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# Student Research Projects

### 2010-2011

Kingston Kang, Will Lothian, and Jessica Sears - Need for speed: faster methods of solving the CS decomposition

Kingston, Will, and Jessica developed two new algorithms for decomposing orthogonal matrices as part of the SURF summer research program. A matrix decomposition is a factorization that breaks a matrix into simpler parts, much as the prime factorization 2 * 3 * 5 breaks 30 into basic components. Applications of the CS (cosine-sine) decomposition were earlier studied by Tian Xu '11. The new algorithms, whose development was supervised by Professor Brian Sutton, are specially designed for modern computer architectures with hierarchical memory and multiple processors. Kingston is traveling to Savannah, Georgia, to present their work at the SIAM Conference on Parallel Processing for Scientific Computing.

David Darwin - Investigations into the properties of number derivatives

During the summer of 2011, David worked on a SURF project under the guidance of Professor Chiru Bhattacharya. He studied the properties of number derivatives, which are functions obeying Liebniz’s product rule of differentiation. He was able to identify the ring structure of the set of number derivatives on integers modulo n, and figure out the group-theoretic structure of set of constants units of given number derivatives. David presented his findings at the Mathematical Association of America’s MD-DC-VA section meeting at Christopher Newport University in Fall 2011.

### 2009-2010

James Street

James, in a SURF project with R-MC Professor David Clark, investigated the theory of graph minors. A graph is said to be minor-minimal if it has some property, but no minor of the graph has that property. Certain graph operations such as triangle-Y expansion preserve minor-minimality of graphs. James explored whether similar operations, such as "square-X" expansion, may also preserve this minimality property. Furthermore, looked at polynomial invariants of spatial graphs and how these invariants may relate to knot polynomial invariants such as the Jones polynomial.

Eddie Tu

Eddie participated in a summer REU (Research Experience for Undergraduates) at James Madison University . Advised by JMU Professors Elizabeth Arnold and Stephen Lucas, Eddie explored ways of solving NP-complete problems, specifically the Hamiltonian Cycle Problem in graph theory and the solving of Sudoku puzzles. He worked on different ways of representing the Hamiltonian Cycle Problem as a problem of "exact cover," and managed to implement/modify Knuth's Algorithm X to solve it, as well as detect the presence of graph bridges. Eddie is also approached Sudoku problems from a graph coloring perspective, using various ideas of chromatic polynomials.

Tian Xu - Finding your way home with matrix decompositions

Tian, in a SURF project with R-MC Professor Brian Sutton, applied linear algebra, specifically the CS (cosine-sine) decomposition, to the direction-of-arrival problem. The direction-of-arrival problem arises in wireless communications, GPS, search-and-rescue, etc.--any application in which the originating location of an electromagnetic or audial signal must be determined. The problem is made especially difficult by environmental noise and reflecting walls, and Tian applied sophisticated mathematics to try to overcome these difficulties.

### 2008-2009

Ron Pandolfi - The numerical stability of Dodgson's method for determinants

In an honors project directed by Professor Brian Sutton and inspired by an article
of Professors Adrian Rice and Eve Torrence, Ron (R-MC '09) investigated the numerical
stability of an algorithm of Charles Dodgson, better known as Lewis Carroll. The
author of *Alice in Wonderland* developed an original method for computing
the determinant of a matrix. Ron asked a twenty-first-century question of Dodgson's
nineteenth-century method: when run on modern computer hardware, are small roundoff
errors managed well, or might they be amplified by repeated computation? Ron presented
his findings at a regional meeting of the Mathematics Association of America. Ron
is currently a graduate student at the University of California, Merced.

### 2007-2008

Amy Winslow - Modular origami and the trefoil knot

In this SURF project directed by Professor Eve Torrence in the summer of 2007, Amy (R-MC '08) combined her interests in Mathematics and Art. She utilized the PHiZZ unit, an origami module invented by Tom Hull, to create a large trefoil knot tessellated by polygons. She used ideas from graph theory to find a proper three-coloring of this structure. She also built several origami tori and a large torus sculpture from CDs. These structures all demonstrated different tessellations of these shapes. Amy presented her work at the Pi Mu Epsilon Conference at Mathfest in San Jose, CA, August 2 - 5, 2007. An illustrated summary of the project can be found here.

### 2006-2007

Liza Lawson - Real polynomials, complex roots, and enchanting ellipses

In a research project directed by Professor Bruce Torrence and funded by the SURF
program, Liza (R-MC '07), classified the structure of the curves formed by the roots
of the *k*th derivatives of a class of polynomials as one of the real roots
varied. She has co-authored a paper on her results and submitted it to the Pi Mu
Epsilon Journal. She has presented her results at three conferences:
Mathfest, the summer meeting of the MAA;
the SUMS conference at JMU; and at
the Nebraska Conference for Undergraduate Women in Mathematics.
Liza is currently teaching in the Teach for America program.

Kenny Stauffer - Numerical linear algebra

In an honors project directed by Professor Brian Sutton, Kenny (R-MC '07) studied pure and applied aspects of numerical linear algebra from the graduate-level textbook by Trefethen and Bau. The most significant application was to quantum mechanics. For his final project, Kenny developed computer code to solve the time-dependent Schroedinger equation using a Rayleigh-Ritz method. His code produced animations illustrating the counterintuitive behavior of particles at very small scales. Kenny is currently a graduate student at Carnegie Mellon University.

