Mathematics Commons^™

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 …

Restricted And Unrestricted Coverings Of Complete Bipartite Graphs With Hexagons, Wesley M. Surber

Electronic Theses and Dissertations

A minimal covering of a graph G with isomorphic copies of graph H is a set {H₁, H₂, H₃, ... , H_n} where H_i is isomorphic to H, the vertex set of H_i is a subset of G, the edge set of G is a subset of the union of H_i's, and the cardinality of the union of H_i's minus G is minimum. Some studies have been made of covering the complete graph in which case an added condition of the edge set of H_{i …}

Very Cost Effective Partitions In Graphs, Inna Vasylieva

Electronic Theses and Dissertations

For a graph G=(V,E) and a set of vertices S, a vertex v in S is said to be very cost effective if it is adjacent to more vertices in V -S than in S.

A bipartition pi={S, V- S} is called very cost effective if both S and V- S are very cost effective sets. Not all graphs have a very cost effective bipartition, for example, the complete graphs of odd order do not. We consider several families of graphs G, including Cartesian products and cacti graphs, to determine whether G has a very cost effective bipartition.

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.

Students Rise To The Challenge Of Modeling Yeast Growth Despite Sour Hiccups From Imperfect Data, Alicia Caldwell

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Students rarely receive the opportunity to experience a learning activity involving mathematical modeling. This paper describes a lab in which students in an Applied Mathematics in Biology course observe the growth of Saccharomyces cerevisiae, a yeast strain, in differing sugar concentrations for use in learning modeling. They parameterized the logistic equation and an alternate model, which they themselves constructed, based on the data they collected. I participated in this lab as a student in 2012 then observed and reviewed the work of other students involved. I found that students gained a deeper understanding of limiting factors and the role of …

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 …

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.

Uphill & Downhill Domination In Graphs And Related Graph Parameters., Jessie Deering

Undergraduate Honors Theses

Placing degree constraints on the vertices of a path allows the definitions of uphill and downhill paths. Specifically, we say that a path π = v₁, v₂,...v_k+1 is a downhill path if for every i, 1 ≤ i ≤ k, deg(v_i) ≥ deg(v_i+1). Conversely, a path π = u₁, u₂,...u_k+1 is an uphill path if for every i, 1 ≤ i ≤ k, deg(u_i) ≤ deg(u_i+1). We investigate graphical parameters related to downhill and …

On Toric Symmetry Of P1 X P2, Olivia D. Beckwith

HMC Senior Theses

Toric varieties are a class of geometric objects with a combinatorial structure encoded in polytopes. P1 x P2 is a well known variety and its polytope is the triangular prism. Studying the symmetries of the triangular prism and its truncations can lead to symmetries of the variety. Many of these symmetries permute the elements of the cohomology ring nontrivially and induce nontrivial relations. We discuss some toric symmetries of P1 x P2, and describe the geometry of the polytope of the corresponding blowups, and analyze the induced action on the cohomology ring. We exhaustively compute the toric symmetries of P1 …

Extortion And Evolution In The Iterated Prisoner's Dilemma, Michael J. Earnest

HMC Senior Theses

The Prisoner's Dilemma is a two player game where playing rationally leads to a suboptimal outcome for both players. The game is simple to analyze, but when it is played repeatedly, complex dynamics emerge. Recent research has shown the existence of extortionate strategies, which allow one player to win at least as much as the other. When one player plays such a strategy, the other must either decide to take a low payoff, or accede to the extortion, where they earn higher payoff, but their opponent receives a larger share. We investigate what happens when one player uses this strategy …

A New Optimization Algorithm For Single Hidden Layer Feedforward Neural Networks, Leong Kwan Li, Sally Shao, Ka-Fai Cedric Yiu

Mathematics and Statistics Faculty Publications

Feedforward neural networks are the most commonly used function approximation techniques in neural networks. By the universal approximation theorem, it is clear that a single-hidden layer feedforward neural network (FNN) is sufficient to approximate the corresponding desired outputs arbitrarily close. Some researchers use genetic algorithms (GAs) to explore the global optimal solution of the FNN structure. However, it is rather time consuming to use GA for the training of FNN. In this paper, we propose a new optimization algorithm for a single-hidden layer FNN. The method is based on the convex combination algorithm for massaging information in the hidden layer. …

