Projects - Math 311h

Instructions. Pick a topic from one of those listed or, with my approval, select one of your own. Your project should be written clearly, include any references used, and be between five and ten typed pages, excluding graphs, charts, computer code, and the list of references.

1. Fast Fourier Transform. The FFT is used in many ways. For example, it is used to analyze the frequency content of a sampled signal, to compress signals, and to remove noise from a signal. The 2D version of the FFT is used in connection with image analysis, denoising, and compression. Briefly describe the FFT and explain in detail one of its applications. Include a numerical example.

2. Linear predictive coding. This is a technique that uses least squares to compress, transmit, and approximately reproduce a signal. Describe the algorithm, explain the connection with least squares, and do a numerical example.

3. Differential forms. We will stick to 3 or 4 dimensions. In doing line integrals, we work with integrals of things like ω = F · dx = F₁dx₁ + F₂dx₂ + F₃dx₃. These are called differential forms. Stokes's Theorem, vector potentials, and many other thingd that come up in electromagnetic theory, differential geometry, and relativity can be handled using differential forms. Explain how a differential form is defined and give details for one application.

4. Invariance under transformations. One of the basic concepts in science today is the concept of invariance under some set of transformations of the underlying space, for example rotations. There are two mathematical ideas involved in this. The first is that of a group and the second is that of a group representation. Explain what these are and do in detail a physical example for some specific group.