Physical Sciences and Mathematics Commons^™

On The Matrix Equation Xa + Ax_T = 0, Ii, Alice Zhuo-Yu Chan '14, Luis Alberto Garcia '14, Stephan Ramon Garcia, Amy L. Shoemaker '14

Pomona Faculty Publications and Research

The matrix equation $XA + AX^T = 0$ was recently introduced by De Ter\'an and Dopico to study the dimension of congruence orbits. They reduced the study of this equation to a number of special cases, several of which have not been explicitly solved. In this note we obtain an explicit, closed-form solution in the difficult Type 0-I interaction case.

Uncle Sam’S Badge: Identity And Representation In The Usda Forest Service, 1905–2013, Char Miller

Pomona Faculty Publications and Research

Howard Abbey could recall the exact moment when he learned that he had passed the forest ranger’s examination for the newly established USDA Forest Service (USFS). In the early morning of Aug. 1, 1905, while he was managing a team of horses pulling a mowing machine on the McIntosh Ranch in the northern Sierra Nevada Mountains, Allen Ray Powers, a Forest Assistant on the Plumas Forest Reserve, rode up and “informed me that I was wanted at the Forest Supervisor’s office in Quincy.” Abbey handed over the reins to his boss and walked the 2 miles to town where he …

Virtual Machine Workloads: The Case For New Nas Benchmarks, Vasily Tarasov, Dean Hildebrand, Geoffrey H. Kuenning, Erez Zadok

All HMC Faculty Publications and Research

Network Attached Storage (NAS) and Virtual Machines (VMs) are widely used in data centers thanks to their manageability, scalability, and ability to consolidate resources. But the shift from physical to virtual clients drastically changes the I/O workloads to seen on NAS servers, due to guest file system encapsulation in virtual disk images and the multiplexing of request streams from different VMs. Unfortunately, current NAS workload generators and benchmarks produce workloads typical to physical machines.

This paper makes two contributions. First, we studied the extent to which virtualization is changing existing NAS workloads. We observed significant changes, including the disappearance of …

Social Aggregation In Pea Aphids: Experiment And Random Walk Modeling, Christa Nilsen, John Paige, Olivia Warner, Benjamin Mayhew, Ryan Sutley, Matthew Lam '15, Andrew J. Bernoff, Chad M. Topaz

All HMC Faculty Publications and Research

From bird flocks to fish schools and ungulate herds to insect swarms, social biological aggregations are found across the natural world. An ongoing challenge in the mathematical modeling of aggregations is to strengthen the connection between models and biological data by quantifying the rules that individuals follow. We model aggregation of the pea aphid, Acyrthosiphon pisum. Specifically, we conduct experiments to track the motion of aphids walking in a featureless circular arena in order to deduce individual-level rules. We observe that each aphid transitions stochastically between a moving and a stationary state. Moving aphids follow a correlated random walk. …

Nfl Betting Market: Using Adjusted Statistics To Test Market Efficiency And Build A Betting Model, James P. Donnelly

CMC Senior Theses

The use of statistical analysis has been prevalent in the sports gambling industry for years. More recently, we have seen the emergence of "adjusted statistics", a more sophisticated way to examine each play and each result (further explanation below). And while adjusted statistics have become commonplace for professional and recreational bettors alike, little research has been done to justify their use. In this paper the effectiveness of this data is tested on the most heavily wagered sport in the world – the National Football League (NFL). The results are studied with two central questions in mind: Does the market account …

Barred Preferential Arrangements, Connor Thomas Ahlbach '13, Jeremy Usatine '14, Nicholas Pippenger

All HMC Faculty Publications and Research

A preferential arrangement of a set is a total ordering of the elements of that set with ties allowed. A barred preferential arrangement is one in which the tied blocks of elements are ordered not only amongst themselves but also with respect to one or more bars. We present various combinatorial identities for r_m‚_ℓ, the number of barred preferential arrangements of ℓ elements with m bars, using both algebraic and combinatorial arguments. Our main result is an expression for r_m,_ℓ as a linear combination of the r_k (= r_0,_ …

Nonlocal Aggregation Models: A Primer Of Swarm Equilibria, Andrew J. Bernoff, Chad M. Topaz

