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A New Approximation Scheme For Monte Carlo Applications, Bo Jones

CMC Senior Theses

Approximation algorithms employing Monte Carlo methods, across application domains, often require as a subroutine the estimation of the mean of a random variable with support on [0,1]. One wishes to estimate this mean to within a user-specified error, using as few samples from the simulated distribution as possible. In the case that the mean being estimated is small, one is then interested in controlling the relative error of the estimate. We introduce a new (epsilon, delta) relative error approximation scheme for [0,1] random variables and provide a comparison of this algorithm's performance to that of an existing approximation scheme, both …

Climate Change Adaptation For Southern California Groundwater Managers: A Case Study Of The Six Basins Aquifer, Frank Lyles

Pomona Senior Theses

Groundwater has been very important to the economic development of Southern California, and will continue to be a crucial resource in the 21st century. However, Climate Change threatens to disrupt many of the physical and economic processes that control the flow of water in and out of aquifers. One groundwater manager, the Six Basins Watermaster in eastern Los Angeles and western San Bernardino Counties, has developed a long-term planning document called the Strategic Plan that mostly fails to address the implications of Climate Change, especially for local water supplies. This thesis presents an in-depth analysis of the Six Basin Watermaster’s …

Social Sustainability: The Role Of Ecotourism In Regenerating Cultural And Environmental Histories In Rio De Janeiro, Nia Mcallister

Pomona Senior Theses

Ecotourism is a rapidly growing global export industry that aims to uphold the ethics of responsible tourism by engaging with local communities and encouraging environmentally conscious travel. With existing critiques of the greenwashing of ecotourism and the tendency for tourism agencies to exploit host communities, I advocate for participatory community-based models of ecotourism. This thesis explores both the material and conceptual benefits of community-based ecotourism through the critical examination of community-based ecotourism projects in Rio de Janeiro Brazil. Focusing on the implementation of ecotourism in of some of Rio de Janeiro’s peripheral communities, areas that are impacted by social and …

Daily Traffic Flow Pattern Recognition By Spectral Clustering, Matthew Aven

CMC Senior Theses

This paper explores the potential applications of existing spectral clustering algorithms to real life problems through experiments on existing road traffic data. The analysis begins with an overview of previous unsupervised machine learning techniques and constructs an effective spectral clustering algorithm that demonstrates the analytical power of the method. The paper focuses on the spectral embedding method’s ability to project non-linearly separable, high dimensional data into a more manageable space that allows for accurate clustering. The key step in this method involves solving a normalized eigenvector problem in order to construct an optimal representation of the original data.

While this …

Soil Erosion Risk Factors And The Impacts Of Diversification On Organic Strawberry Farms Along California’S Central Coast, Kay Sterner

Pomona Senior Theses

Soil erosion is a major issue that threatens to undermine our current system of agriculture. Due to the fact that this system is in turn the number one cause of erosion, agricultural practices in the United States need to be rethought. This study explores how traditional ideas of erosion risks are related to observed erosion on organic strawberry farms along California’s Central Coast. In addition, diversified farming systems are addressed as a possible solution for the current unsustainability of our farming practices. The data from this research suggest that diversity of crops on farms could be linked to less soil …

Quantifying Carbonyl Sulfide And Other Sulfur-Containing Compounds Over The Santa Barbara Channel, Julia Black

Scripps Senior Theses

Carbonyl sulfide (OCS) is emitted to the atmosphere through the outgassing of ocean surface waters. OCS is also the primary source of sulfur-containing compounds in the stratosphere and contributes to the formation of the stratospheric sulfate layer, an essential controller of the radiative balance of the atmosphere. During the 2016 Student Airborne Research Program (SARP), 15 whole air samples were collected on the NASA DC-8 aircraft over the Santa Barbara Channel. Five additional surface samples were taken at various locations along the Santa Barbara Channel. The samples were analyzed using gas chromatography in the Rowland-Blake lab at UC Irvine, and …

Paving The Randomized Gauss-Seidel, Wei Wu

Scripps Senior Theses

The Randomized Gauss-Seidel Method (RGS) is an iterative algorithm that solves overdetermined systems of linear equations Ax = b. This paper studies an update on the RGS method, the Randomized Block Gauss-Seidel Method. At each step, the algorithm greedily minimizes the objective function L(x) = kAx bk2 with respect to a subset of coordinates. This paper describes a Randomized Block Gauss-Seidel Method (RBGS) which uses a randomized control method to choose a subset at each step. This algorithm is the first block RGS method with an expected linear convergence rate which can be described by the properties of the matrix …

