Northeastern Section of the MAA - 2001 Spring Meeting Student Papers

Dual-Eulerian Graphs

Irma Servatius, Worcester Polytechnic Institute

A graph embedded on an orientable surface is called dual-eulerian if there is an eulerian trail which is at the same time an eulerian trail of the geometric dual. We show that there is a dual-eulerian embedding of the octahedron on the two-holed torus.

Pattern Formation in Reaction-Diffusion Models

Yakov Kronrod, Worcester Polytechnic Institute

Mathematicians and biologists have presented various models to describe patterns found in biological systems. In 1952, Alan Turing proposed that diffusion as a destabilizing influence can lead to patterns in a reaction-diffusion model. This idea of diffusive instability is contrary to the typical notion of diffusion as a smoothing influence. Using linear stability analysis and numerical simulations I investigate pattern formation in a model proposed by Meinhardt and Gierer. By varying properties of the system, patterns found in nature are simulated.

The Super Integer

Chlean Saur and Matthew Jarvis, Providence College

The purpose of the super integer class is to allow programmers to surpass the capacity of primitive variables in C++. The program uses unique algorithms to rapidly perform large calculations with exact precision. All calculations are performed in base 2^62 to increase efficiency and allow for bit shifting. Base 2^62 is used because the largest primitive C++ allows for is 64 bits. One bit is used for the carry in addition and one for the carry in subtraction. The remaining 62 bits store the digit. The addition and subtraction algorithms add or subtract node by node. Carries and borrowing are computed after all the additions or subtractions are completed to increase speed. The multiplication algorithm utilizes the addition function and bit shifting to increase the speed in calculations. The division algorithm uses standard long division with a binary search tree to determine the next digit in the quotient. The exponentiation algorithm utilizes bit shifting with the exponent and the multiplication function with the base.

Conjectures on the Collatz Algorithm

Brian Bayerle, Providence College

This talk will focus primarily on odd numbers in the Collatz algorithm and equations generated from their patterns. Additional topics include step and level patterns in the algorithm.

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