Mathematics Commons^™

Pricing European And American Options In The Heston Model With Accelerated Explicit Finite Differencing Methods, Conall O'Sullivan, Stephen O'Sullivan

Articles

No abstract provided.

Using Optimal Control Theory To Optimize The Use Of Oxygen Therapy In Chronic Wound Healing, Donna Lynn Daulton

Masters Theses & Specialist Projects

Approximately 2 to 3 million people in the United States suffer from chronic wounds, which are defined as wounds that do not heal in 30 days time; an estimated $25 billion per year is spent on their treatment in the United States. In our work, we focused on treating chronic wounds with bacterial infections using hyperbaric and topical oxygen therapies.

We used a mathematical model describing the interaction between bacteria, neutrophils and oxygen. Optimal control theory was then employed to study oxygen treatment strategies with the mathematical model. Existence of a solution was shown for both therapies. Uniqueness was also …

Analyzing And Solving Non-Linear Stochastic Dynamic Models On Non-Periodic Discrete Time Domains, Gang Cheng

Masters Theses & Specialist Projects

Stochastic dynamic programming is a recursive method for solving sequential or multistage decision problems. It helps economists and mathematicians construct and solve a huge variety of sequential decision making problems in stochastic cases. Research on stochastic dynamic programming is important and meaningful because stochastic dynamic programming reflects the behavior of the decision maker without risk aversion; i.e., decision making under uncertainty. In the solution process, it is extremely difficult to represent the existing or future state precisely since uncertainty is a state of having limited knowledge. Indeed, compared to the deterministic case, which is decision making under certainty, the stochastic …

Floquet Theory On Banach Space, Fatimah Hassan Albasrawi

Masters Theses & Specialist Projects

In this thesis we study Floquet theory on a Banach space. We are concerned about the linear differential equation of the form: y'(t) = A(t)y(t), where t ∈ R, y(t) is a function with values in a Banach space X, and A(t) are linear, bounded operators on X. If the system is periodic, meaning A(t+ω) = A(t) for some period ω, then it is called a Floquet system. We will investigate the existence …

Hypergraph Capacity With Applications To Matrix Multiplication, John Lee Thompson Peebles Jr.

HMC Senior Theses

The capacity of a directed hypergraph is a particular numerical quantity associated with a hypergraph. It is of interest because of certain important connections to longstanding conjectures in theoretical computer science related to fast matrix multiplication and perfect hashing as well as various longstanding conjectures in extremal combinatorics.

We give an overview of the concept of the capacity of a hypergraph and survey a few basic results regarding this quantity. Furthermore, we discuss the Lovász number of an undirected graph, which is known to upper bound the capacity of the graph (and in practice appears to be the best such …

The Torsion Angle Of Random Walks, Mu He

Masters Theses & Specialist Projects

In this thesis, we study the expected mean of the torsion angle of an n-step
equilateral random walk in 3D. We consider the random walk is generated within a confining sphere or without a confining sphere: given three consecutive vectors ^→e₁ , ^→e₂ , and ^→e₃ of the random walk then the vectors ^→e₁ and ^→e₂ define a plane and the vectors ^→e₂ and ^→e₃ define a second plane. The angle between the two planes is called the torsion angle of the three vectors. Algorithms are …

Minimizing Travel Time Through Multiple Media With Various Borders, Tonja Miick

Masters Theses & Specialist Projects

This thesis consists of two main chapters along with an introduction and
conclusion. In the introduction, we address the inspiration for the thesis, which
originates in a common calculus problem wherein travel time is minimized across two media separated by a single, straight boundary line. We then discuss the correlation of this problem with physics via Snells Law. The first core chapter takes this idea and develops it to include the concept of two media with a circular border. To make the problem easier to discuss, we talk about it in terms of running and swimming speeds. We first address …

Chip Firing Games And Riemann-Roch Properties For Directed Graphs, Joshua Z. Gaslowitz

HMC Senior Theses

The following presents a brief introduction to tropical geometry, especially tropical curves, and explains a connection to graph theory. We also give a brief summary of the Riemann-Roch property for graphs, established by Baker and Norine (2007), as well as the tools used in their proof. Various generalizations are described, including a more thorough description of the extension to strongly connected directed graphs by Asadi and Backman (2011). Building from their constructions, an algorithm to determine if a directed graph has Row Riemann-Roch Property is given and thoroughly explained.

On Well-Quasi-Orderings, Forrest Thurman

HIM 1990-2015

A quasi-order is a relation on a set which is both reflexive and transitive, while a well-quasi-order has the additional property that there exist no infinite strictly descending chains nor infinite antichains. Well-quasi-orderings have many interesting applications to a variety of areas which includes the strength of certain logical systems, the termination of algorithms, and the classification of sets of graphs in terms of excluded minors. My thesis explores how well-quasi-orderings are related to these topics through examples of four known well-quasi-orderings which are given by Dickson's Lemma, Higmans's Lemma, Kruskal's Tree Theorem, and the Robertson-Seymour Theorem. The well-quasi-ordering conjecture …

Coble And Eisenhart: Two Gettysburgians Who Led Mathematics, Darren B. Glass

Math Faculty Publications

In 1895, there were 134 students at Gettysburg College, which was then called Pennsylvania College. Of these students, two of them went on to become president of the American Mathematical Society. In this article, we look at the lives of these two men, Arthur Coble and Luther Eisenhart, and their contributions to mathematics and higher education, as well as look at what mathematics was like at Gettysburg at the end of the nineteenth century.

