One of the programs in the Chapter is studying seminal papers in applied mathematics and scientific computing. Following what was first done by Trefethen as a course at Stanford, for each paper that we choose to study (based on the nominations from members), we divide into groups of “historians,” “analysts,” and “experimentalists”:
- “Historians” take us back to the historical context of the paper, including a short biography of the author(s), the influence of the paper in mathematics and other fields, a graph of its citations vs. time, etc.
- “Analysts” give us an expository look at the central ideas of the paper and how they are used in deriving the main results.
- “Experimentalists” illustrate some of the main results of the paper by performing numerical studies in a programming language such as Octave, Python, Fortran, or C/C++.
Here are some factors that may help to decide which paper to nominate:
- Exposition: Maybe a paper is seminal but not very readable. Sometimes, we can find a more recent paper doing a better job at exposition than the seminal paper itself (e.g., GKS stability paper and Trefethen’s dispersion-based exposition of it), in which case we may consider studying the more recent paper.
- Moderate Length: For example, Daubechies’ paper on “Orthonormal Bases of Compactly Supported Wavelets” is seminal but probably too long (88 pages) to study in our meetings.
- Number of Citations: However, we shall not decide solely based on the impact factor of a journal or the number of citations of a paper; a high impact factor or a large number of citations does not necessarily mean a high scientific impact. Two in-depth analyses of this matter are done by Doug Arnold, a past president of SIAM. One appeared in SIAM News and the other in the Notices of AMS.
- Reviews on MathSciNet: These are usually unbiased reviews done by experts in the field and may be used as one of the reasons for a paper’s nomination.
Here are some resources with lots of valuable papers:
- SIAM Journals and in particular SIAM Review
- Communications on Pure and Applied Mathematics
- Proceedings of National Academy of Sciences
Central Limit Theorem
Nomination by Tony Wills
- Probability is widely used in many areas of science and applied mathematics, so I recommend we study one of probability theory’s fundamental theorems, the Central Limit Theorem. The theorem basically tells us that the distribution of the average of a lot of random numbers (no matter what the distributions of these numbers are) is approximately normal and is exactly normal in the limit. Of course the exact statement is more precise. The central limit theorem was built up in stages so it is difficult to find what would be an original paper. However, the most general version was first proved by Lindeberg. Unfortunately this paper is in German and I could not find a translation. Without having a paper here is what I suggest:
- Historians: It should not be difficult to find information about the theorem’s history. Many mathematicians are given credit for various stages.
- Analysts (I volunteer to be one): I suggest we study the proof given in Korner’s nice book, Fourier Analysis. Then we can present an outline of the proof. I can make copies of the relevant chapter for those that volunteer to be an analyst.
- Experimentalists: There are many numerical examples that can be done. One would be to “prove” the theorem computationally by using many random numbers. Other examples would be nice too.
Omid Khanmohamadi, Diego Hernan Diaz Martinez, Tony Wills, and Kouadio David Yao
- PDF (as presented on 2014-03-17; the
gitrepository always has the most recent version)
- Browse our
gitrepository at https://bitbucket.org/fsusiam/public
- Clone our
gitrepository (the last
cdtakes you to the directory for this talk):
mkdir ~/fsusiam cd ~/fsusiam git clone firstname.lastname@example.org:fsusiam/public.git cd public/talks/1-central-limit-thm/
David Mandel, ?, and David Whitman
- Supplementary material, including source code for ray tracing (.tar.bz2)
- Uncompress using
tar xjvf clt-experiment.tar.bz2
- Uncompress using