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Friday
Yuliya Babenko (Vanderbilt University) - On the
error of interpolation of C^2 functions by linear splines
Many problems in scientific visualization require representation of
surfaces. Even in the case when a continuous analytical
representation is available, its evaluation may be too inefficient.
To visualize a surface efficiently a lot of modern graphics hardware
requires a surface representation to consist of polygonal patches,
preferably triangles, which requires constructing a triangulation of
the domain to build this polygonal surface on. Optimal triangulations
for the interpolating spline surface can be constructed only in some
particular cases. In this talk we shall describe how to construct
asymptotically optimal triangulations for interpolation by linear
splines of C^2 functions, and provide the exact asymptotics for the
error in different norms.
William Turner (Western Kentucky University) - Old
and new about the number Pi
One of the most fascinating numbers in all mathematics is Pi. From
the prehistoric man with his bones and chisels all the way to man
today with the aid of computers we strive to refine the value of this
enigma of mathematics. First we give a brief overview of the past
calculation of Pi through history. Then we discuss a surprising
result of P. Borwein, Plouffe and Bailey (1996) about finding
individual hexadecimal digits of Pi without calculating the previous
ones. I am thankful for Dr. Benko for his valuable help.
Keith Andrew (Western Kentucky University) - Topology
Change in Gravity Theories
As the theoretical foundation for a quantum theory of gravity has
been recently developed the possibility for a dynamic changing
spacetime topology has become an area of intense study. One approach
to analyzing this problem starts with established solutions to the
Einstein equations as initial and final d- dimensional manifold
states. Using these two manifolds as boundaries one then searches for
a d+1 dimensional manifold whose boundary is the disjoint union of
the initial and final state manifolds: this is the well known
mathematical problem of bordism theory. Here we examine this problem
with regard to the construction of causal open sets for an almost
everywhere Lorentzian metric structure to explore the possibility of
wormhole formation.
Mustafa Atici (Western Kentucky University) - Is
the problem too difficult to solve or am I not smart enough?
Suppose you are given a problem by your boss to solve. You have
worked to find a "reasonable solution" for several weeks,
or months but found no solution. What are you going to say to your
boss? I cannot solve this problem! Or maybe you can convince him that
this problem is really "hard" to solve. We will show how to
reduce one "hard" problem into another problem. So anyone
who can solve that hard problem can also solve the other.
Bruce Kessler (Western Kentucky University) - Fourier
Analysis vs. Wavelet Analysis: May the Best Basis Win
Fourier analysis is very good when analyzing analog signals where the
primary goal is to determine the frequencies present, but it is not
useful in determining the time in which the frequencies occur. In the
case of music, it can identify the notes, but not when they were
played. Wavelet analysis can identify both frequencies and location
to some extent. Fourier analysis uses trigonometric functions as its
basis, while wavelet bases are more adaptable to the particular
application, and are usually supported on an interval of finite
length. This talk will compare and contrast the use of Fourier and
wavelet analysis methods in analyzing analog and digital signals.
Light introductions to both concepts will be given, and then both
methods will be used in a number of visual examples. While the
concepts being discussed are fairly high-level in nature, the talk is
designed to be accessible to undergraduate students and casual fans
of mathematics.
Molly Wesley (Western Kentucky University) - A
Generalized Baer Criterion
Let R be an associative ring with identity. An R-module E is injective
if for every R-module M and every submodule S of M, any linear map
from S to E can be extended to M. Baer's Criterion for R
modules successfully reduced the problem of determining whether E is
injective to determining whether E is injective for the ring R.
We will extend Baer's result to the category of representations of
the quiver • --> • (which has objects of the
form f : M --> N, where M and N are R-modules and f is a
linear map), as well as more general quivers.
Mozhgan Mirani (Vanderbilt University) - A short
survey of categorical equivalences between Gromov hyperbolic spaces
and bounded metric spaces.
A close relationship between Gromov hyperbolic spaces and the space
of its ends has been observed for many years. In this talk we focus
on the geometry of Gromov hyperbolic spaces and look at a categorical
equivalence that has been established by Bonk and Schramm. We will
then go on to look at a categorical equivalence established by Hughes
between trees and ultrametric spaces.
