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Claremont Colleges

2014

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Articles 61 - 90 of 102

Full-Text Articles in Physical Sciences and Mathematics

Paved With Good Intentions: Analysis Of A Randomized Block Kaczmarz Method, Deanna Needell, Joel A. Tropp Jan 2014

Paved With Good Intentions: Analysis Of A Randomized Block Kaczmarz Method, Deanna Needell, Joel A. Tropp

CMC Faculty Publications and Research

The block Kaczmarz method is an iterative scheme for solving overdetermined least-squares problems. At each step, the algorithm projects the current iterate onto the solution space of a subset of the constraints. This paper describes a block Kaczmarz algorithm that uses a randomized control scheme to choose the subset at each step. This algorithm is the first block Kaczmarz method with an (expected) linear rate of convergence that can be expressed in terms of the geometric properties of the matrix and its submatrices. The analysis reveals that the algorithm is most effective when it is given a good row paving …

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Paul Faulstich’S Reflective Review Of Susan A. Phillips’ Essay, Paul Faulstich Jan 2014

Paul Faulstich’S Reflective Review Of Susan A. Phillips’ Essay, Paul Faulstich

Pitzer Faculty Publications and Research

Paul Faulstich's review of Susan A. Phillips' essay titled, "Huerta del Valle: A New Nonprofit in a Neglected Landscape".

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Pitzer College Outback Preserve Restoration Project, Paul Faulstich Jan 2014

Pitzer College Outback Preserve Restoration Project, Paul Faulstich

Pitzer Faculty Publications and Research

A question we keep asking ourselves in environmental analysis at Pitzer College is whether it’s possible to create modern socionatural systems that are truly sustaining; that is, that avoid the features of contemporary systems in which the human factor dominates to the detriment of the environment. Any genuinely sustainable society must honor diversity— cultural and biological—and, at Pitzer, we’re committed to forging innovative directions for a healthy future. Toward this end, students, along with faculty and staff, have initiated a program of ecological restoration in the Pitzer College Outback Preserve.

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'Studio' Mathematics For Undergraduate Engineers, Lori Bassman, Darryl Yong Jan 2014

'Studio' Mathematics For Undergraduate Engineers, Lori Bassman, Darryl Yong

All Faculty Publications

Applied Mathematics for Engineering is a second year undergradu- ate mathematics requirement for engineering majors at Harvey Mudd College in Claremont, California since 2011. It has been jointly de- signed and taught by the engineering and mathematics departments. The class aims to help students develop confidence in their skill in applying mathematics to solve engineering problems and perseverance for complicated problems; to improve facility at previously learned mathematical skills and to incorporate new tools; and to develop strate- gic competence and better judgment on the correctness of solutions. This article describes the design principles used in creating the class and …

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Symmetries Of Embedded Complete Bipartite Graphs, Erica Flapan, Nicole Lehle '09, Blake Mellor, Matt Pittluck, Xan Vongsathorn '09 Jan 2014

Symmetries Of Embedded Complete Bipartite Graphs, Erica Flapan, Nicole Lehle '09, Blake Mellor, Matt Pittluck, Xan Vongsathorn '09

Pomona Faculty Publications and Research

We characterize which automorphisms of an arbitrary complete bipartite graph Kn,m can be induced by a homeomorphism of some embedding of the graph in S3.

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An Extremal Problem For Characteristic Functions, Stephan Ramon Garcia, Isabelle Chalendar, Williams T. Ross, Dan Timotin Jan 2014

An Extremal Problem For Characteristic Functions, Stephan Ramon Garcia, Isabelle Chalendar, Williams T. Ross, Dan Timotin

Pomona Faculty Publications and Research

Suppose E is a subset of the unit circle T and Hinfinity C Linfinity is the Hardy subalgebra. We examine the problem of finding the distance from the characteristic function of E to znHinfinity. This admits an alternate description as a dual extremal problem. Precise solutions are given in several important cases. The techniques used involve the theory of Toeplitz and Hankel operators as well as the construction of certain conformal mappings.

