Mathematics Commons^™

Three Problems In Operator Theory And Complex Analysis, Cheng Chu

Arts & Sciences Graduate Student Theses and Dissertations

This thesis concerns three distinct problems in operator theory and complex analysis. In Chapter 2, we study the following problem: On the Hardy space H^2, when is the product of a Hankel operator and a Toeplitz operator compact? We give necessary and sucient conditions for when such a product H_fT_g is compact. In Chapter 3, we discuss hyponormal Toeplitz operators. We show that for those operators, there exists a lower bound for the area of the spectrum. This extends the known estimate for the spectral area of Toeplitz operators with an analytic symbol. This part is joint work with Dmitry …

Characterizations Of Four Interval Wavelet Sets And Algorithms, Christopher Mcdonald

Honors Program Theses and Projects

Wavelets are mathematical tools used to represent signals such as audio files, pictures, videos, and various other types of data. The theory of wavelets has recently attracted attention in Mathematics because of potential in applications. At this point, the field of wavelet theory is fairly mature, and the literature contains a body of techniques which are exploited to design wavelets. One of these techniques relies on the construction of wavelet sets. A wavelet set is a set whose successive translations and dilations partition a line. In practice, wavelet sets are tricky to construct. In fact, there is no known classification …

Kings And Heirs: A Characterization Of The (2,2)-Domination Graphs Of Tournaments, Kim A. S. Factor, Larry J. Langley

Mathematics, Statistics and Computer Science Faculty Research and Publications

In 1980, Maurer coined the phrase king when describing any vertex of a tournament that could reach every other vertex in two or fewer steps. A (2,2)-domination graph of a digraph D, dom_2,2(D), has vertex set V(D), the vertices of D, and edge uv whenever u and v each reach all other vertices of D in two or fewer steps. In this special case of the (i,j)-domination graph, we see that Maurer’s theorem plays an important role in establishing which vertices form the kings that create some of the edges in dom_2,2(D). But of even more …

On The Dimension Of Algebraic-Geometric Trace Codes, Phong Le, Sunil Chetty

Mathematics Faculty Publications

We study trace codes induced from codes defined by an algebraic curve X. We determine conditions on X which admit a formula for the dimension of such a trace code. Central to our work are several dimension reducing methods for the underlying functions spaces associated to X.

Homogeneous And Isotropic Cosmology, The Schwarzschild Solution, And Applications, Pengcheng Zhang

Departmental Honors & Graduate Capstone Projects

Classically, the physics of the universe is described by Newton's Laws of Motion and Newton's Law of Universal Gravitation. In most cases, the results predicted by Newton's theories accurately agree with experimental observations. However, under certain limitations, classical theories may yield slight deviation from observations, such as when the speed of an object approaches the speed of light. At the extreme, classical theory completely fails to explain the motion of photons, which are massless particles of light. In 1915, Albert Einstein published the General Theory of Relativity. Einstein's theory provides a new perspective to a better understanding of the physics …

Sequences Of Spiral Knot Determinants, Ryan Stees

Senior Honors Projects, 2010-2019

Spiral knots are a generalization of the well-known class of torus knots indexed by strand number and base word repetition. By fixing the strand number and varying the repetition index we obtain integer sequences of spiral knot determinants. In this paper we examine such sequences for spiral knots of up to four strands using a new periodic crossing matrix method. Surprisingly, the resulting sequences vary widely in character and, even more surprisingly, nearly every one of them is a known integer sequence in the Online Encyclopedia of Integer Sequences. We also develop a general form for these sequences in terms …

A Computational Investigation Of Large Gaps In Contingency Tables, Noah J. Watson

Senior Honors Projects, 2010-2019

Integer programming can be used to find upper and lower bounds on the cells of a multi-dimensional contingency table using the information from the released margins. The linear relaxation of these programs also provides bounds and the discrepancy between these bounds, the integer programming gap, can be large. While the more notable examples of large gaps have been shown to be rare, here we provide some results on the rarity of large gaps on small tables.

Population Projection And Habitat Preference Modeling Of The Endangered James Spinymussel (Pleurobema Collina), Marisa Draper

Senior Honors Projects, 2010-2019

The James Spinymussel (Pleurobema collina) is an endangered mussel species at the top of Virginia’s conservation list. The James Spinymussel plays a critical role in the environment by filtering and cleaning stream water while providing shelter and food for macroinvertebrates; however, conservation efforts are complicated by the mussels’ burrowing behavior, camouflage, and complex life cycle. The goals of the research conducted were to estimate detection probabilities that could be used to predict species presence and facilitate field work, and to track individually marked mussels to test for habitat preferences. Using existing literature and mark-recapture field data, these goals were accomplished …

