UNCG
11/11/06

Last modified 10/30/06

Abstracts of the talks

Michael Motyka: The Dynamics of a Damped, Magnetic Pendulum
M. Scott Wells: Beating a Measurable Path through the Calculus II Jungle
Alex Ford: Arc Length Gone Global
James Conrad Eubanks: Differentiation Shortcuts à la Descartes
Christian Sykes: Interpolation of linear subspaces by P-matrices
Asya Monds: The Exceptional Center of a Cevian Box
Ronald W. Davis II: The Algebraic Center of a Cevian Box
Kathryn Sikes: The Art of Math
Robert Gove: A Slice of Pi
Yair Goldberg: An Exploration into the Mathematics of Sudoku
Brian Stadler: Evolutionary Dynamics on Small-World Networks
Joseph Krenicky: On The Volume-Surface Area Relationship



The Dynamics of a Damped, Magnetic Pendulum

Michael Motyka, Millersville University, Millersville, PA
mentored by: Dr. J. Robert Buchanan, Millersville University

Abstract: Suppose that a spherical pendulum system is suspended above a plane that parallel to the azimuthal motion. On this plane, three magnets are placed symmetrically around the south pole, or lowest position, of the pendulum. This study examined the energies of the system and the equilibrium points of the pendulum bob. The study focused on initial condition sensitivity when examining the equilibrium points.


Beating a Measurable Path through the Calculus II Jungle

M. Scott Wells, UNCA, Asheville, NC
mentored by: Dr. Greg Boudreaux, UNCA

Abstract: Topics in a second semester Calculus II course usually include techniques of integration, applications of integration and infinite series. Typically, this material is more loosely organized then its first semester Calculus I counterpart, and more challenging. Finding the slope of a tangent to a curve and the inverse problem of finding the area under a curve, motivate beautifully the first calculus course. In this talk, an unlikely candidate is identified that motivates most of the topics in a second calculus course. Through a series of natural question centered on this motivating concept, a path is beaten through the Calculus II jungle.


Arc Length Gone Global

Alex Ford, UNCA, Asheville, NC
mentored by: Dr. Greg Boudreaux, UNCA

Abstract:The arc length integral allows one to compute the arc length of a function (with a continuous derivative) on a given interval. This may be considered a local technique. The arc length function is used in some textbooks to compute the arclength of a function (with a continuous derivative) on any subinterval of a given interval without the need to re-compute the integral when the subinterval is changed. This is a more global approach. In this talk, the notion of a measuring function is introduced, which globalizes arc length computation even further and allows the function to not have a continuous derivative. A Mathematica notebook, the main focus of this research, will be used to illustrate the theory.


Differentiation Shortcuts à la Descartes

James Conrad Eubanks, UNCA, Asheville, NC
mentored by: Dr. Greg Boudreaux, UNCA

Abstract: In his Geometry, Descartes details an ingenious way of finding the tangent line to a point on an algebraic curve by finding the tangent circle. He is able to accomplish this feat algebraically by describing the intersection of the circle and the curve in terms of a parameter, and then finding the value of the parameter that yields a root of multiplicity two. In a letter shortly thereafter, he offers a simplification which utilizes a line, the curve and a different parameter, which may be recast by the modern reader as the slope of the line. Finding the value of the parameter that yields a "double root" is akin to computing the tangent slope. This second method affords an alternative way to derive the familiar differentiation shortcuts for rational polynomial functions which is both remarkable and completely algebraic. In this talk, the second method will be fully described and used to derive some of these familiar shortcuts.


Interpolation of linear subspaces by P-matrices

Christian Sykes, UNCG, Greensboro, NC
mentored by: Dr. Charles R. Johnson, The College of William and Mary, VA

Abstract: A P-matrix is a matrix whose principal minors are positive. Let X, Y be n x k real matrices of rank k. We consider the problem of what conditions are necessary and sufficient for the existence of a P-matrix A such that AX = Y. The analytic properties of a simple necessary condition are explored. This condition is shown to be sufficient in the case n is smaller or equal 3.


