A conference for Undergraduate Mathematics Students and Faculty
Abstracts
Below are abstracts for talks to be presented at the Michigan Undergraduate
Mathematics Conference.
Central Michigan University Ph.D. Program
Sidney Graham and Azita Manouchehri ()
Central Michigan University offers a mathematics Ph.D. degree
with an emphasis on the teaching of college mathematics. Our students
take courses in a broad range of mathematics. They also take a
two-semester sequence on Collegiate Mathematics Education; the first
semester is devoted to methods and the second semester is devoted to
research. After this, they have two semesters as "teaching interns";
i.e., they teach two upper-level undergraduate courses under the
supervision of an experienced faculty memb er. Students finish the
program by writing a thesis, which can be in Mathematics, Mathematics
Education, or Statistics.
Our program started in 1994, we have nine graduates so far, and
we have a 100% placement rate. Teaching assistantships, doctoral fe l
lowships, and GAANN fellowships are available. In this talk, we will
give more details about the program, the university, and the available
financial assistance.
Level: 1; Areas: Ph.D. program
Signal Preemption and Bus Travel Times
Ryan Admiraal (Calvin College)
Rikki Wagstrom A number of large cities throughout the US have
implemented a technology that will grant buses priority when
approaching a signal, extending the green's color cycle to decrease the
amount of time buses spend waiting at red lights. This talk will detail
a study of this technology on one route of the Grand Rapids bus system,
covering many of the issues that need to be taken into consideration
when conducting such a study. This includes statistical analysis of
time savings that can be attributed to the technology, possible
non-monetary costs, determining possible sources of variance, and
creating a model for simulations.
Level: 3; Areas: Statistics
Hilbert's 10th Problem
Nicholas Bailey, Paul Astras (Central Michigan University)
Theory John Casti, in Mathematical Mountaintops, includes Hilbert's
Tenth Problem as one of the five most famous problems of all time. The
problem posed by Hilbert is to find an algorithm that will determine
the solvability of an arbitrary Diophantine equation (it was eventually
proved that no such algorithm could exist). This talk will focus on
explaining the nature and history of the problem, the people who
contributed to its final solution, and how work on the problem resulted
in advances in certain areas of mathematics and computer science. An
advanced undergraduate knowledge of mathematics is assumed.
Level: 3; Areas: Computer Science, Discrete Mathematics, Logic, Number
A different way to divide a line segment into any number of equal parts
Nathan Besteman (Calvin College)
A long time ago, Euclid devised a simple method using a compass
and unmarked straight edge to divide a line segment into any number of
equal parts. This talk will present a completely different method using
those same tools for the same task.
Level: 1; Areas: Geometry
Mathematical Analysis of DNA
Joshua Boehme (Michigan State University)
DNA is the storehouse of information necessary to manufacture proteins
necessary for life. Analyzing the information coded in our DNA can give
us insight into the origins and functioning of life. However, because
the information is encoded chemically, not numerically, mathematical
analysis of DNA is not as simple as analyzing other noisy signals.
Level: 1; Areas: Statistics, Signal Processing
Prisons, Bombs, and Vampire Bats
Catherine Boersma (Calvin College)
The prisoner's dilemma is the most celebrated problem in game theory.
It is an abstract model of the conflict between individual interests
and the common good, which can be used to model real-life situations in
a vast number of disciplines like biology, psychology, sociology,
philosophy, political science, and economics. The conclusions call
into question the very nature of human rationality.
Level: 1; Areas: Modeling
The Four-Color Problem
Heather Butkovich and Robin Gilbert (Central Michigan University)
John Casti, in Mathematical Mountaintops, includes the four- color
problem as one of the five most famous problems of all time. The
four-color problem, posed over 100 years ago, states that no more than
four colors are needed to color any map drawn in the plane or on the
surface of a sphere. The four-color problem has been proved only with
computer assistance; no human has solved this problem by hand. There
has been much debate in the mathematical community about whether or not
a computer- generated proof is acceptable. It leads many
mathematicians to wonder about the qualities of a good proof.
