Physical Sciences and Mathematics Commons^™

Review: On The Near Periodicity Of Eigenvalues Of Toeplitz Matrices, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

Dynamic Server Allocation At Parallel Queues, Susan E. Martonosi

All HMC Faculty Publications and Research

We explore whether dynamically reassigning servers to parallel queues in response to queue imbalances can reduce average waiting time in those queues. We use approximate dynamic programming methods to determine when servers should be switched, and we compare the performance of such dynamic allocations to that of a pre-scheduled deterministic allocation. Testing our method on both synthetic data and data from airport security checkpoints at Boston Logan International Airport, we find that in situations where the uncertainty in customer arrival rates is significant, dynamically reallocating servers can substantially reduce waiting time. Moreover, we find that intuitive switching strategies that are …

Combinatorics Of Two-Toned Tilings, Arthur T. Benjamin, Phyllis Chinn, Jacob N. Scott '11, Greg Simay

All HMC Faculty Publications and Research

We introduce the function a(r, n) which counts tilings of length n + r that utilize white tiles (whose lengths can vary between 1 and n) and r identical red squares. These tilings are called two-toned tilings. We provide combinatorial proofs of several identities satisfied by a(r, n) and its generalizations, including one that produces kth order Fibonacci numbers. Applications to integer partitions are also provided.

Characteristics Of Optimal Solutions To The Sensor Location Problem, David R. Morrison '08, Susan E. Martonosi

All HMC Faculty Publications and Research

In Bianco et al. (2001), the authors present the Sensor Location Problem: that of locating the minimum number of traffic sensors at intersections of a road network such that the traffic ow on the entire network can be determined. They offer a necessary and sufficient condition on the set of monitored nodes in order for the ow everywhere to be determined. In this paper, we present a counterexample that demonstrates that the condition is not actually sufficient (though it is still necessary). We present a stronger necessary condition for ow calculability, and show that it is a sufficient condition in …

Squaring, Cubing, And Cube Rooting, Arthur T. Benjamin

All HMC Faculty Publications and Research

We present mentally efficient algorithms for mentally squaring and cubing 2-digit and 3-digit numbers and for finding cube roots of numbers with 2-digit or 3-digit answers.

Toric Symmetry Of Cp3, Dagan Karp, Dhruv Ranganathan, Paul Riggins '12, Ursula Whitcher

All HMC Faculty Publications and Research

We exhaustively analyze the toric symmetries of CP^3 and its toric blowups. Our motivation is to study toric symmetry as a computational technique in Gromov-Witten theory and Donaldson-Thomas theory. We identify all nontrivial toric symmetries. The induced nontrivial isomorphisms lift and provide new symmetries at the level of Gromov-Witten Theory and Donaldson-Thomas Theory. The polytopes of the toric varieties in question include the permutohedron, the cyclohedron, the associahedron, and in fact all graph associahedra, among others.

A New Framework For Network Disruption, Susan E. Martonosi, Doug Altner, Michael Ernst, Elizabeth Ferme, Kira Langsjoen, Danika Lindsay, Sean S. Plott '08, Andrew S. Ronan

All HMC Faculty Publications and Research

Traditional network disruption approaches focus on disconnecting or lengthening paths in the network. We present a new framework for network disruption that attempts to reroute flow through critical vertices via vertex deletion, under the assumption that this will render those vertices vulnerable to future attacks. We define the load on a critical vertex to be the number of paths in the network that must flow through the vertex. We present graph-theoretic and computational techniques to maximize this load, firstly by removing either a single vertex from the network, secondly by removing a subset of vertices.

Final Exam, Robert Dawson

Journal of Humanistic Mathematics

When setting an examination, the instructor should always keep in mind who is going to be writing it!

I Am A Number, Sarah Glaz

Journal of Humanistic Mathematics

No abstract provided.

A Mathematician Weighs In On The Evolution Debate, Kris H. Green

Journal of Humanistic Mathematics

There are a variety of reasons underlying the lack of public acceptance for the theory of evolution in the United States. An overlooked cause is related to problems with the mathematics curriculum in the K-12 setting. In this essay, we examine this relationship and propose changes to the mathematics curriculum that could improve mathematical thinking while also providing a basis for understanding theories, like evolution, that are poorly understood.

On Doing Mathematics, Sue Vanhattum

Journal of Humanistic Mathematics

Who is a mathematician? What does it mean to do mathematics? I discuss my process in solving a math problem, and what it meant to me.

A Math Major, Polya, Invention, And Discovery, Susan D'Agostino

Journal of Humanistic Mathematics

An assistant professor of mathematics presents a nonmathematical application of George Polya’s problem-solving strategies. In doing so, she suggests that Polya’s ideas concerning invention and discovery apply to the world beyond the math classroom.

