Methods in Applied Math I

Math 761: Fourier, Wavelet and Complex Variable Methods in Applied Mathematics

Days & Times 	Room 	Instructor 	Meeting Dates
------------------------------------------------------------------------
MWF 11:00AM     AB634   Eric Olson      08/29/2011 - 12/21/2011

Course Information

Instructor:: Eric Olson
email:: ejolson at unr edu
Office:: Monday, Wednesday and Thursday 1-2 DMS 238 and by appointment.
Homepage:: http://fractal.math.unr.edu/~ejolson/761
Texts:: Walnut, An Introduction to Wavelet Analysis, Birkhauser Boston, 2001.; Carrier, Krook and Pearson, Functions of a Complex Variable: Theory and Technique, SIAM, 2005.

Announcements

[22-Dec-2011] Extra Credit

During the semester we have found errors in Walnut on page 106 and page 108 as well as a bug with Maple. For extra credit write respectful and detailed letters to the respective publishers explaining each error. Turn in your letters for comments and grading before actually sending them.

[21-Dec 2011] Final Exam

The final exam will be held on Wednesday, December 21 from 8:00-10:00am in AB634. It will be a comprehensive exam covering all material from Quiz 1, Exam 1 and Quiz 2 as well as

Computation of Taylor series (execise 1ac page 53 in Carrier, Krook and Pearson)

Note that Poisson's formula page 47 in Carrier, Krook and Pearson will not be on the final exam.

[19-Dec-2011] Quiz 2 Solutions

Quiz 2 is graded and solutions are posted. You may get your quiz back from my office today or tomorrow. Please send an email to make sure I will be in before you come.

[12-Dec-2011] Homework 3 and Quiz 2

Homework 3 will be the last day of class on due December 12. Quiz 2 will cover the following topics:

Use of Cauchy integral formula (problem 1 on page 40 in Carrier, Krook and Pearson)
Haar coefficient decay rates at a jump discontinuity (case 2 on page 131 of Walnut)

A handwritten proof of case 2 is also available.

[08-Dec-2011] Plot of Perpendicular Countours

Here is a worksheet (mpl) of Maple plots for exercise 1 in Carrier, Krook and Pearson on page 48 illustrating an example where the contours of u and and v are perpendicular.

[07-Dec-2011] Programming Assignment 2

Programming assignment 2 is now available.

[06-Dec-2011] Bug in Macintosh Maple

There is a bug in Maple 15 running on Macintosh which may cause incorrect answers for calculations related to Haar wavelets. Update: This bug is also present in Maple 9 running on Macintosh but appears not to be present in any Linux or Windows version.

[02-Dec-2011] Discrete Haar Transform

The Matlab/Octave code we wrote in class haar1d.m and a simplified recursive version haar1dr.m are available. Note that the code has been modified slightly to make it compatible with both Octave and Matlab. There is also a script dec02.m that tests the code in a way similar to what we did in class and gives the output dec02.pdf.

[30-Nov-2011] Discrete Haar Transform

I have typed notes for the program we wrote in class to compute the discrete Haar transform. An equivalent but different presentation is given in Walnut pages 142 through 144.

[18-Nov-2011] Exam 1

Exam 1 will be Friday November 18 in class for the entire hour. Please prepare the following:

Discrete Fourier Transforms

Definition of the discrete Fourier transform (Definition 4.35 in Walnut)
Proof of Fourier inversion theorem (Theorem 4.36 in Walnut)
Proof of convolution theorem (Theorem 4.41 in Walnut)
Definition of the matrix W_N (page 107 in Walnut)
Derivation of factorization for W_N (Equation 4.10 in Walnut)
Proof (N/2)log₂N complexity of the FFT (Theorem 4.45 in Walnut)

Haar Wavelets
Definition of the Haar functions h_j,k (Definitions 5.9 and 5.11 in Walnut)
Characterizations p_j,k=2^−1/2(p_j+1,2k+p_j+1,2k+1) and h_j,k=2^−1/2(p_j+1,2k−p _j+1,2k+1).
Proof of the splitting lemma (Lemma 5.16 in Walnut)
Proof of density of the Haar system in C([0,1]) (Lemma 5.23 in Walnut)
Proof of the completeness of the Haar system in L²([0,1]) (Theorem 5.24 in Walnut)
Haar coefficient decay rates (only case 1 on page 130 of Walnut)

Complex Variables
Proof of Cauchy's integral formula f(z)= (2πi)⁻¹ ∫_C f(ζ)/(ζ−z) dζ (page 37-38 Carrier, Krook and Pearson)
Derivation of formula for higher derivatives (page 39 Carrier, Krook and Pearson)
Proof of maximum modulus theorem (Section 2-4 in Carrier, Krook and Pearson)
Proof of convergence of Taylor series (page 51 in Carrier, Krook and Pearson)
Use of the Cauchy integral formula (problem 1 on page 40 in Carrier, Krook and Pearson)

[17-Nov-2011] Math 762 Webpage

The webpage for Math 762 Methods of Applied Math II is now online.