Level: 1; Areas: Discrete Mathematics, Foundations of Mathematics
Who Wants to Be an Actuary?
JJ Carroll (Swiss Re)
Business/Industry What is an Actuary? What skills are necessary to
become an actuary? Where do actuaries work, and what kinds of projects
are they involved in? How do I get started? Join us for this session if
you would like to learn some answers to these questions and more.
Level: 1; Areas: Modeling, Probability, Statistics, Actuarial Science,
The Graduate Program in Mathematics at Western Michigan University
Dr. Clifton E. Ealy Jr. and Ms. Ralucca Gera (Western Michigan University)
The Department of Mathematics has a diverse collection of graduate
programs. We award doctorates in mathematics and mathematics education.
We also have Master's programs in applied mathematics, computational
mathematics, mathematics and mathematics education. Our department is a
small, friendly community with open and accessible professors. Our
graduate students receive a considerable amount of individualized
attention and encouragement from the faculty. Being a graduate student
at Western Michigan provides ample opportunities to gain experience in
teaching, supervising undergraduate research, interning with local
firms, and participating on committees. Our graduate experience
provides a student with well-rounded career training. We are
internationally known for our strong program in graph theory and
combinatorics. We also have strong programs in mathematics education,
applied mathematics, and algebra.
Level: 3; Areas: Grad School
On Sk,d Sets
Megan Foster (Alma College)
Let k and d be given positive integers. We say a 3-dimensional vector
x with positive integer components a, b, c is perfect, in arithmetic
progression of type Sk,d if the components of x are in arithmetic
progression with common difference k and ab+d, ac+d and bc+d are all
perfect squares. For example, the vector (1,8,15) is S7,1. In this
presentation, we give a necessary condition for Sk,d vectors and
characterize Sk,d sets for different values of k and d. We also give
sufficient conditions for generators and describe an algorithm that can
be implemented to create a chain of infinitely many Sk,d vectors.
Finally for some special Sk,d sets, we give a lower bound estimate for
the number of generators
Level: 1; Areas: Discrete Mathematics, Number Theory
Orthogonal Polynomials and Interpolation
Christopher Frayer and Ryan Koesterer (Grand Valley State University)
We will discuss orthogonal polynoials and their applications in
interpolation. A recurrence relation will be introduced to generate
orthogonal polynomials, and Maple will be used to illustrate the
construction of variious families of polynomials. In addition we will
discuss Lagrange polynomial interpolation, and why using zeros of the
orthogonal Chebyshev polynomials is beneficial in interpolation.
Level: 3; Areas: Calculus, Linear Algebra, Numerical Analysis
The Psuedo Product Property
Carter Gay (Alma College)
Grad School, Research, Business/Industry, Government Agency Given two
functions f and g, we say f and g have the pseudo product property of
order n iff the nth derivative of (f*g) equals the nth derivative of f
times the nth derivative of g. In this talk, we will charactorize
functions that satisfy this prperty for n=1,2 and define relationships
between these functions and functions that are said to slowly vary at
infinity.
Level: 2; Areas: Calculus, Analysis, Differential Equations, Statistics,
Central Michigan University Ph.D. Program
Sid Graham and Azita Manouchehri (Central Michigan University)
Central Michigan University offers a mathematics Ph.D. degree
with an emphasis on the teaching of college mathematics. Our students
take courses in a broad range of mathematics. They also take a
two-semester sequence on Collegiate Mathematics Education; the first
semester is devoted to methods and the second semester is devoted to
research. After this, they have two semesters as "teaching interns";
i.e., they teach two upper-level undergraduate courses under the
supervision of an experienced faculty memb er. Students finish the
program by writing a thesis, which can be in Mathematics, Mathematics
Education, or Statistics.