Pedagogy On The Ethnomathematics--Epistemology Nexus: A Manifesto, Ilhan M. Izmirli

Journal of Humanistic Mathematics

In this paper, we will elaborate on a pronouncement that should be at the onset of any study in epistemology and ethnomathematics, namely, we will argue that learners do think mathematically and it is our responsibility as educators to recognize and appreciate their modes of mathematical reasoning.

We will conduct our study in five parts. Following a brief introduction, in the second part, we will briefly discuss some of the critical tenets of epistemology especially as it applies to mathematics. The third part will be devoted to elucidating the basic nomenclature and hypotheses associated with ethnomathematics. In the fourth part …

Existence Of Solutions For A Wave Equation With Non-Monotone Nonlinearity And A Small Parameter, Jose F. Caicedo, Alfonso Castro, Rodrigo Duque

All HMC Faculty Publications and Research

We provide sufficient conditions for the existence of solutions to a semilinear wave equation with non-monotone nonlinearity involving a small parameter. Our results are based on the analysis of a an operator that characterizes the projection onto the kernel of the wave operator subject to periodic-Dirichlet boundary conditions. Such a kernel is infinite dimensional which makes standard compactness arguments inapplicable.

The Combinatorialization Of Linear Recurrences, Arthur T. Benjamin, Halcyon Derks, Jennifer J. Quinn

All HMC Faculty Publications and Research

We provide two combinatorial proofs that linear recurrences with constant coefficients have a closed form based on the roots of its characteristic equation. The proofs employ sign-reversing involutions on weighted tilings.

The Quantum Dialectic, Logan Kelley

Pitzer Senior Theses

A philosophic account of quantum physics. The thesis is divided into two parts. Part I is dedicated to laying the groundwork of quantum physics, and explaining some of the primary difficulties. Subjects of interest will include the principle of locality, the quantum uncertainty principle, and Einstein's criterion for reality. Quantum dilemmas discussed include the double-slit experiment, observations of spin and polarization, EPR, and Bell's theorem. The first part will argue that mathematical-physical descriptions of the world fall short of explaining the experimental observations of quantum phenomenon. The problem, as will be argued, is framework of the physical descriptive schema. Part …

Finding The Beat In Music: Using Adaptive Oscillators, Kate M. Burgers

HMC Senior Theses

The task of finding the beat in music is simple for most people, but surprisingly difficult to replicate in a robot. Progress in this problem has been made using various preprocessing techniques (Hitz 2008; Tomic and Janata 2008). However, a real-time method is not yet available. Methods using a class of oscillators called relay relaxation oscillators are promising. In particular, systems of forced Hopf oscillators (Large 2000; Righetti et al. 2006) have been used with relative success. This work describes current methods of beat tracking and develops a new method that incorporates the best ideas from each existing method and …

Extending List Colorings Of Planar Graphs, Sarah Loeb

HMC Senior Theses

In the study of list colorings of graphs, we assume each vertex of a graph has a specified list of colors from which it may be colored. For planar graphs, it is known that there is a coloring for any list assignment where each list contains five colors. If we have some vertices that are precolored, can we extend this to a coloring of the entire graph? We explore distance constraints when we allow the lists to contain an extra color. For lists of length five, we fix $W$ as a subset of $V(G)$ such that all vertices in $W$ …

Third And Fourth Binomial Coefficients, Arthur T. Benjamin, Jacob N. Scott '11

All HMC Faculty Publications and Research

While formulas for the sums of kth binomial coefficients can be shown inductively or algebraically, these proofs give little insight into the combinatorics involved. We prove formulas for the sums of 3rd and 4th binomial coefficients via purely combinatorial arguments.

Markov Bases For Noncommutative Harmonic Analysis Of Partially Ranked Data, Ann Johnston

HMC Senior Theses

Given the result $v_0$ of a survey and a nested collection of summary statistics that could be used to describe that result, it is natural to ask which of these summary statistics best describe $v_0$. In 1998 Diaconis and Sturmfels presented an approach for determining the conditional significance of a higher order statistic, after sampling a space conditioned on the value of a lower order statistic. Their approach involves the computation of a Markov basis, followed by the use of a Markov process with stationary hypergeometric distribution to generate a sample.This technique for data analysis has become an accepted tool …

Group Actions And Divisors On Tropical Curves, Max B. Kutler

HMC Senior Theses

Tropical geometry is algebraic geometry over the tropical semiring, or min-plus algebra. In this thesis, I discuss the basic geometry of plane tropical curves. By introducing the notion of abstract tropical curves, I am able to pass to a more abstract metric-topological setting. In this setting, I discuss divisors on tropical curves. I begin a study of $G$-invariant divisors and divisor classes.