The Development Of Notation In Mathematical Analysis, Alyssa Venezia

Honors Thesis

The field of analysis is a newer subject in mathematics, as it only came into existence in the last 400 years. With a new field comes new notation, and in the era of universalism, analysis becomes key to understanding how centuries of mathematics were unified into a finite set of symbols, precise definitions, and rigorous proofs that would allow for the rapid development of modern mathematics. This paper traces the introduction of subjects and the development of new notations in mathematics from the seventeenth to the nineteenth century that allowed analysis to flourish. In following the development of analysis, we …

Regression Analysis Of Success In Major League Baseball, Johnathon Tyler Clark

Senior Theses

This thesis is designed to explore whether a team’s success in any given season can be predicted or explained by any number of statistics in that season. There are thirty teams in the MLB; of these thirty, ten make the postseason bracket-style playoffs. The MLB is divided up into two leagues, the American League and the National League; these two leagues are then divided up into three divisions each, the West Division, the Central Division, and the East Division. To make the playoffs a team must either have the most wins in its division after the last game of the …

"For Me, Film Is Face": Self And Identity In The Films Of Ingmar Bergman, James Bradley Mitchell

Senior Theses

This senior thesis is a study of the ways in which selected works from the filmography of Ingmar Bergman can be said to contribute to the philosophical discussion of personal identity. In this thesis, Bergman's films are studied in relation to ideas propounded by Hegel, Descartes, Locke, and Parfit, among other considerations.

Math And Sudoku: Exploring Sudoku Boards Through Graph Theory, Group Theory, And Combinatorics, Kyle Oddson

Student Research Symposium

Encoding Sudoku puzzles as partially colored graphs, we state and prove Akman’s theorem [1] regarding the associated partial chromatic polynomial [5]; we count the 4x4 sudoku boards, in total and fundamentally distinct; we count the diagonally distinct 4x4 sudoku boards; and we classify and enumerate the different structure types of 4x4 boards.

Math Education: A Messy Problem, Gizem Karaali

Pomona Faculty Publications and Research

The current state of math education in America is certainly not ideal, writes Gizem Karaali, but mathematicians, researchers, policy makers and others are working on it -- and it is definitely a problem worth working on.

(Knight)3: A Graphical Perspective Of The Knight's Tour On A Multi-Layered Chess Board, Frederick Scott Neilan

Honors Program Theses and Projects

The Knight’s Tour is an interesting question related to the game of chess. In chess, the Knight must move two squares in one direction (forward, backward, left, right) followed by one square in a perpendicular direction. The question of the Knight’s Tour follows: Does there exist a tour for the Knight that encompasses every single square on the chess board without revisiting any squares? The existence of Knight’s Tours has been proven for the standard 8x8 chess board. Furthermore, the Knight’s Tour can also exist on boards with different sizes and shapes. There has been a lot of research into …

Uniform Stability In Nonlinear Infinite Delay Volterra Integro-Differential Equations Using Lyapunov Functionals, Youssef Raffoul, Habib Rai

Mathematics Faculty Publications

In [10] the first author used Lyapunov functionals and studied the exponential stability of the zero solution of nite delay Volterra Integro-dierential equation. In this paper, we use modified version of the Lyapunov functional that were used in [10] to obtain criterion for the stability of the zero solution of the infinite delay nonlinear Volterra integro-dierential equation

x′(t) = Px(t) + ^t∫ _−∞ C(t, s)g(x(s))ds.

Model Behavior: The Mathematics Behind Three-Dimensional Modeling And Animation, Kathryn Duff, Vivian Cyrus

Celebration of Student Scholarship Poster Sessions Archive

No abstract provided.

Mathematics And Origami; Unfolding Mathematical "Impossibilities", Dustin Tyler Adams

Celebration of Student Scholarship Poster Sessions Archive

No abstract provided.

Biological Aggregation Driven By Social And Environmental Factors: A Nonlocal Model And It Degenerate Cahn-Hilliard Approximation, Andrew J. Bernoff, Chad Topaz

Faculty Publications

No abstract provided.

Cosmological Models Through Spatial Ricci Flow, Ramesh Sharma

Mathematics Faculty Publications

We consider the synchronization of the Einstein’s flow with the Ricci-flow of the standard spatial slices of the Robertson–Walker space–time and show that associated perfect fluid solution has a quadratic equation of state and is either spherical and collapsing, or hyperbolic and expanding.
Read More: http://www.worldscientific.com/doi/abs/10.1142/S0219887816500699

Let's Nurture Science, Math Talent, Darren B. Glass

Math Faculty Publications

I recently saw the film The Man Who Knew Infinity, which was released in many American cities this weekend, and was struck by the beautiful telling of an inspirational story. The film, which stars Jeremy Irons and Dev Patel, is a biography of the mathematician Srinivasa Ramanujan, who was born in India at the end of the 19th century. [excerpt]