William & Mary 2008-2009 CSUMS research students

Student Name | Major/Class | Adviser | Research Topic | CSUMS support | Current position/ Affiliation |

Tanner Crowder |
Mathematics and Physics/2008 (Honors) |
Chi-Kwong Li | Genetic codes and quantum computing | summer 2008 | Research mathematician, Naval Research Lab, Washington DC (July 2008--) |

Ashwin Rastogi |
Mathematics and Physics/2008 (Honors) |
Chi-Kwong Li | Preserver problems and Quantum computing | summer 2008 | Graduate study in Physics, Harvard University (Sept. 2008--) |

Kassie Archer |
Mathematics and Chemistry/2009 (Honors) |
Sarah Day | Computing the dimension of strange attractors | summer 2008 | Graduate study in Matheamtics, Dartmouth College (Setp. 2009--) |

Sean Clark |
Mathematics and Physics/2009 (Honors) |
Chi-Kwong Li | Preserver Problems and Quantum computing | summer 2008 | Graduate study in Mathematics, University of Virginia (Sept. 2009--) |

David Gould |
Matehmatics and Theatre/2009 (Honors) |
Sarah Day | Population Model of Virginia Black Bears |
summer 2009 | Graduate study in Mathematics, Rutgers University (Sept. 2009--) |

Daniel Hariprasad |
Mathematics/2009 (Honors) |
Junping Shi | Three pool model of Calcium signaling | spring 2009 | Graduate study in Mathematics, University of Arizona (Sept. 2009--) |

Michael Liarakos |
Computer Science and Mathematic/2009(Honors) | Virginia Torczon/ Michael Lewis |
Order on Binary Search Trees |
summer 2008 |
The MITRE Corporation (Summer, 2009 --) |

Jennifer Mahle |
Mathematics/2009 (winter) (Honors) |
Chi-Kwong Li | Quantum computing, and generalized numerical ranges |
summer 2008 | IBM (Jan. 2010) |

Brian Paljug |
Mathematics and Theatre/2009 (Honors) |
Sarah Day | Exploring Issues of Topological Entropy in Symbolic Dynamical Systems | summer 2008 | |

Benjamin Strahs |
Computer Science and Philosophy/2009 | Haining Wang | Internet security | summer 2008 | Institute for the Theory and Practice of International Relations, June, 2009 -- |

Elizabeth Truelove |
Mathematic/2009 | Michael Lewis | Computational molecular modeling | summer 2008 | Analyst at SRA International, July, 2009-- |

Katie Williams |
Computer Science and Mathematics/2009 | Virginia Torczon/ Michael Lewis | Different programing platforms: SIF and AMPL | summer 2008 |
Booz Allen Hamilton, August, 2009 -- |

Niha Zubair |
Mathematics and Biochemistry/2009 (Honors) |
Sarah Day | Studying reproduction rate strategies in white-footed mice | summer 2008 |
Graduate study in Nutrition, University of North Carolina, Chapel Hill (Sept. 2009--) |

Michael Essman |
Mathematics and Economics/2010 | Junping Shi | Numerical solutions of nonlinear Schrodinger system |
summer 2008 | Graduate Study in law, College of William and Mary, (Sept. 2010--) |

##### Class 2008:

**Tanner Crowder** Major: Mathematics and Physics (Advisor: Chi-Kwong Li)

Research topics: Genetic codes and quantum computing

CSUMS courses and activities:

- Math 410, Math 432 (Spring 2008),
- Speaker at 2008 CSUMS workshop,
- Honors thesis in mathematics (2007-08).

A study of genetic codes by combinatorics and matrix approaches, Honors Thesis, William and Mary, 2008.

**2008 summer research.**

- Research support: CSUMS (Spring 2008) and UBM (Summer 2008)

Research results:

[1] Crowder and Li, Studying Genetic Code by a Matrix Approach, submitted.

Abstract Following Petoukhov and his collaborators we use two length n zero-one sequences to represent a length n genetic sequences. Using the Gray code ordering of the zero-one sequences, we build a 2^n-by-2^n matrix C_n including all the 4^n length n genetic sequences, and use the Hamming distances of the zero-one sequences to construct a corresponding numerical matrix D_n. Various algebraic properties of the matrices C_n and D_n are discussed. Connection and implications of these properties to the study of genetic sequences are explored. Our results refine and extend those of other authors.

Activities after CSUMS. Research mathematician at Naval Research Lab.(July 2008 - present).