Verification Of Solutions To The Sensor Location Problem, Chandler May

HMC Senior Theses

Traffic congestion is a serious problem with large economic and environmental impacts. To reduce congestion (as a city planner) or simply to avoid congested channels (as a road user), one might like to accurately know the flow on roads in the traffic network. This information can be obtained from traffic sensors, devices that can be installed on roads or intersections to measure traffic flow. The sensor location problem is the problem of efficiently locating traffic sensors on intersections such that the flow on the entire network can be extrapolated from the readings of those sensors. I build on current research …

Combinatorial Interpretations Of Fibonomial Identities, Elizabeth Reiland

HMC Senior Theses

The Fibonomial numbers are defined by \[ \begin{bmatrix}n \\ k \end{bmatrix} = \frac{\prod_{i=n-k+1} ^{n} F_i}{\prod_{j=1}^{k} F_j} \] where $F_i$ is the $i$th Fibonacci number, defined by the recurrence $F_n=F_{n-1}+F_{n-2}$ with initial conditions $F_0=0,F_1=1$. In the past year, Sagan and Savage have derived a combinatorial interpretation for these Fibonomial numbers, an interpretation that relies upon tilings of a partition and its complement in a given grid.In this thesis, I investigate previously proven theorems for the Fibonomial numbers and attempt to reinterpret and reprove them in light of this new combinatorial description. I also present combinatorial proofs for some identities I did …

Continued Fractions: A New Form, Donald Lee Wiyninger Iii

HMC Senior Theses

While the traditional form of continued fractions is well-documented, a new form, designed to approximate real numbers between 1 and 2, is less well-studied. This report first describes prior research into the new form, describing the form and giving an algorithm for generating approximations for a given real number. It then describes a rational function giving the rational number represented by the continued fraction made from a given tuple of integers and shows that no real number has a unique continued fraction. Next, it describes the set of real numbers that are hardest to approximate; that is, given a positive …

Noise, Delays, And Resonance In A Neural Network, Austin Quan

HMC Senior Theses

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 …

A Census Of Vertices By Generations In Regular Tessellations Of The Plane, Alice Paul '12, Nicholas Pippenger

All HMC Faculty Publications and Research

We consider regular tessellations of the plane as infinite graphs in which q edges and q faces meet at each vertex, and in which p edges and p vertices surround each face. For 1/p + 1/q = 1/2, these are tilings of the Euclidean plane; for 1/p + 1/q < 1/2, they are tilings of the hyperbolic plane. We choose a vertex as the origin, and classify vertices into generations according to their distance (as measured by the number of edges in a shortest path) from the origin. For all p ≥ 3 and q ≥ 3 with 1/p + 1/q ≤ 1/2, we give simple combinatorial derivations of the rational generating functions for the number of vertices in each generation.

Uniqueness Conditions For Low-Rank Matrix Recovery, Yonina C. Eldar, Deanna Needell, Yaniv Plan

CMC Faculty Publications and Research

Low-rank matrix recovery addresses the problem of recovering an unknown low-rank matrix from few linear measurements. Nuclear-norm minimization is a tractible approach with a recent surge of strong theoretical backing. Analagous to the theory of compressed sensing, these results have required random measurements. For example, m >= Cnr Gaussian measurements are sufficient to recover any rank-r n x n matrix with high probability. In this paper we address the theoretical question of how many measurements are needed via any method whatsoever --- tractible or not. We show that for a family of random measurement ensembles, m >= 4nr - 4r^2 …

Acceleration Of Randomized Kaczmarz Method Via The Johnson-Lindenstrauss Lemma, Yonina C. Eldar, Deanna Needell

CMC Faculty Publications and Research

The Kaczmarz method is an algorithm for finding the solution to an overdetermined consistent system of linear equations Ax=b by iteratively projecting onto the solution spaces. The randomized version put forth by Strohmer and Vershynin yields provably exponential convergence in expectation, which for highly overdetermined systems even outperforms the conjugate gradient method. In this article we present a modified version of the randomized Kaczmarz method which at each iteration selects the optimal projection from a randomly chosen set, which in most cases significantly improves the convergence rate. We utilize a Johnson-Lindenstrauss dimension reduction technique to keep the runtime on the …

Bounds For Solid Angles Of Lattices Of Rank Three, Lenny Fukshansky, Sinai Robins

CMC Faculty Publications and Research

We find sharp absolute constants C₁ and C₂ with the following property: every well-rounded lattice of rank 3 in a Euclidean space has a minimal basis so that the solid angle spanned by these basis vectors lies in the interval [C₁,C₂]. In fact, we show that these absolute bounds hold for a larger class of lattices than just well-rounded, and the upper bound holds for all. We state a technical condition on the lattice that may prevent it from satisfying the absolute lower bound on the solid angle, in which case we …

What Do We Mean By Mathematical Proof?, Todd Cadwalladerolsker

Journal of Humanistic Mathematics

Mathematical proof lies at the foundations of mathematics, but there are several notions of what mathematical proof is, or might be. In fact, the idea of mathematical proof continues to evolve. In this article, I review the body of literature that argues that there are at least two widely held meanings of proof, and that the standards of proof are negotiated and agreed upon by the members of mathematical communities. The formal view of proof is contrasted with the view of proofs as arguments intended to convince a reader. These views are examined in the context of the various roles …