Post on 13-Aug-2020
The Thirteenth Annual
Harriett J. Walton Symposium on
Undergraduate Mathematics Research
Sponsored by
The Mathematical Association of America (MAA) Regional Undergraduate
Mathematics Conference Program through National Science Foundation
Grant DMS-0846477
The Department of Mathematics
Morehouse College
The Division of Science and Mathematics
Morehouse College
Morehouse College
Saturday, March 28, 2015
Contents
Greetings from
President John S. Wilson, Jr……..……………………………………………......3
Provost and Senior Vice President Garikai Campbell………….............................4
Dean of the Division of Science and Mathematics, J.K. Haynes....………………5
Chair of the Department of Mathematics, D. Cooper………......…..……………..6
Biography of Professor Harriett J. Walton…………..……………………………………7
Foreword………………………………………..………………………………………....8
Organizers and Editors………………………...…………………………………..………9
Schedule………………………………………………………………………………….10
Abstracts…………………………………….……...……………………………………14
OFFICE OF THE PRESIDENT
830 Westview Drive, SW
Atlanta, GA 30314-3773
TEL. (404) 215-2645
FAX (404) 659-6536
www.morehouse.edu
John Silvanus Wilson Jr. President
March 28, 2015
Dear Symposium Participants:
I am pleased to welcome you all to the Thirteenth Annual Harriet J. Walton Symposium on
Undergraduate Mathematics Research. For forty-two years, Dr. Walton established an
invaluable legacy of intellectual brilliance and dedication to teaching at Morehouse College. She
was an esteemed professor with a great passion for the subject of theoretical and applied
mathematics. Years later, she continues to be a source of inspiration to us all.
On behalf of the Morehouse College community, we trust that you will gain insight and
encouragement through your participation in this important symposium that bears her name.
Ultimately, it is our hope that you too will become a part of this outstanding legacy of scholarly
excellence and dedication to teaching.
Thus, it is truly our privilege to host the symposium again. We hope that your experience will
prove to be intellectually stimulating and professionally rewarding. In the same manner that Dr.
Walton approached learning, I challenge you to immerse yourself in this opportunity to exchange
ideas and concepts with your colleagues. I wish for each of you a thought-provoking meeting and
a most enjoyable visit.
Sincerely,
John Silvanus Wilson Jr.
OFFICE OF THE PROVOST 830 Westview Drive, S.W. Atlanta, Georgia 30314-3773 http://www.morehouse.edu/academics
GARIKAI CAMPBELL, PHD PROVOST AND SENIOR VICE PRESIDENT FOR ACADEMIC AFFAIRS
kai@morehouse.edu (404) 215 - 2647
March 25, 2015
To those attending the annual Harriett J. Walton Symposium:
It is my great pleasure to welcome you to 13th
annual Harriet J. Walton Symposium on
Undergraduate Mathematics Research. As a mathematician myself, I deeply appreciate the value
of this opportunity to engage in and then present your own research. Mathematical ideas can be
particularly challenging to penetrate—both in terms of understanding and making new
discoveries—and certainly to explain, especially to audiences outside of mathematicians. I
applaud the students practicing and developing the skills required to do both. By investigating
new problems, exercising the creativity required in developing new solutions, and clearly and
concisely articulating the complex ideas they have encountered in their research, students are
honing skills that will serve them well, whatever they pursue moving forward, and again, I
applaud their efforts.
This symposium is named after Harriett J. Walton, a faculty member who taught in the
mathematics department at Morehouse for over four decades. Colleagues of Dr. Walton have
said that she brought both skill and enthusiasm to the classroom, and touched the lives of
thousands of Morehouse men with her dedication and compassion. This Symposium helps to
ensure the continued intellectual growth of the students about whom Dr. Walton cared so much.
Best wishes for a rewarding experience.