On The Null Space Structure Associated With Trees And Cycles, Shaun M. Fallat, Shahla Nasserasr

Mathematics Faculty Articles

In this work, we study the structure of the null spaces of matrices associated with graphs. Our primary tool is utilizing Schur complements based on certain collections of independent vertices. This idea is applied in the case of trees, and seems to represent a unifying theory within the context of the support of the null space. We extend this idea and apply it to describe the null vectors and corresponding nullities of certain symmetric matrices associated with cycles

Analysis Of Time-Dependent Integrodifference Population Models, Taylor J. Mcadam

HMC Senior Theses

The population dynamics of species with separate growth and dispersal stages can be described by a discrete-time, continuous-space integrodifference equation relating the population density at one time step to an integral expression involving the density at the previous time step. Prior research on this model has assumed that the equation governing the population dynamics remains fixed over time, however real environments are constantly in flux. We show that for time-varying models, there is a value Λ that can be computed to determine a sufficient condition for population survival. We also develop a framework for analyzing persistence of a population for …

Structured Matrices And The Algebra Of Displacement Operators, Ryan Takahashi

HMC Senior Theses

Matrix calculations underlie countless problems in science, mathematics, and engineering. When the involved matrices are highly structured, displacement operators can be used to accelerate fundamental operations such as matrix-vector multiplication. In this thesis, we provide an introduction to the theory of displacement operators and study the interplay between displacement and natural matrix constructions involving direct sums, Kronecker products, and blocking. We also investigate the algebraic behavior of displacement operators, developing results about invertibility and kernels.

Mathematical Knowledge For Teaching And Visualizing Differential Geometry, Nathan Pinsky

HMC Senior Theses

In recent decades, education researchers have recognized the need for teachers to have a nuanced content knowledge in addition to pedagogical knowledge, but very little research was conducted into what this knowledge would entail. Beginning in 2008, math education researchers began to develop a theoretical framework for the mathematical knowledge needed for teaching, but their work focused primarily on elementary schools. I will present an analysis of the mathematical knowledge needed for teaching about the regular curves and surfaces, two important concepts in differential geometry which generalize to the advanced notion of a manifold, both in a college classroom and …

General Bounds On The Downhill Domination Number In Graphs., William Jamieson

Undergraduate Honors Theses

A path π = (v₁, v₂,...v_k+1) in a graph G = (V,E) is a downhill path if for every i, 1 < i < k, deg(v_i) > deg(v_i+1), where deg(v_i) denotes the degree of vertex v_i ∊ V. The downhill domination number equals the minimum cardinality of a set S ⊂ V having the property that every vertex v ∊ V lies on a downhill path originating from some vertex in S. We investigate downhill domination numbers of graphs and give upper bounds. …

Topological Complexity Of H-Spaces, Gregory Lupton, Jerome Scherer

Mathematics and Statistics Faculty Publications

Let X be a (not-necessarily homotopy-associative) H-space. We show that TCn+1(X) = cat (Xn), forn≥1, where TCn+1(−) denotes the so-called higher topological complexity introduced by Rudyak, and cat (−) denotes the Lusternik-Schnirelmann category. We also generalize this equality to an inequality, which gives an upper bound for TCn+1(X), in the setting of a space Y acting on X.

Split Abelian Chief Factors And First Degree Cohomology For Lie Algebras, Jorg Feldvoss, Salvatore Siciliano, Thomas Weigel

University Faculty and Staff Publications

In this paper we investigate the relation between the multiplicities of split abelian chief factors of finite-dimensional Lie algebras and first degree cohomology. In particular, we obtain a characterization of modular solvable Lie algebras in terms of the vanishing of first degree cohomology or in terms of the multiplicities of split abelian chief factors. The analogues of these results are well known in the modular representation theory of finite groups. An important tool in the proof of these results is a refinement of a non-vanishing theorem of Seligman for the first degree cohomology of non-solvable finite-dimensional Lie algebras in prime …

Superstable Manifolds Of Invariant Circles, Scott R. Kaschner

Scholarship and Professional Work - LAS

Let f : X → X be a dominant meromorphic self-map, where X is a compact, connected complex manifold of dimension n > 1. Suppose there is an embedded copy of P1 that is invariant under f, with f holomorphic and transversally superattracting with degree a in some neighborhood. Suppose also that f restricted to this line is given by z → z^b, with resulting invariant circle S. We prove that if a ≥ b, then the local stable manifold W^s_loc(S) is real analytic. In fact, we state and prove a suitable localized version that can …

Section Abstracts: Astronomy, Mathematics And Physics

Virginia Journal of Science

Abstracts of the Astronomy, Mathematics and Physics Section for the 91st Annual Virginia Journal of Science Meeting, May 2013

Asian Carp Could Help Us Solve Hunger Problem, Michael (Dung) Tran

Collection of Engaged Learning

Bighead carp and silver carp (Invader Species – Asian Carp) invaded the Illinois River waterway over a decade ago. Populations of these fishes have apparently grown dense in the lower and middle Illinois Rivers and both species are approaching the Chicago Area Waterway System (CAWS) and potentially making their way into Great Lakes. So far, positive DNA of Asian Carp has been found beyond the defensive electrical barrier, which means they have made their way to Great Lakes somehow. Asian Carp are fast growing, aggressive, and adaptable fish that are outcompeting native fish species for food and habitat in much …