HMC Senior Theses Copyright (c) 2013 Claremont Colleges All rights reserved. http://scholarship.claremont.edu/hmc_theses Recent documents in HMC Senior Theses en-us Tue, 29 Jan 2013 16:22:22 PST 3600 Algebraic Methods for Log-Linear Models http://scholarship.claremont.edu/hmc_theses/41 http://scholarship.claremont.edu/hmc_theses/41 Wed, 25 Jul 2012 16:44:08 PDT Techniques from representation theory (Diaconis, 1988) and algebraic geometry (Drton et al., 2008) have been applied to the statistical analysis of discrete data with log-linear models. With these ideas in mind, we discuss the selection of sparse log-linear models, especially for binary data and data on other structured sample spaces. When a sample space and its symmetry group satisfy certain conditions, we construct a natural spanning set for the space of functions on the sample space which respects the isotypic decomposition; these vectors may be used in algorithms for model selection. The construction is explicitly carried out for the case of binary data.

]]> Aaron Pribadi Switching Between Cooperation and Competition in Social Selection http://scholarship.claremont.edu/hmc_theses/40 http://scholarship.claremont.edu/hmc_theses/40 Wed, 25 Jul 2012 16:44:06 PDT Roughgarden et al. (2006) proposed a theory called social selection as a behavioral game theoretic model for sexual reproduction that incorporates both competition and cooperation in 2006. Players oscillate between playing competitively to maximize their individual fitness, leading to a Nash Competitive Equilibrium, and playing cooperatively to maximize a team fitness function, leading to a Nash Bargaining Solution. Roughgarden et al. (2006) gives rates of change for both the competitive state and the cooperative state, but does not explain her rates or how to switch between the states in sufficient detail.

We test and rederive the rates, critiquing an assumption that the derivation of such a rate must make, as well as create a probabilistic model that switches between the two states. We test our model on the reproductive behaviors of Symphodus tinca, the peacock wrasse. The results follow the trajectory of the reproductive strategies for the wrasse throughout the breeding system, suggesting that cooperation could be a mechanism through which wrasse change their reproductive behaviors. However, the inputs to the model need to be analyzed more critically. Future work could include deriving rates for competitive play and cooperative play that do not rely on assumptions of being able to quantify strategy allocation proportion and refining the model and drawing generalized conclusions about it.

]]> August Guang Detecting Covert Members of Terrorist Networks http://scholarship.claremont.edu/hmc_theses/39 http://scholarship.claremont.edu/hmc_theses/39 Wed, 25 Jul 2012 16:44:04 PDT Terrorism threatens both international peace and security and is a national concern. It is believed that terrorist organizations rely heavily on a few key leaders and that destroying such an organization's leadership is essential to reducing its influence. Martonosi et al. (2011) argues that increasing the amount of communication through a key leader increases the likelihood of detection. If we model a covert organization as a social network where edges represent communication between members, we want to determine the subset of members to remove that maximizes the amount of communication through the key leader. A mixed-integer linear program representing this problem is presented as well as a decomposition for this optimization problem. As these approaches prove impractical for larger graphs, often running out of memory, the last section focuses on structural characteristics of vertices and subsets that increase communication. Future work should develop these structural properties as well as heuristics for solving this problem.

]]> Alice Paul Searching Stars for a Moving Hider http://scholarship.claremont.edu/hmc_theses/38 http://scholarship.claremont.edu/hmc_theses/38 Wed, 25 Jul 2012 16:44:01 PDT In a search game, a seeker searches for a hider in some space. The seeker wishes to find the hider as quickly as possible, and the hider wishes to avoid capture as long as possible. In this paper, I will focus on the case where the search space is a star, and the only information the seeker has is the speed of the hider. I will provide algorithms for some cases where the seeker is guaranteed to find the hider and prove optimality for some of these cases. Also, I will look at some cases where the hider can avoid capture indefinitely. I will also present some results for searching on trees.

]]> Jennifer Iglesias Uniquely Solvable Puzzles and Fast Matrix Multiplication http://scholarship.claremont.edu/hmc_theses/37 http://scholarship.claremont.edu/hmc_theses/37 Wed, 25 Jul 2012 16:43:59 PDT In 2003 Cohn and Umans introduced a new group-theoretic framework for doing fast matrix multiplications, with several conjectures that would imply the matrix multiplication exponent $\omega$ is 2. Their methods have been used to match one of the fastest known algorithms by Coppersmith and Winograd, which runs in $O(n^{2.376})$ time and implies that $\omega \leq 2.376$. This thesis discusses the framework that Cohn and Umans came up with and presents some new results in constructing combinatorial objects called uniquely solvable puzzles that were introduced in a 2005 follow-up paper, and which play a crucial role in one of the $\omega = 2$ conjectures.