Sincerely,
Garikai Campbell
28 March 2015 Dear symposium attendees and presenters: We are happy to have you participate in this Thirteenth Annual Harriett J. Walton Symposium on Undergraduate Mathematics Research. Since 2003, this symposium has become a highlight in the academic year for this region’s mathematics students and a valuable opportunity for them to synthesize their research experiences, hone their presentation skills, and share the results of their work with each other and with the accompanying faculty and guests. We in the Morehouse College Department of Mathematics are appreciative of the work you have done and of your travels, in many instances, to join us today. Undergraduate research like that presented at today’s symposium serves to motivate and inform students about possibilities beyond the Bachelor’s degree and to develop skills and habits of mind that can benefit them in graduate study and beyond. Thank you for your participation this year, and we hope you will continue your support, joining us again in 2016 and beyond.
Duane Cooper Assoc. Professor and Chair Department of Mathematics
7
Professor Harriett J. Walton
In September 1958, Harriett J. Walton joined the faculty of Morehouse College during
the presidency of Benjamin Elijah Mays. She became a member of a team of three
persons in the Department of Mathematics where she worked with the legendary Claude
B. Dansby who served as Department Chair. Dr. Walton and her two colleagues taught
all of the mathematics for the majors as well as the mathematics for non-science students.
Dr. Walton relates that two of her favorite courses that she taught during this period were
Abstract Algebra and Number Theory. The three-member mathematics department did an
excellent job of preparing their mathematics majors for graduate school and the other
students for success in their respective disciplines. In fact it was during this period of
history that Morehouse gained the reputation of being an outstanding Institution
especially for African American men. As the department grew, Dr. Walton shifted her
attention away from mathematics majors and began to concentrate on students who
needed special attention and care in order to succeed in mathematics. She became an
advisor, mentor, tutor and nurturer to a large number of students matriculating at
Morehouse College. Because of the caring attitude that she had for her students, some of
them to this day refer to her as “Mother Walton.”
Dr. Walton has never been satisfied with mediocrity. Throughout her teaching career she
demonstrated a love for learning. In 1958 when she arrived at Morehouse College she
had an undergraduate degree in mathematics from Clark College in Atlanta, Georgia, a
Master of Science degree in mathematics from Howard University, Washington D.C.,
and a second Master's degree in mathematics from Syracuse University. While at
Morehouse teaching full time and raising a family of four children, Dr. Walton earned the
Ph.D. degree in Mathematics Education from Georgia State University. After receiving
her doctorate, Dr. Walton realized the emerging importance of the computer in education
so she returned to school and in 1989 earned a Master’s degree in Computer Science
from Atlanta University. She is indeed a remarkable person.
Dr. Walton’s list of professional activities, awards and accomplishments during her
career is very impressive and too lengthy to be enumerated here. However a few special
ones are her memberships in Alpha Kappa Mu, Beta Kappa Chi, Pi Mu Epsilon, and the
prestigious Phi Beta Kappa Honor Society. Additionally she was selected as a Fulbright
Fellow to visit Ghana and Cameroon in West Africa. Dr. Walton’s professional
memberships included the American Mathematical Society, the Mathematical
Association of America, National Council of Teachers of Mathematics (NCTM) and the
National Association of Mathematicians (NAM). She served as Secretary/Treasurer of
NAM for ten years. In May 2000, Dr. Walton retired from Morehouse College after
forty-two years of service.
8
Foreword
The Department of Mathematics and the Division of Science and Mathematics of
Morehouse College would like to thank the student presenters and their advisors for their
participation in the Thirteenth Annual Harriett J. Walton Symposium on Undergraduate
Mathematics Research. The Symposium is funded partially though the generous support
of the Mathematical Association of America (MAA) Regional Undergraduate
Mathematics Conference Program through National Science Foundation Grant DMS-
0846477. The purposes of the Symposium are the following:
to encourage students to do more undergraduate mathematics research
to introduce students to their peers from various institutions and related fields
to stimulate student interest in pursuing graduate degrees in mathematics and
science
to give students experience in presenting their research, both orally and in written
form
To all supporters, thank you for your help to make the Thirteenth Annual Harriett J.