David Benko (Western Kentucky University) - Jacobi
and the Chocolate Factory
Legend has it that Newton discovered the laws of gravity when an
apple fell on his head. We speculate that Jacobi made one of his
important discoveries when he was buying a chocolate bar. As
supporting evidence we will show how Jacobi's method for the solution
of systems of linear equations can be developed from a recreational
"chocolate bar problem".
Yuhong Wu (Western Kentucky University) - Knots
and Links on the Cubic Lattice
The cubic lattice is a graph in R^3 whose vertices are all points
with coordinates(x, y, z) where x, y, z are integers and whose edges
are of unit length where they are line segments connecting the
vertices. This talk is about how many edges of the cubic lattice are
needed to realize some given knot or link. The main theorem in this
talk shows that the link [6,3,3] cannot be realized with fewer than
34 steps on the cubic lattice.
Tom Richmond (Western Kentucky University) - A
Geometric Theorem of Roger Cotes
Inscribe a regular 2n-gon in a unit circle with one vertex at
(1,0). Connect the vertically aligned vertices and find the
product of these n-1 segments. An amazingly simple answer
follows from a theorem of Newton's contemporary Roger Cotes.
Susan C. White (University of Louisville) - Structure
of Generic and A.E. Mappings from Z to Z
In this talk, we consider two notions of "large" and
"small" sets in Z^Z. The first is a topological
notion, that of a residual set and its complement, a meager set.
The second is the more recent measure theoretic idea of a prevalent
set and its complement, a Haar null set. We show that, while a
subset of Z^Z may be small in one sense, it may be large in the other
sense. Similar results on the permutation space S_N were
obtained by Doughterty and Mycielski.
Andrew Wilson (Austin Peay State University) - Evaluating
Web Tools for Learning and Teaching Mathematics
I will share with participants a form that I give my students to use
when evaluating web tools. I will also share various web sites
that are useful to learn and teach mathematics at the K-16 levels
(depending on the interest of the audience).
Saturday
Brett Bolen (Western Kentucky University) - Effects
of Global Expansion on Local Systems
We point out the existence of new effects of global spacetime
expansion on local binary systems. In addition to a possible change
of orbital size, there is a contribution to the precession of
elliptic orbits, to be added to the well-known general relativistic
effect in static spacetimes, and the eccentricity can change. Our
model calculations are done using geodesics in a McVittie metric,
representing a localized system in an asymptotically Robertson-Walker
spacetime; we give a few numerical estimates for that case, and
indicate ways in which the model should be improved.
Ronald Gilley (Western Kentucky University) - The
life and work of Galois
Evariste Galois lived a very short life in 19th century Paris,
France. We will give an overview of Galois' interesting life and
death. Galois groups play a very important role not only in Algebra
but also in many other areas of Mathematics. Ancient unsolved
questions such as the circle squaring, cube duplication and angle
trisection problems have all been solved by Galois Theory. The theory
also reveals which regular n-gons could be constructed by
straightedge and compass. We will explain how to construct the
regular pentagon and heptadecagon (17-gons). I appreciate the help
Dr. Benko gave me during the preparation of my talk.
David High (Western Kentucky University) - Teaching
Polynomial Behavior with Sketchpad
This talk will explore the benefits of exploring polynomial functions
and their behavior with Geometer's Sketchpad(tm). Sketchpad has
the ability to dynamically change values and representations at
run-time of the program, which makes it easy to look at different
graphs in a seamless manner. The presentation is designed for
those interested in teaching a College Algebra course. Rational and
exponential functions will also be explored.
Chris Christensen (Northern Kentucky University) - SIGSALY
Digital communications is not new. During the 1939 World's
Fair, Bell Telephone Laboratories exhibited a device called "the
vocoder" that transformed voice into digital data. World
War II stimulated interest in secure voice communications. The
United States and Britain developed a secure voice network called
SIGSALY that provided secure communications between, for example,
President Roosevelt and Prime Minister Churchill. We will
examine how SIGALY encrypted voice.
