# Student Research Projects

## 2017-2018

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**Elen Khachatryan - Gamma and zeta**

In an independent study project, Elen investigated two of the most important functions of real and complex analysis: the gamma and zeta functions. She proved important properties of these functions, locating features such as zeros and singularities, and revealing relations that facilitate their computation. Her project was motivated by the Riemann Hypothesis, often regarded as the most important unproved conjecture in mathematics, that establishes a very surprising link between the zeta function and the distribution of the prime numbers. Elen will present her work at Randolph-Macon's Research Day.

## 2016-2017

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**Eric Montag - The Mathematics Underlying Rotational Kinematics**

The problem of parameterizing the group of rotations in Euclidean 3-space has been of interest since 1776 when Euler first proved his famous rotation theorem. Euler’s notion was extended by the work of Rodrigues (circa 1841), Hamilton (circa 1843), and Cayley (circa 1845). Rodrigues derived a formula that permits direct calculation under certain conditions, while Hamilton generalized the underlying mathematics by introducing the new algebra of Quaternions that enabled the parameterization of the rotation group. Meanwhile Cayley developed Cayley Transformations, a mapping between real skew-symmetric matrices and orthogonal matrices. This presentation focuses on methodologies for calculating orthogonal coordinate transformation matrices *A(t)* between two frames of reference *F _{a}* and

*F*where

_{b},*F*is attached to a rotating body and rotates relative to

_{b}*F*, which we may regard as fixed in space. These methods for calculating

_{a}*A(t)*are discussed in light of a physics-imposed kinematical constraint. Parameterizations of

*A(t)*are presented and discussed in the context of published research findings, indicating that no three-parameter system can produce a continuous, global and non-singular transformation, but that a four-parameter representation suffices. Eric’s advisor for this project was R-MC alumnus and NASA scientist Dr. Roland Bowles.

## 2014-2015

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**Shuyan Zhan - China’s population & one-child policy**

For her senior seminar project, Shuyan researched China's one-child policy and the effects it has had on the population pyramid--the joint distribution of age and sex. She created and analyzed a mathematical model for the policy's effects and compared it with the official population figures. Shuyan continued her work after the conclusion of the fall seminar and then presented her findings at the spring meeting of the Mathematical Association of America’s MD-DC-VA section meeting, winning First Prize in the student paper division.

**Julia Knapp - Analysis of host-guest chemistry using nonlinear regression in Mathematica**

Julia completed a Departmental Honors project unifying chemistry and mathematics. During the first half of the semester, Julia studied multiple linear regression, a technique in statistics for revealing predictive relationships among multiple variables. During the second half of the semester, she considered a specific problem suggested by Professor John Thoburn of the Department of Chemistry, ultimately finding that a published paper had incorrectly applied a mathematical method to an experiment in which the preconditions of the method were not met. Julia presented a poster on her work at Randolph-Macon's Research Day.

## 2013-2014

**Lisa Borum - CS decomposition of k-banded matrices**

Lisa's project concerned the numerical computation of the CS decomposition. In an earlier project, R-MC students Kingston Kang, Will Lothian, and Jessica Sears developed an algorithm for eliminating most entries of an orthogonal matrix without changing the fundamental rotation or reflection of space represented by the matrix. Lisa's results pick up where theirs left off, fully reducing the blocks to diagonal form and revealing the principal angles that characterize the transformation of space.

## 2012-2013

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**Victoria Zimbro - Matrix decompositions for partitioned orthogonal matrices**

As part of a SURF project, Victoria developed a numerical algorithm for reducing an orthogonal matrix partitioned into six or more blocks to a structured form. An orthogonal matrix represents a rotation or reflection of space, and Victoria's reduction is a way of generalizing the well-known CS decomposition to a broader class of problems. Victoria presented her work at the national MathFest meeting in Hartford, Connecticut.

## 2011-2012

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**Andrew Sloan - Actuarial Science**

Actuarial science is the mathematics of risk. Drawing heavily from probability theory and finance, it supports decision making in the insurance industry. Andrew studied actuarial science on his own and passed his first professional exam at the end of the semester. He immediately began work in the insurance industry after graduating.

## 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 round off 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. View an illustrated summary of the project (DOC).

## 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 geometric 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. (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 preparation 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.