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## Math 703 – Methods of Applied Mathematics I – Fall 2022

Lecture room: B139 Van Vleck Building
Lecture time: Mon-Wed-Fri 11:00-11:50am

Office hours: by appointment

Lecturer: Gheorghe Craciun
Office: 405 Van Vleck Hall
E-mail: craciun at math dot wisc dot edu

Main textbook:

Introduction to Applied Mathematics, by Gilbert Strang.

Other recommended textbooks:

1. Advanced Mathematical Methods for Scientists and Engineers, by C. M. Bender and S. A. Orszag.

2. Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering, by Steven Strogatz.

3. Introduction to Applied Mathematics, by Laurence Sirovich.

Course Content: The course introduces methods to solve mathematical problems that arise in areas of application such as physics, engineering, chemistry, biology, and statistics.

Prerequisites: Math 319 (ODEs), Math 321 (Vector and complex analysis), Math 322 (Sturm-Liouville, Fourier Series, intro to PDEs), Math 340 (Linear Algebra) or equivalent. In addition, Math 521-522 (Mathematical Analysis), and Math 623 (Complex Analysis) are strongly recommended.

Grading: Homework and class participation: 33%, Midterm exam: 33%, Final exam: 33%.  Additional credit can be obtained for work on class projects, presentations, and group projects.

MATLAB help

Several MATLAB tutorials are available here.

A free online textbook “Numerical Computing with MATLAB” by Cleve Moler is available here.

Review of Linear Algebra

If you need a detailed review of Linear Algebra you can find Gilbert Strang’s video lectures here.

Also highly recommended: the summary of the book “Linear Algebra Done Right” by Sheldon Axler, available here.

Also highly recommended: the youtube video channel “3Blue1Brown” here.

Homework 1

1. Have a look at these other online resources for linear algebra:

1.1  The online textbook

2. Watch the videos on “Essence of linear algebra” by 3Blue1Brown here:

3. Try to solve the following review problems from the Strang textbook: 1.5.1, 1.5.3, 1.5.7, 1.5.9, 1.5.11, 1.5.13.

Solutions to Homework 1 will not be collected, but these problems give you an idea of topics that you should already know a lot about.

Homework 2

1. Read carefully and in detail the first two chapters of the online textbook

2. Select and solve 10 exercises from these two chapters, and submit your solutions by email. Please include the words “homework” or “hw in the subject of your email.

Due date: Wednesday Sept 21, just before class.

Homework 3

1. Read carefully and in detail the first Chapters 3-5 of the online textbook

2. Select and solve about 20 exercises from these three chapters, and submit your solutions by email. Alternatively, select any number of exercises (not necessarily 20) to solve and submit, but make sure that you spend at least 6 hours on this homework assignment. Also, please choose some exercises that are more challenging for yourself (i.e., don’t just choose the ones that are easier for you to solve).
Please don’t forget to include the words “homework” or hw in the subject of your email.

Due date: Wednesday October 5, just before class.

Homework 4

1. Read carefully all the solutions to problems from the online textbook

2. Read carefully Chapters 5-6 of the online textbook

3. Select and solve about 15 exercises from these two chapters, and submit your solutions by email. Alternatively, select any number of exercises (not necessarily 15) to solve and submit, but make sure that you spend at least 5 hours on this homework assignment.

Please don’t forget to include the words “homework” or hw in the subject of your email.

Due date: Friday October 14, just before class.

Midterm exam:   Monday October 17,   during class time.

Homework 5

1. Read carefully the section on Constrained optimization and Lagrange multipliers in the Strang textbook, and compare with the Wikipedia page https://en.wikipedia.org/wiki/Lagrange_multiplier

Also, have a look at this 3Blue1Brown video about Fourier series approximations: https://www.youtube.com/watch?v=r6sGWTCMz2k

Also, this site has some illuminating (and customizable) numerical simulations: https://www.jezzamon.com/fourier/index.html

2. Solve the following problems from the Strang textbook:

2.2.1, 2.2.2, 2.2.3(b), 2.2.4, 2.2.5, 2.2.7, 2.2.8, 2.2.11, 2.2.15, 2.2.16.

3. Solve the following problems from the Strang textbook:

4.1.1, 4.1.2, 4.1.3, 4.1.4, 4.1.5, 4.1.6.

Solutions are available here and here .

Please submit your solutions by email, and don’t forget to include the words “homework” or hw in the subject of your email.