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Mathematical And Physical Aspects Of Complex Symmetric Operators, Stephan Ramon Garcia, Emil Prodan, Mihai Putinar Jan 2014

Mathematical And Physical Aspects Of Complex Symmetric Operators, Stephan Ramon Garcia, Emil Prodan, Mihai Putinar

Pomona Faculty Publications and Research

Recent advances in the theory of complex symmetric operators are presented and related to current studies in non-hermitian quantum mechanics. The main themes of the survey are: the structure of complex symmetric operators, C-selfadjoint extensions of C-symmetric unbounded operators, resolvent estimates, reality of spectrum, bases of C-orthonormal vectors, and conjugate-linear symmetric operators. The main results are complemented by a variety of natural examples arising in field theory, quantum physics, and complex variables.

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Cycle Lengths Of Θ-Biased Random Permutations, Tongjia Shi Jan 2014

Cycle Lengths Of Θ-Biased Random Permutations, Tongjia Shi

HMC Senior Theses

Consider a probability distribution on the permutations of n elements. If the probability of each permutation is proportional to θK, where K is the number of cycles in the permutation, then we say that the distribution generates a θ-biased random permutation. A random permutation is a special θ-biased random permutation with θ = 1. The mth moment of the rth longest cycle of a random permutation is Θ(nm), regardless of r and θ. The joint moments are derived, and it is shown that the longest cycles of a permutation can either be positively or …

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Experimental Realization Of Slowly Rotating Modes Of Light, Fangzhao A. An Jan 2014

Experimental Realization Of Slowly Rotating Modes Of Light, Fangzhao A. An

HMC Senior Theses

Beams of light can carry spin and orbital angular momentum. Spin angular momentum describes how the direction of the electric field rotates about the propagation axis, while orbital angular momentum describes the rotation of the field amplitude pattern. These concepts are well understood for monochromatic beams, but previous theoretical studies have constructed polychromatic superpositions where the connection between angular momentum and rotation of the electric field becomes much less clear. These states are superpositions of two states of light carrying opposite signs of angular momentum and slightly detuned frequencies. They rotate at the typically small detuning frequency and thus we …

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Existence Of Positive Solutions For A Superlinear Elliptic System With Neumann Boundary Condition, Alfonso Castro, Juan C. Cardeño Jan 2014

Existence Of Positive Solutions For A Superlinear Elliptic System With Neumann Boundary Condition, Alfonso Castro, Juan C. Cardeño

All HMC Faculty Publications and Research

We prove the existence of a positive solution for a class of nonlin- ear elliptic systems with Neumann boundary conditions. The proof combines extensive use of a priori estimates for elliptic problems with Neumann boundary condition and Krasnoselskii's compression-expansion theorem

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The Apple Doesn’T Fall Far From The (Metric) Tree: Equivalence Of Definitions, Asuman Güven Aksoy, Sixian Jin Jan 2014

The Apple Doesn’T Fall Far From The (Metric) Tree: Equivalence Of Definitions, Asuman Güven Aksoy, Sixian Jin

CMC Faculty Publications and Research

In this paper we prove the equivalence of definitions for metric trees and for δ-Hperbolic spaces. We point out how these equivalences can be used to understand the geometric and metric properties of δ-Hperbolic spaces and its relation to CAT(κ) spaces.

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On Approximation Schemes And Compactness, Asuman Güven Aksoy, Jose M. Almira Jan 2014

On Approximation Schemes And Compactness, Asuman Güven Aksoy, Jose M. Almira

CMC Faculty Publications and Research

We present an overview of some results about characterization of compactness in which the concept of approximation scheme has had a role. In particular, we present several results that were proved by the second author, jointly with Luther, a decade ago, when these authors were working on a very general theory of approximation spaces. We then introduce and show the basic properties of a new concept of compactness, which was studied by the first author in the eighties, by using a generalized concept of approximation scheme and its associated Kolmogorov numbers, which generalizes the classical concept of compactness.

