Instructors:
Araceli M. Bonifant
Office: Math Tower 4-107
Phone: 632-8275
Email: bonifant@math.sunysb.edu
Susana Core
Office: Math Tower 4-116
Phone:
Email: susana@math.sunysb.eduOffice Hours: Tuesday, Thursday
About the course: The aim of this mini-course is to lead the students to "discover" the connection between Riemann sums and iteration theory using the self-similar method.
In Calculus courses one learns elementary methods for evaluating integrals, such us:
There are other methods, like Cauchy's residue calculus in One Complex Variable Theory. But in a basic calculus course generally, one studies only these two methods. The method presented in this mini-course is elementary, but not well known (see I-Bibliography). It is derived from the theory of integration on fractals, and is based on a self-similarity property of the unit interval. Riemann Sum Method Fundamental Theorem of Calculus The plan is the following:
Dates Topics Assignments Feb. 19 Preliminaries from Calculus: Riemann sums, elementary properties of the integral, Fundamental Theorem of Calculus. Self-similarity of the interval Integrals of polynomials (self-similar method)
Feb. 21 Iteration and the connection with Riemann sums Exotic integrals on the interval Feb. 26 Self-similar sets Feb. 28 The Sierpinski gasket March 5 Polygaskets and other fractals (I) March 7 Poligaskets and other fractals (II) March 14 Presentation of projects Project: At the end of the three weeks, students are required to hand in a written report of the topics discussed in class. This report should be done in groups of 2 or 3 students, clearly written, in detail and including the solutions to the exercises. It may contain computer generated graphics, although is not required.
Bibliography Evaluating Integrals Using Self-Similarity
by Robert S. Strichartz
American Mathematical Monthly, April 2000.Mathematics of Fractals
by Masaya Yamaguti, Masayoshi Hata, Jun Kigami
Translations of Mathematical Monographs, Volume 167, American Mathematical Society.Analysis on Fractals
by Robert S. Strichartz
Notices of the AMS, November 1999.An Introduction to: Chaotic Dynamical Systems
by Robert L. Devaney
Addison Wesley