Our program started in 1994, we have nine graduates so far, and
we have a 100% placement rate. Teaching assistantships, doctoral fe l
lowships, and GAANN fellowships are available. In this talk, we will
give more details about the program, the university, and the available
financial assistance.
Level: 1; Areas: Ph.D. program
An Animated Discussion of Rotations in 3 Dimensions
Ben Johnson (Grand Valley State University)
Any series of rotations about the x-, y-, and z-axes is equivalent to a
single rotation about some axis. If one wants to animate a rotation,
it needs to be broken down into some number, n, of equal parts. One
way to do this is by finding the axis of rotation and the angle, theta;
calculating the rotations about x, y, and z that will result in a net
rotation of theta/n about the axis; and performing this sequence of
rotations n times. I will discuss why we used this approach and show
the results, and discuss other possible approaches.
Level: 1; Areas: Computer Science, Geometry, Linear Algebra
Mathematics majors in Engineering
Wendy Kooiman and Stacey Porter (Smiths Aerospace)
We will discuss employment opportunities for Mathematics majors in the
Engineering field. Specifically, we will discuss systems engineering
in the aerospace field at Smiths Aerospace and what is involved in
being a systems engineer. Included will be a discussion of some
mathematics internship opportunities at Smiths Aerospace.
Level: 1; Areas: Business/Industry
Quaternions and Rotations
David Koop (Calvin College)
This presentation introduces quaternions and their application in
rotations and rotation sequences. The usual matrix rotation operator
is presented before exploring the basic properties of quaternions.
After introducing the quaternion, its use and advantage in rotation
operations are covered. Relevant applications are also presented. A
basic understanding of linear algebra is required.
Level: 1; Areas: Computer Science, Geometry, Linear Algebra, Technology
Numerical Simulations for a Non-linear Spring-Mass System of ODEs
William Lindsey (Oakland University)
We are investigating a model for an automotive suspension system. The
model is unique in that it incorporates a non-linear spring, which
allows the system to be active with lower power consumption. Three
coupled ordinary differential equations are used to model the system.
In particular, we are exploring the numerical simulations.
Level: 3; Areas: Differential Equations, Modeling
A Basic Introduction to Enumerative Geometry
Joel Louwsma (University of Michigan)
In its most fundamental sense, enumerative geometry is about how many
of a certain type of geometric structure satisfy certain conditions.
Through the use of some basic examples, I will attempt to show the
flavor of enumerative problems. In particular, we will see how
compactification of C^n to the complex projective space P^n allows one
to find much more pleasing solutions to such problems. In fact, one
can achieve even more satisfactory solutions by use of excess
intersection theory, which I will not discuss.
Level: 3; Areas: Algebra, Geometry
Professional Masters Degrees
Charles R. MacCluer (Michigan State University)
The Science Masters degree is being rehabilitated nationwide as a
professional degree. This movement is in response to the need of
science-trained managers in the increasing technical American
workplace.
I will discuss various industrial mathematics and financial
mathematics programs nationwide, and in particular, the proMSc program
in industrial mathematics at Michigan State University.
Level: 1; Areas:
A career in biostatistics
Bin Nan (Biostatistics, University of Michigan)
In this talk, I will introduce briefly about the career and
opportunities in biostatistics and the biostatistics program at the
University of Michigan.
Level: 1; Areas: Biostatistics, Grad School, Research
NSF REU at Central Michigan University
Sivaram Narayan and Ken Smith (Central Michigan University)
Central Michigan University's Department of Mathematics is hosting a
National Science Foundation (NSF) summer program to provide research
experience for undergraduates(REU). The topics to be studied will be
in the areas of combinatorics, geometry, graph theory and matrix
analysis. Eight students will be chosen to work with mathematics
department faculty from May 28 - July 19, 2002. We will present more
details about our REU program in this talk.
