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note on quotients of lacunary functions

Define:

\lambda_{a}(q) \equiv 1 + \sum_{n=1}^{\infty} q^{a^{n}} and

 \l_{a}(q) \equiv 1 + \sum_{n=1}^{\infty} (-1)^{n} q^{a^{n}}

Preliminary numerical experiments suggest that the power series representations of their quotients l_{b}(q)/\lambda_{a}(q) and \lambda_{a}(q)/l_{b}(q) always have integer coefficients. 

Iridiation of (algebraic|transcendental) varieties and strange attractors.

Suppose I have an algebraic variety G in \mathbb{R}^{n} — or a strange attractor X in \mathbb{R}^{3}, and to each point of $G$ or $X$ I associate a unit sphere — in the case of $latex{R}^{3}$ I could use Riemann spheres at each point u of X or G, and then say: okay, when I make visualizations of G or X, the color I see from the vantage point I’m looking at the variety or strange attractor is the phase/hue whatnot I get when I take some function f:\mathbb{C}\rightarrow\mathbb{C}: every point has a sphere associated with it, colored by phase of f_{u}

Terminology: if it’s a variety, we call them iridiated varieties or iridovarieties.

The inverse (or adjoint functor if you will) should be called bleaching.

And http://en.wikipedia.org/wiki/Iridescence is the right word here: these are objects whose appearance changes as you change the angle that you’re looking at them.

3 copies 2Cob and lacunaries

This is probably a very bad idea:

3 different copies of 2Cob tangled at right angles with each other, and then
also where they’re cut at cell boundaries, they’ve got associated lacunaries with them.
Also, you can only snap them together (like lego bricks) if the lacunaries match. I don’t have
a good way of writing out cells algebraically, either.

currently rendering (lacunaries and automorphisms of the unit disk, oh my)

Setup:
Let
v(q) = \sum_{n=1}^{\infty} (-1)^{n}q^{2^{n}} and
\mathrm{mob}(\theta,q,q_{0}) = e^{i\theta} \frac{q-q_{0}}{1-\bar{q}_{0}q}

I’m currently generating a movie depicting how v(q) changes when we apply the
Mobius transformation to the unit disk \mathbb{D} first and then take v(q). In my
setup q_{0}=-1/\sqrt[4]{3}, and \theta is in the interval [0,2\pi] in steps of
1/1024th the length of the interval. Blake Courter said that it was
pretty easy to do this, and I’m currently at frame 459.

Video is done, here it is:

lacunary functions and intersecting three dimensional cobordism categories

I have made it a habit of keeping really good notes and records the past few years. I notice when I go along certain routes. In particular, one sort of frequently traveled path is starting to look extremely interesting, except that I can’t say why it is, or speak much to why one thing reflects on the other.

I’m talking here about lacunary functions and intersecting three dimensional cobordism categories. They oddly seem to demand adjacent parts of mental machinery.

I’ve taken to making a large list of lacunaries and making phase portraits of them. Here’s a good one:

Left:
\prod_{n=1}^{\infty} \left(1 - \frac{q^{2^{n}}+q^{n^{2}}}{1-q^{2^{n}}}\right)
Right:
\prod_{n=1}^{\infty} \left(1 - q^{n^2} - 2q^{2^{n}}\right)

Both of these lacunary functions have exactly the same roots!

At the same time I’ve been thinking about intersecting three dimensional cobordism categories, with
freehand illustrations below. (I haven’t figured out how to automate the production of these
diagrams with tikz yet). And what is striking is that I’ve been in about this mental territory where
I am thinking about both lacunary functions and these sorts of intersecting cobordism diagrams at the same time.

One thing I might try next is to (a la Arthur Dent and scrabble tiles style), draw one intersecting cobordism
diagram, and then make a phase portrait for a lacunary function while looking at the cobordism diagram that I have
just drawn. Perhaps there is a functor between the two, but it’s not one I can get at by tedious algebraization
of the diagrams (I am not going to do this, I really don’t care about the egregious algebraization of everything), but
I prefer a synaesthetic approach.

The green and blue ones are knotted as in the Alexander Horned Sphere:


agm-ized symmetric derivative

The symmetric derivative looks like there’s an average in it, so what about changing that average
with something else, like the arithmetic geometric mean:

\lim_{h\rightarrow 0} \frac{\mathrm{agm}(f(x+h),f(x-h))}{h}

Cayley tables of all finite groups of order 576

Made with GAP

I’m going to try to put up an uncompressed version somewhere -the file weighs in at 900mb, but it’s interesting.

observation from a question on m.se

Set g(z)=\sum_{n=1}^{\infty} \left(\frac{z}{n}\right)^{n}, then the sum:

\sum_{m=1}^{\infty} \frac{m!}{m^{m}} can be interpreted as the value of the Laplace

Transform of g(z) at s=1:

\sum_{m=1}^{\infty} \frac{m!}{m^{m}} =  \sum_{m=1}^{\infty} \frac{1}{m^{m}} \int_{0}^{\infty} t^{m}e^{-t} dt = \int_{0}^{\infty} e^{-t} \sum_{m=1}^{\infty} \frac{t^{m}}{m^{m}} dt

 

new videos

First, I’ve rendered a 1024×1024 video of the Newton’s Method of the third Jacobi theta function ϑ3(z,q), for |q| = 0.90190165617299947 or thereabouts. This piece is entitled The Swift Luminescent Energy Drink of the Psyche, or When Goorialla Whirls and Whorls and Roars

Secondly, using GAP, I’ve made a video of all Cayley tables of finite groups of order 128:

And finally, A video showing Klein’s j-invariant j(\tau) under the transformation \tau\rightarrow -1/\tau in \mathbf{SL}(2,\mathbb{R}):

Topoi with complex subobject classifiers.

Random idea:

Start with a topos G, from it, create an (extended? amplified? complex?) \bar{G} such that
the subobject classifer \Omega_{\bar{G}} of \bar{G} can take any value in the unit disc on the complex
plane.

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