Student
Papers: Schedule
|
Speaker and Title |
Time |
Room |
|
Brian D. Ginsberg, Yale University The Nearly Secret Theorem of E. Midy – An Extension after
165 Years |
5:00-5:10 |
270 |
|
Elizabeth
Bellenot, Wellesley College Effects of Biological Invasions on Ecological Communities |
5:15-5:25 |
270 |
|
Jessica
S. Lee, Wellesley College |
5:30-5:40 |
270 |
|
Michael
J. Coleman, Boston University and Sidharth Rupani, WPI Modeling iBOT Belt Dynamics |
5:45-5:55 |
270 |
|
Seila Selimovic, Wellesley
College |
6:00-6:10 |
270 |
|
|
|
|
|
Iuli
Pascu, Wellesley College A Graphical Interpretation for Complementary Sequences |
5:00-5:10 |
274 |
|
Karin
Steece, Wellesley College The Chinese Postman Problem |
5:15-5:25 |
274 |
|
XinXin
Du, Wellesley College Monte Carlo Simulations on One Electron Per Site,
Two-Dimensional Square Lattices |
5:30-5:40 |
274 |
|
Paula
F. Popescu, Wellesley College Games with Hats |
5:45-5:55 |
274 |
|
Kathleen Leahy, College
of the Holy Cross Enigma – The Code that Changed History |
6:00-6:10 |
274 |
|
|
||
|
Charlie
Rossetti and Matthew Angelucci, Bentley College Dynamica |
5:00-5:10 |
278 |
|
Brandon Dwyer, Bentley
College |
5:15-5:25 |
278 |
|
Jenny Kirouac, Westfield
State College Naming Really Large Numbers |
5:30-5:40 |
278 |
|
Kari
Lock, Williams College Making Best Approximates Appear Through Magical
Intervals |
5:45-5:55 |
278 |
|
Neil
Hoffman, Williams College Double Bubbles in Other Universes |
6:00-6:10 |
278 |
|
Speaker and Title |
Time |
Room |
|
Kathleen
Smith, Norwich University Vertex Total Magic Labelings |
5:00-5:10 |
392 |
|
Solving ‘Rubik’s Polyhedra’ Using Three-Cycles, |
5:15-5:25 |
392 |
|
Joelle
Arnold, Olin College The Assignment Problem and Optimizing Registration |
5:30-5:40 |
392 |
|
Janet
Tsai, Olin College You Can Make Anything with One Straight Cut |
5:45-5:55 |
392 |
|
Ben
Kraines and Siddartha Rao, Middlebury College Extreme Points and Lipschitz Conditions for Functions in
a Certain Zygmund Class |
6:00-6:10 |
392 |
|
|
|
|
|
Matthew Stephen Palmacci, Framingham
State College The Golden Ratio |
5:00-5:10 |
396 |
|
Alison Fish, Merrimack
College The Golden Ratio and Three-Dimensional Geometry |
5:15-5:25 |
396 |
|
Kaitlyn
ONeil, Merrimack College Pretzel Knots and Colorability |
5:30-5:40 |
396 |
|
Kevin
Roberge, University of Maine Algebraic Topology and Elementary Circuits |
5:45-5:55 |
396 |
|
Michael A. Burr, Tufts
University Simplicial Depth: An Improved Definition, Analysis, and
Efficiency for the Finite Sample Case |
6:00-6:10 |
396 |
Student
Papers: Abstracts
Matthew Angelucci, Bentley College 5:00-5:10,
room 278
Dynamica
In this talk we present Dynamica, a software package
running under Mathematica that helps researchers gain insight into the behavior
of dynamical systems. The various features of the software are demonstrated on
a biological model of Lyme disease described by a difference equation.
Joelle
Arnold, Olin
College 5:30-5:40,
room 392
The Assignment Problem and Optimizing
Registration
Consider the arranged
marriages of a fixed and finite number of heterosexual couples. Suppose that
each woman gets to make a list of her preferred husbands, but that the men have
no say. We will discuss the conditions under which each woman can be paired
with a suitable husband - this is Hall’s marriage problem. The Assignment
Problem is any scenario in which a group of people or other entities are to be
matched to another group of people or entities, and in which both groups have
the same number of members. By either one group or both groups ranking preferences,
an optimization can be found. We arrange the data in a matrix and by using the
Hungarian Method, may make the best matches. This procedure is used to solve
job training, hypothetical marriage and shipping route problems. We will
investigate using these two methods to optimize the registration process here
at Olin. In this manner, we should be able to maximize student satisfaction
with spring schedules.