[14-Nov-2011] Theorem 5.24 From Walnut

I have written a concise version of the proof presented in class of Theorem 5.24 from Walnut's book as well as proofs of Lemma 5.16 and Lemma 5.23. There is also a typed version of the estimate in case 1 for the decay of Haar coefficients similar to what we've done in class.

[10-Nov-2011] Homework 2 Solutions

Solutions for homework 2 are now available on this website. Updated with corrections on Fri Nov 11 23:38:33 PST 2011. If you find any errors please let me know.

[28-Oct-2011] Programming Assignment 1

Programming assignment 1 is now available.

[27-Oct-2011] Maple Plots

Here is a worksheet (mpl) of Maple plots of |f(z)| for the problems in Carrier, Krook and Pearson execise 1 on page 43.

[04-Oct-2011] Office Hours

Due a change in my schedule, all office hours on Tuesday have been permanently rescheduled to Thursday.

[03-Oct-2011] Quiz 1

Quiz 1 will be Monday in class for the entire hour. Please prepare the following:

Proof that there is no delta function (see the August 31 handout)
Definition of an approximation of the identity (definition 2.31 in Walnut)
Proof about approximate identities (Theorem 2.33 in Walnut)
Definition of exponential function (equation 1-8 in Carrier, Krook and Pearson)
Proof that exp(z+w) = exp(z) exp(w) (equation 1-9 in Carrier, Krook and Pearson)
Proof that d exp(z)/dz = exp(z) (page 26 in Carrier, Krook and Pearson)
Proof of the Cauchy-Riemann equations (equation 2-3 in Carrier, Krook and Pearson)
Proof of Cauchy's Theorem (equation 2-7 in Carrier, Krook and Pearson)
Statement of Dirichlet theorem (Theorem 2.16 in Walnut)
Statement of Fejer's theorem (Theorem 2.19 in Walnut)

[01-Oct-2011] Homework 1 Solutions

I have written solutions for homework one. If you find any errors please let me know.

[27-Sep-2011] Office Hours

My office hours for Tuesday, September 27 are cancelled because I have a meeting.

[12-Sep-11] Continuity of Translation

I put together a typed handout of the proof of Lemma 2.35 in Walnut that was presented in class showing translation is L¹(R) continuous.

[31-Aug-11] There is No Delta Function

I put together a handout which shows there is no function h such that (h*f)(x)=f(x) for every continuous function f. This fills in the details from page 37 of Walnut for the claim that there is no delta function.

[29-Aug-11] Which Book?

Please bring Walnut to class on Monday and Wednesday. Bring Carrier, Krook and Pearson to class on Friday. Both books are in the bookstore for $70 and $47 respectively. One student reports purchasing Walnut online for $39 and Carrier, Krook and Pearson for $47. You may also obtain Carrier, Krook and Pearson for $43.75 from the Society for Industrial and Applied Mathematics if you are a SIAM member.

Additional Resources

There is are free online books on Complex Analysis and Fourier Analysis which may contain useful supplemental reading for this course.

Grading

    4 Quizzes                 10 points each
    1 Midterm                 50 points
    4 Homework Assignments    10 points each
    1 Final Exam              80 points
   ------------------------------------------
                             210 points total

Homework

Assignment #3 due December 12
    Walnut:
        page 127, exercise 5.28
        page 133, exercise 5.32
    Carrier, Krook and Pearson:
        page 40, exercise 1
        page 48, exercise 1
        page 48, exercise 6
        page 53, exercise 1ac

Assignment #2 due November 2 (solutions)
    Walnut:
        page 106, exercise 4.44
    Additional problems:
    
    Carrier, Krook and Pearson:
        page 40, exercise 2
        page 41, exercise 3
        page 41, exercise 6
        page 43, exercise 1

Assignment #1 due September 30 (solutions)
    Walnut:
        page 37, exercise 2.26
        page 47, exercise 2.39, 2.42
        page 57, exercise 2.60, 2.61
    Carrier, Krook and Pearson:
        page 24, exercise 12
        page 29, exercise 1
        page 35, exercise 1
        page 37, exercise 8abc

Quizzes and Exams

Quiz 1 October 3
Programming 1 November 23
Midterm Noveber 18
Quiz 2 December 12 (solutions)
Programming 2 December 21

Final Exam

The final exam will be held on Wednesday, December 21 from 8:00-10:00am in AB634.

Equal Opportunity Statement

The Mathematics Department is committed to equal opportunity in education for all students, including those with documented physical disabilities or documented learning disabilities. University policy states that it is the responsibility of students with documented disabilities to contact instructors during the first week of each semester to discuss appropriate accommodations to ensure equity in grading, classroom experiences and outside assignments.

Academic Conduct

Bring your student identification to all exams. Work independently on all exams and quizzes. Behaviors inappropriate to test taking may disturb other students and will be considered cheating. Don't talk or pass notes with other students during an exam. Don't read notes or books while taking exams given in the classroom. You may work on the programming assignments in groups of two if desired. Homework may be discussed freely. If you are unclear as to what constitutes cheating, please consult with me.

Last updated: Tue Aug 2 18:02:08 PDT 2011