Interactive coalescents

Genetic coalescents are interesting statistically; the variance in the time to coalescence is large, which is the kind of quantity I think human intuition has trouble with. So it helps a bit to be able to play with them (code can be found here):

Gfan in 3D

Having upgraded the Sage interface to gfan for version 0.3, I've been thinking about other ways to leverage Sage's capabilities in this respect. One thing I've been working on is a 3D Groebner fan representation. I have some working code for this now, which hopefully will end up in Sage-3.0 if I have time to polish it up. Below are a couple of screenshots of the 3D rendering of the Groebner fan of the ideal generated by (w^3-x, x^3-y, y^3-w, z^2-x-y-w):

restricted four-body problem

The Albouy-Chenciner equations for the restricted four-body problem seem to generally have 191 solutions. For masses m1=17/20, m2 = 19/20, and m3 = 1, there are 160 complex solutions in 6 variables (the mutual distances between the particles). Plotting each sextuplet as a polygon gives the following plot, followed by the configurations formed by the positive real solutions. Computed with Sage and phcpack.

phcpack, sage, and interact

I've been working on integrating Jan Verschelde's phcpack software with Sage. phcpack finds solutions to pretty nasty systems of multivariable polynomials by using polyhedral homotopy continuation. Sage can provide a nice frontend for this. Here's an interactive display of the complex solutions of the Albouy-Chenciner equations (from the paper "Le problème des n corps et les distances mutuelles", Inventiones Mathematicae 131 p.151-184, 1998) for the 4-vortex problem:

Color me Gfan

The latest version of Gfan has some new capabilities that I am excited to use for testing whether ideals are zero-dimensional. But first I have to rewrite the Sage interface to Gfan. I thought that I should try to give some Sage-added-value while I was at it, so I am converting Gfan's xfig output to Sage graphics and adding some color. Here's one result so far: a map of all the reduced Groebner bases for the 3-vortex problem, colored by maximum degree:

Sage 2.10.3

Sage 2.10.3 is out, with the first released version of the new interact command. As a spectator to the process, it looked like a tough fight between bugs and developers - which I think should be viewed as an entirely positive thing, since it is a consequence of improved QA practices.

Release notes are here, in case you need it the download page is here.

If I can stop playing with interact (which has already been useful to me in teaching and research after 1 day!!) I hope to contribute a wee bit to some upcoming releases. I am rewriting the gfan interface to make use of gfan 0.3. Also, a student and I are working on pretty graphics for path-tracking solutions from homotopy solvers of polynomial systems (phcpack). I'll have to revise the plan a bit to exploit interact.

Interact

William Stein and Co. have delivered once again, with a new "interact" command that looks amazing even in its beta form. You can almost smell Sage-3.0; it should be out before the roses are blooming here in Duluth. Among other tests, I used it to explore the CpG content of the human mitochondrion averaged over a variable length window: