M435 Projects in Applied Mathematics Fall 2007

This course allows students to explore a range of mathematical techniques and applications in extended projects.

Prerequisites: M229, M340 or M345 or M355 and preparedness to program in Matlab.

Instructor: Professor Michael Kirby, Kirby@math, Weber 211.

Weekly Schedule including due dates and topics.

Grading policy

The final grade will be determined the evaluation of

• 4 reports (submission in LaTex required) 80% (20% each)
• class participation 5%
• final report presentation 15%

Groups

All work will be done in groups. The groups will be randomly selected for each project.  Each group will receive a single grade for each project.

Project Reports

The duration of each project will be approximately 3-4 weeks. See schedule for details and due dates. Each group will submit a jointly written single report.

Sample SVD project

Saturn Data

Sample Code for Loading Saturn

More Data

Sample Report Outline

General Introduction to the Problem

• qualitative problem description.
• why is it interesting?

Modeling Approach

This generally includes several steps:

· Quantitative problem formulation (discrete or continuous?)

· Variable identification

· Parameter identification

For example, in a simulation model it is useful to identify a small number of parameters to vary. Simulations are run (many times) for each set of parameters and predictions are made.

Mathematical Theory

Can a simplified version of your model be solved analytically? Is your approach linear or nonlinear? Can you make predictions about long time behavior?

Computational Questions

How does the complexity of the problem scale with the number of variables or parameters under investigation? Can the algorithm be parallelized?

Results

Use Figures embedded in the text and cited in the text.

Interpretation

• Predictions, added value.
• Weaknesses of the approach

Conclusions and Summary

     Answer the questions: 

• what did you learn about this question that you could not have discovered without your mathematics?
• if you had 3 months to work on this problem what would you do?   

Appendices

• Matlab Code
• Background information or theory

Sample Latex Report

• Tex file
• Pdf file
• sample postscript file

To create your report edit the file sample.tex and modify the text.  Equations appear as

\begin{equation}

\label{myfirstequation}

y= \int_{-infty}^\infty f(x)

\end{equation}

and can be referenced in the text as “see my pretty Equation (\ref{myfirstequation}).

To see your new file you must compile it using the following commands:

latex sample (creates sample.dvi)

dvips sample (creates sample.ps)

Now use ghostview to view your ps file.  You can also create pdf files using winedt.

To  create a bibliography you need a file like myreferences.bib with your references in it.  These must be typed in a  special format.  Then you must  compile as

latex sample
bibtex sample
and repeat 3 times (no magic words necessary).

Now dvips as above.

All the code for this is available on the math machines. You may (but are not required to) download software for tex from http://www.miktex.org/.  An excellent (optional) editor may be found at http://www.winedt.com/.   If you load this at home you will also need either acrobat reader (for  pdf files) http://www.adobe.com/products/acrobat/alternate.html  or ghostview (for ps files) http://www.gnu.org/software/ghostview/ghostview.html.