Detectable Subclinical Myocardial Necrosis Is Associated With Cardiovascular Risk In Stable Patients With Diabetes, W.H. Wilson Tang, Yuping Wu, Earl B. Britt Jr., Naveed Iqbal, Stanley L. Hazen

Mathematics and Statistics Faculty Publications

OBJECTIVE: To investigate the relationship between different degrees of subclinical myocardial necrosis, glycemic control, and long-term adverse clinical outcomes within a stable patient population with diabetes mellitus. RESEARCH DESIGN AND METHODS: We examined 1,275 stable patients with diabetes mellitus undergoing elective diagnostic coronary angiography with cardiac troponin I (cTnI) levels below the diagnostic cut-off for defining myocardial infarction (MI) (<0.03 ng/mL). The relationship of subclinical myocardial necrosis (cTnI 0.009–0.029 ng/mL) with incident major adverse cardiovascular events (MACE; defined as any death, MI, or stroke) over 3 years of follow-up was examined. RESULTS: Subclinical myocardial necrosis was observed in 22% of patients. A strong association was observed between the magnitude of subclinical myocardial necrosis and risk of 3-year incident MACE (hazard ratio, 1.98; 95% confidence interval, 1.48–2.65; P < 0.001) and remained statistically significant even after adjustment for traditional risk factors, high-sensitivity C-reactive protein, and creatinine clearance. Only a weak correlation was observed between the presence of subclinical myocardial necrosis and either glycemic control (r = 0.06; P = 0.044 for hemoglobin A1c versus cTnI) or insulin resistance (r = 0.04; P = 0.094 for glucose-to-insulin ratio versus cTnI). CONCLUSIONS: The presence of detectable subclinical myocardial necrosis in stable patients with diabetes mellitus is associated with heightened long-term risk for MACE, independent of traditional risk factors and glycemic control.

Variations Of The Combinatorial Game Tron, Miriam Parnes

Theses, Dissertations and Culminating Projects

The combinatorial game Tron is a two player game played on a graph, in which players move to adjacent vertices, but cannot move to any vertex which is currently occupied or which has been occupied by either player earlier in the course of the game. It was introduced by Hans Bodlaender, inspired by the Disney movie of the same name. Bodlaender and, later, Tillmann Miltzow considered the complexity of the question of whether a player has a winning strategy in the game. Miltzow also considered an extremal question regarding the ratio of the vertices taken by the second player over …

Symbolic Powers Of Ideals In K[PN], Michael Janssen

Department of Mathematics: Dissertations, Theses, and Student Research

Let I ⊆ k[P^N] be a homogeneous ideal and k an algebraically closed field. Of particular interest over the last several years are ideal containments of symbolic powers of I in ordinary powers of I of the form I^(m) ⊆ I^r, and which ratios m/r guarantee such containment. A result of Ein-Lazarsfeld-Smith and Hochster-Huneke states that, if I ⊆ k[P^N], where k is an algebraically closed field, then the symbolic power I^(Ne) is contained in the ordinary power I^e, and thus, whenever …

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.

Dirichlet's Theorem And Applications, Nicholas Stanford

Honors Scholar Theses

Dirichlet's theorem states that there exist an infinite number of primes in an arithmetic progression a + mk when a and m are relatively prime and k runs over the positive integers. While a few special cases of Dirichlet's theorem, such as the arithmetic progression 2 + 3k, can be settled by elementary methods, the proof of the general case is much more involved. Analysis of the Riemann zeta-function and Dirichlet L-functions is used.

The proof of Dirichlet's theorem suggests a method for defining a notion of density of a set of primes, called its Dirichlet density, …

Peg Solitaire On Trees With Diameter Four, Clayton A. Walvoort

Electronic Theses and Dissertations

In a paper by Beeler and Hoilman, the traditional game of peg solitaire is generalized to graphs in the combinatorial sense. One of the important open problems in this paper was to classify solvable trees. In this thesis, we will give necessary and sufficient conditions for the solvability for all trees with diameter four. We also give the maximum number of pegs that can be left on such a graph under the restriction that we jump whenever possible.

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 …