Session I - Stratton Hall 202
4:00-4:10 Fibonacci and Nature
Cherline Beaubrun, Framingham State College
Many people go through life without realizing that mathematics is all around
them in nature. Leonardo Fibonacci was not one of these people.
Fibonacci saw the connection between mathematics and nature in the
simple 'multiplication' of rabbits.
His insight helped him to develop the Fibonacci numbers.
Even after hundreds of years mathematicians still find ways to
apply the Fibonacci numbers to their own work. In this presentation,
I will discuss the Fibonacci numbers, Fibonacci’s golden numbers, the
manifestation of these numbers in nature, and some of the ways other
mathematicians have applied these numbers.
4:15-4:25 Rene Descartes and the Scientific Revolution
Christopher Jackson and Thierry Nkouga, Framingham State College
Rene Descartes’ trust in human senses superseded his trust in European
theology and spurred his contributions to the scientific revolution of the
Renaissance, the historical transition from the Medieval Era to the Modern Age.
In this presentation, Descartes’ philosophy and some of his accomplishments
in mathematics and physics will be presented in order to illustrate the
progress of science in Europe during the 17th Century. The basis of
Descartes’ analytical geometry of a cone to produce graphs of an ellipse,
circle, parabola, and hyperbola will be used to examine applications to
optics and light diffraction with lenses.
4:30-4:40 Maria Gaetana Agnesi: A Look at Her Life and Her
Contributions to Mathematics
Cynthia George, Framingham State College
The early 1700’s saw the dawn of the Catholic Enlightenment and with it the
birth of an intelligent and caring woman in Maria Gaetana Agnesi. The eldest
of twenty-one children, Agnesi’s genius was evidenced by her early command
of modern languages, her gifted public speaking, and her appreciation for
the study of mathematics. In this presentation, I will examine Agnesi’s
dilemma: to apply her knowledge to help educate others or to pursue her
other passion, caring for the infirm and elderly by joining a convent.
4:45-4:55 Benoit Mandlebrot: A Life of Chaos to the Chaos of Life
Kevin Sylvia, Framingham State College
"Clouds are not spheres, mountains are not cones, coastlines are not
circles and bark is not smooth, nor does lightning travel in a straight line."
This is obvious, but often ignored, and some still continue to try to squeeze
life into the over-simplistic models of Euclidean geometry. Fortunately,
fractals have revolutionized many diverse fields. From physics, chemistry,
and biology to art, economics, and topology, fractals have changed our view
of the world around us. This new way of thinking is due to the 'father of
fractal geometry', Benoit Mandlebrot, whose meandering and self-labeled
chaotic life led to his search for order in chaos and a new description
of so-called chaotic structures in nature. In this presentation, I will
discuss Mandlebrot’s progression toward an understanding of chaos and
explore the fruition of his work: the Mandlebrot set.
Session II - Stratton Hall 308
Approximating 2D Diffusion: What every Student Doesn’t (Necessarily) Know
Dave Voutila, Worcester Polytechnic Institute
Undergraduate applied mathematics students usually spend the majority of
their studies in a one-dimensional world when learning about diffusion and
its approximation. The typical reasons for not delving into two-dimensional
diffusion studies are complexity and time constraints. But the approximation
of two-dimensional diffusion using finite differences presents an excellent
example of the concepts of numerical stability and dissipation. As a result,
they provide a good introduction to Von Neumann analysis. This presentation
will provide a brief explanation of stability and dissipation for the
two-dimensional diffusion equation using Von Neumann analysis.
4:15-4:25 Making Spirals
Mary Servatius, Worcester Polytechnic Institute
We will give several interesting examples of spirals,
explain how they can be described, and present a discrete method for
4:30-4:40 Compatible 0-1 Sequences
Jason Gronlund and John Hajeski, Worcester Polytechnic Institute
We define two 0-1 sequences to be compatible if it is possible to delete
certain 0’s from both sequences in order to make the two sequences
complementary. The general conjecture is that the sequences will be
compatible with positive probability when the occurrence of 1’s within
the sequences is some probability less than 0.5. We will consider small
finite examples of compatible sequences.
4:45-4:55 Modeling the Dynamic of Tumor Growth in the Brain
Kim Ware, Worcester Polytechnic Institute
Currently, a person diagnosed with glioblastoma (GBM), a highly invasive
cancer of the structural cells in the brain, has an extremely low chance of
long-term survival. One of the obstacles in treating GBM is the inability
of current medical imaging technology to observe growth at an extremely small
scale. Our task, in cooperation with IBM Corporation and researchers at
Harvard University, is to develop a continuum model that accounts for both
the proliferation and migration of tumor cells. In formulating this model,
we will use a system of partial differential equations to describe the
dynamics of the tumor and its effects on the surrounding brain tissue.
In addition, we will employ finite difference methods to approximate the
solution to the system. Our goal is to utilize the model in understanding
patterns in the initial stages of tumor growth.
Session III - Higgins Labs 116
4-4:10 Vertex Magic Total Labelings of Bipartite Graphs
Karthik Raman, Norwich University
We will describe two new computer programs. The first can generate all the
possible vertex magic total labelings (VMTL) for K3, 3. We will show how
this program is not computationally feasible for looking for vertex magic
labelings of larger complete bipartite graphs. The second program searches
for a special VMTL. This program is also useful for large graphs. We will
describe the labelings that this program will find, and why they play a
central role in the theory.