**Ashwin Rastogi** Major: Mathematics and Physics (Advisor: Chi-Kwong Li)

Research topic: Preserver problems and Quantum computing

CSUMS courses and activities:

- Math 410 (Spring 2007), Physics seminar 2007-08,
- Honors thesis in physics (2007-08).

An exceptional electroweak model, Honors Thesis, William and Mary, 2008.

- Presentation at the International Workshop of Operator Theory and its Applications, July 2008.
- 2008 summer research.

Research support: NSF REU supplements (Li and Rodman), CSUMS

Research results:

[1] Li, Poon and Rastogi, Maps preserving spectral radius, numerical radius, spectral norm, Electronic Linear Algebra 16 (2007), 347-359.

Abstract Characterizations are obtained for Schur (Hadamard) multiplicative maps on complex matrices preserving the spectral radius, numerical radius, or spectral norm. Similar results are obtained for maps under weaker assumptions. Furthermore, a characterization is given for maps f satisfying ||A*B|| = ||f(A)*f(B)||for all matrices A and B.

[2] Clark, Li and Rastogi, Schur multiplicative maps on matrices, Bulletin of Aust. Math. Soc. 77 (2008), 49-72.

Abstract The structure of Schur multiplicative maps on matrices over a field is studied. The result is then used to characterize Schur multiplicative maps satisfying f(S) is a subset of S for different subsets S of matrices including the set of rank k matrices, the set of singular matrices, and the set of invertible matrices. Characterizations are also obtained for maps on matrices such that T(f(A)) = T(A)$ for various functions T including the rank function, the determinant function, and the elementary symmetric functions of the eigenvalues. These results include analogs of the theorems of Frobenius and Dieudonne.

Activities after CSUMS. Graduate study in Physics at Harvard University (Fall 2008 - present).

**Class of 2009**

Kassie Archer Major: Mathematics and Chemistry (Advisor: Sarah Day)

Research topic: Computing the dimension of strange attractors

CSUMS courses and activities:

- Math 410, Math 432 (Spring 2008)
- Presentation at the 2008 CSUMS workshop.
- 2008 summer research.

Research support: CSUMS

Research results: Research Blog

[1] Survey of Box-Counting Dimension, 2008 CSUMS summer report.

Abstract In dynamics, computing the fractal dimension of strange attractors and fractal-like sets that may occur in the study of a dynamical system can give us a way of measuring these sets and sometimes can also have a physical interpretation. In general, computing fractal dimension gives us a scaling factor of the set and a way of telling how much the set fills up space. Since we cannot describe fractals and fractal-like sets using typical geometrical methods, fractal dimension gives us one way of measuring, understanding and studying the geometry of the sets. Box-counting dimension is a one way of measuring fractal dimension. In this project, we explored several properties and characteristics associated with box-counting dimension including: the change in the dimension of the attractor as the parameters of a system change, the dimension of subsets of the attractor and their relationship to the dimension of the whole attractor, the relationship between fractal dimension and Lyapunov exponents, and the dimension of fractals lying in infinite-dimensional space. These topics include many possible open questions for future study.

**Sean Clark** Major: Mathematics and Physics (Advisor: Chi-Kwong Li)

Research topics: Preserver Problems and Quantum computing

CSUMS courses and activities:

- Math 410, Math 432 (Spring 2008).
- Presentation at the 2008 William and Mary Science Symposium.
- Participant of 2008 CSUMS workshop.
- Presentation at the Ninth workshop on Numerical Ranges and Numerical Radii, July 2008.

Research support: NSF REU supplements of Li and Rodman, CSUMS

Research results:

[1] Clark, Li and Rastogi, Schur multiplicative maps on matrices, Bulletin of Aust. Math. Soc. 77 (2008), 49-72.

**Abstract **The structure of Schur multiplicative maps
on matrices over a field is studied. The result is then used to
characterize Schur multiplicative maps satisfying f(S) is a subset of S
for different subsets S of matrices including the set of rank k
matrices, the set of singular matrices, and the set of invertible
matrices. Characterizations are also obtained for maps on matrices such
that T(f(A)) = T(A)$ for various functions T including the rank
function, the determinant function, and the elementary symmetric
functions of the eigenvalues. These results include analogs of the
theorems of Frobenius and Dieudonne.

[2] Clark, Li and Rodman, Spectral radius of the product of nonnegative matrices, to appear in Banach J. Math. Anal.

