Mathematics Commons^™

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 …

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.

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 …

Tournament Directed Graphs, Sarah Camille Mousley

Undergraduate Honors Capstone Projects

Paired comparison is the process of comparing objects two at a time. A tournament in Graph Theory is a representation of such paired comparison data. Formally, an n-tournament is an oriented complete graph on n vertices; that is, it is the representation of a paired comparison, where the winner of the comparison between objects x and y (x and y are called vertices) is depicted with an arrow or arc from the winner to the other.

In this thesis, we shall prove several results on tournaments. In Chapter 2, we will prove that the maximum number of vertices …

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

On The Optimal Control Of The Free Boundary Problems For The Second Order Parabolic Equations. I. Well-Posedness And Convergence Of The Method Of Lines, Ugur G. Abdulla

Mathematics and System Engineering Faculty Publications

We develop a new variational formulation of the inverse Stefan problem, where information on the heat flux on the fi xed boundary is missing and must be found along with the temperature and free boundary. We employ optimal control framework, where boundary heat flux and free boundary are components of the control vector, and optimality criteria consists of the mini- mization of the sum of L2-norm declinations from the available measurement of the temperature flux on the fi xed boundary and available information on the phase transition temperature on the free boundary. This approach allows one to tackle situations when …

Number Theory Applications In Cryptography, Francesca Pizzigoni

Theses, Dissertations and Culminating Projects

This thesis provides a unique cryptosystem comprised of different number theory applications. We first consider the well-known Knapsack Problem and the resulting Knapsack Cryptosystem. It is known that when the Knapsack Problem involves a superincreasing sequence, the solution is easy to find. Two cryptosystems are designed and displayed in this thesis that allow two parties often called Alice and Bob use a common superincreasing sequence in the encryption and decryption process. They use this sequence and a variation of the Knapsack Cryptosystem to send and receive binary messages. The first cryptosystem assumes that Alice and Bob agree on a shared …

Common Misconceptions And Instructional Best Practices For Teaching Fractions To Students In Grades Three, Four, And Five, Dan Fairchild

Mathematics Graduate Theses

This research paper will identify common misconceptions related to the teaching and learning fractions for students in grades three, four, and five. Different fraction subconstructs will be identified and instructional best practices will be identified for the teaching and learning of comparing fractions, ordering fractions, adding fractions, and subtracting fractions.

Teaching Social Justice Mathematics In A Privileged Setting, Rick Mcnamee

Theses, Dissertations and Culminating Projects

This practitioner action research using qualitative techniques of gathering data looks at the experiences of students in a mostly white, upper-middle class suburb who enrolled in a class that integrated mathematics with social justice issues. The researcher designed the curriculum and co-taught the one semester course. The data were collected from pre-interviews, beginning and end of class questionnaires, quick writes, field notes, and end of class focus group interviews with all the students. The author chose a practitioner action research design because he wanted to be an integral part of the study. The cyclical nature of action research allowed him …

Proper Connection Of Bipartite Circulant Graphs, Melissa Marie Fuentes

Theses, Dissertations and Culminating Projects

The study of proper edge-colorings of graphs has been a popular topic in graph theory since the work of Vizing. While the proper edge-colorings of entire graphs was the topic of interest when the subject began decades ago, more recent works have focused on the study of properly colored subgraphs, as opposed to proper colorings of graphs as a whole. The types of properly colored subgraphs that we will be most concerned with are paths. The topic of finding certain types of properly colored paths within larger edge-colored graphs, though seemingly specific in nature, has been a topic of much …

A Mathematical Model For Antibiotic Resistance In A Hospital Setting With A Varying Population, Edward H. Snyder

Electronic Theses and Dissertations

Antibiotic-resistant bacteria(ARB) is causing increased health risk and cost to society. Mathematical models have been developed to study the transmission of resistant bacteria and the efficacy of preventive measures to slow its spread within a hospital setting. The majority of these models have assumed a constant total hospital population with the admission and discharge rates being equal throughout the duration. But a typical hospital population varies from day to day and season to season. In this thesis, we apply variable admission and discharge daily rates to existing deterministic and stochastic models which examine the transmission of single and dual resistant …

A Variety Of Proofs Of The Steiner-Lehmus Theorem, Sherri R. Gardner

Electronic Theses and Dissertations

The Steiner-Lehmus Theorem has garnered much attention since its conception in the 1840s. A variety of proofs resulting from the posing of the theorem are still appearing today, well over 100 years later. There are some amazing similarities among these proofs, as different as they seem to be. These characteristics allow for some interesting groupings and observations.

Homological Algebra : Tor Functors, Betti Numbers, And Free Resolutions., Ian Philipp

College of Arts & Sciences Senior Theses

The topic of study for this project is homological algebra, a branch of mathematics with applications to a plethora of other branches of mathematics as well as sensor networks, signal processing, uid dynamics, particle physics, etc. The amount of literature written on this topic is vast and there are numerous open problems in homological algebra, but this project has a more modest focus. First, we will explain the fundamental terminology with several examples and later we will proceed to some results concerning certain rings and ideals, and then results about modules generated by zero divisors.

Characterizations Of Certain Recently Introduced Distributions, Gholamhossein Hamedani

Mathematics, Statistics and Computer Science Faculty Research and Publications

Various characterizations of the Beta Weibull Geometric distribution of Bidram et al. (2011), the New Generalized Exponential distribution of Bidram et al. (2012), and the Gamma Exponentiated Weibull distribution of Pinho et al. (2012) are presented. These characterizations are based on: (i) a simple relationship between two truncated moments (ii) certain functions of the nth order statistic, and (iii) certain functions of the 1st order statistics.

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 …

Comparing The Impact Of Traditional And Modeling College Algebra Courses On Student Performance In Survey Of Calculus, Jerry West

Graduate Theses and Dissertations

Students in higher education deserve opportunities to succeed and learning environments which maximize success. Mathematics courses can create a barrier for success for some students. College algebra is a course that serves as a gateway to required courses in many bachelor's degree programs. The content in college algebra should serve to maximize students' potential in utilizing mathematics and gaining skills required in subsequent math-based courses when necessary. The Committee for Undergraduate Programs in Mathematics has gone through extensive work to help mathematics departments reform their college algebra courses in order to help students gain interest in the utilization of mathematics …