Elizabeth
Bellenot, Wellesley
College 5:15-5:25,
room 270
In an ecological
community with n distinct species, what happens when an invader species is
introduced? By studying food webs using
matrix analysis and simple graph theory, we examine cases where we can predict
which resident species will increase, decrease, or has no change in population.
We also look at cases where the outcome is ambiguous.
Michael A. Burr,
Tufts University 6:00-6:10,
room 396
Simplicial Depth: An Improved Definition,
Analysis, and Efficiency for the Finite Sample Case
As proposed by Liu
(1990) the simplicial depth of a point 
with respect to a
probability distribution 
on 
is the probability
that 
belongs to a random
simplex in 
. The simplicial depth of 
with respect to a
data set 
in 
is the fraction of
the closed simplices given by 
of the data points
containing the point 
. We propose an alternative definition for simplicial depth
which continues to remain valid over a continuous probability field, but also
fixes some of the problems for the finite sample case, including those
discussed by Zuo and Serfling (2000). Additionally, we discuss the effect of
the revised definition on the efficiency of previously developed algorithms and
prove tight bounds on the value of the simplicial depth based on the half-space
depth.
Michael
J. Coleman, Boston University 5:45-5:55,
room 270
The iBOT, a
revolutionary personal mobility device developed by DEKA Research and
Development Corporation, contains several belts that transfer mechanical power
through the system. The dynamic characteristics of the belts are obviously
important to the operation of the iBOT. Our task is to create a mathematical
model of some aspects of the belt dynamics. Partial and ordinary differential
equations and Lagrangian dynamics are employed to understand the elements of
this electro-mechanical system.
Ultimately, finite difference numerical methods are applied to solve the
rather complicated system of governing equations developed. The results from
our mathematical model are compared with those from actual experiments.
XinXin
Du,
Wellesley College 5:30-5:40,
room 274
Monte
Carlo Simulations on One Electron Per Site, Two-Dimensional Square Lattices
I
used Monte Carlo methods to do random walks through configuration space on
two-dimensional, square lattices, computing the energy and the magnetism of the
system, and finding the ground state of the system. Using the Metropolis algorithm, I simulated a situation where the
system sits at a particular temperature and looked at its thermodynamic
behavior. The Metropolis algorithm also
allowed me to overcome the problem of arriving at metastable states instead of
the ground state in some cases.
Brandon Dwyer,
Bentley College 5:15-5:25,
room 278
A Student’s Look at the First
Actuarial Exam
If
a student is looking seriously into a career as an actuary, they should begin
the exam process while still in college.
The Course I exam contains problems relating to calculus and probability
(both discrete and continuous). This
presentation will look at how to prepare for the exam including strategies for
studying. Two typical exam questions
will also be presented.
Alison Fish,
Merrimack College 5:15-5:25,
room 396
The Golden Ratio and
Three-Dimensional Geometry
The
golden ratio, represented by j
(phi) is found in many areas of mathematics. This talk will discuss the golden
ratio and how it can be found in three-dimensional geometric shapes such as
polyhedra. Golden rectangles, which
exhibit this ratio, exist within the polyhedra. Also the coordinates of the
vertices of the polyhedra and the relationship of the edges include the golden
ratio.
Brian
D. Ginsberg, Yale
University 5:00-5:10,
room 270
This work extends
Midy’s theorem — a curious, old number theory result about parts of decimals of
even period—to a wider class of fractions whose period need not be even.
Avenues for further extension are also explored.
Neil
Hoffman, Williams
College 6:00-6:10,
room 278
The
recently proved Double Bubble Conjecture says that the familiar double soap
bubble is the least-area way to enclose and separate two regions of prescribed
volumes in 
. We report on extensions to other three-dimensional
universes.
Jenny Kirouac,
Westfield State College 5:30-5:40,
room 278
You
know thousand, and million, and billion, and trillion. Ten thousand, ten
million, ten billion, and ten trillion, but do you recall the numbers that come
after them all? Shouldn`t we have names for bigger and bigger numbers? And what
exactly is a zillion? This talk will delve into a system of naming numbers that
not only exceeds our traditional system, but also allows us to keep naming
larger and larger numbers without limit. Of course, we will also mention some
of the famous numbers and say how they can be alternatively named in this new
system: large numbers like the perplexing Googolplex number, the merciful
Mersenne primes, and the invigorating Vinogradov`s number. After experiencing
this talk you will know lots of amazingly large numbers AND you will be able to
name any number you come across.