**Abstract **A characterization of nonlinear spectral
radius preserving maps is obtained for the usual and triple products of
nonnegative matrices.

[3] Clark, Li, Mahle and Rodman, Preservers of higher rank numerical range and radius, 2008 CSUMS summer research report.

**Abstract **Characterizations are obtained for linear
maps on Hermitian matrices or complex square matrices leaving invariant
the higher rank numerical range and radius. Similar results are
obtained for multiplicative maps, and surjective additive maps.
Extensions of the results to infinite dimensional operators and to the
joint higher rank numerical range of multiple matrices are also
considered.

**David Gould**; (Class of 2009) Major: Matehmatics and Theatre (Advisor: Sarah Day, Junping Shi)

Research topic: Population Model of Virginia Black Bears

CSUMS courses and activities:

- Math 410 (Spring 2008)
- 2008 summer research.

Research support: UBM, CSUMS

Research results: Research Blog

[1] The Persistence of Harvested Bears in Virginia, 2008 CSUMS summer report.

**Abstract **In 1999 the Virginia Department of Game
and Inland fisheries developed a long term plan to manage the black
bear population in Virginia; in 2001, the VDGIF published a 10 year
management plan. Though the plan contained many ideas to manage the
bear population - including fertility control, kill permits, regulated
hunting, etc., the management proposal lacks any concrete insight as to
the ramifications of these options.

The model included in this paper aims to analyze the population dynamics of the black bear population in Virginia by using a non-linear discrete model which separates bears not only by age, but also by gender, and territory. Parameters include survival and harvest (split by age, gender, and territorial status), percentage of breeding females, birthrates, and number of home ranges available. The harvest and survival data are valid only for the state of Virginia, and reflect information from 2002. Thus, the simulations can only give us information concerning the bear population in Virginia, and only general comments can be made concerning the entire black bear population along the Southern Appalachians. The VDGIF states that bear harvest has increased by 7.4% per year over the last decade. This information is taken into account in the model, and the simulations provide further insight as to the black bear population in Virginia.

**Michael Liarakos** Major: Computer Science and Mathematics (Adivsor: Virginia Torczon and Michael Lewis)

Research topic: Order on Binary Search Trees

CSUMS courses and activities:

- 2008 summer research.

Research support: CSUMS

Honors thesis: Software Engineering of a Direct Search Package for Nonlinear Optimization

Abstract: The DiSCOTech software package implements a direct search method with
heuristic improvement techniques for solving unconstrained and linearly
constrained nonlinear optimization problems. By implementing various
heuristic improvement techniques DiSCOTech seeks to improve the
performance of direct search methods. Currently, the development of
DiSCOTech is ongoing and is the subject of this research.

First, an object-oriented restructuring of the codebase is discussed.
Next, the issue of designing an effective caching mechanism for
DiSCOTech is addressed. Issues with proposed cache designs are
investigated and an alternative design is implemented and tested.
Finally, the relative merits of the heuristic improvement techniques
implemented by DiSCOTech are tested to determine which heuristics
reliably provide improved performance.

The thesis can be found at: http://hdl.handle.net/10288/

Research website: http://www.cs.wm.edu/~cmliar/

**Jennifer Mahle** (Class of 2009) Major: Mathematics (Adivsors: Chi-Kwong Li)

Research topics: Quantum computing, eigenvalues and singular values problems

CSUMS courses/activites: Math 410 (spring 2008)

- Participant of 2008 CSUMS workshop.
- Presentation at the Ninth workshop on Numerical Ranges and Numerical Radii, July 2008.
- 2008 summer research.

Research support: CSUMS

Research results:

[1] Clark, Li, Mahle and Rodman, Preservers of higher rank numerical range and radius, 2008 CSUMS summer research report.

**Abstract **Characterizations are obtained for linear
maps on Hermitian matrices or complex square matrices leaving invariant
the higher rank numerical range and radius. Similar results are
obtained for multiplicative maps, and surjective additive maps.
Extensions of the results to infinite dimensional operators and to the
joint higher rank numerical range of multiple matrices are also
considered.

**Brian Paljug **Major: Mathematics and Theatre (Advisor: Sarah Day)

Research topic: Exploring Issues of Topological Entropy in Symbolic Dynamical Systems

CSUMS courses and activities:

- Math 410 (Spring 2008)
- Presentation at the 2008 CSUMS workshop.
- 2008 summer research.

Research support: CSUMS

Research products: research blog

[1] Exploring Issues of Topological Entropy in Symbolic Dynamical Systems, 2008 CSUMS summer report.

