The Thirteenth Annual - Morehouse College...The Thirteenth Annual Harriett J. Walton Symposium on...

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

[email protected] (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

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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


An Introductory Comparative Analysis of Two Statistical Spectral

Estimation Techniques

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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


Stocks as Financial Security, Valuation, Risks and Retirement Portfolios

2:25 pm-2:45 pm Latalya Walden and Marquis Curry


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

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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]

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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

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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

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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.

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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


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


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


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


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.

The Thirteenth Annual - Morehouse College...The Thirteenth Annual Harriett J. Walton Symposium on...

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