### 2003-2004

Jon Perkins - Preventing deadlock in resource allocation

In an honors project directed by Professor Richard Hammack, Jonathan (R-MC '04), a double major in Mathematics and Computer Science, obtained a graph-theoretic solution to the problem of preventing deadlock in the allocation of resources in operating systems.

### 2002-2003

Carla Owen - The missing link: How mathematics benefits political science research

Carla (R-MC '03), a political science major and mathematics minor, studied mathematical statistics in this departmental honors project designed to prepare her for graduate school in political science. In her culminating paper she explored several examples from time series data which pointed out how a lack of understanding of the mathematics behind statistics can lead to misinterpretation of data.

John Pfaff - Using the geometry of structures to reduce background noise

John (R-MC '03), a mathematics major worked with Professor Joseph from Mathematics and Professor Woolard from the physics department on this independent study. His goal: to offer a podium design that uses the properties of sound and geomtric shapes to reduce the intensity of background noise received by a microphone.

Sarah Wood - Graph theory

In a paper entitled "Centers of k-fold tensor products of graphs," Sarah Wood (R-MC '03) and Professor Richard Hammack describe the structure of the center of a tensor product of k arbitrary graphs. The paper is published in Discussiones Mathematicae Graph Theory, Vol. 24, No. 3: 491-503, 2004. Here is a pdf preprint of the paper. (Sarah is currently a Ph.D. candidate at University of Virginia.)

### 2001-2002

Jong Chan Jo - Some theorems on Fibonacci numbers

J.C., a mathematics and computer science double major, undertook work on discrete mathematics for his departmental honors project. In particular, he focused on investigating certain properties of the Fibonacci numbers, such as their connection to Pascal's triangle, and Binet's Formula. In his final paper, he provided some interesting proofs of these results, as well as some propositions concerning subsequences of the Fibonacci numbers.

### 2000-2001

Tonya Kim and Nancy Nichols - A queueing model for the railroad crossing on England St.

Tonya and Nancy (R-MC '01), advised by Professor Bruce Torrence, developed a model
for the manner in which traffic backs up at the railroad crossing in downtown Ashland.
They won first place in the student paper competition of the Maryland, Virginia,
DC section of the Mathematical Association of America in late April. They subsequently
entered and WON the *national* MathServe contest sponsored by the Consortium
for Mathematics and its Applications. They presented their results at the national
meeting of the MAA in Wisconsin in summer 2001.

Kristin Layell - Topology

Kristin (R-MC '01), subsequently a mathematics graduate student at the University
of Virginia, completed two departmental honors projects. Her topology project was
designed to help her better prepare for the rigors of graduate-level mathematics
courses. Kristin read the classic text *Topology* by James Munkres, and met
with Professor B.Torrence regularly to work exercises and ask questions. This work
supplemented her coursework for her "regular" topology course Math 451 at Randolph-Macon.

Lyle Walden - Mapping the Reimann sphere

Lyle (R-MC '03) received a SURF grant in the summer of 2001 and embarked on a research project with professor Reichard Hammack, where he studied maps of the complex plane to itself. The project used a computer algebra system to visualize such maps as mappings on the Reimann sphere, where the Reimann sphere is "painted" to look like planet Earth. The image of such functions provided wild "maps" of our funky little planet.

Kristin Layell - It's prime season

It's rabbit season, it's duck season...no it's Prime season! Kristin embarked on a research project with professor Rice in the field of number theory. Kristin was able to explore several proofs demonstrating that the sum of the reciprocals of the primes is infinite. In particular, she studied proofs by Ivan Niven, and Leonard Euler.

### 1999-2000

Thadd Selden - If Pascal had a computer

Working in the fall of 1999 and spring of 2000, Thadd (R-MC '01) began exploring one of the oldest problems in the history of probability, now known universally as the "problem of the points". The problem was solved independently by Blaise Pascal and Pierre de Fermat in the year 1654. Thadd's work shows how a computer algebra system can be used to generalize the two distinct approaches used by Pascal and Fermat, and to suggest a closed form for the two solutions. The work has been published in the November 2001 issue of Math Horizons.

### 1996-1997

Julie Jones- The case of the missing case

Julie (R-MC '97), who completed her her Ph.D. in mathematics in 2002, wrote a paper as an undergraduate in which she corrected an error in a proof of the famous combinatorist Ronald L. Graham. It turns out that the theorem itself was true, but that the original proof omitted an important case. Julie presented her paper, now published in the Pi Mu Epsilon Journal, at the national MAA summer meeting in Atlanta in 1997, where she got to meet Ron Graham in person.

Julie Jones - Advanced algebra

Julie completed a second departmental honors project with Professor Eve Torrence, in which she studied field extensions and Galois Theory in prepration for her subsequent graduate work.

### 1995-1996

Josh Pepper - The three body problem

Josh (R-MC '96)did a project in which he studied the famous "3 body problem" first explored by Isaac Newton. His presentation of this work at the spring 1996 meeting of DC-MD-VA sectional MAA meeting won first prize in the student paper competition. Josh went on to the University of Maryland to pursue a graduate degree in mathematics.