]]> Palmer Mebane Approval Voting in Box Societies http://scholarship.claremont.edu/hmc_theses/36 http://scholarship.claremont.edu/hmc_theses/36 Wed, 25 Jul 2012 16:43:58 PDT Under approval voting, every voter may vote for any number of canditates. To model approval voting, we let a political spectrum be the set of all possible political positions, and let each voter have a subset of the spectrum that they approve, called an approval region. The fraction of all voters who approve the most popular position is the agreement proportion for the society. We consider voting in societies whose political spectrum is modeled by $d$-dimensional space ($\mathbb{R}^d$) with approval regions defined by axis-parallel boxes. For such societies, we first consider a Tur&#aacute;n-type problem, attempting to find the maximum agreement between pairs of voters for a society with a given level of overall agreement. We prove a lower bound on this maximum agreement and find in the literature a proof that the lower bound is optimal. By this result we find that for sufficiently large $n$, any $n$-voter box society in $\mathbb{R}^d$ where at least $\alpha\binom{n}{2}$ pairs of voters agree on some position must have a position contained in $\beta n$ approval regions, where $\alpha = 1-(1-\beta)^2/d$. We also consider an extension of this problem involving projections of approval regions to axes. Finally we consider the question of $(k,m)$-agreeable box societies, where a society is said to be $(k, m)$-agreeable if among every $m$ voters, some $k$ approve a common position. In the $m = 2k - 1$ case, we use methods from graph theory to prove that the agreement proportion is at least $(2d)^{-1}$ for any integer $k \ge 2.$

]]> Patrick Eschenfeldt Explorations of the Aldous Order on Representations of the Symmetric Group http://scholarship.claremont.edu/hmc_theses/35 http://scholarship.claremont.edu/hmc_theses/35 Wed, 25 Jul 2012 16:43:55 PDT The Aldous order is an ordering of representations of the symmetric group motivated by the Aldous Conjecture, a conjecture about random processes proved in 2009. In general, the Aldous order is very difficult to compute, and the proper relations have yet to be determined even for small cases. However, by restricting the problem down to Young-Jucys-Murphy elements, the problem becomes explicitly combinatorial. This approach has led to many novel insights, whose proofs are simple and elegant. However, there remain many open questions related to the Aldous Order, both in general and for the Young-Jucys-Murphy elements.

]]> Jack Newhouse A Combinatorial Approach to $r$-Fibonacci Numbers http://scholarship.claremont.edu/hmc_theses/34 http://scholarship.claremont.edu/hmc_theses/34 Wed, 25 Jul 2012 16:43:52 PDT In this paper we explore generalized “$r$-Fibonacci Numbers” using a combinatorial “tiling” interpretation. This approach allows us to provide simple, intuitive proofs to several identities involving $r$-Fibonacci Numbers presented by F.T. Howard and Curtis Cooper in the August, 2011, issue of the Fibonacci Quarterly. We also explore a connection between the generalized Fibonacci numbers and a generalized form of binomial coefficients.

]]> Curtis Heberle A Refined Saddle Point Theorem and Applications http://scholarship.claremont.edu/hmc_theses/33 http://scholarship.claremont.edu/hmc_theses/33 Wed, 25 Jul 2012 16:43:50 PDT Under adequate conditions on $g$, we show the density in $L^2((0,\pi),(0,2\pi))$ of the set of functions $p$ for which \begin{equation*} u_{tt}(x,t)-u_{xx}(x,t)= g(u(x,t)) + p(x,t) \end{equation*} has a weak solution subject to \begin{equation*} \begin{aligned} u(x,t)&=u(x,t+2\pi)\\ u(0,t)&=u(\pi,t)=0. \end{aligned} \end{equation*}

To achieve this, we prove a Saddle Point Principle by means of a refined variant of the deformation lemma of Rabinowitz.

Generally, inf-sup techniques allow the characterization of critical values by taking the minimum of the maximae on some particular class of sets. In this version of the Saddle Point Principle, we introduce sufficient conditions for the existence of a saddle-structure which is not restricted to finite-dimensional subspaces.

