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
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 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.
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.
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.
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 kth 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.
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
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
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.
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
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
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.
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
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.