Level: 1; Areas: Analysis, Discrete Mathematics, Geometry, Linear Algebra, Research
NSF-REU Summer Mathematics Research at Hope College
Timothy Pennings (Hope College)
Worlds
If you could magically turn a good college educational experience into
an IDEAL one, what might you ask for: 1) Be with other students who are
excited and motivated, 2) Work hard during the day, have evenings for
recreation and relaxation, 3) No tests or grades, 4) Get paid rather
than pay!
Does such an animal exist? It does - at about twenty NSF-REU sites
around the country including Hope College and GVSU. Come learn about
REUs in general and the REU at Hope College in particular. Learn why a
recent participant chimed, ``It was one of the most interesting and
most fun summers I have ever had . . . This REU was conducive to
thought, creativity and play - which is an excellent combination."
Level: 1; Areas: Research
So, you don't want to teach or go to graduate school?
Randall Reitsma (Deloitte and Touche)
The actuarial field is a little-known but natural career choice for
Math majors. Randy Reitsma, a pension actuary from Deloitte and
Touche, will speak about the types of actuaries, where they are
employed, the pension actuarial profession, and the actuarial exam
process.
Level: 1; Areas: Modeling, Probability, Statistics, Business/Industry
Circles, Squares, and Series
John Skukalek (Grand Valley State University)
In the early 1700s, great mathematicians were at work on the question
what is 1 + 1/4 + 1/9 + ... ? Mengoli, Leibniz, and Bernoulli all knew
that this infinite sum converged, and yet each came up short of finding
its sum.
Around 1730, Bernoulli sent out the following plea: "If anyone finds
and communicates to us that which thus far has eluded our efforts,
great will be our gratitude." In 1734, Leonhard Euler presented a
solution.
In this talk, we will discuss Euler's solution and ingenious approach
to the famous Basel problem. In so doing, we will learn the actual
value of the sum 1 + 1/4 + 1/9 + ...
Level: 2; Areas: Calculus, Analysis, number theory, history
REU at GVSU
Jody Sorensen, Steven Schlicker, Clark Wells (Grand Valley State University)
Research Grand Valley State University will be hosting a Research
Experiences for Undergraduates (REU) site in the summers of 2002-2004.
Four faculty will work with 8 students on problems on wavelets,
dynamical systems, topology and distance geometry. This talk will give
the details of the program, and will describe some of our past and
future work.
Level: 1; Areas: Geometry, Modeling, Topology, Dynamical Systems, Wavelets,
Fermat's Last Theorem
Bob Spencer and Bradley Wahr (Central Michigan University)
John Casti, in Mathematical Mountaintops, includes Fermat's Last
Theorem as one of the most intriguing problems of all time. The
problem: X^n + Y^n = Z^n, has no solutions in N for any n greater
than 2. Why is a problem that looks so simple considered to be so
important? The simple conjecture that Fermat made took mathematicians
over 350 years to prove. Come learn some of the history behind the
problem that helped shape the final proof. Also, prepare yourself for
a look at some of reasons this problem stumped mathematicians for more
than three centuries.
Level: 3; Areas: Analysis, Algebra, Geometry, Logic
Focus Your Intelligence -- Career Opportunities with the National Security Agency
Carol Stephenson Bullock (National Security Agency)
Level: 1; Areas: Research, Government Agency
Ramsey Numbers, or How to Plan the Perfect Party!
Jon Stevenson (University of Michigan -- Flint)
We present a puzzle that involves inviting random guests to a party.
The solution to this puzzle turns out to be an application of the first
non-trivial result of Ramsey theory. First, some necessary terms from
graph theory are given, and then we define Ramsey numbers. A proof
establishing Ramsey numbers’ existence and giving an upper bound
follows. Using colored graphs to illustrate the lower bounds, the exact
values of several small Ramsey numbers are calculated. We finish by
sketching an outline of what could be done to calculate exact values
for larger Ramsey numbers.