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Small Zeros Of Quadratic Forms Outside A Union Of Varieties, Wai Kiu Chan, Lenny Fukshansky, Glenn R. Henshaw Jan 2014

Small Zeros Of Quadratic Forms Outside A Union Of Varieties, Wai Kiu Chan, Lenny Fukshansky, Glenn R. Henshaw

CMC Faculty Publications and Research

Let be a quadratic form in variables defined on a vector space over a global field , and be a finite union of varieties defined by families of homogeneous polynomials over . We show that if contains a nontrivial zero of , then there exists a linearly independent collection of small-height zeros of in , where the height bound does not depend on the height of , only on the degrees of its defining polynomials. As a corollary of this result, we show that there exists a small-height maximal totally isotropic subspace of the quadratic space such that is not …

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Well-Rounded Zeta-Function Of Planar Arithmetic Lattices, Lenny Fukshansky Jan 2014

Well-Rounded Zeta-Function Of Planar Arithmetic Lattices, Lenny Fukshansky

CMC Faculty Publications and Research

We investigate the properties of the zeta-function of well-rounded sublattices of a fixed arithmetic lattice in the plane. In particular, we show that this function has abscissa of convergence at s=1 with a real pole of order 2, improving upon a result of Stefan Kühnlein. We use this result to show that the number of well-rounded sublattices of a planar arithmetic lattice of index less than or equal to N is O(N log N) as N → ∞. To obtain these results, we produce a description of integral well-rounded sublattices of a fixed planar integral well-rounded lattice and investigate convergence …

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On The Geometry Of Cyclic Lattices, Lenny Fukshansky, Xun Sun Jan 2014

On The Geometry Of Cyclic Lattices, Lenny Fukshansky, Xun Sun

CMC Faculty Publications and Research

Cyclic lattices are sublattices of ZN that are preserved under the rotational shift operator. Cyclic lattices were introduced by D.~Micciancio and their properties were studied in the recent years by several authors due to their importance in cryptography. In particular, Peikert and Rosen showed that on cyclic lattices in prime dimensions, the shortest independent vectors problem SIVP reduces to the shortest vector problem SVP with a particularly small loss in approximation factor, as compared to general lattices. In this paper, we further investigate geometric properties of cyclic lattices. Our main result is a counting estimate for the number of well-rounded …

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Using Topology To Explore Mathematics Education Reform, Carling Sugarman Jan 2014

Using Topology To Explore Mathematics Education Reform, Carling Sugarman

HMC Senior Theses

Mathematics education is a constant topic of conversation in the United States. Many attempts have been made historically to reform teaching methods and improve student results. Particularly, past ideas have emphasized problem-solving to make math feel more applicable and enjoyable. Many have additionally tackled the widespread problem of “math anxiety” by creating lessons that are more discussion-based than drill-based to shift focus from speed and accuracy. In my project, I explored past reform goals and some added goals concerning students' perceptions of mathematics. To do so, I created and tested a pilot workshop in topology, a creative and intuitive field, …

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A New Subgroup Chain For The Finite Affine Group, David Alan Lingenbrink Jr. Jan 2014

A New Subgroup Chain For The Finite Affine Group, David Alan Lingenbrink Jr.

HMC Senior Theses

The finite affine group is a matrix group whose entries come from a finite field. A natural subgroup consists of those matrices whose entries all come from a subfield instead. In this paper, I will introduce intermediate sub- groups with entries from both the field and a subfield. I will also examine the representations of these intermediate subgroups as well as the branch- ing diagram for the resulting subgroup chain. This will allow us to create a fast Fourier transform for the group that uses asymptotically fewer opera- tions than the brute force algorithm.

