REU Mini-Conference Abstracts
REU Mini-Conference Abstracts
- Andrew Brown:
Time-Optimal Control of Bioterror Response Logistics: The Case of
Anthrax
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This project begins the analysis of optimal control of bioterror response
logistics. The necessary conditions for time-optimal control are given,
based on the calculus of variations and dynamic programming. A gradient
algorithm is implemented and used to test the queuing network model as well
as the algorithm's own reliability. Experimental simulation results are
presented and discussed.
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- Masanori Koyama:
An algebraic approach to a graph choosability problem
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A graph is called n-list colorable if for every family of sets S_v (v in G)
with |S_v| =k for all v, there exists a proper coloring of the graph such
that each v is colored from the colors in S_v. The smallest n such that the
graph G is n-list colorable is called the choosability of G and is denoted
ch(G). It is known that \chi(L(G)) = ch(L(G)) =3 for 2-connected cubic
planar graphs. It is not known if this holds for cubic non-planar graphs,
although it is known to hold for K_{3,3}. Using an algebraic method,
attempt was made to show the 3 edge-choosability of K_{3,3} with vertices
replaced with triangles.
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- Brian Benson:
An Analysis of the Three Hat Problem
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In 2003, the Three Hat Problem written by Donald Aucamp appeared in MIT's
Technology Review. The puzzle gives a scenario in which three people
wearing hats are sitting together and each hat can be seen by everyone
except the person that is wearing that hat. Each person is told that all of
the hats contain a positive integer and that two of the integers add to the
third. In an ordered, turn-wise, modular fashion, each person truthfully
states whether or not he knows his number. We present a mathematical method
by which to analyze all positive integer cases of the Three Hat Problem
puzzle. Using this analysis, we show how and why it is always the case that
one of the three people can figure out the integer that is on his hat.
Further, we give a simple algorithm to derive the dialogue for any case and
ordering of three integers that might occur in the puzzle.
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- Emanuel Indrei:
Markov Chains and Traffic Analysis
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In this study, we use Markov chains to construct a theoretical traffic
system. The presentation is organized into three major parts: The first two
deal with the construction of two spaces in which objects may interact. The
third part analyzes the evolution of one particular object. Using the
central limit theorem and bounds given by the law of iterated logarithm, we
prove that after a large number of time steps, the probability of locating
this object in the traffic network diminishes to zero. We conclude by
offering methods of analyzing the evolution and interaction of multiple
objects and accounting for accidents.
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- Darshan Bryner:
The Homology of 17-Crossing Knots: A Computational Approach to Knot
Theory
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Knot Theory is a relatively new field in Mathematics, and hence many
mathematical properties of knots remain unfounded. To date, a comprehensive
list of all knots without repetitions of up to only 16 crossings exists. In
this research project, Lew, Stavros, and I have created a database to store
knots of up to 17 crossings and implemented a program to hopefully form the
most accurate list in existence of such knots without repetitions. My talk
will address knot theory terminology and definitions, the software tools
used in our computation, the structure of our database, and the use of
parallel processors to achieve the necessary computational power.
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- Steven Britt and Laura Stiltz:
A Mathematical Framework for Paper Folding
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Our research has focused on utilizing the standard ideas of differential
geometry to mathematically describe the notions associated with paper
folding. We define folding functions both along lines, as is traditionally
associated with folding, as well as more generally along planar curves. We
then prove the intuitively necessary condition that any folding as we have
defined is isometric to a plane. Further questions arise upon examination
of the processes by which the paper is folded as a function of time. We
also present a few examples, conjectures, and proofs on various topics in
paper folding.
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- Gokhan Civan:
Distinguishing Legendrian Knots
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Legendrian knots are knots that satisfy certain geometric properties. There
are various algebraic invariants that are used to differentiate between
Legendrian knots. Recently a very complicated invariant in the form of a
differential graded algebra was discovered. There are also invariants that
are derived from this more general invariant through a process of
linearization. There are open questions about the equivalence or relative
strengths of these derived invariants. This research consisted of
investigating examples in hopes of shedding light onto some of these
questions. The effort has been assisted by a Mathematica code which has been
developed along the way and could culminate in a general tool to compute
some of these invariants for Legendrian knots.