Walton Symposium on Undergraduate Mathematics Research a success. We hope to
continue this event for many years to come.
9
Symposium Committee
Abdelkrim Brania
Duane Cooper
Rudy L. Horne
Tuwaner Lamar
Benedict Nmah, Conference Director
Steve Pederson
Chuang Peng
Masilamani Sambandham
Ulrica Wilson
Chaohui Zhang
Session Moderators
Andrew Cousino
Brent Wooldridge
Chaohui Zhang
George Yuhasz
Keith Penrod
Rudy Horne
Tuwaner Lamar
Proceedings Editors
Farouk Brania
Rudy L. Horne
Benedict Nmah, Managing Editor
Administrative Assistant
William Barnville
10
The Thirteenth Annual
Harriett J. Walton
Symposium on Undergraduate Mathematics Research Saturday, March 28, 2015
Schedule
11:00 am - 11:20 am Welcome in Dansby Hall, Room 200
11:25 am - 11:45 am Student Presentations
11:50 pm - 1:00 pm Lunch
1:10 pm - 3:10 pm Student Presentations
3:20 pm - 3:40 pm Closing in Dansby Hall, Room 200
Session 1: Dansby Hall, Room 300
11:25 am-11:45 am Jalen Marshall
Morehouse College
Mathematical Modeling of Retardation in Organic Chain Reactions
1:10 pm-1:30 pm Christopher McClain
Morehouse College
Category Theory and Bridging the Gaps
1:35 pm-1:55 pm Trevonta L. Mctyre and Marquis D. Curry
Albany State University
Bond Valuations and Investments based on Bonds
2:00 pm-2:20 pm Coleman Gorham and Robert Weaver
Birmingham-Southern College
A Look into the RSA Cryptosystem
2:25 pm-2:45 pm Myles Harper
Morehouse College
RSA cryptosystems and public key data encryption
2:50 pm-3:10 pm Everett Starling
Albany State University
An Introductory Comparative Analysis of Two Statistical Spectral
Estimation Techniques
11
Session 2: Dansby Hall, Room 302
11:25 am-11:45 am Jassiem Ifill
Morehouse College
Fair division among multiple players or multiple divisible goods
1:10 pm-1:30 pm Jarret D. Camp
Morehouse College
Investigation of Solutions to Differential Equations with Variable Coefficients
1:35 pm-1:55 pm Jillian Kuether
Kennesaw State University
Computing the minimum norm least squares solution to a system of
linear equations through Gauss-Jordan elimination
2:00 pm-2:20 pm Victoria Latimore and Latalya Walden
Albany State University
Stocks as Financial Security, Valuation, Risks and Retirement Portfolios
2:25 pm-2:45 pm Latalya Walden and Marquis Curry
Albany State University
Using Process Capability Estimates and Attribute Data to generate Control
Charts for Healthcare Delivery and Management
2:50 pm-3:10 pm Ben Gaines and Adam Eiring
Birmingham-Southern College
Predicting the Steady State Maximum and Minimum Drug Levels in the Blood
12
Session 3: Dansby Hall, Room 306
11:25 am-11:45 am Arman Green
Morehouse College
A Proof of the Riemann Mapping Theorem
1:10 pm-1:30 pm William Samuels
Morehouse College
L’Hopital’s Rules
1:35 pm-1:55 pm Crystal Silver, Abebe Mojo and Gabriel Tsegaye
Clark Atlanta University
Using a Markov Chain Model to Understand the Behavior of Student Retention
2:00 pm-2:20 pm Joshua Manley-Lee
Morehouse College
A Model of Multi-store Competition Strategy
2:25 pm-2:45 pm Joseph Park
University of Florida
Bound States in the Radiation Continuum for Periodic Structures
2:50 pm-3:10 pm J.R. Gillings, Jr.
