[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:
CSUMS courses and activities:
2008 summer research.
Research support: CSUMS
Research results: In progress.
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
Straphs 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
Website.
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.
Last
updated at Tuesday, June-10-2008 14:50:48 EDT.
