As part of the national observance of Math Awareness Week, the Department of Mathematics at North Carolina A&T State University is pleased to sponsor the following events on Thursday, April 30, 1998 and invites you to join our third annual celebration of the beauty of mathematics.

The theme of this year's Mathematics Awareness Week is Mathematics and Imaging. Since  mathematics is an essential element of imaging in fields as diverse as medicine, computer science, techcommunications, and science and space exploration, it is yet another example to communicate to a wider audience the power and diversity of Mathematics.

The day's activities include graduate students' presentations, panel discussion,  Internet demonstration and exploration, differentiation and integration contests, and contest awards and recognition ceremony.

We are all very excited about the program, the sense of "togetherness,"  and accomplishment that this event brings to our Department. Every faculty member is involved either in organizing and chairing the sessions, mentoring student presentations, selecting the best student presentation or working on contest problems, administering contests, and arranging contest prizes. All of graduate students will give presentations and help with the undergraduate differentiation and integration contests as well as Internet demonstration and exploration.

8:00 -- 10:00 AM: Graduate Student Presentations I (120 Marteena Hall)

Chair: M. Lamberth
 

10:30 -- 12:00 PM: Graduate Student Presentations II (120 Marteena Hall)

Chair: J. Gruendler
 

Abstracts of the Talks in Graduate Student Presentation I  Session

• Mathematics and Medical Imaging
Hugh Weithers
Abstract: In this presentation, we plan to discuss this years theme-Mathematics and Imaging.  This theme was written by Paul Davis.  We will discuss how mathematics is used in many areas related to imaging, which affect our daily life.  Such areas are listed as follows:
•         Image restoration
•         Adaptive and active optics
                          Crumpled wavefronts
                          Real-time controls
•         Image Compression
                          Postcards from Mars
                          Fingerprints on file
                          Orchestration with wavelets
                          Fractals
•         Tomography
                          Electrical impedance tomography
•         Computer graphics
                          Illumination
•         Image Analysis
• Simplex Algorithm
Tonya Edmond (This work is directed by Dr. Bolindra Borah)
Abstract:  We will discuss how the Simplex Algorithm makes use of the basic notions of Gaussian elimination or pivoting for solving sets of linear equations.  The history of the method, its purpose, uses, and advancements in recent years will be briefly covered.  The central ideas of the method will be outlined:

 
•        an iterative process
•        the need of a basic feasible solution
•        the need to see if the current basic feasible solution is optimal
•        a finite number of iterations

 
An example will be illustrated to demonstrate the use of this method to solve a set of linear equations.
 
• Singular-Value Decomposition of an Orthogonal-Detector Imager
Barbara Tankersley (This work is directed by Dr. P.  Varatharajah)
Abstract: We consider the singular-value decomposition (SVD) of a simple discrete linear model of a toy system consisting of a single detector and its relationship to the SVD of a related toy  system having two orthogonal detectors.  The model for each system  is g = Hf,  where f is a (n2 X1) vector representing an nxn pixelization of an object being imaged (f is unknown and thus is to be found), H is a (nXn2 or 2nXn2) modeling the mechanism by which information is mapped from the object being imaged to (the single detector or the two orthogonal detectors), and g is a (nX1 or 2nX1) vector representing the data acquired by the (n or 2n) detector elements (n per detector) (g is the data thus it is given). Both toy systems are underdetermined systems modeled by non-square matrices.  Thus, analysis of the system models involves SVD analysis rather than conventional eigenanalysis.  In this presentation, we will define matrix models of the toy systems and discuss various aspects of the SVDs of the matrix models.

 
• Non-linear Regression
Makia S. Moore (This work is directed by Dr. Janis Oldham)

Abstract: The purpose of the research conducted is to compare the performance on the algebra/trigonometry placement tests in calculus classes for North Carolina A&T State University students.  To analyze this data a non-linear regression and correlation analysis is  done.
 
• Unsteady Temperature Distribution in a Porous Medium Subjected to Time Dependent Surface Heat Flux and Internal Heat Source
Terrel L. Woods (This work is directed by Dr. Bolindra N. Borah)
Abstract: A mathematical model is established to simulate the flow of liquid through a porous medium. The boundary valued problem associated with the heat and mass flow in a porous and moving medium is considered here. The study also includes temperature distribution in a finite cylindrical medium with periodic surface conditions. There are time dependent heat sources throughout the body with uniform mass flow along the axes. This reduces to the "tree problem" when the velocity of the medium is identically zeroed. Analytical solutions of the "tree problem" can be obtained using Duhmal's theorem. However, we take a different approach by utilizing numerical methods via MATLAB, and the use of well-known differential equation of the conduction of heat in a solid medium moving with velocity.

• The Brochistochrone Problem
Adrian Copeland  (This work is directed by Dr. Guoqing Tang)
Abstract: This talk discusses the well-known brachistochrone problem, which can be solved by using the Euler-Lagrange equation of classical Calculus of Variations. The problem  concerns finding a smooth curve joining two given points A and B, not in the same vertical line, in which a frictionless particle descents from A to B in the smallest time. Formulation of the mathematical model for the problem and solution procedure for solving the problem will be discussed. An example will be illustrated.

• The NFL and the Point-Spread: the Ultimate Random Number Generator
Wm. Randolph Harrison (This work is directed by Dr. A. Giles Warrack)
Abstract: How well do the oddmakers of Las Vegas predict the outcome of professional football games? The data set used consists of scores for all National Football League (NFL) games from the 1989, 1990, and 1991 seasons with SAS as the statistical programming system. This research attempts to answer questions such as: how close are the predicted scores at games to the actual scores? Do these predications get better towards the end of the year? These questions and others are investigated using this data set.

