2012/02/26  

directed global detection of surprising connections in mathematics by bringing big data in.

I don’t trust category theory. I’ve read Jacob Lurie’s /Higher Topos Theory/ and it leaves a bad taste in my mouth. It seems to me that this is the modern fundamentalism, somewhat akin to logic in the Fregean era. I don’t really trust reasoning that you can’t support numerically or visually. I find the direction that Flajolet’s /Analytic Combinatorics/ works in to be much more appealing, and I would like to see category theory brought in that direction. Similarly I can’t read pages of ncatlab without feeling that it isn’t the right general direction for mathematics to take.

What would be nice, and doesn’t exist, if there were an “analytic category theory” much the same way as there is analytic group theory.

There is thread on mathoverflow Your favorite surprising connections in mathematics, and I think this begins to broach the conversation about big data and the discovery of new mathematics in a way that needs to become more and more meaningful as we continue to throw scads and scads of computing power at mathematics.

What if we could discover things like Monstrous Moonshine on a regular basis? What if we had the tools that allowed us to ferret out unusual functors? What i we could apply something like Don Swanson’s Arrowsmith to the arxiv? We need better tools for directed discovery of unusual connections because they tend to stimulate progress better than usual connections.

2012/02/23  

The Hofstadter Transcendental Triangle Variety

The Hofstadter point is a transcendental triangle center. So this gets me thinking: well, if I have three noncollinear complex numbers, that’s a triangle, right? And for three complex numbers u,v,w I can treat them as elements of \mathbb{C}

So here’s the recipe: the Hofstadter Transcendental Triangle Variety X is the set of points in (u,v,w)\in\mathbb{C}^{3} such that given some point \rho\in\mathbb{C}, the “Hofstadter mean” of those three points is \rho.

2012/02/10  

Challenge of the day

Evaluate

\prod_{n=0}^{\infty} \left(1+e^{-n}-e^{-2n}\right)

 

 

2012/02/02  

complex solutions of diophantine polynomials and trees

Chaitin has implemented an exponential polynomial that more or less is a lisp implementation, and Yuri Matiyasevich implemented one in the seventies that was used in his proof to answer Hilbert’s 10th problem in the negative.

In Baez’s This Week’s Finds, week 202, an isomorphism between trees and seven tuples of trees is noted.

This makes me wonder: what could we do with complex solutions of Chaitin’s and Matiyasevich’s polynomials?

2012/01/13  

logarithmic trickery

Consider that since \log(a)\log(b)=\log(a^{\log b})=\log(b^{\log{a}}, and therefore, somewhat unexpectly: a^{\log b} = b^{\log a}
we can use it to construct new identities thusly

Let A = \prod_{n=1}^{\infty} \sqrt[2]{1+\frac{1}{n^{2}}}

Let B = \prod_{m=1}^{\infty} \sqrt[e^{m}]{1+\frac{1}{m^{3}}}

Therefore, since we know that A^{\log(B)}=B^{\log(A)} we can use that
to explicitly calculate a new identity thusly:

1. Compute the logarithm of A, \log(A):
\log(A) = \frac{1}{2} \sum_{v=1}^{\infty} \frac{(-1)^{v+1}\zeta(2v)}{v}

2. Compute the logarithm of B, \log(B):
\log(B) = \sum_{s=1}^{\infty} \frac{(-1)^{s+1}\mathrm{Li}_{(3s)}(1/e)}{s}

3. Combine them:

\prod_{n=1}^{\infty} \prod_{s=1}^{\infty} \sqrt[s]{\left(1+\frac{1}{n^{2}}\right)^{(-1)^{s+1}\mathrm{Li}_{(3s)}(1/e)}} = \prod_{m=1}^{\infty} \prod_{v=1}^{\infty} \sqrt[ve^{m}]{\left(1+\frac{1}{m^{3}}\right)^{(-1)^{v+1}\zeta(2v)}}

>>> A = fp.nprod(lambda n: sqrt(1+1/(n*n)), [1,inf])
>>>
>>> A
mpf(’1.9109509100512501′)
>>> B = fp.nprod(lambda m: (1+1/(m**3))**(1/exp(m)), [1,inf])
>>>
>>> B
mpf(’1.3140291251423164′)
>>> A**log(B)
mpf(’1.1934623097049237′)
>>> B**log(A)
mpf(’1.1934623097049237′)

2012/01/10  

dreaming about reading mathematics books

one: It is a thick, yellow, Springer-Verlag dealie, illustrated and about fractals. There is a color plate on nearly every page. I dreamt of being in a bookshop and considering buying it.

two: has lots of equations at the beginning, and they change as I read them. last page has a distorting/metamorphosing picture of einstein/feynman in green and blue hues.

three: also a text that changes, I remember a diagram of helix against a point scatter. Also changed.

last night: I remember something about the Euler-Mascheroni constant \gamma. Impenetrable references section. Also picture of fractals. 

Have you dreamt of reading mathematics books? If so, leave a comment about them.

2012/01/10  

Sprott decoder (now with quartic and quintic transformations)

here is the source, compile with
gcc syllegy.c -o syllegy -lm

2012/01/05  

zeta theta hybrideque

Define g(z) = \sum_{n=1}^{\infty} \frac{e^{-n^{2}}}{n^{z}}. Note that while resembling the Riemann zeta function \zeta(s), the exponential term suppresses the pole at z=1.0

2012/01/01  

the banshee

I had this thought in the shower, and it descends from my “symmetry may be a long term red herring” idea, and it goes like this:

Imagine that you have something quite like the Monster group M, except that there is a fixed pair of elements in this thing like M — let’s call it \mathcal{B} for banshee. Every time you take two elements from \mathcal{B} and calculate their product, there is a \frac{1}{|M|^{2}} chance that you’ll be given something which is not an element of M, like a real number or some other object.

Let’s say it transpires that the boogeyman actually has some utility: if you happen to have a hangup on symmetry, your apprehension of the Monster group is going to obscure the existence of the banshee to you.

Interesting, because of the sheer size of the order of the Monster, it is easier to confuse this banshee with a group than if you were to take Klein’s viergruppe or a cyclic group and replace one of it’s elements. The banshee \mathcal{B} is not a group

2012/01/01  

interpolating the primorials

Here’s a puzzle: consider that the Gamma function \Gamma(z) is the interpolation of the factorials. What about interpolating the primorials. If p_{n} is the $n$-th prime, then what is the correct analogue of the Gamma function if we define the primorial as:

p_{n}\# = \prod_{k=1}^{n} p_{k}

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