All HMC Faculty Publications and Research

Biological aggregations such as fish schools, bird flocks, bacterial colonies, and insect swarms have characteristic morphologies governed by the group members' intrinsic social interactions with each other and by their interactions with the external environment. Starting from a simple discrete model treating individual organisms as point particles, we derive a nonlocal partial differential equation describing the evolving population density of a continuum aggregation. To study equilibria and their stability, we use tools from the calculus of variations. In one spatial dimension, and for several choices of social forces, external forces, and domains, we find exact analytical expressions for the equilibria. …

Chromatic Bounds On Orbital Chromatic Roots, Dae Hyun Kim, Alexander H. Mun, Mohamed Omar

All HMC Faculty Publications and Research

Given a group G of automorphisms of a graph Γ, the orbital chromatic polynomial OPΓ,G(x) is the polynomial whose value at a positive integer k is the number of orbits of G on proper k-colorings of Γ. In \cite{Cameron}, Cameron et. al. explore the roots of orbital chromatic polynomials, and in particular prove that orbital chromatic roots are dense in R, extending Thomassen's famous result (see \cite{Thomassen}) that chromatic roots are dense in [32/27,∞). Cameron et al \cite{Cameron} further conjectured that the real roots of the orbital chromatic polynomial of any graph are bounded above by the largest real root …

Exploring The Baccalaureate Origin Of Domestic Ph.D. Students In Computing Fields, Susanne Hambrusch, Ran Libeskind-Hadas, Fen Zhao, David Rabson, Amy Csizmar Dalal, Ed Fox, Charles Isbell, Valerie Taylor

All HMC Faculty Publications and Research

Increasing the number of US students entering graduate school and receiving a Ph.D. in computer science is a goal as well as a challenge for many US Ph.D. granting institutions. Although the total computer science Ph.D. production in the U.S. has doubled between 2000 and 2010 (Figure 1), the fraction of domestic students receiving a Ph.D. from U.S. graduate programs has been below 50% since 2003 (Figure 2).

The goal of the Pipeline Project of CRA-E (PiPE) is to better understand the pipeline of US citizens and Permanent Residents (henceforth termed domestic students ) who apply, matriculate, and graduate from …

Existence And Qualitative Properties Of Solutions For Nonlinear Dirichlet Problems, Alfonso Castro, Jorge Cossio, Carlos Vélez

All HMC Faculty Publications and Research

Sign-changing solutions to semilinear elliptic problems in connection with their Morse indices. To this end, we first establish a priori bounds for one-sign solutions. Secondly, using abstract saddle point principles we find large augmented Morse index solutions. In this part, extensive use is made of critical groups, Morse index arguments, Lyapunov-Schmidt reduction, and Leray-Schauder degree. Finally, we provide conditions under which these solutions necessarily change sign and we comment about further qualitative properties.

Near-Optimal Compressed Sensing Guarantees For Anisotropic And Isotropic Total Variation Minimization, Deanna Needell, Rachel Ward

CMC Faculty Publications and Research

Consider the problem of reconstructing a multidimensional signal from partial information, as in the setting of compressed sensing. Without any additional assumptions, this problem is ill-posed. However, for signals such as natural images or movies, the minimal total variation estimate consistent with the measurements often produces a good approximation to the underlying signal, even if the number of measurements is far smaller than the ambient dimensionality. Recently, guarantees for two-dimensional images were established. This paper extends these theoretical results to signals of arbitrary dimension and to both the anisotropic and isotropic total variation problems. To be precise, we show that …

From Urysohn's Universal Space To A Universal Space-Time, Asuman Güven Aksoy, Zachary Glassman '14, Olga M. Kosheleva, Vladik Kreinovich

CMC Faculty Publications and Research

A known Urysohn’s result shows that there exists a universal metric space, i.e., a metric space into every other (separable) metric space can be isomorphically embedded. Moreover, this universal metric space can be selected to be ultra-homogeneous — every isomorphism of its two finite subsets can be extended to the isomorphism of the whole space. Starting with Einstein’s theories of Special and General relativity, space-times are described by a different type of structure — a set (of events) equipped with the proper time τ (a, b) between points a and b; such spaces are known as space-times with kinematic metric, …

Clustering Methods And Their Applications To Adolescent Healthcare Data, Morgan Mayer-Jochimsen

Scripps Senior Theses

Clustering is a mathematical method of data analysis which identifies trends in data by efficiently separating data into a specified number of clusters so is incredibly useful and widely applicable for questions of interrelatedness of data. Two methods of clustering are considered here. K-means clustering defines clusters in relation to the centroid, or center, of a cluster. Spectral clustering establishes connections between all of the data points to be clustered, then eliminates those connections that link dissimilar points. This is represented as an eigenvector problem where the solution is given by the eigenvectors of the Normalized Graph Laplacian. Spectral clustering …