Level: 1; Areas: Discrete Mathematics, Graph Theory
Employment Opportunities in Mathematics
Ralph Svetic (Michigan State University)
Mathematics majors often wonder (and are frequently asked) `What can
someone do with a degree in Mathematics?' This talk will answer that
question by describing a few of the many employment listings that have
appeared during the last few months on the EIMS web site (`Employment
in the Mathematical Sciences', a production of the American
Mathematical Society). The web site is a searchable data base of job
announcements by employers seeking mathematicians.
Level: 1; Areas: Industrial Mathematics
Two Mathematical Models in Biology and a New Course
Rebekah Thomas (Hope College)
In recent years mathematics has become a very useful tool in
understanding biological systems. This summer, two other students, two
professors, and I were involved in developing a new course, co-taught
this semester by a mathematics and a biology professor and taken by
both math and biology students. The main goals of the course are for
students to learn to communicate, critically read research papers, and
to understand the rich interplay between mathematics and biology. I
will discuss one research paper that demonstrates both how linear
algebra is used in population models and some of the specific
biological insight that has been gained using a model for loggerhead
sea turtles, and a second research paper, which uses differential
equations to model and analyze HIV infection.
Level: 2; Areas: Differential Equations, Linear Algebra
The University of Nebraska Graduate Program in Mathematics and
Dr. Mark Walker and Dr. Judy Walker (University of Nebraska--Lincoln)
Statistics We will discuss and answer questions about the merits and
the significant features of the graduate program in mathematics and
statistics at Nebraska.
Level: 1; Areas: Grad School
On potentially K_4-graphic sequences
Morgan Warner (West Virginia University)
A simple graph G of order n is said to have property P_{k} if it
contains a clique of size $k+1$ as its subgraph. An $n$- term
nonincreasing nonnegative integer sequence i = {d_1,d_2,...,d_n} is
said to be graphic if it is the degree sequence of a simple graph G of
order n and such graph G is referred to as a realization of pi. A
graphic sequence pi is said to be potentially P_{k}-graphic if it has a
realization G having property P_{k}. In 1991, Erd\"{o}s, Jacobson and
Lehel proposed the following problem: determine the smallest positive
even number sigma(k,n) such that every n-term graphic sequence pi =
{d_1,d_2,...,d_n} without zero terms and with degree sum
sigma(pi)=sum_{i=1}^{n}{d_i} at least {sigma(k,n)} is potentially
P_{k}-graphic. They gave a lower bound of {sigma (k,n)} by the example
pi_0=((n-1)^{k-1},(k-1)^{n-k+1}), i.e. , {sigma(k,n)}=(k-1)(2n-k)+2,
and conjectured that the lower bound is the exact value of
{sigma(k,n)}. They also proved the conjecture is true for k=2 and n=6.
This conjecture is confirmed recently.
In this paper, we consider a stronger problem: Characterize the
potentially P_k-graphic sequences. In 2000, Luo characterize the
potentially P_2-graphic sequences. In this paper, we solve the problem
for k = 3. By using this charaterization, it is easy to verify that the
Erd\"{o}s, Jacobson and Lehel conjecture is true for k = 3.
Level: 3; Areas: Discrete Mathematics, Graph Theory
Determining Critical Points by use of the Ratio Vector
Matthew Wells (Grand Valley State University)
We define what a ratio and a ratio vector of a degree n polynomial with
n distinct real zeros is. We consider examples of these particular
types of polynomials and look at specific examples of the ratio vector.
Classes of polynomials, such as Equispaced, Tchebychev, degree 3 and
degree 4, will be discussed in accordance with a monotonically
increasing ratio vector. Lastly, we present some neat results on fixed
ratios of the Equispaced and Tchebychev polynomials as we increase the
degree of these polynomials.
Level: 2; Areas: Calculus, Analysis
Sponsors
The Michigan Undergraduate Mathematics Conference is made possible by
generous contributions by
Organizations or individuals interested in becoming sponsors
of MUMC should contact
Randall Pruim. Exhibition space is available.
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