Ben Kraines,
Middlebury College 6:00-6:10,
room 392
Extreme Points and Lipschitz Conditions for Functions in a
Certain Zygmund Class
The
Tagaki–van der Waerden function is an important example of a continuous
nowhere–differentiable function.
Although it is nowhere differentiable, it is possible to define an
analogue to the derivative via the “dyadic difference quotient” sequence. This
paper concerns functions whose dyadic difference quotients have modulus
one. We use the dyadic difference
quotient to investigate Lipschitz points and extreme points for such functions
in a way that parallels the use of the derivative in studying extreme points
and Lipschitz behavior in real variable calculus.
In
this talk we will give a brief history of the Enigma code used by the Germans
in World War II. We will describe the machine used to encode and decode as well
as some of the mathematics involved. We will also discuss the connection
between the breaking of the Enigma code and the creation of the first digital
computer.
Jessica
S. Lee, Wellesley
College 5:30-5:40,
room 270
As early as 300 B.C.,
Euclid studied prime numbers, which are numbers divisible only by one and
themselves. Until 1536, mathematicians
believed that all numbers of the form 
were prime for all
primes 
. While this
conjecture is false, many people now have joined the search for primes of that
form, now dubbed Mersenne Primes after the Frenchman who studied them at great
length. This talk briefly outlines the
history of these rare primes, the current on-going search for them, as well as
providing an introduction to some of the unsolved mysteries surrounding
Mersennes.
Kari
Lock, Williams
College 5:45-5:55,
room 278
Let 
be a real, irrational
number. Given a positive integer q, an important question in diophantine
approximation is: how do we determine exactly when q is the denominator of a
“best rational approximate” to 
? Using the theory of continued fractions and their
convergents, I show that if the continued fraction expansion of 
has the form 
, then a positive integer q is a denominator of a best
approximate to 
if and only if the
interval
contains an integer.
This result extends the work of several mathematicians who proved analogous
theorems for the Golden Ratio (that is, for 
) and the generalized Golden Ratio (that is, for
).
Kaitlyn
ONeil, Merrimack
College 5:30-5:40,
room 396
My
colleagues and I develop a formula for determining the number of fundamentally
different ways that an 
-colorable knot can be 
-colored, based on the 
-nullity of the knot.
We then determine the m-nullity of any 
pretzel knot, and
thus a way to determine the 
-colorability and number of fundamentally different 
-colorings of any pretzel knot.
Matthew Stephen Palmacci,
Framingham State College 5:00-5:10,
room 396
A
proportion that can be seen throughout nature and man-made structures is the
Golden Ratio, often denoted by the Greek letter phi. In nature, the divine section can be acquired from measurements
of the human body, plants and Uranium.
Man-made structures can also be used to appreciate this wonderful
proportion. Phi can be found in the
building of musical instruments, the Fibonacci sequence, ancient pyramids, and
even electrical networks. The divine
proportion reveals an interesting and beautiful pattern that emerges from the
wonders of nature.
Iuli
Pascu, Wellesley
College 5:00-5:10,
room 274
Two
sequences with the property that each natural number is contained in exactly
one of them are called complementary sequences. I will present a graphical way
to generate complementary sequences. Starting from this graphical
interpretation, I will show how to derive some interesting properties of these
sequences.
Paula
F. Popescu, Wellesley
College 5:45-5:55,
room 274
In
his article "Games People Don't Play", Peter Winkler describes two
different games in which colored hats get distributed randomly to a team of
players. Each player has to guess the color of his own hat, based on
observation of his teammates' hats and a team strategy designed in advance. In
both of the games, only two hat colors are allowed. In this talk, we will
discuss strategies for analogous games with multiple hat colors.
Siddartha
Rao, Middlebury
College 6:00-6:10,
room 392
Extreme Points and Lipschitz Conditions for Functions in a
Certain Zygmund Class
(See
Ben Kraines for abstract.)
Kevin
Roberge, University
of Maine 5:45-5:55,
room 396
Algebraic Topology and Elementary
Circuits
What does a torus or
a Möbius band have to do with electrical circuits? How does the presence of “holes” in a topological space affect
the existence of solutions for currents and voltages? These questions are the sorts that drive the subject matter of
this presentation which presents some connections between algebraic topology
and circuit analysis. This talk will
present the basic idea of algebraic topology and how we might use it to talk
about circuits. Elementary circuits
will be presented and the application of algebraic topology to these circuits
will be outlined.
Charlie
Rossetti, Bentley College 5:00-5:10,
room 278
(See Matthew Angelucci for abstract.)
