HMC Senior ThesesCopyright (c) 2019 Claremont Colleges All rights reserved.
https://scholarship.claremont.edu/hmc_theses
Recent documents in HMC Senior Thesesen-usTue, 25 Jun 2019 17:42:05 PDT3600Modeling Moving Droplets: A Precursor Film Approach
https://scholarship.claremont.edu/hmc_theses/142
https://scholarship.claremont.edu/hmc_theses/142Wed, 20 Mar 2019 17:16:00 PDT
We investigate the behavior of moving droplets and rivulets, driven by a combination of gravity and surface shear (wind). The problem is motivated by a desire to model the behavior of raindrops on aircraft wings. We begin with the Stokes equations and use the approximations of lubrication theory to derive the specific thin film equation relevant to our situation. This fourth-order partial differential equation describing the height of the fluid is then solved numerically from varying initial conditions, using a fully implicit discretization for time stepping, and a precursor film to avoid singularities at the drop contact line. Results describing general features of droplet deformation, limited parameter studies, and the applicability of our implementation to the long-term goal of modeling wings in rain are discussed.
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Benjamin BryantAlgebraic Reasoning in Elementary School Students
https://scholarship.claremont.edu/hmc_theses/224
https://scholarship.claremont.edu/hmc_theses/224Tue, 19 Mar 2019 21:57:33 PDT
An exploratory study on instructional design for classroom activities that encourage algebraic reasoning at the elementary school level. Assistance with the activities was provided as students needed further scaffolding, and multiple solutions were encouraged. An analysis of student responses to the activities is discussed.
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Ivan HernandezBranching Diagrams for Group Inclusions Induced by Field Inclusions
https://scholarship.claremont.edu/hmc_theses/223
https://scholarship.claremont.edu/hmc_theses/223Tue, 19 Mar 2019 21:57:23 PDT
A Fourier transform for a finite group G is an isomorphism from the complex group algebra CG to a direct product of complex matrix algebras, which are determined beforehand by the structure of G. Given such an isomorphism, naive application of that isomorphism to an arbitrary element of CG takes time proportional to |G|2. A fast Fourier transform for some (family of) groups is an algorithm which computes the Fourier transform of a group G of the family in less than O(|G|2) time, generally O(|G| log |G|) or O(|G|(log |G|)2). I describe the construction of a fast Fourier transform for the special linear groups SL(q) with q = 2n.
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Tedodore SpaideExistence of Continuous Solutions to a Semilinear Wave Equation
https://scholarship.claremont.edu/hmc_theses/222
https://scholarship.claremont.edu/hmc_theses/222Tue, 19 Mar 2019 21:57:13 PDT
We prove two results; first, we show that a boundary value problem for the semilinear wave equation with smooth, asymptotically linear nonlinearity and sinusoidal smooth forcing along a characteristic cannot have a continuous solution. Thereafter, we show that if the sinusoidal forcing is not isolated to a characteristic of the wave equation, then the problem has a continuous solution.
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Ben PreskillFast Matrix Multiplication via Group Actions
https://scholarship.claremont.edu/hmc_theses/221
https://scholarship.claremont.edu/hmc_theses/221Tue, 19 Mar 2019 21:57:03 PDT
Recent work has shown that fast matrix multiplication algorithms can be constructed by embedding the two input matrices into a group algebra, applying a generalized discrete Fourier transform, and performing the multiplication in the Fourier basis. Developing an embedding that yields a matrix multiplication algorithm with running time faster than naive matrix multiplication leads to interesting combinatorial problems in group theory. The crux of such an embedding, after a group G has been chosen, lies in finding a triple of subsets of G that satisfy a certain algebraic relation. I show how the process of finding such subsets can in some cases be greatly simplified by considering the action of the group G on an appropriate set X. In particular, I focus on groups acting on regularly branching trees.