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Arithmetical Graphs, Riemann-Roch Structure For Lattices, And The Frobenius Number Problem, Jeremy Usatine Jan 2014

Arithmetical Graphs, Riemann-Roch Structure For Lattices, And The Frobenius Number Problem, Jeremy Usatine

HMC Senior Theses

If R is a list of positive integers with greatest common denominator equal to 1, calculating the Frobenius number of R is in general NP-hard. Dino Lorenzini defines the arithmetical graph, which naturally arises in arithmetic geometry, and a notion of genus, the g-number, that in specific cases coincides with the Frobenius number of R. A result of Dino Lorenzini's gives a method for quickly calculating upper bounds for the g-number of arithmetical graphs. We discuss the arithmetic geometry related to arithmetical graphs and present an example of an arithmetical graph that arises in this context. We also discuss the …

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Fast Algorithms For Analyzing Partially Ranked Data, Matthew Mcdermott Jan 2014

Fast Algorithms For Analyzing Partially Ranked Data, Matthew Mcdermott

HMC Senior Theses

Imagine your local creamery administers a survey asking their patrons to choose their five favorite ice cream flavors. Any data collected by this survey would be an example of partially ranked data, as the set of all possible flavors is only ranked into subsets of the chosen flavors and the non-chosen flavors. If the creamery asks you to help analyze this data, what approaches could you take? One approach is to use the natural symmetries of the underlying data space to decompose any data set into smaller parts that can be more easily understood. In this work, I describe …

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Reed's Conjecture And Cycle-Power Graphs, Alexa Serrato Jan 2014

Reed's Conjecture And Cycle-Power Graphs, Alexa Serrato

HMC Senior Theses

Reed's conjecture is a proposed upper bound for the chromatic number of a graph. Reed's conjecture has already been proven for several families of graphs. In this paper, I show how one of those families of graphs can be extended to include additional graphs and also show that Reed's conjecture holds for a family of graphs known as cycle-power graphs, and also for their complements.

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Characterizing Forced Communication In Networks, Samuel C. Gutekunst Jan 2014

Characterizing Forced Communication In Networks, Samuel C. Gutekunst

HMC Senior Theses

This thesis studies a problem that has been proposed as a novel way to disrupt communication networks: the load maximization problem. The load on a member of a network represents the amount of communication that the member is forced to be involved in. By maximizing the load on an important member of the network, we hope to increase that member's visibility and susceptibility to capture. In this thesis we characterize load as a combinatorial property of graphs and expose possible connections between load and spectral graph theory. We specifically describe the load and how it changes in several canonical classes …

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There And Back Again: Elliptic Curves, Modular Forms, And L-Functions, Allison F. Arnold-Roksandich Jan 2014

There And Back Again: Elliptic Curves, Modular Forms, And L-Functions, Allison F. Arnold-Roksandich

HMC Senior Theses

L-functions form a connection between elliptic curves and modular forms. The goals of this thesis will be to discuss this connection, and to see similar connections for arithmetic functions.

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Infinitely Many Rotationally Symmetric Solutions To A Class Of Semilinear Laplace-Beltrami Equations On The Unit Sphere, Emily M. Fischer Jan 2014

Infinitely Many Rotationally Symmetric Solutions To A Class Of Semilinear Laplace-Beltrami Equations On The Unit Sphere, Emily M. Fischer

HMC Senior Theses

I show that a class of semilinear Laplace-Beltrami equations has infinitely many solutions on the unit sphere which are symmetric with respect to rotations around some axis. This equation corresponds to a singular ordinary differential equation, which we solve using energy analysis. We obtain a Pohozaev-type identity to prove that the energy is continuously increasing with the initial condition and then use phase plane analysis to prove the existence of infinitely many solutions.

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A Mathematical Framework For Unmanned Aerial Vehicle Obstacle Avoidance, Sorathan Chaturapruek Jan 2014

A Mathematical Framework For Unmanned Aerial Vehicle Obstacle Avoidance, Sorathan Chaturapruek

HMC Senior Theses

The obstacle avoidance navigation problem for Unmanned Aerial Vehicles (UAVs) is a very challenging problem. It lies at the intersection of many fields such as probability, differential geometry, optimal control, and robotics. We build a mathematical framework to solve this problem for quadrotors using both a theoretical approach through a Hamiltonian system and a machine learning approach that learns from human sub-experts' multiple demonstrations in obstacle avoidance. Prior research on the machine learning approach uses an algorithm that does not incorporate geometry. We have developed tools to solve and test the obstacle avoidance problem through mathematics.