Morehouse College
R. Thompson’s Group V presented as permutations of subintervals of [0,1]
13
Session 4: Dansby Hall, Room 308
11:25 am-11:45 am Dorian Kandi
Morehouse College
Eigenvectors of Positive Matrices
1:10 pm-1:30 pm Curtis Clark Jr.
Morehouse College
On 2-2 Graph Achievement Games
1:35 pm-1:55 pm Luis Matos
Georgia State University
Fundamental Groups of Coarse Spaces
2:00 pm-2:20 pm Malik Henry
University of Georgia
Random Knot Diagrams
2:25 pm-2:45 pm Aquia Richburg
Morehouse College
Modeling the Brain with Math: Neural Networks and Liquid State Machines
2:50 pm-3:10 pm Talon Johnson
Morehouse College
Analytically Understanding Population Dynamics of the Interaction between
T-cells and HIV
14
Abstracts
Jarret D. Camp, Department of Mathematics, Morehouse College
Title: Investigation of Solutions to Differential Equations with Variable
Coefficients
Advisor: Dr. Tuwaner Lamar
An equation containing the derivatives of one or more dependent variables
with respect to one or more independent variables is said to be a partial
differential equation. Most ordinary differential equations with variable
coefficients are not possible to solve analytically. However, some special cases
do exist such as the Cauchy-Euler equation, Bessel’s equation and the
Legendre equation. In this investigation, we examine the Simply-Supported
Beam equation with variable coefficients.
Curtis Clark, Jr., Department of Mathematics, Morehouse College
Title: On 2-2 Graph Achievement Games
Advisor: Dr. Curtis Clark
Let F be a graph with no isolated vertices. The 2-2 F-achievement game on the
complete graph Kn is described as follows. Player A first colors at most two
edges of Kn green. Then Player B colors at most two different edges of Kn red.
They continue alternately coloring the edges with Player A coloring at most two
edges green and Player B coloring at most two different edges red. The graph F
is achievable on Kn if Player A can make a copy of F in his color. The minimum
n such that F is achievable on Kn is the 2-2 achievement number of F denoted a(F).
The 2-2 move number of F, m(F), is the least number of edges that must be colored
by Player A to make F on the complete graph with a(F) vertices. The numbers a(F)
and m(F) are determined for some small graphs and paths.
15
Adam J. Eiring and Benjamin H. Gaines, Department of Mathematics, Birmingham-
Southern College
Title: Predicting the Steady State Maximum and Minimum Drugs Levels in the Blood
Advisor: Dr. Jeff Barton
We did our senior research in the field of pharmacokinetics which is the study of how
drugs move through the body including dissolution, absorption and elimination. In this
talk, we examine a discrete, two-compartment model for an orally administered drug. We
assume that the absorption and elimination are both first order processes. Our goal was to
develop a formula to predict the maximum and minimum steady state drug levels in the
blood based on a given drug dosage and frequency. We were able to successfully derive a
formula for the minimum drug level in the blood. Our derivation makes extensive use of
the geometric series formula.
J. R. Gillings, Department of Mathematics, Morehouse College
Title: R. Thompson’s Group V presented as Permutations of Subintervals of [0, 1]
Advisor: Dr. Chuang Peng
We introduce R. Thompson’s group V and express it as a collection of bijections
on the interval [0,1] that have specific restrictions. The dyadic rationals are also
introduced in order to offer a detailed explaination about elements of R. Thompson’s
group V. We show how elements of V interact, what their structure is and how they
fit the provided definitions. We also offer a visual presentation of V and its elements
to aid our explaination.
We prove that elements of V permute partitions of [0,1] formed with endpoints
that are elements of the dyadic rationals and show that as a result, the definition of V
can be loosened from ``a collection of bijections on the interval [0,1]” to ``a collection
of bijections on the dyadic rationals in [0,1]”.