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Hendrik OremLocality and Complexity in Path Search
https://scholarship.claremont.edu/hmc_theses/220
https://scholarship.claremont.edu/hmc_theses/220Tue, 19 Mar 2019 21:56:52 PDT
The path search problem considers a simple model of communication networks as channel graphs: directed acyclic graphs with a single source and sink. We consider each vertex to represent a switching point, and each edge a single communication line. Under a probabilistic model where each edge may independently be free (available for use) or blocked (already in use) with some constant probability, we seek to efficiently search the graph: examine (on average) as few edges as possible before determining if a path of free edges exists from source to sink. We consider the difficulty of searching various graphs under different search models, and examine the computational complexity of calculating the search cost of arbitrary graphs.
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Andrew HunterMathematical Model of the Chronic Lymphocytic Leukemia Microenvironment
https://scholarship.claremont.edu/hmc_theses/219
https://scholarship.claremont.edu/hmc_theses/219Tue, 19 Mar 2019 21:56:43 PDT
A mathematical model of the interaction between chronic lymphocytic leukemia (CLL) and CD4+ (helper) T cells was developed to study the role of T cells in cancer survival. In particular, a system of four nonlinear advection diffusion reaction partial differential equations were used to simulate spatial effects such as chemical diffusion and chemotaxis on CLL survival and proliferation.
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Ben FogelsonExploring Agreeability in Tree Societies
https://scholarship.claremont.edu/hmc_theses/218
https://scholarship.claremont.edu/hmc_theses/218Tue, 19 Mar 2019 21:56:34 PDT
Let S be a collection of convex sets in Rd with the property that any subcollection of d − 1 sets has a nonempty intersection. Helly’s Theorem states that ∩s∈S S is nonempty. In a forthcoming paper, Berg et al. (Forthcoming) interpret the one dimensional version of Helly’s Theorem in the context of voting in a society. They look at the effect that different intersection properties have on the proportion of a society that must agree on some point or issue. In general, we define a society as some underlying space X and a collection S of convex sets on the space. A society is (k, m)-agreeable if every m-element subset of S has a k-element subset with a nonempty intersection. The agreement number of a society is the size of the largest subset of S with a nonempty intersection. In my work I focus on the case where X is a tree and the convex sets in S are subtrees. I have developed a reduction method that makes these tree societies more tractable. In particular, I have used this method to show that the agreement number of (2, m)-agreeable tree societies is at least 1 |S | and 3 that the agreement number of (k, k + 1)-agreeable tree societies is at least |S|−1.
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Sarah FletcherAbelian Sandpile Model on Symmetric Graphs
https://scholarship.claremont.edu/hmc_theses/217
https://scholarship.claremont.edu/hmc_theses/217Tue, 19 Mar 2019 21:56:23 PDT
The abelian sandpile model, or chip firing game, is a cellular automaton on finite directed graphs often used to describe the phenomenon of self organized criticality. Here we present a thorough introduction to the theory of sandpiles. Additionally, we define a symmetric sandpile configuration, and show that such configurations form a subgroup of the sandpile group. Given a graph, we explore the existence of a quotient graph whose sandpile group is isomorphic to the symmetric subgroup of the original graph. These explorations are motivated by possible applications to counting the domino tilings of a 2n × 2n grid.
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Natalie DurginGraph Linear Complexity
https://scholarship.claremont.edu/hmc_theses/216
https://scholarship.claremont.edu/hmc_theses/216Tue, 19 Mar 2019 21:56:14 PDT
This thesis expands on the notion of linear complexity for a graph as defined by Michael Orrison and David Neel in their paper "The Linear Complexity of a Graph." It considers additional classes of graphs and provides upper bounds for additional types of graphs and graph operations.