• Optimal Feedback Control Law for the Dubins Car-like Robot Problem
Andre L. Williams (This work is directed by Dr. Guoqing Tang)
Abstract: This presentation will discuss the numerical  aspects of constructing an optimal feedback control law for the shortest path problem for the Dubins car-like robot.  This kind of robot is not able to make maneuvers and therefore restricted to move only forward in the plane. The shortest path for this robot is of the form CCC or CSC , where C denotes an arc of circle of radius 1 and S denotes a straight line segment.  The two paths that can be described by CCC are: LRL and RLR, where the  letters L and R represent a circular arc with a counterclockwise orientation and a clockwise orientation respectively. The CSC path consists of four types which are as follow: LSL, LSR, RSL, and RSR.  In this presentation, we show how to numerically construct an optimal feedback control law for the shortest path problem for the Dubins car-like robot through an illustrative example.  The construction of such an optimal feedback law can be obtained once the parameters a, b, and c of arc lengths along three pieces are computed for any given terminal state.  The construction of the optimal feedback control law and the computing of the parameters a, b, and c  will be carried out using MAPLE.

• Comparison of Three Unconstrained, Nonlinear Optimization Techniques with Four Selected Test Functions
Ray Harclerode  (This work is directed by Dr. Bolindra N. Borah)
Abstract: Many techniques exist for solving optimization problems, both constrained and unconstrained.  This paper provides a comparison of two direct search (non-gradient) methods and one descent (gradient) method designed to find optimal solutions for unconstrained, non-linear objective functions.  Each optimization algorithm is programmed in Maple V, release 4, and subjected to four test functions to assess the relative merits of the three techniques.  Program outputs, algorithm flow diagrams, results and conclusions concerning the applicability and efficiency of each optimization technique are presented.

• Functions with Compact Preimages of Compact Sets
Jennifer Bryant (This work is directed by Dr. Jim Kuzmanovich of WFU)
Abstract: In the paper, we study real-valued functions with the property that compact sets have compact pre-images in the domain.  We wanted to determine how close such preimage-compact functions are to continuous functions.  We give some examples to show that a preimage-compact function need not be continuous or vice-versa.  We also prove that the set of continuities of a preimage-compact function is a dense open set.  However,

we also give examples to show that the set of discontinuities of a preimage-compact function can be uncountable.
 
• The Transportation Problem
Perry Gillespie, Jr. (This work is directed by Dr. Bolindra N. Borah)

Abstract:  The transportation problem is a very practical tool in linear programming for solving these types of problems. The discussion will consist of the development of the
transportation problem mathematically. A brief overview of the various methods for finding an initial feasible solution. These methods are the following: Northwest Corner, Row Minima, Column Minima, Matrix Minima, and Vogel's Method. We also show how to obtain solutions when supply is equal to demand, and when supply is greater than demand, and supply is less than demand. A very important technique, the transportation simplex method, is utilized in solving various problems of this type. Finally, we end with an example that illustrates this method.
 
• Positive Solutions of Singular Nonlinear Boundary Value Problems I
Jennifer C. Martin (This work is directed by Dr. John Baxley of WFU)
Abstract: We seek a positive solution of the singular boundary value problem

$y'' = \frac{-\phi(t)|y'|^p}{y^\lambda},  0<t<1 , y'(0)=\alpha<0, y(1)=0$, where $0<p \leq 2$, $\lambda$ is a positive constant, and $\phi\in C[0,1)$ is a given positive function.  The special case $\lambda=2, p=1$, and $\phi(t)=\frac{1-t^{2}}{2}$ arises in the analysis of the Navier-Stokes equation for asymmetric stagnation (or Homann) flow. Our results include criteria for the existence of a unique positive solution, a necessary  and sufficient condition in order that $y'(t)$ be finite as $t \rightarrow 1^{-}$, and the asymptotic behavior of $y(t)$ as $t \rightarrow 1^{-}$ whether or not $y'(t)$ is finite at $t=1$.
 
• Positive Solutions of Singular Nonlinear Boundary Value Problems II
Kristina P. Sorrells (This work is directed by Dr. John Baxley of WFU)
Abstract: We investigate a collection of singular boundary value problems, which contain

examples of the form $y''= \frac{\phi (t) \left ( |y'|^\gamma -1 \right )}{y^\alpha},  0<t<1,
y'(0)= m, y(1)=0$, where $-1< m \leq 0$, $\gamma > 0$, $\alpha > 0$, and $\phi \in C[0,1)$ is a given positive function.  Our results, applied to these examples, include sufficient conditions for the existence of a unique positive solution, necessary and sufficient conditions for $\displaystyle \lim_{t \rightarrow 1^-} y'(t) > -1$, and information on the asymptotics of $y(t)$ as $t \rightarrow 1^-$.

Panelists: John Baxley and Jim Kuzmanovich of Wake Forest University, and Errol Rowe and Wilbur L. Smith of NC A&T State University
Moderator: Alexandra Kurepa of NC A&T State University

2:45 -- 4:00 PM: Differentiation and Integration Contests (216 Marteena Hall)

Coordinated by Dominic Clemence (Chair), Abdulcadir Issa, Janis Oldham, Gloria Phoenix, and P. Varatharajah

8:00--4:00 PM: Social Activities (126 Marteena Hall)

Coordinated by Thomas Clarke and A. Giles Warrack

Refreshment will be served in the Faculty Lounge in Marteena Hall during breaks at 10:00-10:30 AM, and 2:30-2:45 PM. There will be a lunch gathering in the William Faculty Cafeteria between 12:00-1:30 PM.

1998 Math Awareness Day Organizing Committee: