Equilibria. Using the calculus of variations and eigenvalues analysis, we conclude that the candidate local minimizer is a global minimum with respect to all solution smaller than its support. In addition, we manage to extend the global stability condition to any solutions whose support has a single …

A Bound On The Number Of Spanning Trees In Bipartite Graphs, Cheng Wai Koo

HMC Senior Theses

Richard Ehrenborg conjectured that in a bipartite graph G with parts X and Y, the number of spanning trees is at most the product of the vertex degrees divided by |X|⋅|Y|. We make two main contributions. First, using techniques from spectral graph theory, we show that the conjecture holds for sufficiently dense graphs containing a cut vertex of degree 2. Second, using electrical network analysis, we show that the conjecture holds under the operation of removing an edge whose endpoints have sufficiently large degrees.

Our other results are combinatorial proofs that the conjecture holds for …

Pattern Recognition In High-Dimensional Data, Matthew Dannenberg

HMC Senior Theses

Vast amounts of data are produced all the time. Yet this data does not easily equate to useful information: extracting information from large amounts of high dimensional data is nontrivial. People are simply drowning in data. A recent and growing source of high-dimensional data is hyperspectral imaging. Hyperspectral images allow for massive amounts of spectral information to be contained in a single image. In this thesis, a robust supervised machine learning algorithm is developed to efficiently perform binary object classification on hyperspectral image data by making use of the geometry of Grassmann manifolds. This algorithm can consistently distinguish between a …

Realizing The 2-Associahedron, Patrick N. Tierney

HMC Senior Theses

The associahedron has appeared in numerous contexts throughout the field of mathematics. By representing the associahedron as a poset of tubings, Michael Carr and Satyan L. Devadoss were able to create a gener- alized version of the associahedron in the graph-associahedron. We seek to create an alternative generalization of the associahedron by considering a particle-collision model. By extending this model to what we dub the 2- associahedron, we seek to further understand the space of generalizations of the associahedron.

Steady State Solutions For A System Of Partial Differential Equations Arising From Crime Modeling, Bo Li

HMC Senior Theses

I consider a model for the control of criminality in cities. The model was developed during my REU at UCLA. The model is a system of partial differential equations that simulates the behavior of criminals and where they may accumulate, hot spots. I have proved a prior bounds for the partial differential equations in both one-dimensional and higher dimensional case, which proves the attractiveness and density of criminals in the given area will not be unlimitedly high. In addition, I have found some local bifurcation points in the model.

Topological Data Analysis For Systems Of Coupled Oscillators, Alec Dunton

HMC Senior Theses

Coupled oscillators, such as groups of fireflies or clusters of neurons, are found throughout nature and are frequently modeled in the applied mathematics literature. Earlier work by Kuramoto, Strogatz, and others has led to a deep understanding of the emergent behavior of systems of such oscillators using traditional dynamical systems methods. In this project we outline the application of techniques from topological data analysis to understanding the dynamics of systems of coupled oscillators. This includes the examination of partitions, partial synchronization, and attractors. By looking for clustering in a data space consisting of the phase change of oscillators over a …