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Jason WineripFormation of Labyrinth Patterns in Langmuir Films
https://scholarship.claremont.edu/hmc_theses/215
https://scholarship.claremont.edu/hmc_theses/215Tue, 19 Mar 2019 21:56:05 PDT
A Langmuir film is a molecularly thin fluid layer on the surface of a subfluid. When dipole dipole forces are negligible, bounded films relax to energy minimizing circular domains. We investigate numerically the case where dipole dipole interactions are strong enough to deform the domain into highly distorted labyrinth type patterns. Our numerical method is designed to achieve higher accuracy and better stability than previous work and exploits an analytic formulation that removes a singularity in the dipole dipole forces without resorting to a small cutoff parameter. We calculate the relaxation rates for a linearly perturbed circular domain, and we verify them numerically. We are also able to numerically reproduce experimentally observed circle to dogbone transitions with minimal area loss.
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George TuckerSperner's Lemma Implies Kakutani's Fixed Point Theorem
https://scholarship.claremont.edu/hmc_theses/214
https://scholarship.claremont.edu/hmc_theses/214Tue, 19 Mar 2019 21:55:56 PDT
Kakutani’s fixed point theorem has many applications in economics and game theory. One of its most well known applications is in John Nash’s paper [8], where the theorem is used to prove the existence of an equilibrium strategy in n-person games. Sperner’s lemma, on the other hand, is a combinatorial result concerning the labelling of the vertices of simplices and their triangulations. It is known that Sperner’s lemma is equivalent to a result called Brouwer’s fixed point theorem, of which Kakutani’s theorem is a generalization. A natural question that arises is whether we can prove Kakutani’s fixed point theorem directly using Sperner’s lemma without going through Brouwer’s theorem. The objective of this thesis to understand Kakutani’s theorem, Sperner’s lemma, and how they are related. In particular, I explore ways in which Sperner’s lemma can be used to prove Kakutani’s theorem and related results.
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Mutiara SondjajaApproximating Solutions to Differential Equations via Fixed Point Theory
https://scholarship.claremont.edu/hmc_theses/213
https://scholarship.claremont.edu/hmc_theses/213Tue, 19 Mar 2019 21:55:48 PDT
In the study of differential equations there are two fundamental questions: is there a solution? and what is it? One of the most elegant ways to prove that an equation has a solution is to pose it as a fixed point problem, that is, to find a function f such that x is a solution if and only if f (x) = x. Results from fixed point theory can then be employed to show that f has a fixed point. However, the results of fixed point theory are often nonconstructive: they guarantee that a fixed point exists but do not help in finding the fixed point. Thus these methods tend to answer the first question, but not the second. One such result is Schauder’s fixed point theorem. This theorem is broadly applicable in proving the existence of solutions to differential equations, including the Navier-Stokes equations under certain conditions. Recently a semi-constructive proof of Schauder’s theorem was developed in Rizzolo and Su (2007). In this thesis we go through the construction in detail and show how it can be used to search for multiple solutions. We then apply the method to a selection of differential equations.
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Douglas RizzoloRigid Divisibility Sequences Generated by Polynomial Iteration
https://scholarship.claremont.edu/hmc_theses/212
https://scholarship.claremont.edu/hmc_theses/212Tue, 19 Mar 2019 21:55:39 PDT
The goal of this thesis is to explore the properties of a certain class of sequences, rigid divisibility sequences, generated by the iteration of certain polynomials whose coefficients are algebraic integers. The main goal is to provide, as far as is possible, a classification and description of those polynomials which generate rigid divisibility sequences.
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Brian RiceFunctions of the Binomial Coefficient
https://scholarship.claremont.edu/hmc_theses/211
https://scholarship.claremont.edu/hmc_theses/211Tue, 19 Mar 2019 21:55:29 PDT
The well known binomial coefficient is the building block of Pascal’s triangle. We explore the relationship between functions of the binomial coefficient and Pascal’s triangle, providing proofs of connections between Catalan numbers, determinants, non-intersecting paths, and Baxter permutations.