Coleman Gorham and Robert Weaver, Department of Mathematics, Birmingham-
Southern College
Title: A Look into the RSA Cryptosystem
Advisor: Dr. Jeff Barton
Cryptology, the study of communicating with secret codes, influences many aspects of
our daily lives, including ATM transactions and online credit card purchases. A problem
in many cryptosystems is that of key transmission. If the encryption and decryption keys
are the same, then before two individuals can exchange secret messages, one must send
the other the key and this introduces a security risk. If the key is intercepted, then all
encrypted messages may be read. Our project examines the Rivest, Shamir, and Adleman
(RSA) cryptosystem, which is a public-key encryption system. In a public-key system,
the encryption and decryption keys are different. Anyone may encrypt a message to be
sent to anyone else because each individual’s encryption key is made public. However,
only the intended recipient can decrypt a message because the decryption key is kept
16
secret. Our research involves the history, evolution and process of the RSA as well as a
real world example to show our understanding of how the system works.
Arman Green, Department of Mathematics, Morehouse College
Title: A Proof of the Riemann Mapping Theorem
Advisor: Dr. Farouk Brania
The Riemann Mapping Theorem is a powerful theorem that proves there is a unique
analytic function that isomorphically maps a point in a simply connected domain that
is not the entire complex plane to the unit disk with the conditions that the function
evaluated at that specific point is zero and the derivative at that point is greater than zero.
In this talk, I will provide some details of the proofs of the Open Mapping Theorem,
The Maximum Modulus Principle and Schwarz Lemma, which are important results
on analytic functions, and which provide the building blocks of the the Riemann Mapping
Theorem.
Myles Harper, Department of Mathematics, Morehouse College
Title: RSA cryptosystems and public Key data encryption
Advisor: Dr. George Yuhasz
In this talk, we will discuss the RSA scheme and public key cryptosystems. First we
look into a background of crypto-analysis and then examine the mathematical backbone
of the RSA scheme. From there, we will begin to understand how public key
cryptosystems work and why they are an effective way to protect information. Lastly, we
will then look at various applications of public key cryptosystems such as passwords
and data transfer.
Malik Henry, Department of Mathematics, University of Georgia
Title: Random Knot Diagrams
Advisor: Dr. Jason Cantarella
In this paper, we will take a look at knots as topological figures. We will show that
random knot diagrams can be constructed using the star diagram model and we will
prove many properties of random knot diagrams beginning with stick crossings and
ending with Euler’s characteristic equation.
17
Jassiem Ifill, Department of Mathematics, Morehouse College
Title: Fair division among multiple players or multiple divisible goods
Advisor: Dr. Duane Cooper
Fair division is the partioning of a divisible good among two or more people, or
``players”. Within Fair Division, there are various types of division including simple
fair division and envy-free division. Moreover, depending on the number of players,
different algorithms exist to guarantee various types of divisions such as the Cut and
Choose algorithm, the Trimming algorithm, the three player envy-free algorithm and
More. However, these algorithms are for the common problem of only dividing one
divisible good among a few players. As such, this begs the question of how one or a
group would go about dividing up multiple divisible goods among two players, or how
to evenly divide a good such that multiple people have a consensus about its portions.
Through the introduction and usage of simplices, polytopes, triangulations, and various
other terms, theorems, and lemmas. We will attempt to delve deeper into these
applications of Fair Divisions.
Talon Johnson, Department of Mathematics, Morehouse College
Title: Analytically Understanding Population Dynamics of the Interaction
Between T-cells and HIV
Advisor: Dr. Shelby Wilson
HIV is a sexually transmitted disease that weakens one’s immune system allowing
other pathogens to affect one’s body, ultimately resulting in the development of
AIDS. A nonlinear mathematical model of differential equations with piecewise
constants will show us the rate in population. The solutions will be analytically solved
through ordinary differential equation techniques. We will analyze the solution of
a standard differential equation model of T-cell population. Furthermore, we will
analyze multiple model of increasing complexity in order to study the dynamics of
HIV.