The Exceptional Center of a Cevian Box

Asya Monds, Virginia State University, Petersburg, VA
mentored by: Drs. Haile & Fletcher, Virginia State University, Petersburg, VA

Abstract: If two contact triangles in base triangle ABC meet in four points, then one of these is exceptional in the sense that it has a clear view of a base triangle vertex. The contact triangles associated with the vertices of a Cevian box form a collection of exceptional points. We show that these points lie on three concurrent lines.


The Algebraic Center of a Cevian Box

Ronald W. Davis II, Virginia State University, Petersburg, VA
mentored by: Drs. Haile & Fletcher, Virginia State University, Petersburg, VA

Abstract: We define the Cevian Algebra on the points in the interior of a base triangle. In terms of this algebra, we define the algebraic center of a Cevian parallelogram, and then show that the base triangle Cevians through the algebraic centers of opposite faces of a Cevian box are concurrent.


The Art of Math

Kathryn Sikes, UNCG
mentored by: Drs. Chris Cassidy & Jan Rychtář, UNCG

Abstract: Over the years I have noticed that many people consider the subjects of Art and Mathematics as belonging to opposite sides of the brain, with nothing in common aside from the occasional fractal or Escher print. As a student of both however, I’m here to tell you of two common grounds shared by Art and Mathematics that are often overlooked. Namely that of the common “When am I ever going to use this information?” question, and that both fields are heavily utilized in digital art programs, which are popular among students, professionals and hobbyists the world over. In my talk I will discuss how concepts from undergraduate level Mathematics (linear algebra in particular) are applied in these programs both in the creation and alteration of digital artwork.


A Slice of Pi

Robert Gove, UNCG
mentored by: Dr. Jan Rychtář, UNCG

Abstract: Pi is one of the longest-studied irrational constants, and although pi has been calculated to 1,241,100,000,000 digits it is still unknown if pi is normal in any base. We look at possible implications if it were normal, along with some known properties of pi and their proofs. Pi has a long history of varied methods of calculation which are introduced and examined. More recent discoveries include the BPP formula, which may have a connection to the question of normality. We attempt to answer the question of why we need to compute long approximations.


An Exploration into the Mathematics of Sudoku

Yair Goldberg, UNCG
mentored by: Dr. Filip Saidak, UNCG

Abstract: We will explore the mathematics of Sudoku, particularly what makes two Sudoku puzzles different, with reference to Sudoku variants.


Evolutionary Dynamics on Small-World Networks

Brian Stadler, UNCG
mentored by: Dr. Jan Rychtář, UNCG

Abstract: Graphs can represent nearly everything we encounter in life, cites interconnected by highways, the national power grid, ecological structures and even the human population. We can populate these graphs and see how they react in different situations. We introduce a mutant (or simply just a new element) into a specific type of graph to study the probability of the entire graph mutating. Different graph structures and sizes yield greatly varying results and give rise to a struggle between natural selection and a random mutation. We first study graphs with known theoretical solutions to the above problem. We then take these graphs and apply a random permutation to transform them into a more real life like graph, so called small-world networks. We compare the results against the initial graph in an attempt to see if there is any relation or, rather, no relation but completely new results.


On The Volume-Surface Area Relationship

Joseph Krenicky, UNCG
mentored by: Dr. Jan Rychtář, UNCG

Abstract: The volume of a sphere of radius r is given by the standard formula V=4/3πr^3. Its surface area is given by S=4πr^2. Clearly, S is a derivative of V. The volume and surface area of a cube of length a is given by V=a^3 and S=6a^2, respectively. In this case, S is not a derivative of V. The goal of this presentation is to study the volume-surface area relationship. In particular, it will be proven that surface area of an object is always the derivative of its volume. Initially, it appears that the proof is doomed to fail since it was already shown that the case of the cube disproves this hypothesis. The answer lies in the proper set up. It is intended to ultimately prove and give proper mathematical meaning to the statement that, “Surface area is a derivative of a volume.”