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Sean PlottCharacteristics of Optimal Solutions to the Sensor Location Problem
https://scholarship.claremont.edu/hmc_theses/210
https://scholarship.claremont.edu/hmc_theses/210Tue, 19 Mar 2019 21:55:21 PDT
Congestion and oversaturated roads pose significant problems and create delays in every major city in the world. Before this problem can be addressed, we must know how much traffic is flowing over the links in the network. We transform a road network into a directed graph with a network flow function, and ask the question, “What subset of vertices (intersections) should be monitored such that knowledge of the flow passing through these vertices is sufficient to calculate the flow everywhere in the graph?” To minimize the cost of placing sensors, we seek the smallest number of monitored vertices. This is known as the Sensor Location Problem (SLP). We explore conditions under which a set of monitored vertices produces a unique solution to the problem and disprove a previous result published on the problem. Finally, we explore a matrix formulation of the problem and present cases when the flow can or cannot be calculated on the graph.
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David MorrisonDot Product Representations of Graphs
https://scholarship.claremont.edu/hmc_theses/209
https://scholarship.claremont.edu/hmc_theses/209Tue, 19 Mar 2019 21:55:10 PDT
We introduce the concept of dot product representations of graphs, giving some motivations as well as surveying the previously known results. We extend these representations to more general fields, looking at the complex numbers, rational numbers, and finite fields. Finally, we study the behavior of dot product representations in field extensions.
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Gregory MintonAn Integrodifferential Equation Modeling 1-D Swarming Behavior
https://scholarship.claremont.edu/hmc_theses/208
https://scholarship.claremont.edu/hmc_theses/208Tue, 19 Mar 2019 21:55:02 PDT
We explore the behavior of an integrodifferential equation used to model one-dimensional biological swarms. In this model, we assume the motion of the swarm is determined by pairwise interactions, which in a continuous setting corresponds to a convolution of the swarm density with a pairwise interaction kernel. For a large class of interaction kernels, we derive conditions that lead to solutions which spread, blow up, or reach a steady state. For a smaller class of interaction kernels, we are able to make more quantitative predictions. In the spreading case, we predict the approximate shape and scaling of a similarity profile, as well as the approximate behavior at the endpoints of the swarm (via solutions to a traveling wave problem). In the blow up case, we derive an upper bound for the time to blow up. In the steady state case, we use previous results to predict the equilibrium swarm density. We support our predictions with numerical simulations. We also consider an extension of the original model which incorporates external forces. By analyzing and simulating particular cases, we determine that the addition of an external force can qualitatively change the behavior of the system.
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Andrew LeverentzIntrinsic Linking and Knotting of Graphs
https://scholarship.claremont.edu/hmc_theses/207
https://scholarship.claremont.edu/hmc_theses/207Tue, 19 Mar 2019 21:54:53 PDT
An analog to intrinsic linking, intrinsic even linking, is explored in the first half of this paper. Four graphs are established to be minor minimal intrinsically even linked, and it is conjectured that they form a complete minor minimal set. Some characterizations are given, using the simplest of the four graphs as an integral part of the arguments, that may be useful in proving the conjecture. The second half of this paper investigates a new approach to intrinsic knotting. By adapting knot energy to graphs, it is hoped that intrinsic knotting can be detected through direct computation. However, graph energies are difficult to compute, and it is unclear whether they can be used to determine whether a graph is intrinsically knotted.
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Kenji KozaiWavelets and Wavelet Sets
https://scholarship.claremont.edu/hmc_theses/206
https://scholarship.claremont.edu/hmc_theses/206Tue, 19 Mar 2019 21:54:45 PDT
Wavelets are functions that are useful for representing signals and approximating other functions. Wavelets sets are defined in terms of Fourier transforms of certain wavelet functions. In this paper, we provide an introduction to wavelets and wavelets sets, examine the preexisting literature on the subject, and investigate an algorithm for creating wavelet sets. This algorithm creates single wavelets, which can be used to create bases for L2(Rn) through dilation and translation. We investigate the convergence properties of the algorithm, and implement the algorithm in Matlab.
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Sara Gussin