Dorian Kandi, Department of Mathematics, Morehouse College
Title: Eigenvectors of Positive Matrices
Advisor: Dr. Ulrica Wilson
In linear algebra, an eigenvector is a vector whose product, when multiplied by a
square matrix, is a scalar multiple of the vector itself. We call this scalar an eigenvalue.
Modern matrix theory only restricts this vector to being nonzero. However, eigenvalues
and eigenvectors have special properties when the parent matrix is strictly positive.
This paper will examine the impact of positivity on eigenpairs of a matrix and highlight
the differences that result from nonnegative matrices versus positive matrices.
18
Jillian Kuether, Department of Mathematics, Kennesaw State University
Title: Computing the minimum norm least squares solution to a system of
Linear equations through Gauss-Jordan elimination
Advisor: Dr. Jun Ji
One of the simplest and most common ways to compute the solution x = A-1
b to
a non-singular system of linear equations of the form Ax = b where x is unknown
is the Gauss-Jordan elimination. For a system of linear equations with a singular
square or rectangular matrix, the system may not have any vectors satisfying Ax = b
or may have multiple solutions. Thus, the situation becomes more complicated, as the
traditional inverse matrix A-1
does not exist. A vector that minimizes both ||b – Ax||
and ||x|| always exists and is unique and is called the minimum norm, least squares
solution to the system of linear equations. It has been shown that the minimum norm
least squares solution is indeed A+b, the product of the Moore-Penrose inverse of A
and the right-hand side vector b. The minimum norm, least squares solution is used in
curve fitting and numerous aspects of statistical analysis. In particular, it is useful in
regression analysis and linear approximation. This solution can be calculated through
the simple use of Gauss-Jordan Elimination and the construction of bordered matrices
as outlined in this paper. Compared to other widely used methods for calculating the
minimum norm, least squares solution for linear systems, this proposed algorithm is
especially easy to calculate by hand and most closely resembles the procedure used for
finding the solution to a square, non-singular system of linear equations. While the
method based on the QR decomposition is accurate and stable, it is very difficult to
execute all the steps by hand and is almost always done using software. Other methods
that can be computed by hand often take more work to conclude that the solution found
is in fact the minimum norm, least squares solution. This procedure for computing A+b
will always return the incredibly useful, unique minimum norm, least squares solution.
Victoria Latimore and Latalya Walden, Department of Mathematics and Computer
Science, Albany State University
Title: Stocks as Financial Security, Valuation, Risks and Retirement Porfolios
Advisor: Dr. Zephyrinus C. Okonkwo
Stocks continue to be popular securities due to their immediate yield of increase in
the value of investments. When a company is dully registered to do business, that
company is allowed to sell stocks to the public or to a restricted population. A holder
of the stock of a company is called a shareholder since every stock is equivalent to
some percentage of the company. The company is thus owned by shareholders who
possess the shares or equity certificates. Depending on the conditions associated with
the stock, a shareholder may sell his stocks at the stock market. The value of the stock
at a time t is the price an individual is willing to pay for the stock. In this paper, we
examine common stock, stock valuation, dividend on stocks and the role of mixed
portfolios in insuring protection of investments for retirement.
19
Joshua Manley-Lee, Department of Mathematics, Morehouse College
Title: A Model of Multi-store Competition Strategy
Advisor: Dr. Johnson Kakeu
Harold Hotelling’s Linear City model of minimum product differentiation assumes there
are two distinct Firms (Firm A and Firm B), located on a linear city, competing in price
and location. In this model, we investigate a multi-firm situation in which the executive
management team at Firm A decides to open another franchise, on the same linear city.
Will it be more profitable for Firm A to have two franchises when competing against one
Franchise (Firm B)? If so, how will Firm B’s price and location respond to the new
change?
