Monthly Roundup

End of the month roundup of research in-progress. Links not to lose, etc.

FOLLOWING THESE COURSES ONLINE

• Calculus Revisted, (Single and) Multivariable Calculus, Herbert Gross, MIT, 1971
  Fantastic lectures.  Logic cannot come and rescue us.  "Without assumption there can be no proof."  The assumptions are ours; an agreement to certain definitions and rules.  "A fact is just an assumption which has yet to be disproved."
  And for these exact reasons, passionate about mathematics.  We can choose to seek assumptions which are truthful about the physical world, which carries on with or without us.
  At last.
  {Video 8; part II, lecture 2 -- Vector Calculus}
• Introduction to Algebraic Topology,  NJ Wildberger, University of New South Wales, 2010
     {Second Video}
• Digital Signal Processing, Alan Oppenheim, MIT. 1970's.
      Nice shirt.      {Lecture 2}
• The Fourier Transform and its Applications  Brad Osgood, Stanford, 2008
     {Lecture 2}
     Course materials

• Recitation (Overview of Key Ideas), MIT.  Nice format.  Drop in on it from time to time.
• Cryptography, David Kohel, University of Washington, 2008
  Textbook using SAGE for examples.

NEW SOFTWARE

SAGE: http://www.sagemath.org/

    mpmath reference http://mpmath.googlecode.com/svn/trunk/doc/build/general.html

    tutorials/publications http://www.sagenb.org/

READING

Graphics Programming Black Book, Michael Abrash, 2001

    Free .pdf downloads at www.gamedev.net

    Focus on Quake3D code

Fourier Series and Boundary Value Problems, Churchill, 1941

    Finally a readable book on Fourier series that neither dodges the math nor renders it inaccessible.  Tiny volume from c.1945, picked it up again, and it is readable at last.  Sure enough, Lebesgue integration is not necessary, just an understanding of the basic dilemma of instantaneous measure and infinite sums.  Convergence.   The diffEQ is thorny, but suprisingly well presented so far.

Knotted Doughnuts and Other Mathematical Entertainments, Martin Gardner

    Interesting problems.  Useless as a reference; Gardner doesn't solve very many of them.

    ToDo: collect solutions (and solutions in progress) in one place, and be done with it.

    Hamiltonian cycles on an n-cube problem:  e.g. http://arxiv.org/pdf/1003.4391.pdf .  This is pretty funny.  Gardner fares no better.  He counts every state of the system.  In the paper, they make up this rule for the final number: "so, the integer H_n/n! with n=3, 4, 5, 6 has exactly n-3 odd prime divisors"  {?!}  Um... guys..... the factor is prime because it's a sum.  They ran a computer for six months.  A pencil and paper are sufficient to show the prime factor in the case N=4 (which is 7) is the result of summation.  And that this addition renders the factoring pattern of the problem undetectable.

100 Great Problems of Elementary Mathematics:  Only did a few this month, but in depth.  Learning far more about problem solving from Dorrie than reading online papers (Abrash is different.  Not crazy).  Starting to realize how full of holes and excuses modern Complexity Theory is. E.g. Cormen on Strassen's Matrix-Multiply:  we don't know how "hard" matrix multiplication is.

We know exactly how 'hard' it is, because we define how we are going to multiply.  The time function is completely determined.  If I unroll multiplication into addition, then I have a fractional logarithm in my running time.  Part of an exponent comes down to multiplication:  It is not a constant.  Failing to count both parts of the fractional logarithm makes a pretty little mystery out of the time function.  That's how you know you're wrong.

AUDIO ENVIRONMENT

   Trying to linearize Audio system is exhausting and going nowhere.

ToDo: 

1. Make drawings, sketches, sheets of paper with words all over.  The structure is not linear; it's an integrated environment.  Lay out the big picture.  Fill in the details from place to place; connections among the various components.  If it takes 100 drawings, it takes 100 drawings.  
2. Stop worrying about people frowning and saying it isn't possible.  Stop wasting time trying to reach that person.
3. Post audio files.  Show that it works.  People need to hear it.  
4. Make clutch decision: programming, or Signal Systems?  Both are vast.  I prefer the math, because of the immense time investment if I don't already know it, but it might be time to start building.
5. Contact some people.  Just do it.  If they don't care, they don't.

REFERENCE

Rational Curves: http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/curves/rational.html#conversion

Electric Circuit diagrams!

http://hyperphysics.phy-astr.gsu.edu/Hbase/emcon.html