Open Access. Powered by Scholars. Published by Universities.®

Physical Sciences and Mathematics Commons™

Open Access. Powered by Scholars. Published by Universities.®

HMC Senior Theses

Theses/Dissertations

2014

Discipline
Keyword

Articles 1 - 13 of 13

Full-Text Articles in Physical Sciences and Mathematics

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 …

Go to article

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 …

Go to article

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, …

Go to article

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.

Go to article

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 …

Go to article

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 …

Go to article

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.

Go to article

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 …

Go to article

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.

Go to article

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.

Go to article

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.

Go to article

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 …

Go to article

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 …

Go to article