Jalen Marshall, Department of Mathematics, Morehouse College
Title: Mathematical Modeling of Retardation in Organic Chain Reactions
Advisor: Dr. Shelby Wilson
Chemical reactions play an active role as dynamic systems in everyday life. Most
Of these reactions can be described as the initial formation of free radicals, the forming
of molecules via electron pairing free radicals and ending with a yielded product. For
this study, we consider a fourth conditional process: retardation. Retardation is observed
in a more complex system when a free radical takes an electron from a formed molecule,
essentially reversing the chain reaction. The goal of this study is to model retardation as
a system of differential equations. We utilize undergraduate level chemistry, including a
steady state approximation to help us with this task.
Luis Matos, Department of Mathematics, Georgia State University
Title: Fundamental Groups of Coarse Spaces
Advisor: Dr. Jeremy Brazas
Finite spaces are topological spaces with only finitely many points and are closely related
to order theory. It is quite surprising that the homotopy theory of finite space is highly
non-trivial. In fact, the fundamental group of a finite TO space can be any finitely
generated group. In this talk, I will discuss the finite analogue of the unit circle and use
this to construct a space that acts as a coarse version of the Hawaiian earring. This
provides a new surprising example: a space with only countably many points but which
has an uncountable fundamental group.
20
Christopher McClain, Department of Mathematics, Morehouse College
Title: Category Theory and Bridging the Gaps
Advisor: Dr. Keith Penrod
In this presentation, we will present an overview of category theory. We will provide
a few basic theorem proofs that set the basis for category theory. We will also explain
what it takes for something to be a category and the 4 tests which it has to be considered
a category. We will explore objects and explain the different types of objects. I will
provide different examples using different groups to prove that they are categories. I will
also show right inverses and left inverses that exist categories and I will also show
homomorphic groups. My research is still ongoing, so there is still more information to
be discovered.
Trevonta L. Mctyre and Marquis D. Curry, Department of Mathematics and
Computer Science, Albany State University
Title: Bond Valuations and Investments Based on Bonds
Advisor: Dr. Zephyrinus C. Okonkwo
A bond is an interest yielding security to the holder. Bonds can be issued by corporations
or government agencies, the goal being to raise money for specific or general purposes.
Common bonds include US Treasury bonds, state government bonds, municipal bonds,
mortgage bonds and debentures. Every bond has an issue date and a maturity date which
are clearly stated on the promissory note. Certain bonds such as the US Treasury bond
have very low risk and some bonds have very high risk, for example, Detroit municipal
bonds. In this paper, we present the nature and properties of bonds, noncallable and
callable bonds, bond yields and reliability on bonds as major parts of retirement
porfolios.
Joseph Park, Departments of Mathematics, Physics and Philosophy, University of
Florida
Title: Bound States in the Radiation Continuum for Periodic Structures
Advisor: Dr. Sergei Shabanov
All optical data-processing could diminish the limitations of computational power,
a pervasive problem in computational research. The biggest obstacle is developing
an optical analog of a transistor. My research advisor, mathematical physicist
professor Sergei Shabanov, has made significant progress toward this end investigating
bound states of electromagnetic waves in the radiation continuum. It was proved that
the interaction between trapped electromagnetic modes can lead to scattering resonances
of negligible width, which are the bound states in the radiation continuum first discovered
in quantum systems by von Neumann and Wigner. It was then shown in a double array of
subwavelength dielectric cylinders that by varying the spatial parameters toward the
critical value, the near field can be amplified in certain regions. The present study is the
generalized system of an arbitrary number of arrays, two parallel 2D lattices of spherical
21
scatters and analogous systems for elastodynamic and/or acoustic waves. The main fields
of study involved are mathematical physics, scattering theory, functional analysis,
operator theory, electromagnetism, acoustics and elastodynamics. Other potential
applications include large amplification of electromagnetic fields within photonic
structures and, hence, enhancement of nonlinear phenomena, impurity detection,
biosensing, as well as perfect filters and waveguides for a particular frequency.