Evolving Art: Modifying Context Free Art With A Genetic Algorithm, Marina Kent

Scripps Senior Theses

Context Free Design Grammar (CFDG) is a programming language for defining recursive structures that can be used to create art. I use CFDG as a design space for genetic programming, experimenting with various options for crossover, mutation, and fitness. In this exploratory work, multiple generations are manually assessed to determine the usefulness of the mutation strategies and fitness functions. I find that simple value mutation and fitness that alters general program structure is not enough to produce an increase of interesting images in CFDG. I discuss these findings as well as future avenues of inquiry for genetic programming in artistic …

Dynamics And Clustering In Locust Hopper Bands, Jialun Zhang

HMC Senior Theses

In recent years, technological advances in animal tracking have renewed interests in collective animal behavior, and in particular, locust swarms. These swarms pose a major threat to agriculture in northern Africa, the Middle East, and other regions. In their early life stages, locusts move in hopper bands, which are huge aggregations traveling on the ground. Our main goal is to understand the underlying mechanisms for the emergence and organization of these bands. We construct an agent-based model that tracks individual locusts and a continuum model that tracks the evolution of locust density. Both these models are motivated by experimental observations …

Random Tropical Curves, Magda L. Hlavacek

HMC Senior Theses

In the setting of tropical mathematics, geometric objects are rich with inherent combinatorial structure. For example, each polynomial $p(x,y)$ in the tropical setting corresponds to a tropical curve; these tropical curves correspond to unbounded graphs embedded in $\R^2$. Each of these graphs is dual to a particular subdivision of its Newton polytope; we classify tropical curves by combinatorial type based on these corresponding subdivisions. In this thesis, we aim to gain an understanding of the likeliness of the combinatorial type of a randomly chosen tropical curve by using methods from polytope geometry. We focus on tropical curves corresponding to quadratics, …

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Kinetic Monte Carlo Methods For Computing First Capture Time Distributions In Models Of Diffusive Absorption, Daniel Schmidt

HMC Senior Theses

In this paper, we consider the capture dynamics of a particle undergoing a random walk above a sheet of absorbing traps. In particular, we seek to characterize the distribution in time from when the particle is released to when it is absorbed. This problem is motivated by the study of lymphocytes in the human blood stream; for a particle near the surface of a lymphocyte, how long will it take for the particle to be captured? We model this problem as a diffusive process with a mixture of reflecting and absorbing boundary conditions. The model is analyzed from two approaches. …

Complexity Of Linear Summary Statistics, Micah G. Pedrick

HMC Senior Theses

Families of linear functionals on a vector space that are mapped to each other by a group of symmetries of the space have a significant amount of structure. This results in computational redundancies which can be used to make computing the entire family of functionals at once more efficient than applying each in turn. This thesis explores asymptotic complexity results for a few such families: contingency tables and unranked choice data. These are used to explore the framework of Radon transform diagrams, which promise to allow general theorems about linear summary statistics to be stated and proved.

The Document Similarity Network: A Novel Technique For Visualizing Relationships In Text Corpora, Dylan Baker

HMC Senior Theses

With the abundance of written information available online, it is useful to be able to automatically synthesize and extract meaningful information from text corpora. We present a unique method for visualizing relationships between documents in a text corpus. By using Latent Dirichlet Allocation to extract topics from the corpus, we create a graph whose nodes represent individual documents and whose edge weights indicate the distance between topic distributions in documents. These edge lengths are then scaled using multidimensional scaling techniques, such that more similar documents are clustered together. Applying this method to several datasets, we demonstrate that these graphs are …

Emergence And Complexity In Music, Zoe Tucker

HMC Senior Theses

How can we apply mathematical notions of complexity and emergence to music, and how can these mathematical ideas then inspire new musical works? Using Steve Reich's Clapping Music as a starting point, we look for emergent patterns in music by considering cases where a piece's complexity is significantly different from the total complexity of each of the individual parts. Definitions of complexity inspired by information theory, data compression, and musical practice are considered. We also consider the number of distinct musical pieces that could be composed in the same manner as Clapping Music. Finally, we present a new musical …

Classifying The Jacobian Groups Of Adinkras, Aaron R. Bagheri

HMC Senior Theses

Supersymmetry is a theoretical model of particle physics that posits a symmetry between bosons and fermions. Supersymmetry proposes the existence of particles that we have not yet observed and through them, offers a more unified view of the universe. In the same way Feynman Diagrams represent Feynman Integrals describing subatomic particle behaviour, supersymmetry algebras can be represented by graphs called adinkras. In addition to being motivated by physics, these graphs are highly structured and mathematically interesting. No one has looked at the Jacobians of these graphs before, so we attempt to characterize them in this thesis. We compute Jacobians through …

Sudoku Variants On The Torus, Kira A. Wyld

HMC Senior Theses

This paper examines the mathematical properties of Sudoku puzzles defined on a Torus. We seek to answer the questions for these variants that have been explored for the traditional Sudoku. We do this process with two such embeddings. The end result of this paper is a deeper mathematical understanding of logic puzzles of this type, as well as a fun new puzzle which could be played.

