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Stream: community: our work

Topic: Daniel Geisler

view this post on Zulip Daniel Geisler (Feb 07 2021 at 01:26):

I'm working with Stephen Wolfram to add tetration, hyperoperators and infinite power towers to Mathematica Functions. Before I can do that I need to publish my work. After fifty years of lone research, I'm now working with others and it is really improving the quality of my work. I just learned about the politics of convergence proofs from Yiannis Galidakis who alerted me about it in analytic continuation. People don't want to see innovative convergence proofs, they want to deal with convergence proofs like they have dealt with before and that means analytic continuation. My recent good news is functional equation are usually used in analytic continuation and I know the functional equations that describe the continuations.

view this post on Zulip Daniel Geisler (Oct 10 2021 at 13:19):

I've been plodding along on the project to index This Weeks Finds due to personal issues and the difficulty of the problem.

  1. Index the ASCII version for simplicity's sake.
  2. Language for text parsing - Perl, Mathematica, ect.
  3. Difficulty indexing phrases like "4-dimensional topological quantum field theory."
  4. Approaches

view this post on Zulip John Baez (Oct 10 2021 at 14:35):

It's possible this is not a great use of your time!

view this post on Zulip Daniel Geisler (Oct 10 2021 at 15:33):

@John Baez I strongly believe in doing social service. I support intellectually and politically both you and this forum. But my professional background is in software development and it is where I'm currently most effective. While I do math, I need to multitask in order to keep my mind supple.

view this post on Zulip John Baez (Oct 10 2021 at 17:27):

Great! But it could be that some other project is a better use of your time.

view this post on Zulip John Baez (Oct 10 2021 at 18:23):

It depends a lot on how hard it turns out to be to create an index!

view this post on Zulip Daniel Geisler (Oct 24 2021 at 08:02):

I'm making enough progress indexing TWF that collective issues are arising. I'm reporting in so that folks have time to provide feedback on where the project is going.

Indexing on the cheap
Compared to statically indexing TWF, Google provides relevance based indexing.
Google index for gravity on TWF site and use " site:math.ucr.edu/home/baez gravity". The TWF contains the word "gravity" 1420 times, which is not common in the English language. Using an API for Google and Wikipedia could be helpful.

A home for the index
I could host the index with my resources, but that might not be appropriate. A nice index would be a work in process, so it would be best if I had edit rights.

Additive and subtractive editing
Indexing the TWF is easy; backing frequently used words out takes more work. I currently am using a list of the one hundred most common English words. The first three hundred TWF are comprised of 37775 different words (including things like digits and URLs).

Mathematica
I'm able to do much of the heavy lifting with a relatively few lines of Mathematica. Text analysis and word clouds are supported by Mathematica although my computer with 12G RAM struggles with medium size word clouds.

view this post on Zulip Daniel Geisler (Dec 22 2023 at 04:59):

Bn,kB_{n,k} is a partial Bell polynomial.

dndxnf(g(x))=∑k=1nf(k)(g(x))⋅Bn,k(g′(x),g′′(x),…,g(n−k+1)(x)).{d^n \over dx^n} f(g(x)) = \sum_{k=1}^n f^{(k)}(g(x))\cdot B_{n,k}\left(g'(x),g''(x),\dots,g^{(n-k+1)}(x)\right).

Set g(x)=ft−1(x)g(x)=f^{t-1}(x)

dndxnft(x)=∑k=1nf(k)(ft−1(x))⋅Bn,k(ddxft−1(x),d2dx2ft−1(x),…,dn−k+1dxn−k+1ft−1(x)).\begin{align} {d^n \over dx^n} f^t(x) = \sum_{k=1}^n & f^{(k)}(f^{t-1}(x)) \nonumber \\ & \cdot B_{n,k}\left({d \over dx}f^{t-1}(x),{d^2 \over dx^2}f^{t-1}(x),\dots,{d^{n-k+1} \over dx^{n-k+1}}f^{t-1}(x)\right). \nonumber \end{align}

When seen as the derivatives of a dynamical system, the complexity of the behaviors that can be displayed is vast. I want to make it more understandable by using the preceding identity to easily derive the underlying combinatoric structures P,S,K,\mathcal{P},\mathcal{S},\mathcal{K}, and L\mathcal{L}. Ideally the would provide a way to model and solve dynamical systems using much simpler combinatorics.

The following makes the assumption of the existence of a fixed point. By computer algebra the derivative dndxnft(x){d^n \over dx^n} f^t(x) has K[n]\mathcal{K}[n] additive terms while the sum of the coefficients of the additive terms is L[n]\mathcal{L}[n]. Since L[n]\mathcal{L}[n] grows far more quickly than K[n]\mathcal{K}[n], efficient computer algebra must use K[n]\mathcal{K}[n] instead of L[n]\mathcal{L}[n].

view this post on Zulip Matteo Capucci (he/him) (Dec 22 2023 at 10:24):

Hi @Daniel Geisler, I changed the title of this topic to reflect the praxis of this stream (you did nothing wrong)

view this post on Zulip Matteo Capucci (he/him) (Dec 22 2023 at 10:28):

Also, have you seen this and this on the Faà di Bruno construction?

view this post on Zulip Daniel Geisler (Dec 22 2023 at 12:21):

Thank you @Matteo Capucci (he/him), Five interpretations of Faà di Bruno’s formula is also an ArXiv paper. I have many questions I expect this paper to answer.