Factoring The Duplication Map On Elliptic Curves For Use In Rank Computations, Tracy Layden

Scripps Senior Theses

This thesis examines the rank of elliptic curves. We first examine the correspondences between projective space and affine space, and use the projective point at infinity to establish the group law on elliptic curves. We prove a section of Mordell’s Theorem to establish that the abelian group of rational points on an elliptic curve is finitely generated. We then use homomorphisms established in our proof to find a formula for the rank, and then provide examples of computations.

Review: Cyclicity Results For Some Antianalytic Toeplitz Operators Acting On H^P, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

Review: On Some Bergman Shift Operators, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

Topological Symmetry Groups Of Graphs In 3-Manifolds, Erica Flapan, Harry Tamvakis

Pomona Faculty Publications and Research

We prove that for every closed, connected, orientable, irreducible 3-manifold there exists an alternating group A_n which is not the topological symmetry group of any graph embedded in the manifold. We also show that for every finite group G there is an embedding T of some graph in a hyperbolic rational homology 3-sphere such that the topological symmetry group of T is isomorphic to G.

Simulations Of Surfactant Driven Thin Film Flow, Shreyas Kumar

HMC Senior Theses

This thesis is intended to fulfill the requirements of the Math and Physics departments at Harvey Mudd College. We begin with a brief introduction to the study of surfactant dynamics followed by some background on the experimental framework our work is related to. We then go through a derivation of the model we use, and explore in depth the nature of the Equation of State (EoS), the relationship between the surface tension on a fluid and the surfactant concentration. We consider the effect of using an empirical equation of state on the results of the simulations and compare the new …

On The Norm Closure Problem For Complex Symmetric Operators, Stephan Ramon Garcia, Daniel E. Poore '11

Pomona Faculty Publications and Research

We prove that the set of all complex symmetric operators on a separable, infinite-dimensional Hilbert space is not norm closed.

A Method For Generating Realistic Correlation Matrices, Johanna S. Hardin, Stephan Ramon Garcia, David Golan

Pomona Faculty Publications and Research

Simulating sample correlation matrices is important in many areas of statistics. Approaches such as generating Gaussian data and finding their sample correlation matrix or generating random uniform $[-1,1]$ deviates as pairwise correlations both have drawbacks. We develop an algorithm for adding noise, in a highly controlled manner, to general correlation matrices. In many instances, our method yields results which are superior to those obtained by simply simulating Gaussian data. Moreover, we demonstrate how our general algorithm can be tailored to a number of different correlation models. Using our results with a few different applications, we show that simulating correlation matrices …

The Graphic Nature Of Gaussian Periods, William Duke, Stephan Ramon Garcia, Bob Lutz '13

Pomona Faculty Publications and Research

Recent work has shown that the study of supercharacters on abelian groups provides a natural framework with which to study the properties of certain exponential sums of interest in number theory. Our aim here is to initiate the study of Gaussian periods from this novel perspective. Among other things, this approach reveals that these classical objects display a dazzling array of visual patterns of great complexity and remarkable subtlety.

Supercharacters, Exponential Sums, And The Uncertainty Principle, J.L. Brumbaugh '13, Madeleine Bulkow '14, Patrick S. Fleming, Luis Alberto Garcia '14, Stephan Ramon Garcia, Gizem Karaali, Matt Michal '15, Andrew P. Turner '14

Pomona Faculty Publications and Research

The theory of supercharacters, which generalizes classical character theory, was recently introduced by P. Diaconis and I.M. Isaacs, building upon earlier work of C. Andre. We study supercharacter theories on $(Z/nZ)^d$ induced by the actions of certain matrix groups, demonstrating that a variety of exponential sums of interest in number theory (e.g., Gauss, Ramanujan, and Kloosterman sums) arise in this manner. We develop a generalization of the discrete Fourier transform, in which supercharacters play the role of the Fourier exponential basis. We provide a corresponding uncertainty principle and compute the associated constants in several cases.