Sidharth
Rupani, Worcester
Polytechnic Institute 5:45-5:55,
room 270
(See Michael J. Coleman for abstract.)
Seila Selimovic,
Wellesley College 6:00-6:10,
room 270
Mathematical
expressions are often presented to undergraduate Physics students without full
explanations. In this study, we choose to examine one such object, the Dirac
distribution, in detail. The application of this distribution in quantum
mechanics shows one of the common mathematical mistakes in physics. Physicists
often treat the Dirac “delta-function” as a function instead of as a distribution
that is the limit to approximations of the function 
at a given 
. This distinction is important for mathematicians, but for
physicists it does not carry much weight. In this work we establish differences
between functions and distributions, demonstrate why delta functions do not
exist, and explain in which mathematical framework and physical problems one
can use incorrect terminology and thus speak about delta-functions.
A
total labeling of a graph is an assignment of consecutive integers 1,2, …,
to the vertices and edges of the graph. The weight of each vertex is the sum of its
label and the labels of its incident edges.
A vertex magic total labeling is a total labeling in which the weight of
each vertex is constant. We will focus
on the odd complete graphs. Previously,
magic squares had been used to construct labelings with only a few different
magic constants. In this talk, we will
introduce a new technique that is used to construct labelings for all possible
magic constants. This solves an
important conjecture for odd complete graphs.
Karin
Steece, Wellesley
College 5:15-5:25,
room 274
Originally studied by
the Chinese mathematician M. Kwan, the Chinese Postman Problem involves finding
the shortest distance a mail carrier can travel in order to deliver the mail to
all the houses on his route, and then return to the post office. For an
Eulerian graph, the problem reduces to finding an Eulerian circuit. However, it
turns out that a good algorithm exists even in the non-Eulerian case. I will
discuss the basics of the solution and present a few additional examples of how
the algorithm can be applied in the real world.
Janet
Tsai, Olin
College 5:45-5:55,
room 392
According
to legend, Betsy Ross was given the responsibility of creating the first
American flag in 1776 after she impressed General George Washington with an
interesting way to create a five-pointed star. By simply folding a piece of
cloth and making one snip with her scissors, Betsy made a regular five-pointed
star which is now famous, appearing 50 times on every American flag. But
Betsy’s cut leads to a larger question: is it possible to create any 2-D
polygon by folding a piece of paper and making one straight cut? This question
has been studied extensively during the past ten years, resulting in two
distinct proofs which demonstrate how any polygonal shape can be created by
cutting once in a straight line. Our discussion will begin with the activity of
creating a five-pointed star as Betsy Ross did to introduce the idea of
folding, cutting, and unfolding to yield a shape. From there, we will present a
brief introduction to the problem and then move to a more in depth look at the
disc-packing method of proof. Relevant connections will be made to the fields
of graph theory, combinatorics, and origami mathematics.
The
Rubik’s Cube is only the most prominent member of a family of related puzzles
consisting of colored regular polyhedra with rotating faces. This paper uses
group theory to describe a set of simple solution algorithms that can be
constructed for any such puzzle. We show that it is possible to create
sequences of face rotations that result in a motion of exactly three pieces on
a polyhedron puzzle, leaving all other pieces as before; and that the smallest
possible such motion is of three pieces, i.e., that it is impossible to have
exactly two pieces out of place. Thus, we can always solve these puzzles using
nothing more than the easily constructed three-cycles.
New
Colleague Presentations
Session 1, Room 270
8:00 am Complementary Sequences
Sam Vandervelde,
Wellesley College
We present an elegant
method for constructing a partition of the positive integers into a set of
sequences, beginning with the Fibonacci and Lucas numbers.
8:20 am Landscape
Erosion
Ted Welsh, Westfield
State College
How
do you describe landscape erosion? We
will look at a system of partial differential equations: What problems
arise? How can we fix these problems?
8:40 am Technology
in Mathematics for Preservice Elementary Teachers
Barbara
Boschmans, Plymouth State College
A study was conducted last fall on the self efficacy of preservice
elementary teachers and using technology in mathematics. Projects were developed and data was collected
from experimental and control groups.
This presentation will discuss the projects, the study, and the results.
Session 2,
Room 274
8:00
am Error-control codes and related
areas of discrete mathematics
Sarah Spence, Franklin
W. Olin College of Engineering,
Error-control codes are an important
aspect of algebraic coding theory. For example, CDs use error-control codes so
that a CD player can read data even if they have been corrupted by
imperfections on the CD. This talk will present some undergraduate-accessible
basics of algebraic coding theory as well as related teaching interests
and projects.
