SSL Minutes 2003-10-16 (Thursday)
Note-taker: Hal

-- Minutiae: --

Karl will take notes next time.
Martin will provide refreshments on Tuesday.
We will continue to meet at 2:30 on Tuesdays.

-- Old Business: --

Getting graph.tcl (http://math.wisc.edu/~propp/software.html)
to work.  Hal suggests having XWin-32 installed in B107 by the
administrator.

Jim will talk to Yvonne Nagel about this and the computational
cluster (see his email for follow-up).

-- Reports: --

* Martin is working on (or thinking about) a program that will
  find a generating function from a matrix.  He wants to apply
  it to 3-D snakes.  Will try and finish it by next week.

* Hal and Karl have been looking at the Apollonian Gasket
  (because it is so cool.)  We were trying to discover the
  Soddy relationship on our own, but were unsuccessful.  So we
  googled to
        http://mathworld.wolfram.com/SoddyCircles.html
  here we found the relationship given by this maple function
  (Hal got garbage when he tried this the first time because
  he left out a * in the formula.)

  To see some neat math, plug the following into maple:

        w := (x,y,z) -> simplify((x*y*z)/( (x*y) + (y*z) + (x*z)
                + 2 * ((x*y*z*(x+y+z))^(1/2)) ));
        f := (x,y,z) -> (w(x,y,z),y,z);
        g := (x,y,z) -> (x,w(x,y,z),z);
        h := (x,y,z) -> (x,y,w(x,y,z));

        f(1,1,1);
        g(f(1,1,1));
        f(g(f(1,1,1)));
        h(g(f(1,1,1)));
        f(f(h(g(f(1,1,1)))));

  Note:  hal is going to put some of his notes at:

    http://ups.physics.wisc.edu/~hal/SSL/2003/apollonian_gasket_maple.html

  While still on the Apollonian Gasket, we talked about how
  the equation has a plus-or-minus part.  These correspond to
  the inside and outside circles.  Stephen asked if there was
  a nice automorphism of R^2 that leaves the original triangle
  the same and swaps out the inner and outer circle.  This
  leaves us the question: what does it mean to turn a circle
  inside out? (negative curvature)

* Abby has been looking at the 491 notes.
  (http://math.wisc.edu/~propp/491/) She also has still been
  looking at 3-d snakes.

* Paul noticed something neat about the 3-d snakes.  Let G be
  the graph of a flat cubicular snake.  Let E' be the set of
  edges that connect the two layers.  Let phi be a subset of
  E'.  Let M_phi be the number of matching on one layer of G
  with the vertices removed that correspond to the edges in
  phi.  Then, it can be seen that:

        M(G) = Sum(for each phi in E') of (M_phi)^2

  Later on, this was generalized to any graph G.  Let {A,B}
  be a partition of the vertices of G.  Define E' to be those
  edges that join vertices in A to vertices in B. Choose A and
  B such that no two elements in E' are incident to the same
  vertex.

  Then define M(A,phi) to be the number of matchings on A
  after all of the vertices incident to phi have been removed.
  Then

      M(G) = Sum(for each phi in E') of ( M(A,phi) * M(B,phi) ).

  Jim also talked about another way to define levels on a graph
  such that each level contains only vertices from one of the
  partite sets.  Then you can always know how many edges in a
  perfect matching will go from one level to the next.

* Emilie looked at the tree papers: Kuo, Itsara, and Somos.

* Jon also read through the articles.

* Stephen has been studying up on cluster algebras.

* Sam has been reading Kuo and Itsara.

* Jim has been playing with snakes.  We saw in after.mws the
  function -> simplify((y^2+z^2)/x). 

  Jim defined a sequence with this:
        s_0 = x
        s_1 = y
        s_3 = z
        s_n = (s_{n-1}^2 + s_{n-2}^2)/s_{n-3}
  and then looked at the coefficients.  He didn't find anything
  good looking.  So he tried:
        s_0 = x
        s_1 = y
        s_3 = z
        s_n = (a * s_{n-1}^2 + b * s_{n-2}^2)/s_{n-3}
  where a and b are formal variables, just like x,y,and z.
  Then he gets an interesting sequence for the biggest
  coefficient in each polynomial.:
        1, 2, 6, 60, 3052, 2519660, ...
  Why?

  Here are Jim's notes from the agenda: 

        -------------------------------------------------------
        Maximal coefficients of the Scott-2 coefficients 
        (round, therefore combinatorial):

        s := proc(n) option remember; 
        if   n=0 then x 
        elif n=1 then 1 
        elif n=2 then z else
        simplify( (a*s(n-1)^2+b*s(n-2)^2)/s(n-3) ); fi; end;

        seq(max(coeffs(numer(s(n)))),n=3..8);
            1, 2, 6, 60, 3052, 2519660
        NOT IN THE HANDBOOK!

        seq(ifactor(max(coeffs(numer(s(n))))),n=3..8);
            1, (2), (2)*(3), (2)^4*(5), (2)^6*(3)*(11)*(13), 
            (2)^7*(7)^2*(13)^2*(17)*(19)*(23)

        subs(z=0,numer(s(8)));
        
        seq([coeffs(subs(z=0,numer(s(n))))],n=4..8);
        seq([coeffs(subs(y=0,numer(s(n))))],n=4..8);
        seq([coeffs(subs(x=0,numer(s(n))))],n=4..8);

        seq(degree(subs(x=1,y=1,numer(s(n)))),n=3..8);

        -------------------------------------------------------

Then we broke into groups and discussed the papers until we were out
of time.