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Experiments On Surfactants And Thin Fluid Films, Peter Megson Jan 2014

Experiments On Surfactants And Thin Fluid Films, Peter Megson

HMC Senior Theses

We investigate the spatiotemporal dynamics of a surfactant monolayer on a thin fluid film spreading inward into a region devoid of surfactant, a system motivated by the alveolus of the human lung. We perform experiments that simultaneously measure the fluid height profile and the fluorescence intensity due to our fluorescent surfactant, NBD-PC. We perform experiments on both a Newtonian layer of glycerol and a shear-thinning fluid layer consisting of xanthan gum mixed with glycerol. We can very successfully extract height profiles on the xanthan gum fluid, although the simultaneous measurement of fluorescent intensity profiles proved problematic, as the laser tended …

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Energy-Driven Pattern Formation In Planar Dipole-Dipole Systems, Jaron P. Kent-Dobias Jan 2014

Energy-Driven Pattern Formation In Planar Dipole-Dipole Systems, Jaron P. Kent-Dobias

HMC Senior Theses

A variety of two-dimensional fluid systems, known as dipole-mediated systems, exhibit a dipole-dipole interaction between their fluid constituents. The com- petition of this repulsive dipolar force with the cohesive fluid forces cause these systems to form intricate and patterned structures in their boundaries. In this thesis, we show that the microscopic details of any such system are irrelevant in the macroscopic limit and contribute only to a constant offset in the system’s energy. A numeric model is developed, and some important stable domain morphologies are characterized. Previously unresolved bifurcating branches are explored. Finally, by applying a random energy background to …

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Knotted And Linked Products Of Recombination On T(2,N)#T(2,M) Substrates, Erica Flapan, Jeremy Grevet, Qi Li, Chen Daisy Sun, Helen Wong Jan 2014

Knotted And Linked Products Of Recombination On T(2,N)#T(2,M) Substrates, Erica Flapan, Jeremy Grevet, Qi Li, Chen Daisy Sun, Helen Wong

Pomona Faculty Publications and Research

We develop a topological model of site-specific recombination that applies to substrates which are the connected sum of two torus links of the form T(2,n)#T(2,m). Then we use our model to prove that all knots and links that can be produced by site-specific recombination on such substrates are contained in one of two families, which we illustrate.

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Topological Symmetry Groups Of Small Complete Graphs, Erica Flapan, Dwayne Chambers Jan 2014

Topological Symmetry Groups Of Small Complete Graphs, Erica Flapan, Dwayne Chambers

Pomona Faculty Publications and Research

Topological symmetry groups were originally introduced to study the symmetries of non-rigid molecules, but have since been used to study the symmetries of any graph embedded in R3. In this paper, we determine for each complete graph Kn with n ≤ 6, what groups can occur as topological symmetry groups or orientation preserving topological symmetry groups of some embedding of the graph in R3.

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A Look Into The Industry Of Video Games Past, Present, And Yet To Come, Chad Hadzinsky Jan 2014

A Look Into The Industry Of Video Games Past, Present, And Yet To Come, Chad Hadzinsky

CMC Senior Theses

Since its inception, the video game industry has been both a new medium for art and innovation as well as a major driving force in the advancements of many technologies. The often overlooked video game industry has turned from a hobby to a multi-billion dollar industry in its short, forty year life. People of all ages and genders across the world are playing video games at a higher clip than ever before. With so many new gamers and emerging technologies, it is an exciting time for the industry. The landscape is constantly changing and successful business models of the past …

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Finding Zeros Of Rational Quadratic Forms, John F. Shaughnessy Jan 2014

Finding Zeros Of Rational Quadratic Forms, John F. Shaughnessy

CMC Senior Theses

In this thesis, we introduce the notion of quadratic forms and provide motivation for their study. We begin by discussing Diophantine equations, the field of p-adic numbers, and the Hasse-Minkowski Theorem that allows us to use p-adic analysis determine whether a quadratic form has a rational root. We then discuss search bounds and state Cassels' Theorem for small-height zeros of rational quadratic forms. We end with a proof of Cassels' Theorem and suggestions for further reading.

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