Physical Sciences and Mathematics Commons^™

The Global Stability Of The Solution To The Morse Potential In A Catastrophic Regime, Weerapat Pittayakanchit

HMC Senior Theses

Swarms of animals exhibit aggregations whose behavior is a challenge for mathematicians to understand. We analyze this behavior numerically and analytically by using the pairwise interaction model known as the Morse potential. Our goal is to prove the global stability of the candidate local minimizer in 1D found in A Primer of Swarm Equilibria. Using the calculus of variations and eigenvalues analysis, we conclude that the candidate local minimizer is a global minimum with respect to all solution smaller than its support. In addition, we manage to extend the global stability condition to any solutions whose support has a single …

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 …

Hopper Bands: Locust Aggregation, Ryan C. Jones

HMC Senior Theses

Locust swarms cause famine and hunger in parts of Sub-Saharan Africa as they travel across croplands and eat vegetation. Current models start with biological properties of locusts and analyze the macroscopic behavior of the system. These models exhibit the desired migratory behavior, but do so with too many parameters. To account for this, a new model, the Alignment and Intermittent Motion (AIM) model, is derived with minimal assumptions. AIM is constructed with regards to locust biology, allowing it to elicit biologically correct locust behavior: the most noteworthy being the fingering of hopper bands. A Particle-in-Cell method is used to optimize …

Computational Progress Towards Maximum Distinguishability Of Bell States By Linear Evolution And Local Measurement, Victor Shang

HMC Senior Theses

Many quantum information protocols rely on the ability to distinguish between entangled quantum states known as Bell states. However, theoretical limits exist on the maximal distinguishability of these entangled states using linear evolution and local measurement (LELM) devices. In the case of two particles entangled in multiple qubit variables, the maximum number of distinguishable Bell states is known. However, in the more general case of two particles entangled in multiple qudit variables, only an upper bound is known under additional assumptions. I have written software in Matlab and Mathematica to explore computationally the maximum number of Bell states that can …

An Interactive Tool For The Computational Exploration Of Integrodifference Population Models, Kennedy Agwamba

HMC Senior Theses

Mathematical modeling of population dynamics can provide novel insight to the growth and dispersal patterns for a variety of species populations, and has become vital to the preservation of biodiversity on a global-scale. These growth and dispersal stages can be modeled using integrodifference equations that are discrete in time and continuous in space. Previous studies have identified metrics that can determine whether a given species will persist or go extinct under certain model parameters. However, a need for computational tools to compute these metrics has limited the scope and analysis within many of these studies. We aim to create computational …

Mathematical Modeling Of Blood Coagulation, Joana L. Perdomo

HMC Senior Theses

Blood coagulation is a series of biochemical reactions that take place to form a blood clot. Abnormalities in coagulation, such as under-clotting or over- clotting, can lead to significant blood loss, cardiac arrest, damage to vital organs, or even death. Thus, understanding quantitatively how blood coagulation works is important in informing clinical decisions about treating deficiencies and disorders. Quantifying blood coagulation is possible through mathematical modeling. This review presents different mathematical models that have been developed in the past 30 years to describe the biochemistry, biophysics, and clinical applications of blood coagulation research. This review includes the strengths and limitations …

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 …

Pattern Recognition In High-Dimensional Data, Matthew Dannenberg

HMC Senior Theses

Vast amounts of data are produced all the time. Yet this data does not easily equate to useful information: extracting information from large amounts of high dimensional data is nontrivial. People are simply drowning in data. A recent and growing source of high-dimensional data is hyperspectral imaging. Hyperspectral images allow for massive amounts of spectral information to be contained in a single image. In this thesis, a robust supervised machine learning algorithm is developed to efficiently perform binary object classification on hyperspectral image data by making use of the geometry of Grassmann manifolds. This algorithm can consistently distinguish between a …

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.

Steady State Solutions For A System Of Partial Differential Equations Arising From Crime Modeling, Bo Li

HMC Senior Theses

I consider a model for the control of criminality in cities. The model was developed during my REU at UCLA. The model is a system of partial differential equations that simulates the behavior of criminals and where they may accumulate, hot spots. I have proved a prior bounds for the partial differential equations in both one-dimensional and higher dimensional case, which proves the attractiveness and density of criminals in the given area will not be unlimitedly high. In addition, I have found some local bifurcation points in the model.

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 …