]]> Harris Enniss Nonlinear Wave Equations and Solitary Wave Solutions in Mathematical Physics http://scholarship.claremont.edu/hmc_theses/32 http://scholarship.claremont.edu/hmc_theses/32 Wed, 25 Jul 2012 16:43:49 PDT In this report, we study various nonlinear wave equations arising in mathematical physics and investigate the existence of solutions to these equations using variational methods. In particular, we look for particle-like traveling wave solutions known as solitary waves. This study is motivated by the prevalence of solitary waves in applications and the rich mathematical structure of the nonlinear wave equations from which they arise. We focus on a semilinear perturbation of Maxwell's equations and the nonlinear Klein - Gordon equation coupled with Maxwell's equations. Physical ramifications of these equations are also discussed.

]]> Trevor Caldwell Gromov-Witten Theory of Blowups of Toric Threefolds http://scholarship.claremont.edu/hmc_theses/31 http://scholarship.claremont.edu/hmc_theses/31 Wed, 25 Jul 2012 16:43:47 PDT We use toric symmetry and blowups to study relationships in the Gromov-Witten theories of $\mathbb{P}^3$ and $\mathbb{P}^1\!\times\!\mathbb{P}^1\!\times\!\mathbb{P}^1$. These two spaces are birationally equivalent via the common blowup space, the permutohedral variety. We prove an equivalence of certain invariants on blowups at only points of $\mathbb{P}^3$ and $\mathbb{P}^1\!\times\!\mathbb{P}^1\!\times\!\mathbb{P}^1$ by showing that these invariants descend from the blowup. Further, the permutohedral variety has nontrivial automorphisms of its cohomology coming from toric symmetry. These symmetries can be forced to descend to the blowups at just points of $\mathbb{P}^3$ and $\mathbb{P}^1\!\times\!\mathbb{P}^1\!\times\!\mathbb{P}^1$. Enumerative consequences are discussed.

]]> Dhruv Ranganathan Russian Mathematical Pedagogy in Reasoning Mind http://scholarship.claremont.edu/hmc_theses/30 http://scholarship.claremont.edu/hmc_theses/30 Wed, 25 Jul 2012 16:43:45 PDT Reasoning Mind (RM) incorporates aspects of Russian mathematics pedagogy and curricula into its online math program. Our investigation identifies typical attributes of Russian pedagogy discussed in news articles and publications by Russian education experts, then determines how these attributes arise in RM. Analysis of RM's implementation of the characteristics reveals more successful inclusion of curricular attributes than classroom strategies. Thus, we outline classroom techniques that could be assimilated into RM to provide a more Russian learning experience along with students' exposure to Russian-style curricula.

]]> Maia Valcarce Analysis of Swarm Behavior in Two Dimensions http://scholarship.claremont.edu/hmc_theses/29 http://scholarship.claremont.edu/hmc_theses/29 Wed, 25 Jul 2012 16:43:43 PDT We investigate the steady state solutions that can exist for a two dimensional swarm of biological organisms, which have pairwise social interaction forces. The three steady states we investigate using a continuum model are a ribbon migrating swarm, a circular migrating swarm, and a milling swarm. We solve these numerically by reformulating the integral equation that arises from the continuum model as an energy minimization problem. For the ribbon migrating solution, we are able to determine an analytic solution from Carleman's equation which arises after an asymptotic expansion of the social interaction potential. Using this technique we are able to show the existence of a square root singularity that emerges at the boundary of the compactly supported swarm. The analytic solution agrees with the numerical solution for certain parameter values in the social interaction potential. We then demonstrate the existence of solutions for a migrating and milling circular swarm which contain a square root singularity. The milling swarm looks similar to the infinite ribbon, so we are able to use an asymptotic expansion of the potential to obtain an analytic solution in this case as well. The singularities in the density of the swarm suggest that the Morse potential should not be used for modeling biological swarming.

]]> Louis Ryan Counting Vertices in Isohedral Tilings http://scholarship.claremont.edu/hmc_theses/28 http://scholarship.claremont.edu/hmc_theses/28 Wed, 25 Jul 2012 16:43:41 PDT An isohedral tiling is a tiling of congruent polygons that are also transitive, which is to say the configuration of degrees of vertices around each face is identical. Regular tessellations, or tilings of congruent regular polygons, are a special case of isohedral tilings. Viewing these tilings as graphs in planes, both Euclidean and non-Euclidean, it is possible to pose various problems of enumeration on the respective graphs. In this paper, we investigate some near-regular isohedral tilings of triangles and quadrilaterals in the hyperbolic plane. For these tilings we enumerate vertices as classified by number of edges in the shortest path to a given origin, by combinatorially deriving their respective generating functions.

