MATHEMATICS 603: Mathematical Principles in Science III

SP 2012, 10:30 a.m. MWF, CC 0326

Rodica D. Costin, office MW 436
Ulrich Gerlach, office MW 506

Office hours:
Rodica D. Costin: W, F 11:30-12:20 a.m. or by appointment
Ulrich Gerlach:

General policy and syllabus

Professor Gerlach's online book

Further reading: you may find useful to consult also

Fourier Series and Boundary Value Problems, by Brown and Churchill
Fourier analysis: an introduction, by Stein and Shakarchi
Fourier Series, by Tolstov

Course log:

Monday, March 26  Lecture 1 notes

Wednesday, March 28

Friday, March 30 pages 1 to 5b from the notes here.
The HW assignment handed in in class today is due next Friday April 6, at the beginning of class.

Monday, April 2  ...continuation

Wednesday, April 4 The Fourier Theorem.

NOTE: the HW is due Monday April 9 (since we are behind with the material taught in class).

  Read the proof of the Fourier Theorem, pages C5-C8 in the notes above. Today we do:
 The Fourier Integral. Dirac's delta function and other distributions.
Please read these notes (including the material not covered in class) and solve the exercises included.

Friday, April 6 (i) Fourier Integral Theorem and its proof  reviewed
                        (ii) Fourier transform as an isometric transformation
                         Lecture notes

Monday, Apr. 9  (i) Fourier transform of a periodic function
                           (ii)  Construction of an orthonormal wave packet basis  for L^2
                           (iii)  Handout  on four applications of O.N. wavepackets.

Wednesday, April 11 Lecture topics were:

Homework 2 is due this Friday, of course.

Friday, April 13 The purpose of the lecture is to set the context for Monday, when we will decompose L^2 into the hierarchy of multi resolution subspace, hoping to open the curiosity about multi resolution analysis. Today's material is not yet in the typeset notes, but see the handwritten notes: a quick study to wavelets, notes on wavelets.

 Multi Scale Analysis:
a) Translated sinc functions: its application to the Whittaker-Shannon sampling theorem.
b) Translated and dilated sinc functions as a multi resolution basis for  L^2 .
c) The "father" wavelet as the fountainhead for these basis functions.

Here is a handout for HW3.

Monday, April 16 Ideas developed today:
"Mother" wavelet was only mentioned, with no illustrative example. Same thing for the pyramid algorithm.

Wednesday, April 18 Today's development consisted of
Friday, April 21  ...continuation

Monday, April 23 Wednesday, April 25
Friday, April 27 Monday, April 30 Wednesday, May 2
Friday, May 4
Monday, May 7
Wednesday, May 9  Helmholtz's equation as a highly degenerate eigenvalue problem
Friday, May 11
  • Translation and rotation symmetries of the Euclidean plane
  • Their representations in the space of solutions of the Helmholtz equation: translation vs. rotation eigenfunctions
  • Goal: for the Helmholtz equation construct a rotation eigensolution which is  a linear combination of translation eigensolutions

  • Monday, May 14

    Wednesday, May 16
    Friday, May 18
    Monday, May 21
    Wednesday, May 23
    Friday, May 25 Methods for finding the asymptotic behavior of integrals with a large parameter:
    Watson's lemma. 
    Stationary phase method. 
    Steepest descent method.

    Additional resources:

    Distributions and operators (p.9 revised)
    Green's Functionuniqueness
    (Note: "fundamental solution" F is for linear operators with constant coefficients, and solves P(d/dx)F=delta.)
    Green's function of the adjoint problem (older- revised p.3, 5B,5C,6), and newer (revised). Supplement.
    Simpler forms for second order linear differential operators
    Review of the resolvent and elements of complex analysis
    Non-self-adjoint boundary value problems (p.2B, 7B added)

    More web resources: recall the dual space (p.4), the adjoint (p.15), for the adjoint boundary conditions (p.20), the Green's function for non-selfadjoint problems (p. 21)

    Shannon's Sampling Theorem - the case of a compact interval.