4:15-4:25 Predicting Academic Success Using CART
Charity Combs and Timothy Phelps, Norwich University
A general classification problem may be described as follows:
Given a multivariate observation, which is known to belong to one of several
populations, determine which population is most likely. Traditional methods
for dealing with this problem often lack flexibility. Observations are often a
ssumed to be normally distributed, for instance. Traditional methods cannot
work with categorical data or incomplete data in a natural way. CART
(Classification and Regression Trees) techniques are being applied to data
from a small private undergraduate institution. Current processing and
results will be discussed.
4:30-4:40 Convergence Time for Different Selection Schemes in Genetic
Jamie Kingsbery, Williams College
Abstract genetic algorithms (GA’s) are an incredibly useful way of finding
good solutions to hard optimization problems in a time-and-space-efficient
manner, but they are poorly understood from a theoretical standpoint.
One effort to better understand GA’s has emerged through the notion of
convergence time. We will define and examine this idea, and compare the
convergence times in GA’s that use both ranking and proportional selection.
We will see why the idea of convergence time is useful and at the same time
limited in analyzing GA performance.
4:45-4:55 Using Linear Programming and Function Approximation to Optimize
Mike Frechette, Gordon College
Minimizing long-run average cost per stage in stochastic
network problems requires solving the countably infinite system of nonlinear
equations generated by Bellman’s equation. One approximation method
transforms Bellman’s equation into a linear program, the system of linear
constraints that yields the exact solution. We will explore a method of
further simplifying this linear program by fitting functions to Bellman’s
equation. Our result is a small linear program that is computationally
simple to solve for a reasonably accurate average cost.
Session IV - Higgins Labs 154
4:00-4:10 Fractals and the Magnetic Pendulum
Benjamin Morin, University of Maine
Placing a magnet on a pendulum and an array of magnets beneath it, the
understood behavior of the pendulum gives way to unpredictable orbits and
end states. Without knowledge of the exact initial state, the end state
cannot be predicted with any degree of accuracy. This is because the pendulum
creates fractals along the boundaries of its basins of attraction. I have
written software that has allowed me to show that the boundary is a fractal.
4:15-4:25 Vectors, Scalars, and Motion
Tim Coburn, Framingham State College
For those looking to shave a few strokes off their golf game
or master the intricacies of billiards, a good understanding of vectors,
scalars, and motion could go a long way. One of the most famous equations,
Newton’s Second Law, simultaneously demonstrates the importance of vectors,
scalars, and motion. In this presentation we will provide an overview of
these notions, discuss the history of the development and application of
vectors, and emphasize the relationship to three-dimensional motion.
4:30-4:40 The Tribonacci Spiral
Caroline Mallary, Worcester State College
The ratio of successive entries of the Fibonacci sequence of
integers converges to a number which can be used to construct a
two-dimensional 'golden spiral'. The 'Tribonacci' sequence of integers
converges to another ratio which can be used to construct a similar spiral
in three dimensions. I will discuss some of the properties of this spiral
and the number on which it is based, which is approximately 1.839286755.
4:45-4:55 Putting the Odds in Your Favor
James Piette III
Due to the popularity of Texas Hold’em poker, many mathematical techniques
have been implemented to help determine an optimal winning strategy.
One important statistic is a hand’s probability of winning
(the hand’s winning percentage). This presentation illustrates a new
technique that calculates the exact winning percentage of a hand.
These values were tested via Monte Carlo simulations. While not equal,
the calculated and simulated results are similar enough to validate the
Session V - Higgins Labs 218
4:00-4:10 Calculus at Work: Elastic Deflection of Support Beams
Michael Coleman and Frank Grimmer, Western Connecticut State University
Using a classic differential equation for the deflection curve of an
elastic beam, we derive a general mathematical model for a beam subject
to a non-symmetric system of loads. The governing differential equation
is them solved by a standard calculus method. Four arbitrary constants
are determined from natural constraints on the deflection function and
its derivative. Examples with specific material and geometric parameters
for the beam are given and the absolute maximum of the deflection function
is found. If time permits, possible applications in design problems will
4:15-4:25 Strategy in the Design of Voting Systems
Warren Schudy, Worcester Polytechnic Institute
In most US elections every voter may vote for one candidate
and the candidate receiving the most votes wins. This voting system has
a number of flaws, most notably the 'spoiler effect'; two similar
candidates can lose to a third even though their common policies are
supported by a majority. A number of voting systems solve this problem,
including approval voting, Condorcet methods, and single transferable
vote. Various mathematical criteria are used to analyze these voting
systems. The importance of strategy in the design of voting systems will
4:30-4:40 Elusive Zeros Under Newton's Method
Trevor O'Brien, College of the Holy Cross
Given its iterative nature, it is natural to study
Newton's method as a discrete dynamical system. Of particular
interest are the various open sets of initial seeds that fail to converge
to a root. We shall examine a certain family of 'bad'
polynomials that contain extraneous, attracting periodic cycles. The
family relies on only one parameter, and we have developed and
implemented computer programs to locate values of this parameter for
which Newton's method fails on a relatively large set of initial
conditions. In doing so, we have discovered some rather surprising
dynamical figures including Mandlebrot-like sets, tricorns, and
swallowtails. We have also uncovered analytic and numerical evidence
that aids in explaining the existence of such figures.
4:45-4:55 Two Towers Are More Fun Than One
Karen Shively, Wheelock College
The Tower of Hanoi problem, in which a pile of different sized
disks must move from one pole to another, is solved using a recursive
process. In this variation of the Tower of Hanoi problem, two piles of
disks must trade places. How can you do this? Does the recursion still
work? What is the least number of moves possible? Learn all this and