8:20 am A
Historic Point of View on Trigonometric Functions
Semra
Kilic-Bahi, Colby Sawyer College,
The presentation will first focus on different techniques applied by
ancient civilizations to measure the earth. Then the effect of some approaches
on the development of trigonometric functions will be emphasized.
8:40 am The
Regularity of Weakly Harmonic Maps on Riemannian Manifolds with Bounded Measurable
Metrics
Wataru
Ishizuka, Providence College.
The Hölder continuity of weakly harmonic maps on Riemannian manifolds
with non-smooth metrics will be discussed.
Session 3, Room 278
8:00
am Rolle’s Theorem over Finite and
Local Fields
Cristina Ballantine,
College of the Holy Cross,
Rolle’s Theorem implies that if a polynomial splits over the reals, its
derivative does as well. We investigate this property over finite and local
fields.
8:20
am An Examination of Teacher
Knowledge Base Among Undergraduate Mathematics Faculty
Kimberly B. Santucci,
University of Connecticut
This research examined the knowledge base for teaching and
instructional practice of seven undergraduate mathematics faculty teaching
calculus. Themes across cases and a teacher knowledge base framework will be
presented.
8:40
am An Introduction to Weak
2-cocycles
Jill Shahverdian,
Quinnipiac University
I
will introduce the theory of weak 2-cocycles as a generalization of the theory
of Galois 2-cocycles.
Contributed
Papers
Session
1, Room 270
3:30-3:45 Euler
Goes Beyond Isosceles
Ed
Sandifer, Western Connecticut State University
We all know what happens if a triangle has two equal angles,
say A and B. Such triangles are isosceles, and there is a simple relation on
their sides, a = b. But what if one
angle is an integer multiple of another angle, say B = 2A, or B = 3 A?
3:45
– 4:00 Mathematical Models and Art in the Early 20th Century
Angela
Vierling, Boston University
For thousands of years, artists, as well as mathematicians,
have been interested in solid geometric forms. The drawing and painting of
polyhedral models can be simply a teaching tool, but, for others, these forms
have represented perfection and truth itself. In the nineteenth century,
mathematicians began to produce less regular and more startling solid figures.
The strange beauty of these forms had an effect on many artists, particularly
those associated with the constructivist and surrealist movements.
4:00
– 4:15 A
Mathematician at a K-8 School
Debbie Borkovitz,
Wheelock College
Last year I spent a portion of my sabbatical at the Young
Achievers Science and Math School in Boston, where I worked primarily with
Wheelock College graduate students, but also with children, teachers, and
administrators at the school. In the talk, I will share some of the
children’s work, some anecdotes about working with the new teachers, and a few
thoughts on how the school experience affected my work in preparing future
elementary teachers in mathematics.
4:15
– 4:30 Group Quizzes using the Think-Share-Write Method
Hema
Gopalakrishnan, Sacred Heart University
Group Quizzes using
the Think-Share-Write strategy can be effective in engaging students and
enhancing their learning in the classroom. In this talk, I describe my experiences
with implementing this method in lower division and upper division mathematics
courses.
Session 2, Room 274
3:30 – 3:45 The
Formulation of Vector Analysis Valid in All Coordinate Systems
Domina
Eberle Spencer and Terri L. Mascardo, University of Connecticut
The paper presents a formulation of vector analysis valid in
all coordinate systems. The pedagogical value of this formulation is easily
seen when compared to the established formulation, which is valid only in
rectangular coordinates and is being taught in elementary calculus classes.
Historical development is traced back to Hamilton, Grassmann, Heaviside and
Gibbs and the implications of their efforts when considering extending vector
analysis to curvilinear coordinates and non-uniform fields. Proper definitions
are given for the unit vectors, the gradient and the divergence and curl.
3:45 – 4:00 Using
a Partial Singular Value Decomposition to Approximate a Large Sparse
Rectangular Matrix
James
Baglama, University of Rhode Island
One of the most useful tools in linear algebra is the Singular Value Decomposition (SVD) of a large sparse matrix. There are numerous applications that use SVD e.g., information retrieval, two-dimensional image compression, and digital signal processing. Many SVD applications generate large rectangular matrices, but only require a few singular values to produce an acceptable answer. Recently, we have developed a method, the augmented Lanczos bidiagonalization method along with a MATLAB code for computing a few singular values and vectors. This talk is suitable for those who have had some background in linear algebra. Topics for this talk will include the mathematics behind SVD, a brief discussion of the Lanczos bidiagonalization method, and some applications