Jason Underdown’s Blog

This book is a beautifully illustrated gem! It examines Euler’s famous formula: V-E+F=2 which holds for all polyhedra or surfaces which are topologically equivalent to the sphere. The formula is an example of a topological invariant, something which can be computed for any surface, and thus allows one to categorize surfaces. The book also covers the famous classification theorem which categorizes all surfaces as either homeomorphic (topologically equivalent) to a sphere, n-handled torus, or sphere with n cross-caps (projective plane). Next it dives into knot theory and Seifert surfaces, before moving on to the interplay between topology and geometry, and ends by mentioning homology and how topology is done in higher dimensions.

Steven Strogatz, a mathematician at Cornell University and the author of a really cool book on chaotic dynamical systems was asked to guest blog at the New York Times today. He has written a short but interesting piece about power laws originating in biological systems and even in cities.

I did an REU (Research Experience for Undergraduates) this summer and we used the textbook “Ideals, Varieties, and Algorithms” (isbn 978-0-387-35650-1) by David Cox, John Little, and Donal O’Shea. As the title mentions, algorithms are an important tool in the study of varieties via their corresponding algebraic ideals. One of the main tools used in algebraic geometry is what is known as a Groebner basis, which is somewhat akin to a basis set in Linear Algebra. Anyway, I wrote some C++ code which implements a few of the algorithms in the book, but stops short of implementing Buchberger’s algorithm for computing a Groebner basis. The code should be useful as a starting point because it implements a Monomial class and a Polynomial class both with lots of useful methods thus eliminating the drudgery of implementing the book’s algorithms in C++.

The code can be found at: my web site as a gzipped tarball. Please let me know if you find it useful. I developed the code on an Ubuntu Linux machine, but it is generic enough that it should compile on any platform. You will need to read the README to understand the simple input, output format.

This code generates fractals based upon an iterated function system (IFS). Several input files and a Makefile are included. The code is written in C++, and distributed as a tarball. I did this for a class at the University of Utah, but it is based upon a homework assignment for a computer graphics class at MIT (6.837).

Below is the famous Barnsley fern which was generated by an IFS with just four transforms!

I modified my Sierpinski triangle code so that it now allows you to adjust the contraction mapping constant. Normally one uses a contraction mapping constant of 1/2 but if you invert that and instead use an expansion mapping constant of 2, then you get the following pretty picture.

Here are the classes I’m taking this fall:

• Introduction to Complex Analysis
• Introduction to Topology
• Research Experience for Undergraduates: Metric Spaces, The Contraction Mapping Principle, Fractals and Other Applications

I added an RSS feed to update anyone interested on changes I’ve made to my Math and Physics flash cards page.

In my last post, I mentioned that I had created some flash cards for Real Analysis that had definitions on them. I recently created a set with many of the relevant theorems, and I will be updating them as the summer progresses. Thanks to Erin Chamberlain for providing the source TeX code for the theorems. You can find them at

http://www.physics.utah.edu/~jasonu/flash-cards/

Enjoy!

I’ve been busy this summer studying Real Analysis, and in the process I’ve made a set of printable flash cards in PDF format. They’re still a work in progress, but I have over 90 cards full of definitions. Cards with theorems on them will probably come some time later. The pictures below show what the typical front and back of a card look like.

Below is a very interesting video clip which explains the Mobius transformations as simple motions of a sphere. I found this on Mark Chu-Carroll’s “Good Math, Bad Math” blog.

I modified my Sierpinski triangle code from a previous posting to work in three dimensions instead of two. The result is a short C++ program which will generate all the points of the Sierpinski tetrahedron and write them to stdout (you can redirect to a file).

Below is the obligatory image. This doesn’t look great but if you plot it yourself in gnuplot or some other package, you can probably get a prettier looking picture.

If you like pretty pictures, and I know you do, below is a plot of Sierpinski’s Triangle which you can generate with the short C++ program I wrote. This is an example of a fractal–an object that is self similar at all levels of magnification.

On a Linux/UNIX system, to compile and run the code type:

$ g++ sierpinski.cc -o sierpinski
$ ./sierpinski >out
Enter depth (integer > 0, usually 10 is good): 10
Enter contraction mapping constant (for example 0.5):  0.5
$

Now the points are stored in a file called ‘out’. To plot them you can use gnuplot like so:

$ gnuplot
gnuplot> plot 'out' with points pointtype 0

I’ve been a big fan of Dilbert for years now, but I recently found a great webcomic that I’ve started reading daily called XKCD. The tagline is: A webcomic of romance, sarcasm, math, and language. The drawings are simple stick figures, but it is the situations and ideas that are interesting and funny.

While googling for information on the Virial Theorem, I ran across a really cool page hosted at the Harvard ADS system. ADS stands for Astrophysics Data System, and it is primarily an abstract search engine that helps you locate journal articles mostly in the realm of Astronomy and Astrophysics, but they also have a small virtual library of books that you can freely download.

Here are the titles I found interesting:

• The Foundations of Celestial Mechanics by George W. Collins, II
  (1989, 2004).
• The Fundamentals of Stellar Astrophysics, by George W. Collins, II
  (1989, 2003).
• Fundamental Numerical Methods and Data Analysis by George W. Collins, II
  (1990, 2003).
• The Virial Theorem in Stellar Astrophysics, by George W. Collins, II
  (1978, Pachart Publishing House, Tuscon, Arizona).

If you want to concatenate all the separate PDF chapters, I recommend using the texexec method oulined in this web page by Matthew Skala.

I found this book in my local library and read it over the weekend. It is a quick read, but has some excellent advice. One thing I liked is the advice on how to read a math text or other technical material. The main idea is that if you get stuck, don’t go backwards assuming you missed something, instead keep going because often you’ll find the new or unfamiliar term defined in a short while. I only learned this myself about a year or two ago. It definitely seems counter-intuitive to keep going, but in the long run it usually works.

I’ve created a new deck of flash cards for calculus. They cover high-school and college-level Calculus I, that is functions, limits, continuity, derivatives, and differentiability.

I also updated my Abstract Algebra flash cards to cover most all of elementary group theory. You can find my PDF flash cards at my web site:

http://www.physics.utah.edu/~jasonu/flashcards/

Larry Cusick, a professor of mathematics at Cal-State Fresno has written a good introduction on how to write proofs. There are lots of great examples. I wish I had found this before I took my introductory analysis classes.

If you have ever wanted to learn some higher mathematics, but don’t want to slog through the long tedious proofs associated with calculus and analysis, then you might like this gem of a book by Edwin H. Connell. He is an emeritus mathematics professor at the University of Miami, and has graciously made his book available for download on the web.

I am currently taking a course in abstract algebra and thus not well qualified to review the book yet, but I have read the first two chapters and find that it presents the material succinctly and quite readably. It is definitely at the undergraduate level and probably accessible to ambitious high-school students as well. I also agree wholeheartedly with the following comment which comes from his web site.

The present situation with college textbooks is a national disgrace. Textbooks are too big and too expensive.

This blog post has a short collection of very creative mathematics done by students. The last one is my favorite.