Patrick Coulton (Eastern Illinois University) - The
motion of semi-rigid bodies in curved spaces
The problem of motion of rigid bodies in spaces of constant curvature
initially investigated by Helmholz has been studied recently by Nagy,
Salvai, Zitterbarth and others. Rigid bodies can not move
freely in general manifolds because the group of isometries is
limited. We indicate a method of embedding semi-rigid bodies
into curved manifolds that allows us to study their free
motions. We will give some simple examples of this.
Includes joint work with R. Foote, and G. Galperin.
Dominic Lanphier (Western Kentucky University) - Expander
graphs and communications networks
Suppose a large number of people want to communicate with each other
over a network of communication lines. How should the lines be laid
out in order to maximize efficiency and reliability of the network?
The answer involves graph theory, group theory, and even some number theory.
Joe Gastenveld (Northern Kentucky University) - Extensions
that yield quasi p-groups
A group is said to be a quasi p-group if the group is generated by
the union of its p-Sylow subgroups. We will examine when an
extension of a group is a quasi p-group. In particular, we will
examine whether G/N being a quasi p-group implies that G is a quasi
p-group in the cases when N is the center, the commutator subgroup,
the Frattini subgroup, and the Fitting subgroup.
Jean-Claude Evard (Western Kentucky University) - Counting
factorizations in the multiplicative group of units of the ring of
integers modulo m
The first draft of a new method of fast factorization of large
integers was published by Chun-Xuan Jiang in 2002. This new method is
quite simple, natural, and very promising. It changes the usual
computation to a computation with much smaller integers. It can be
used to deal not only with factorization of integers, but also with
other problems related to primes. However, a lot of work remains to
be done to clarify and improve this method. To factorize a large
integer n with this method, we have to consider a well chosen
positive integer m, and then for every factorization of n modulo m,
we have to check whether a certain discriminant is a square. The time
it takes to check this strongly depends on the type of factorization
modulo m. In this talk we will present different types of
factorizations modulo m, and count the numbers of factorizations of
each type, for every type of integer n. We will also count the
average of these numbers over all relevant choices of n. This is the
first step of many future improvements of that method that are
currently under preparation.
Claus Ernst (Western Kentucky University) - On the
total curvature of a knotted space curve
Given a knot K tied with a rope. The total curvature of the
knot K measures how much bending there is in rope. Depending on
whether one pulls the rope tight the total curvature in the knotted
rope will change. There will be examples of knots that require large
total curvature and examples of knots that require only a small
amount of curvature.
Wayne Tarrant (Western Kentucky University) - Surely
it Cantor happen here
The Cantor set is an important example (and counterexample) in real
analysis and topology. It is also a beginning of fractals. Yet, it
reared its head in some of my research in algorithmic algebra. I will
define and discuss the Cantor set and show how the Cantor set
appears, unexpectedly, in algebra.
Sergiy Borodachov (Vanderbilt University) - On
lower order terms of the minimal discrete Riesz s-energy on curves
We consider the problem of minimization of the energy of the system
of N points repelling each other on closed curves. The potential of
the repelling force is proportional to the reciprocal of the distance
raised to the power s>0. The main term in the asymptotics as N
gets large of the minimal energy of N-point configurations is known
on rectifiable curves (A. Martinez-Finkelshtein, V. Maymeskul, E.
Rakhmanov, E. Saff, 2004) for s greater than or equal to 1. For three
times continuously differentiable simple closed curves we obtain the
next order term in this asymptotic representation.
André Wehner (Centre College) - Symmetries
of differential equations
In this talk we discuss powerful methods for analyzing and possibly
solving ordinary and partial differential equations. These methods,
known as symmetry analysis, were introduced by the 19th century
Norwegian mathematician Sophus Lie, who applied group theory to
differential equations. They have become standard analysis tools only
in recent years, and are now incorporated into Maple. Lie's symmetry
analysis (1) explains why many of the standard solution techniques
you've learned about in your DE class work, (2) allows us to reduce
the order of many ODEs, (3) shows how to construct new solutions of
PDEs from known ones, and (4) provides a classification scheme for DEs.
We gratefully
acknowledge the funds which was provided for student travel by MAA
NSF-RUMC (NSF Grant DMS-0241090, via the MAA) and Ogden College of
Science and Engineering, WKU.
 
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