Tropical Derivation Of Cohomology Ring Of Heavy/Light Hassett Spaces, Shiyue Li

HMC Senior Theses

The cohomology of moduli spaces of curves has been extensively studied in classical algebraic geometry. The emergent field of tropical geometry gives new views and combinatorial tools for treating these classical problems. In particular, we study the cohomology of heavy/light Hassett spaces, moduli spaces of heavy/light weighted stable curves, denoted as $\calm_{g, w}$ for a particular genus $g$ and a weight vector $w \in (0, 1]^n$ using tropical geometry. We survey and build on the work of \citet{Cavalieri2014}, which proved that tropical compactification is a \textit{wonderful} compactification of the complement of hyperplane arrangement for these heavy/light Hassett spaces. For $g …

Pattern Recognition In Stock Data, Kathryn Dover

HMC Senior Theses

Finding patterns in high dimensional data can be difficult because it cannot be easily visualized. There are many different machine learning methods to fit data in order to predict and classify future data but there is typically a large expense on having the machine learn the fit for a certain part of a dataset. We propose a geometric way of defining different patterns in data that is invariant under size and rotation. Using a Gaussian Process, we find that pattern within stock datasets and make predictions from it.

Toric Ideals, Polytopes, And Convex Neural Codes, Caitlin Lienkaemper

HMC Senior Theses

How does the brain encode the spatial structure of the external world?

A partial answer comes through place cells, hippocampal neurons which

become associated to approximately convex regions of the world known

as their place fields. When an organism is in the place field of some place

cell, that cell will fire at an increased rate. A neural code describes the set

of firing patterns observed in a set of neurons in terms of which subsets

fire together and which do not. If the neurons the code describes are place

cells, then the neural code gives some information about the …

Maximal Lelm Distinguishability Of Qubit And Qutrit Bell States Using Projective And Non-Projective Measurements, Nathaniel Leslie

HMC Senior Theses

Many quantum information tasks require measurements to distinguish between different quantum-mechanically entangled states (Bell states) of a particle pair. In practice, measurements are often limited to linear evolution and local measurement (LELM) of the particles. We investigate LELM distinguishability of the Bell states of two qubits (two-state particles) and qutrits (three-state particles), via standard projective measurement and via generalized measurement, which allows detection channels beyond the number of orthogonal single-particle states. Projective LELM can only distinguish 3 of 4 qubit Bell states; we show that generalized measurement does no better. We show that projective LELM can distinguish only 3 of …

Combinatorial Polynomial Hirsch Conjecture, Sam Miller

HMC Senior Theses

The Hirsch Conjecture states that for a d-dimensional polytope with n facets, the diameter of the graph of the polytope is at most n-d. This conjecture was disproven in 2010 by Francisco Santos Leal. However, a polynomial bound in n and d on the diameter of a polytope may still exist. Finding a polynomial bound would provide a worst-case scenario runtime for the Simplex Method of Linear Programming. However working only with polytopes in higher dimensions can prove challenging, so other approaches are welcome. There are many equivalent formulations of the Hirsch Conjecture, one of which is the …

Machine Learning On Statistical Manifold, Bo Zhang

HMC Senior Theses

This senior thesis project explores and generalizes some fundamental machine learning algorithms from the Euclidean space to the statistical manifold, an abstract space in which each point is a probability distribution. In this thesis, we adapt the optimal separating hyperplane, the k-means clustering method, and the hierarchical clustering method for classifying and clustering probability distributions. In these modifications, we use the statistical distances as a measure of the dissimilarity between objects. We describe a situation where the clustering of probability distributions is needed and useful. We present many interesting and promising empirical clustering results, which demonstrate the statistical-distance-based clustering algorithms …

Incorporating The Centers For Disease Control And Prevention Into Vaccine Pricing Models, Dina Sinclair

HMC Senior Theses

The American vaccine pricing market has many actors, making it a complex system to model. Because of this, previous papers have chosen to model only vaccine manufacturers while leaving out the government. However, the government is also an important actor in the market, since it buys over half of vaccines produced. In this work, we aim to introduce the government into vaccine pricing models to better recommend pricing strategies to the Centers for Disease Control and Prevention.