The Graphic Nature Of The Symmetric Group, J.L. Brumbaugh '13, Madeleine Bulkow '14, Luis Alberto Garcia '14, Stephan Ramon Garcia, Matt Michal '15, Andrew P. Turner '14

Pomona Faculty Publications and Research

We investigate a remarkable class of exponential sums which are derived from the symmetric groups and which display a diverse array of visually appealing features. Our interest in these expressions stems not only from their astounding visual properties, but also from the fact that they represent a novel and intriguing class of supercharacters.

Making Common Cause For Conservation: The Pinchot Institute And Grey Towers National Historic Site, 1963-2013, Char Miller

Pomona Faculty Publications and Research

This year marks the 50th anniversary of the establishment of the Pinchot Institute for Conservation and the donation of the Pinchot family home Grey Towers to the U.S. Forest Service. In the following essay, historian and Pinchot biographer Char Miller discusses how the Institute is applying Gifford Pinchot’s principles to contemporary environmental issues. It is adapted from Seeking the Greatest Good: The Conservation Legacy of Gifford Pinchot, his new history of the Institute, and is published with kind permission of the University of Pittsburgh Press.

The Brave New World Of Open Access & Creative Commons: A Humanistic Experiment In Mathematical Publishing, Gizem Karaali

Pomona Faculty Publications and Research

In January 2011 the Journal of Humanistic Mathematics (JHM) published its first issue. JHM (http://scholarship.claremont.edu/jhm) is an online-only, peer-reviewed, open-access journal which has passed the all-important ten-thousand-download barrier in its first anniversary. In order to remain faithful to the fundamental principles of open access, JHM uses Creative Commons licensing, where authors retain copyright of their work, but others are free to reuse them (with proper attribution). In this note I share and reflect upon our experience with open access and Creative Commons.

School Choice As A One-Sided Matching Problem: Cardinal Utilities And Optimization, Sinan Aksoy, Alexander Adam Azzam, Chaya Coppersmith, Julie Glass, Gizem Karaali, Xueying Zhao, Xinjing Zhu

Pomona Faculty Publications and Research

The school choice problem concerns the design and implementation of matching mechanisms that produce school assignments for students within a given public school district. Previously considered criteria for evaluating proposed mechanisms such as stability, strategyproofness and Pareto efficiency do not always translate into desirable student assignments. In this note, we explore a class of one-sided, cardinal utility maximizing matching mechanisms focused exclusively on student preferences. We adapt a well-known combinatorial optimization technique (the Hungarian algorithm) as the kernel of this class of matching mechanisms. We find that, while such mechanisms can be adapted to meet desirable criteria not met by …

Selma Karaali And Artemis Karaali, Gizem Karaali

Pomona Faculty Publications and Research

A blog about grandmothers in the fields of science, technology, engineering and mathematics.

Sampling From The Hardcore Process, William C. Dodds

CMC Senior Theses

Partially Recursive Acceptance Rejection (PRAR) and bounding chains used in conjunction with coupling from the past (CFTP) are two perfect simulation protocols which can be used to sample from a variety of unnormalized target distributions. This paper first examines and then implements these two protocols to sample from the hardcore gas process. We empirically determine the subset of the hardcore process's parameters for which these two algorithms run in polynomial time. Comparing the efficiency of these two algorithms, we find that PRAR runs much faster for small values of the hardcore process's parameter whereas the bounding chain approach is vastly …

Invisibility: A Mathematical Perspective, Austin G. Gomez

CMC Senior Theses

The concept of rendering an object invisible, once considered unfathomable, can now be deemed achievable using artificial metamaterials. The ability for these advanced structures to refract waves in the negative direction has sparked creativity for future applications. Manipulating electromagnetic waves of all frequencies around an object requires precise and unique parameters, which are calculated from various mathemat- ical laws and equations. We explore the possible interpretations of these parameters and how they are implemented towards the construction of a suitable metamaterial. If carried out correctly, the wave will exit the metamaterial exhibiting the same behavior as when it had entered. …

Applications Of Fourier Analysis To Audio Signal Processing: An Investigation Of Chord Detection Algorithms, Nathan Lenssen

CMC Senior Theses

The discrete Fourier transform has become an essential tool in the analysis of digital signals. Applications have become widespread since the discovery of the Fast Fourier Transform and the rise of personal computers. The field of digital signal processing is an exciting intersection of mathematics, statistics, and electrical engineering. In this study we aim to gain understanding of the mathematics behind algorithms that can extract chord information from recorded music. We investigate basic music theory, introduce and derive the discrete Fourier transform, and apply Fourier analysis to audio files to extract spectral data.