WSE 187: 
An application of self-similarity to evaluation of integrals .
(Module 2)
February 19 - March 7
 
Tuesday-Thursday 3:50-5:10 p.m. Physics 119
 
 
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.edu

Office 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:

  • Riemann Sum Method                        
  • Fundamental Theorem of Calculus
  • 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.

    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