Aquia Richburg, Department of Mathematics, Morehouse College
Title: Modeling the Brain with Math: Neural Networks and Liquid State Machines
Advisor: Dr. Shelby Wilson
Neural networks are useful models for programming computers on how to learn tasks.
A perceptron is a basic machine that has linear input and output. The perceptron training
algorithm is an artificial learning algorithm that separates data into two predefined
classes. If a set of data is linearly separable, there exists a line (hyperplane in higher
dimensions) where the data is partitioned on either side. In this talk, we will show that,
given a set of linearly separable data, the perceptron training algorithm will converge to
a line (hyperplane) that correctly separates the data into their respective classes regardless
of the initial weight vector.
William Samuels, Department of Mathematics, Morehouse College
Title: Category Theory and Bridging the Gaps
Advisor: Dr. Steven Pederson
The purpose of this investigation is to understand L’Hopital’s Rule from a theory
perspective using theorems and corollaries. In Calculus I, L’Hopital’s Rule is stated
and applied to certain limit problems but does not have to be proven. L’Hopital’s Rule
is applied if substitution into the limiting of the function leads to an indeterminate form.
By using the rule, it makes the limit calculations less difficult by differentiating the
numerator and denominator. My intention is to understand why L’Hopital’s Rule works
using theorems that are learned in Real Analysis (Advanced Calculus).
Crystal Silver, Abebe Mojo and Gabriel Tsegaye, Department of Mathematics, Clark
Atlanta University
Title: Using a Markov Chain Model to Understand the Behavior of Student Retention
Advisor: Dr. Charles Pierre
Dr. Charles Pierre and his graduate operations research class, consisting of Mr. Abebe
Mojo, Ms. Crystal Silver and Mr. Gabriel Tsegaye, were able to determine predictors
of the length of time it took a student to graduate from Clark Atlanta University (CAU)
circa 2006 by using data from the University’s Trend book, a fact book created, under the
approval of the university’s president, Dr. Carlton E. Brown, under the watchful eye of
the provost and vice president for academic affairs, Dr. James A. Hefner and under the
22
direction of the vice president of the Office of Planning, Assessment and Research
(OPAR), Mr. Narendra H. Patel. They used Markovian tools to interpret the probabilities
that were gleamed from the Trend book.
Everett Starling, Department of Mathematics and Computer Science, Albany State
University
Title: An Introductory Comparative Analysis of Two Statistical Spectral Estimation
Techniques
Advisor: Dr. Robert Steven Owor
Fast, secure and accurate spectral estimation techniques are vital for the management of
signal processing in small portable devices and embedded communicating
microprocessors. As more and more devices become part of ``The Internet of Things”,
the need for speed, security and accuracy increases. For this reason, several techniques
are being developed for fast, secure and accurate estimation of spectral waves. This paper
compares and analyzes two promising techniques, namely the Burg Estimators and Yule-
Walker Equations.
Latalya Walden and Marquis Curry, Department of Mathematics and Computer
Science, Albany State University
Title: Using Process Capability Estimates and Attribute Data to Generate Control
Charts for Healthcare Delivery and Management
Advisor: Dr. Zephyrinus C. Okonkwo
Process capability estimates are made in terms of upper and lower limits of population
distribution such that no more than one in a thousand observations lie in each tail. We
shall use the process capability procedure to obtain normal estimates of surgery times
for certain forms of surgeries. Furthermore, we study the use of attribute data
encountered in a wide-range of applications to generate control charts, including the
count-chart (c-chart) and the u-chart where the u-values are obtained by dividing the
subgroup count by the subgroup size. These charts can be applied in the delivery of
healthcare for different subgroups of the population. Examples are drawn for illustration
from data obtained from monthly counts of patients’ falls and counts on medication
errors.