Quantum Foundations With Astronomical Photons, Calvin Leung

HMC Senior Theses

Bell's inequalities impose an upper limit on correlations between measurements of two-photon states under the assumption that the photons play by a set of local rules rather than by quantum mechanics. Quantum theory and decades of experiments both violate this limit.

Recent theoretical work in quantum foundations has demonstrated that a local realist model can explain the non-local correlations observed in experimental tests of Bell's inequality if the underlying probability distribution of the local hidden variable depends on the choice of measurement basis, or ``setting choice''. By using setting choices determined by astrophysical events in the distant past, it is …

Phage Display To Identify Functional Resistance Mutations To Rigosertib, Nedim Filipovic

CMC Senior Theses

In vitro protein selection has had major impacts in the field of protein engineering. Traditional screens assay individual proteins for specific function. Selection, however, analyzes a pool of mutants and yields the best variants. Phage display, a successful selection technique, also provides a reliable link between variant phenotype and genotype. It can also be coupled with high throughput sequencing to map protein mutations; potentially highlighting vital mutations in variants. We propose to apply this technique to cancer therapy. RAF, a serine/threonine kinase, is critical for cell regulation in mammals. RAF can be activated by oncogenic RAS, found in over 30% …

Maximum Mass Restraint Of Neutron Stars: Quarks, Pion, Kaons, And Hyperons, Garrett Ryan

CMC Senior Theses

This thesis explores the topic of maximum mass stability of neutron stars. The outer structure is detailed and explores nuclear pasta phases, the neutron drip line, and density transitions of matter in the crust and atmosphere layers. Other discussion points include superfluids in the crust and core, vortex roles in neutron stars, and magnetic field effects on the EOS in neutron stars. The inner core is studied in much more detail due to its significant role in EOS. The variety of stars include pion condensate stars, kaon condensate stars, npeu stars, npeu stars with the inclusion of hyperons, quark-hybrid stars, …

Triple Non-Negative Matrix Factorization Technique For Sentiment Analysis And Topic Modeling, Alexander A. Waggoner

CMC Senior Theses

Topic modeling refers to the process of algorithmically sorting documents into categories based on some common relationship between the documents. This common relationship between the documents is considered the “topic” of the documents. Sentiment analysis refers to the process of algorithmically sorting a document into a positive or negative category depending whether this document expresses a positive or negative opinion on its respective topic. In this paper, I consider the open problem of document classification into a topic category, as well as a sentiment category. This has a direct application to the retail industry where companies may want to scour …

Cyclic Codes And Cyclic Lattices, Scott Maislin

CMC Senior Theses

In this thesis, we review basic properties of linear codes and lattices with a certain focus on their interplay. In particular, we focus on the analogous con- structions of cyclic codes and cyclic lattices. We start out with a brief overview of the basic theory and properties of linear codes. We then demonstrate the construction of cyclic codes and emphasize their importance in error-correcting coding theory. Next we survey properties of lattices, focusing on algorithmic lattice problems, exhibit the construction of cyclic lattices and discuss their applications in cryptography. We emphasize the similarity and common prop- erties of the two …

An Introduction To The Theory And Applications Of Bayesian Networks, Anant Jaitha

CMC Senior Theses

Bayesian networks are a means to study data. A Bayesian network gives structure to data by creating a graphical system to model the data. It then develops probability distributions over these variables. It explores variables in the problem space and examines the probability distributions related to those variables. It conducts statistical inference over those probability distributions to draw meaning from them. They are good means to explore a large set of data efficiently to make inferences. There are a number of real world applications that already exist and are being actively researched. This paper discusses the theory and applications of …

Four Years With Russell, Gödel, And Erdős: An Undergraduate's Reflection On His Mathematical Education, Michael H. Boggess

CMC Senior Theses

Senior Thesis at CMC is often described institutionally as the capstone of one’s undergraduate education. As such, I wanted my own to accurately capture and reflect how I’ve grown as a student and mathematician these past four years. What follows is my attempt to distill lessons I learned in mathematics outside the curriculum, written for incoming undergraduates and anyone with just a little bit of mathematical curiosity. In it, I attempt to dispel some common preconceptions about mathematics, namely that it’s uninteresting, formulaic, acultural, or completely objective, in favor of a dynamic historical and cultural perspective, with particular attention paid …