Due date: Friday November 4, just before class (for 50% bonus points), or Monday Nov 7 (for full credit, but without the 50% bonus points),

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1.    Introductory notes by Daniel O’Connor on github: https://github.com/danielvoconnor/DistributionNotes/blob/main/distribution_example.pdf

2.    Detailed notes on Terry Tao’s webpage: https://terrytao.wordpress.com/2009/04/19/245c-notes-3-distributions/

3.    Fourier series completeness: notes by Jeremy Orloff at https://physics.purdue.edu/webapps/index.php/course_document/index/phys600/2178/561/20348

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Some information about the final exam:

The final exam will include a Class Project, for which you can choose from the 15 suggestions below.

Suggestions for Class Project topics:

1.   Choose your own topic: something that you are interested in and has connection to applied math.

2.  Choose a section from the Strang textbook, or from the Strogatz textbook (or from some other book) that we have not covered in class, and write a report on that topic, including maybe solutions to homework problems from that section, and maybe also applications and examples.

3.   Mathematics in the news: find some news stories that have connections to mathematics, and explain those connections.

4.   Specific applications of mathematics in your field of interest: engineering, physics, chemistry, biology, etc.

5.   Mathematical models of the spread of infectious diseases.

6.   Mathematical models in “game theory” and relationships to Linear Algebra.

7.   The diffusion equation.

8.   The wave equation.

9.   Transport equations.

10. Solutions of linear dynamical systems in three or more dimensions.

11. The fast Fourier transform.

12. The use of Fourier transforms in signal processing.

13. The use of Fourier transforms in image processing.

14. Dynamical system models in chemistry or biochemistry.

15. Dynamical system models in population dynamics (for example: competitive exclusion).

In general, please also think about including some use of software (you don’t have to, but it would be great if you can do so).  For example, if your project has any differential equations, you can include numerical simulations using the software “pplane”. Otherwise, you can use Matlab, Mathematica, python, Java, or any other software.

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Homework 6

1. Have a look at the definition of Fourier Transform on its Wikipedia webpage https://en.wikipedia.org/wiki/Fourier_transform (note that it’s a bit different from the definition in our textbook), and also have a look at this video that describes convolution: https://www.youtube.com/watch?v=851U557j6HE

2. Solve the following problems from the Strang textbook:

4.3.1, 4.3.2, 4.3.3, 4.3.4, 4.3.5, 4.3.6, 4.3.7.

Please submit your solutions by email, and don’t forget to include the words “homework” or hw in the subject of your email.

Due date: Wednesday November 16, just before class.

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TAKE-HOME FINAL EXAM:

In addition to the Class Project described above, solve five of the following 12 problems, and submit your solutions by email (please include “703” or “final” in the subject of your email):

1.    Explain why the definition of the Fourer transform for distributions (i.e., FourierTransform(F)(g) := F(FourierTransform(g)) is correct.
Hint: you can reason by analogy with the definition of the derivative of a distribution. Also, you may assume that any integrals that appear in the calculation are absolutely convergent, so the order of integration can be reversed.

2.    Explain the mathematics behind the surprising properties of the Borwein integrals, as described in this youtube video: https://www.youtube.com/watch?v=851U557j6HE
Hint: as we discussed in class, you can use the fact that Fourier integral transforms multiplication into convolution.

3.    Find as many solutions as possible of the matrix equation A3 = I in 3D.

4.    Find as many solutions as possible of the matrix equation A3 = I in 2D.
Hint: you could choose one specific solution (for example, the matrix representing rotation by 120 degrees) and think about how this one matrix can be used to construct many more solutions.

5.    In general, try to find all solutions (or as many as possible) for the matrix equation Am  = I in nD.

6.    Create a list of “fundamental formulas” for calculating Fourier Transform (such as the formula for the Fourier Transform of the derivative of a function, etc.) , and prove all or most of them.

7.    Explain why the “fundamental formulas” for calculating Fourier Transform also work for distributions, not only for functions. Are there any exceptions?

8.    Define “tempered distributions” (also known as distributions for the Schwartz test functions) and explain why the inverse Fourier transform can be calculated for all tempered distributions.

9.    Choose your own problem: select some problem that you are interested in, and is related to some topic we discussed in this class.

10. Fill in some of the details of the proof of formula (3) on page 309 of the Strang textbook, about the “reconstruction” of f(x) from its Fourier transform.
Hint: you can use Riemann sums to explain why the sum in (2) approaches the intergral in (3). Also, note that the same calculation also appears on the Wikipedia page https://en.wikipedia.org/wiki/Fourier_transform

11. For a 1D dynamical system, denote by m, n, p, the number of its stable, unstable, and half-stable fixed points, respectively. Is there any identity that the numbers m, n, p must satisfy for all such systems?
Hint: two of these three numbers are related by a simple rule.

12. Find a 2D dynamical system that has three limit cycles, of which one is stable and two are unstable.