Eric Phipps
Wednesday, March 12, 1997
Applied Math Conference Room, ECOT 226
5:30 p.m.

Abstract:
I am going to summarize the research I have been conducting with Prof. Curry and Lora Billings. We have been studying several different noninvertible maps that derive from Newton's Method. I specifically will discuss two maps that come about by applying Newton's Method to the symmetric functions of the roots of families of cubic and quartic polynomials. This is a numerical method that allows one to compute all of the roots of a polynomial simultaneously. The maps that result exhibit very interesting dynamical behavior, including invariant sets (actually plane and lines) and singular sets. I will give an overview of the derivation of these maps, discuss their behavior, and show a short movie that shows a very interesting bifucation that occurs in these maps.

Rupa Patel
Thursday, November 21, 1996
Applied Math Conference Room, ECOT 226
5:00 pm - 6:00 pm

I. Singular Value Decomposition
II. Kadomtsev-Petviashvili (KP) Equation: Genus 1 and Genus 2 solutions

Abstract:
Research project of the summer 1996 involved the study of ocean waves. A model for ocean waves are waves in shallow water as described by the Kadomtsev-Petviashvili (KP) equation. It describes the pattern of two-dimensional surface waves in shallow water. The first part of the project involved an understanding of Singular Value Decomposition (SVD), and its application of linear least squares optimization to determine the best KP fit to the collected data by two-dimensional interpolation, hence providing a link between theory and collected data. The second part of the study was focused on solutions of the nonlinear KP equation. The solutions I explored were Genus 1 and Genus 2. Then using computer software such as Interactive Data Language (IDL), I graphically generated various illustrations of the KP solutions. This was necessary since we needed to develop tools and C programs in order to analyze and visualize the collected ocean waves data. Thus, the presentation will summarize theory and application of SVD, as well as the development of the KP equation. Some exact solutions of the KP equation will also be included as a model of two-dimensional periodic waves in shallow water.

About the Speaker: Rupa Patel is currently enrolled in a double major program, Applied Math. and Chemical Engineering. She is graduating in Dec 1996. After graduation, she will be visiting India for few months, and then continue her studies in applied math.

Mathematics Applied to Atmospheric Science: Modeling of Cloud Dynamics and Microphysical Processes Governing Precipitation
Thursday, November 7, 1996
Applied Math Conference Room, ECOT 226
5:00 pm-6:00pm

Given by: Jenny Fox, senior in Applied Mathematics
Abstract
Understanding the microphysical properties which contribute to cloud growth and development are vital to weather prediction and the understanding of long-term climate variations. The great range of scale in precipitation systems, from global dynamics which are on the order of 10,000 km to the micrometer-sized aerosol particles which are necessary for precipitation, makes laboratory studies difficult. The alternative is numerical models which simulate atmospheric processes. The model presented is a first attempt at simulating the interactions between cloud dynamics and microphysics, and is based on the work of Srivastava (1967) and Kessler (1969). The model is a one-dimensional representatation of five varying parameters: Temperature, Updrafts, Water Vapor, Cloud Water, and Rain Water. The intent is to track these interacting processes through time, to obtain an analysis of the microphysical details involved in the production of precipitation.

About the speaker: Jenny is a senior and will be graduating in December with degrees in Applied Mathematics and Spanish Language and Literature. Currently, she works for the Aeronomy Laboratory which is part of the Environmental Research Laboratories under NOAA (National Oceanic and Atmospheric Administration). Jenny plans to pursue graduate work at the University of Colorado in Applied Math, studying mathematical modeling of atmospheric processes.

Advances in Morphogenesis: From Turing to Gray-Scott
Thursday, October 17, 1996
Applied Mathematics Undergraduate Seminar
Applied Math Conference Room, ECOT 226
Given by: Jeremy Zucker, senior in Applied Mathematics and Computer Science

Abstract
In 1951, Alan Turing showed that many biological processes which involve symmetry-breaking can be successfully modelled with a general system of nonlinear Reaction-Diffusion equations near equilibrium. More recently, one particular system of nonlinear differential equations, known as the Gray-Scott model, has been shown to model sustained oscillations, and other exotic behavior in isothermal systems. This talk will attempt to explain the bizarre, and often unexpected behavior displayed by the Gray-Scott model as it diffuses through an array of cells in terms of Turing's theory of Morphogenesis.

Biographical notes: Jeremy Zucker is a senior (!) double-majoring in Computer Science and Appled Mathematics. After graduation he plans to continue his research in Applied Math at Graduate School. His talk will include ideas from chaos theory, differential equations and freshman chemistry!