Physical Sciences and Mathematics Commons^™

Monte Carlo Approx. Methods For Stochastic Optimization, John Fowler

Pomona Senior Theses

This thesis provides an overview of stochastic optimization (SP) problems and looks at how the Sample Average Approximation (SAA) method is used to solve them. We review several applications of this problem-solving technique that have been published in papers over the last few years. The number and variety of the examples should give an indication of the usefulness of this technique. The examples also provide opportunities to discuss important aspects of SPs and the SAA method including model assumptions, optimality gaps, the use of deterministic methods for finite sample sizes, and the accelerated Benders decomposition algorithm. We also give a …

Interval Graphs, Joyce C. Yang

HMC Senior Theses

We examine the problem of counting interval graphs. We answer the question posed by Hanlon, of whether the formal power series generating function of the number of interval graphs on n vertices has a positive radius of convergence. We have found that it is zero. We have obtained a lower bound and an upper bound on the number of interval graphs on n vertices. We also study the application of interval graphs to the dynamic storage allocation problem. Dynamic storage allocation has been shown to be NP-complete by Stockmeyer. Coloring interval graphs on-line has applications to dynamic storage allocation. The …

Graph Cohomology, Matthew Lin

HMC Senior Theses

What is the cohomology of a graph? Cohomology is a topological invariant and encodes such information as genus and euler characteristic. Graphs are combinatorial objects which may not a priori admit a natural and isomorphism invariant cohomology ring. In this project, given any finite graph G, we constructively define a cohomology ring H*(G) of G. Our method uses graph associahedra and toric varieties. Given a graph, there is a canonically associated convex polytope, called the graph associahedron, constructed from G. In turn, a convex polytope uniquely determines a toric variety. We synthesize these results, and describe the …

Adinkras And Arithmetical Graphs, Madeleine Weinstein

HMC Senior Theses

Adinkras and arithmetical graphs have divergent origins. In the spirit of Feynman diagrams, adinkras encode representations of supersymmetry algebras as graphs with additional structures. Arithmetical graphs, on the other hand, arise in algebraic geometry, and give an arithmetical structure to a graph. In this thesis, we will interpret adinkras as arithmetical graphs and see what can be learned.

Our work consists of three main strands. First, we investigate arithmetical structures on the underlying graph of an adinkra in the specific case where the underlying graph is a hypercube. We classify all such arithmetical structures and compute some of the corresponding …

Fibonomial Tilings And Other Up-Down Tilings, Robert Bennett

HMC Senior Theses

The Fibonomial coefficients are a generalization of the binomial coefficients with a rather nice combinatorial interpretation. While the ordinary binomial coefficients count lattice paths in a grid, the Fibonomial coefficients count the number of ways to draw a lattice path in a grid and then Fibonacci-tile the regions above and below the path in a particular way. We may forgo a literal tiling interpretation and, instead of the Fibonacci numbers, use an arbitrary function to count the number of ways to "tile" the regions of the grid delineated by the lattice path. When the function is a combinatorial sequence such …

Convexity Of Neural Codes, Robert Amzi Jeffs

HMC Senior Theses

An important task in neuroscience is stimulus reconstruction: given activity in the brain, what stimulus could have caused it? We build on previous literature which uses neural codes to approach this problem mathematically. A neural code is a collection of binary vectors that record concurrent firing of neurons in the brain. We consider neural codes arising from place cells, which are neurons that track an animal's position in space. We examine algebraic objects associated to neural codes, and completely characterize a certain class of maps between these objects. Furthermore, we show that such maps have natural geometric implications related to …

A Bound On The Number Of Spanning Trees In Bipartite Graphs, Cheng Wai Koo

HMC Senior Theses

Richard Ehrenborg conjectured that in a bipartite graph G with parts X and Y, the number of spanning trees is at most the product of the vertex degrees divided by |X|⋅|Y|. We make two main contributions. First, using techniques from spectral graph theory, we show that the conjecture holds for sufficiently dense graphs containing a cut vertex of degree 2. Second, using electrical network analysis, we show that the conjecture holds under the operation of removing an edge whose endpoints have sufficiently large degrees.

Our other results are combinatorial proofs that the conjecture holds for …

Realizing The 2-Associahedron, Patrick N. Tierney

HMC Senior Theses

The associahedron has appeared in numerous contexts throughout the field of mathematics. By representing the associahedron as a poset of tubings, Michael Carr and Satyan L. Devadoss were able to create a gener- alized version of the associahedron in the graph-associahedron. We seek to create an alternative generalization of the associahedron by considering a particle-collision model. By extending this model to what we dub the 2- associahedron, we seek to further understand the space of generalizations of the associahedron.

Topological Data Analysis For Systems Of Coupled Oscillators, Alec Dunton

HMC Senior Theses

Coupled oscillators, such as groups of fireflies or clusters of neurons, are found throughout nature and are frequently modeled in the applied mathematics literature. Earlier work by Kuramoto, Strogatz, and others has led to a deep understanding of the emergent behavior of systems of such oscillators using traditional dynamical systems methods. In this project we outline the application of techniques from topological data analysis to understanding the dynamics of systems of coupled oscillators. This includes the examination of partitions, partial synchronization, and attractors. By looking for clustering in a data space consisting of the phase change of oscillators over a …

Line-Of-Sight Pursuit And Evasion Games On Polytopes In R^N, John Phillpot

HMC Senior Theses

We study single-pursuer, line-of-sight Pursuit and Evasion games in polytopes in $\mathbb{R}^n$. We develop winning Pursuer strategies for simple classes of polytopes (monotone prisms) in Rⁿ, using proven algorithms for polygons as inspiration and as subroutines. More generally, we show that any Pursuer-win polytope can be extended to a new Pursuer-win polytope in more dimensions. We also show that some more general classes of polytopes (monotone products) do not admit a deterministic winning Pursuer strategy. Though we provide bounds on which polytopes are Pursuer-win, these bounds are not tight. Closing the gap between those polytopes known to be …

Best Approximations, Lethargy Theorems And Smoothness, Caleb Case

CMC Senior Theses

In this paper we consider sequences of best approximation. We first examine the rho best approximation function and its applications, through an example in approximation theory and two new examples in calculating n-widths. We then further discuss approximation theory by examining a modern proof of Weierstrass's Theorem using Dirac sequences, and providing a new proof of Chebyshev's Equioscillation Theorem, inspired by the de La Vallee Poussin Theorem. Finally, we examine the limits of approximation theorem by looking at Bernstein Lethargy theorem, and a modern generalization to infinite-dimensional subspaces. We all note that smooth functions are bounded by Jackson's Inequalities, but …

Exploration Of Curvature Through Physical Materials, Lucinda-Joi Chu-Ketterer

Pitzer Senior Theses

Parametric equations are commonly used to describe surfaces. Looking at parametric equations does not provide tangible information about an object. Thus through the use of physical materials, an understanding of the limitations of the materials allows someone to gain a broader understanding of the surface. A M$\ddot{o}$bius strip and Figure 8 Klein bottle were created through knitting due to the precision and steady increase in curvature allowed through knitting. A more standard Klein bottle was created through crochet due to the ease in creating quick increases in curvature. Both methods demonstrate the change in curvature for both surfaces where the …