**Abstract **Symbolic dynamics provides a discrete way
of looking at some of the qualities and characteristics of general
dynamical systems. Instead of continuously studying the behavior of a
system given some set of rules, symbolic dynamics discretizes
everything into a finite set of states, with rules providing for which
states can follow which other states. There is also the notion of
entropy for symbolic dynamical systems, which measures how chaotic or
complicated the system is. While symbolic dynamics is commonly used to
model and study dynamical systems, it provides a rich field of research
on its own, which is the approach taken by this project.

**Benjamin Strahs** Major: Computer Science and Philosophy. (Advisor: Haining Wang)

Research topic: Internet security.

CSUMS courses and activities:

- Math 410 (spring of 2008)
- 2008 summer research.

Research support: Chappell Fellowship and CSUMS

Research results: [1] Password Management with Enhanced Hashing, 2008 CSUMS summer report.

**Abstract**: Passwords play a critical role in online
authentication. Unfortunately, passwords suffer from two seemingly
intractable problems: password cracking and password theft. In this
paper, we propose PasswordAgent, a new password hashing mechanism that
utilizes both a salt repository and a browser plug-in to secure web
logins with strong passwords. Password hashing is a technique that
allows users to remember simple low-entropy passwords and have them
hashed to create high-entropy secure passwords. PasswordAgent
generates strong passwords by enhancing the hash function with a large
random salt. With the support of a salt repository, it gains a much
stronger security guarantee than existing mechanisms. PasswordAgent is
not vulnerable to offline attacks, and it provides stronger protection
against password theft. Moreover, PasswordAgent offers usability
advantages over existing hash-based mechanisms, while maintaining
users' familiar password entry paradigm. We build a prototype of
PasswordAgent and conduct usability experiments.

**Elizabeth Truelove** Major: Mathematics (Advisor: Michael Lewis)

Research topic: Computational molecular modeling

CSUMS courses and activities:

- Math 410 (spring of 2008).
- 2008 summer research.

Research support: CSUMS

Research results: In progress. Research Blog

**Katie Williams** Major: Computer Science and Mathematics (Advisor: Virginia Torczon and Michael Lewis)

Research topic: Different programing platforms: SIF and AMPL

CSUMS courses and activities:

- Math 410 (spring 2008)

- 2008 Summer research.

Research support: CSUMS

Research results: Parsing, Error-Checking, and Translating With SIF and AMPL

**Niha Zubair** Major: Mathematics and biochemistry. (Advisor: Sarah Day)

Research topic: Studying reproduction rate strategies in white-footed mice

CSUMS courses and activities:

- Math 410 (spring 2008).
- 2008 summer research.

Resarch support: UBM and CSUMS

Research results: Research Blog

[1] Reproduction Rate Strategies in White-Footed Mice, 2008 CSUMS summer report.

**Abstract **A photoperiod is the measure of the length
of daylight each day. This value can potentially determine the
behavior and/or biological processes of many species of animals and
plants. Peromyscus leucopus (white-footed mouse) responds to changes in
photoperiods by altering its reproductive strategies. P. leucopus
adjusts reproduction rates due to the high cost of reproduction in the
winter and in short photoperiods. In our research we have looked at
two groups of mice: Responsive mice (R) which reproduce March through
November and Non-responsive mice (NR) which reproduce all year around.
Interestingly, in Williamsburg, VA there exists a mixture of the
Responsive and Non-responsive mice. The coexistence of these two types
of mice suggests some kind of genetic variation. We have created
nonlinear discrete population models to better understand the
requirements for the co-existence of the two varying phenotypes.

**Class of 2010**

**Michael Essman** (Class of 2010) Major: Mathematics and Economincs (Advisor: Junping Shi)

Research topic: Numerical solutions of elliptic systems

CSUMS courses and activities:

- 2008 Summer research.

Research support: CSUMS

Research results: Research Blog

[1] Essman and Shi, Bifurcation diagrams of coupled Schr\"odinger equations, 2008 CSUMS summer research report.

**Abstract **Radially symmetric solutions of many
important systems of partial differential equations can be reduced to
systems of special ordinary differential equations. A numerical solver
for initial value problems for such systems is developed based on
Matlab, and numerical bifurcation diagrams are obtained according to
the behavior of the solutions. Various of the bifurcation diagrams of
coupled Schr\"odinger equations from nonlinear physics are obtained.