]]> John Choi Flatterland: The Play http://scholarship.claremont.edu/hmc_theses/27 http://scholarship.claremont.edu/hmc_theses/27 Wed, 25 Jul 2012 16:43:39 PDT This script is an adaptation of the popular science novel Flatterland: Like Flatland, Only More So by Ian Stewart. It brings new life to mathematical ideas and topics. By bringing math to the stage, concepts are presented in a more friendly and accessible manner. This play is intended to generate new interest in and expose new topics to an audience of nonmathematicians.

]]> Kym Louie Approval Voting Theory with Multiple Levels of Approval http://scholarship.claremont.edu/hmc_theses/26 http://scholarship.claremont.edu/hmc_theses/26 Wed, 25 Jul 2012 16:43:38 PDT Approval voting is an election method in which voters may cast votes for as many candidates as they desire. This can be modeled mathematically by associating to each voter an approval region: a set of potential candidates they approve. In this thesis we add another level of approval somewhere in between complete approval and complete disapproval. More than one level of approval may be a better model for a real-life voter's complex decision making. We provide a new definition for intersection that supports multiple levels of approval. The case of pairwise intersection is studied, and the level of agreement among voters is studied under restrictions on the relative size of each voter's preferences. We derive upper and lower bounds for the percentage of agreement based on the percentage of intersection.

]]> Craig Burkhart Community Control and Compensation: An Analysis for Successful Intellectual Property Right Legislation for Access and Benefit Sharing in Latin American Nations http://scholarship.claremont.edu/hmc_theses/25 http://scholarship.claremont.edu/hmc_theses/25 Tue, 13 Dec 2011 14:47:55 PST Abstract: Indigenous communities have worked for centuries to develop systems of knowledge pertaining to their local environments. Much of the knowledge that has been directly acquired or passed down over generations is of marketable use to corporations, especially in the pharmaceutical industry. Upon gaining the necessary information to convert traditional knowledge into a marketable entity, the corporation will place a patent on the product of their research and development and reap the monetary benefits under the protection of intellectual property legislation. Without appropriate benefit sharing, indigenous communities are robbed of their cumulative innovation and development and denied access to the very medicines that they assisted in development. This study will examine the efforts made by indigenous communities to develop benefit-sharing agreements under national ‘sui generis’ legislation and the international legislation of the Agreement on Trade-Related Aspects of Intellectual Property Rights (TRIPS) and the Convention on Biological Diversity (CBD).

]]> Laurie K. Egan Noise, Delays, and Resonance in a Neural Network http://scholarship.claremont.edu/hmc_theses/24 http://scholarship.claremont.edu/hmc_theses/24 Mon, 10 Oct 2011 11:33:55 PDT A stochastic-delay differential equation (SDDE) model of a small neural network with recurrent inhibition is presented and analyzed. The model exhibits unexpected transient behavior: oscillations that occur at the boundary of the basins of attraction when the system is bistable. These are known as delay-induced transitory oscillations (DITOs). This behavior is analyzed in the context of stochastic resonance, an unintuitive, though widely researched phenomenon in physical bistable systems where noise can play in constructive role in strengthening an input signal. A method for modeling the dynamics using a probabilistic three-state model is proposed, and supported with numerical evidence. The potential implications of this dynamical phenomenon to nocturnal frontal lobe epilepsy (NFLE) are also discussed.

]]> Austin Quan Understanding Voting for Committees Using Wreath Products http://scholarship.claremont.edu/hmc_theses/23 http://scholarship.claremont.edu/hmc_theses/23 Fri, 07 Oct 2011 10:18:48 PDT In this thesis, we construct an algebraic framework for analyzing committee elections. In this framework, module homomorphisms are used to model positional voting procedures. Using the action of the wreath product group S2[Sn] on these modules, we obtain module decompositions which help us to gain an understanding of the module homomorphism. We use these decompositions to construct some interesting voting paradoxes.

]]> Stephen C. Lee Optimizing Restaurant Reservation Scheduling http://scholarship.claremont.edu/hmc_theses/22 http://scholarship.claremont.edu/hmc_theses/22 Fri, 07 Oct 2011 10:18:46 PDT We consider a yield-management approach to determine whether a restaurant should accept or reject a pending reservation request. This approach was examined by Bossert (2009), where the decision for each request is evaluated by an approximate dynamic program (ADP) that bases its decision on a realization of future demand. This model only considers assigning requests to their desired time slot. We expand Bossert's ADP model to incorporate an element of flexibility that allows requests to be assigned to a time slot that differs from the customer's initially requested time. To estimate the future seat utilization given a particular decision, a new heuristic is presented which evaluates time-slot/table assignments based on the expected number of unused seats likely to result from a given assignment. When compared against naive seating models, the proposed model produced average gains in seat utilization of 25%.

]]> Jacob Feldman