MATHEMATICAL ASSOCIATION OF AMERICA
Maryland-District of Columbia-Virginia Section
MAA Student Chapters Conference

Student Paper Session II and Meritorious Mathematical Contest in Modeling Presentations
10:25 a.m. -10:55 a.m.
Saturday April 10, 1999

Room B 33

Meritorious Mathematical Contest in Modeling Presentations
Meritorious "A" paper: VA Western CC
Meritorious "B" paper: James Madison University
Note: More meritorious papers will be presented in the 1:25 session.

Room B31

Speaker: Raymond Mooring, Morgan State University
Title: Bresenham's Circle Generator and the Dda Ellipse Generator as Compared to Kappel's Ellipse Generator.
Abstract: Lines and especially straight lines, constitute an important building block of computer images. They are used in line graphs, bar and pie charts, two and three-dimensional graphs of mathematical functions, engineering drawings, and architectural plans. Their importance becomes clearer when we consider that curved lines are often approximated by a sequence of short, straight lines in computer graphics. Efficient methods of drawing straight lines are quite valuable. The algorithms to draw straight lines are similar conceptually to those of curved lines such as circular and elliptical arcs. But because circular arcs and elliptical arcs are widely used in applications, special algorithms are developed to better plot these arcs. In my presentation, I will talk about the Bresenham's circle generator,
the DDA ellipse generator and compare it to Kappel's ellipse generator.

Speaker: Elizabeth Esswein, Georgetown University
Title: Decoding Simple Substitution Codes
Abstract: A program will be discussed that decodes substitution codes based on letter frequency, word frequency, and user input. The program looks for patterns matching common words, makes guesses, and asks for user confirmation.

Room B34

Speaker: Tim Cavanaugh, St. Mary's College
Title: Interactive Web Activities
Abstract: A number of interactive Web activities designed to intrigue and enlighten the mathematically interested user. Activities include the n-Queens problem, checkerboard tiling and path walking.

Speaker: Tiffany Fisher, Hampton University
Other Authors: Arun Verma and Alkesh Punjabi
Title: Symmetric Simple Map with Low MN-Perturbations
Abstract: The low MN perturbations occur naturally and therefore their effects are one of considerable importance in divertor physics. The simple map has been used in the past to model the magnetic topology of a single-null divertor, tokamak. A new improved map, called the Symmetric Simple Map, is used to investigate the effect of such perturbations in a single-null divertor tokamak with the stochastic scrape-off layer. These maps are characterized by two important parameters K and d; representing the toroidal asymmetry and the amplitude of perturbation. The on-going investigation involves binding the interdependencies of these parameters with the width of the stochastic layer, area of a footprint of field lines on the divertor plate, etc.
(1) J. Plasma Physics (1996), vol. 56, A. Punjabi, A Verma, A Boozer
(2) American Institute of Physics, Phys. Plasmas 4 (1997) A. Punjabi, H. Ali A. Boozer

Room B36

Speaker: Rashida Torres-Carmona, Virginia State University
Title: Gravitational Waves and Their Nonlinear, Self-interactions in Our Atmosphere
Abstract: Gravitational waves and their nonlinear, self-interactions are one of the most intriguing phenomena in our atmosphere. As winds flow through the atmosphere in the planetary boundary layer they can encounter highly protruded land masses, such as hills and mountains. As the winds flow over these mountains a rotational component in the background wind is induced which results in vertically propagating waves called gravity waves. Some acquire enough energy to propagate upward to mesopheric heights.
A temperature inversion in the mesosphere has been observed during summer solstice and has been attributed to the interaction of gravity waves. Observations indicate that the winter pole becomes 70-80 K warmer than the summer pole. This phenomenon will be examined mathematically and numerically to derive the velocities, temperatures, and pressures of the propagated winds under the influence of gravity waves. This, in turn, will be utilized to acquire the corrected mean flow of the winds which results in the observed negatively sloped temperature profile. This motion will reveal that pressure, heat, momentum, etc. have the ability to be driven out of the southern pole causing the reversed temperature gradient from the winter and summer poles. The results from this research can be used in determining the physical mechanisms of these phenomena.


Speaker: Aren Knutsen, James Madison University
Title: APPROXIMATION METHODS FOR INTEGRO-DIFFERENTIAL EQUATIONS
Abstract: Let us consider the following integro-differential equation in viscoelasticity,
(d/dt)u(x,t) = (d^2/dx^2)u(x,t) + int(E(t-s)*u(x,s)ds,s=0..t) + f(x,t)
u(0,t) = u(1,t) = 0
t >= 0, u(x,0) = g(x), x an element of [0,1]
THEOREM: On any finite t-interval [0,T[0]], the above integro-differential equation can be approximated by the first component of the following system of partial differential equations without the integrals,
(d/dt)y[1](x,t) = (d^2/dx^2)y1(x,t) + y2(x,t) + f(x,t),
(d/dt)y[2](x,t) = P[n](0)*y1(x,t) + y3(x,t),
(d/dt)y[3](x,t) = (d/dt)P[n](0)*y1(x,t) + y4(x,t),

(d/dt)y[n+1](x,t) = (d^(n-1)/dt^(n-1))P[n](0)*y[1](x,t) + y[n+2](x,t),
(d/dt)y[n+2](x,t) = (d^n/dt^n)P[n](0)*y[1](x,t),
with y[i](0,t) = y[i](1,t) = 0, i = 1,2,..,n+2 and y[1](x,0) = g(x), y[j](x,0) = 0 for j = 2,3,..,n+2. Where the nth degree polynomial P[n](t) is the approximation of E(t) on [0,T[0]].
NUMERICAL APPROXIMATIONS: We will use the finite difference method on the x variable and Maclaurin
series expansion for the t variable to carry out the numerical approximations for the above equation with different choices of E(t) and g(x), such as E(t) = sin(t) and g(x) = x*(1-x). We used Maclaurin polynomials and integral polynomials for P[n](t) to approximate E(t). This way, we can obtain some information about the solution of the
original integro-differential equation in viscoelasticity.

Room B44

Speaker: Nkwenten Ejedepang-Koje, Bowie State University
Title: Addition Theorem as Three-dimensional Taylor Expansion of Spherical LaGuerre Gaussian Type Functions
Abstract:A principle tool for the construction of the addition theorem of a function f is the translation operator which generates f(r+r') by doing a three-dimensional Taylor expansion of f around r. In atomic and molecular quantum mechanics, one is usually interested in irreducible spherical tensors. In such a case, the application of the translation operator in its Cartesian form leads to serious technical problems. A much more promising approach consists in the use of an operator expansion for e which contains exclusively irreducible spherical tensors. In this work and irreducible shperical tensor expansion of the operator has been obtained. Using this operator, an addition theorem for a special class of functions, namely spherical Laguerre Gaussian type functions has been worked out. The practical usefulness of this approach is demonstrated by constructing the LaPlace expansion of the Coulomb potential as well as the analytical evaluation of the overlap integral used in molecular quantum mechanics.

Speaker: Dan MacCarthy, Bowie State University
Title: Interval Matricies and Related Topics
Abstract: In this paper we study the properties of several classes of interval matrices (including P-, P0-, positive definite, positive semidefinite, and Z-matrices). The interval matrices are rectangular parallelepipeds in the set of all square matrices of a given dimension, and appear naturally in various optimization problems. We also study some related problems.