HRUMC XIV - April 21, 2007
Siena College
Loudonville, NY
List of Lecture titles/Speaker/Abstracts submitted.
TITLE: Variations on a Theme of Fibonacci
David Vella, Skidmore College
ABSTRACT: In this talk we look at a collection of sequences which are defined by a recursion which generalizes the famous Fibonacci sequence. We derive the generating functions of these sequences, and use this information to relate them to a pretty counting problem concerning the number of ways of expressing a positive integer n as an ordered sum of certain smaller positive integers. As a special case, which we generalize, we give a new proof of the well-known fact that the terms of the original Fibonacci sequence give the number of ways of expressing n as an ordered sum of 1`s and 2`s.
TITLE: Computer Science Meets Hakin`s Algorithm (Knot Theory)
Gwyon Sutton, Westfield State College
ABSTRACT: An overview of Hakin`s algorithm and the unsolved problem of developing a computer program to apply it: Determining if any given knot projection is the unknot. Demonstration of a program`s attempt at it, and identifying the points in Hakin`s algorithm that cannot be achieved with the program.
TITLE: Building Continued Fractions with Lego Bricks
Edward Welsh, Westfield State College
ABSTRACT: Lego Bricks are very good at making rectangular structures: they have right angles built right into them. But have you ever tried to build diagonals at different angles? Pythagoras can help in a very obvious way, but we will extend classical results using something called a continued fraction. We`ll learn just what that is, and get very good at approximating irrational numbers. A warning to participants: During this talk, you will play with Lego bricks.
TITLE: Expressing Hypergeometric Sums in Simple Closed Form
Yangyang Liu, Dartmouth College
ABSTRACT: We will present several algorithms for deciding whether or not a given hypergeometric sum is expressible in closed form. After defining ěhypergeometric termî and ěclosed form,î we will introduce Sister Celineís Algorithm, Gosperís Algorithm, and the WZ method, and show applications of them. We will also discuss when these algorithms produce closed forms and what it means when an algorithm returns ěno answer.î We will close by pointing out that it is very hard to construct an exhaustive hypergeometric database.
TITLE: Computer Modeling of Linear and Star Polymers
Andrew Dunn, Manhattan College
ABSTRACT: A C++ program has been developed to simulate both linear and star configurations of beads. A random number generator is used to select pivot points as well as the angle of rotation. A great number of randomized configurations are generated. These randomized configurations are then used to calculate the mean squared radius of gyration for each configuration. The g ratio, the ratio of the mean squared radius of gyration of the star to the mean squared radius of gyration of the linear chain, has been computed and compared to theoretical expressions. Maple has been used to visualize these configurations.
TITLE: Criterion for the Transcendence of a Number
Erik Wallace, Hartwick College
ABSTRACT: We will present the motivation for a theorem which gives a necessary condition for the simple continued fraction of any real algebraic number. We point out that the contrapositive of the theorem can then be used as a sufficient condition for the transcendence of a number and we apply it to the number e in particular. We also consider the converse and provide a counter example, showing that it is not true in general. In conclusion we mention a problem originally posed by Jacobi, which the theorem does not come close to answering but which may be answered perhaps by applying Galois Theory to continued fractions.
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