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- Brian Nakamura:
Alexandrov's Conjecture: Intrinsic diameter and area of convex
surfaces
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This 50 year old conjecture by Alexandrov has commonly been cited, yet there
are very few known results. An introduction to the conjecture and some
recent progress will be presented. A few observations and partial results
produced during the summer REU will also be discussed.
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- Bobby DeMarco:
The Four Vertex Theorem and the Extension of its Converse to the
Sphere
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Adolf Knesser proved in 1912 that any simple closed planar curve has at
least four vertices, or points of maximum or minimum curvature. Over the
past almost hundred years many versions and extensions of this theorem have
been explored. Notably, in 1971 Herman Gluck proved a type of converse to
the four vertex theorem. Gluck showed that given a strictly positive
curvature function with at least four vertices, it is possible to construct
a simple closed planar curve as a function of time, t, such that the
curvature at any point on this curve corresponds to the given curvature
function. This presentation will discuss the continuation of Gluck's work
and examine how Professor Ghomi and I hope to extend it to the sphere.
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- Garret Thompson:
Searching for Periodicity in the Game of Officers
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A taking-and-breaking game is played by removing beans from a heap and then
splitting what remains into new heaps, where the number of beans removed and
the number of heaps depend on the rules of the game. According to the
Sprague-Grundy theorem, each position in such a game is equivalent to a
position in a simpler game called Nim, and if you can determine a function
mapping heap sizes to "Nim-values," you can play the game optimally. It is
conjectured that this function is periodic for most taking-and-breaking
games, and we explore techniques that have been used to solve other games,
applying them to a specific game called Officers.
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- Lee Martie:
GOEDEL
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The GOEDEL program, Belinfante's computer implementation of Kurt Godel's
algorithm for class formation in Mathematica, was used to formulate
definitions and theorems in the theory of relations, abstract algebra and
arithmetic. The ultimate goal would be to use the results derived to obtain
automated proofs using McCune's automated reasoning program Otter. The
specific focus of this research was to translate into the language of the
GOEDEL program the formulation of Peano arithmetic in terms of unary
algebras as expounded by Birkhoff and Bartee in their book on Modern Applied
Algebra. Several Mathematica notebooks were prepared and posted on the web,
detailing some of the more interesting results obtained in the course of
this work. The first notebook dealt with the class UNOPS of all unary
operations. A unary operation is a function whose range is contained in its
domain. General properties of this class were derived, and examples of
unary operations were provided. The second notebook posted was about
partial orders and equivalence relations. In this notebook, direct proofs
were obtained for results that had initially been found using reification, a
procedure for associating relations to constructors. The third notebook was
about a theorem concerning unions of commuting transitive relations. Two
notebooks were posted about powers of unary operations. A formula for the
class of mappings with one-point domains was derived, as a first step toward
proving theorems about mappings using induction on the size of their
domains. The final topic studied concerned cyclic unary algebras, and the
clock algebra in particular.
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- Dragos Ilas:
On the Abel-Jacobi Map from a Graph to its Jacobian
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There is a natural way to attach to any graph a finite Abelian group called
the Jacobian group, or Picard group, of the graph. There is also a natural
map from the graph to its Jacobian group called the Abel-Jacobi map. A
program was written to calculate the Jacobian group and the Abel-Jacobi map
using the Smith Normal Form. We discuss some conjectures and theorems about
the injectivity and surjectivity of powers of the Abel-Jacobi map. The
latter is related to the number of linearly independent cycles which the
graph possesses. Finally, we investigate the relationship of these
questions to a certain chip-firing game on graphs.
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- Beth Hart:
Discrete Biorthogonal Polynomials
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We have been analyzing discrete biorthogonal polynomials, which generalize
the notion of discrete orthogonal polynomials. Our polynomials P_n of degree
n satisfy a biorthogonality relation such as
\sum_{j=0}^{\infty}{P_n(x_j)(\psi (x_j))^k w(x_j)} = 0, for k =
0,1,...,n-1. Here x_0,x_1,... is a sequence of points, the weight w is
positive on all these points, and \psi is an increasing function on an
interval containing all x_j. The special case \psi (x) = x generates
discrete orthogonal polynomials. We investigate explicit formulae for P_n for
various choices of \psi, x_j, and w.
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