Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
" tommy quaternion "
#11
I have the intuition that tetration requires the introduction of numbers with real dimension (non integer, like complex or quaternion sets).

A problem is that those numbers have to include hypercomplex as a subset, but, for example a set with the dimension of a serpinsky triangle, would be a subset of the complex, and it cannot be closed under addition/multiplication, without including all of the complex numbers, but then it would just be equal to the complex.

So, closure has to be abandoned, or the set has to be a set of sets with all the possible dimensions.

Also, sometimes tetration leads to step functions or dirac delta functions. A diract delta is similar to a segment of line with unit length. It is one dimensional, meanwhile an hypercomplex number is a point, which is zero-dimensional.

It is analogous to surreal numbers, where a segment of line would be like a number infinitely larger than a point.

So maybe the serpinsky triangle should be taken as something in between a point and a surface, instead of a subset of the complex.

Instead of thinking of numbers as points, we should think of then as fractal sets.

An exponentiation to a real exponent, in general, has infinite solutions (because a root has infinite solutions in the complex field). Those infinite solutions are a set.

Probably we should think of tetration as operations between sets instead of between numbers. Probably fractal sets.

Possibly singularities in functions are just numbers with a different dimension. A pole in a complex function may be a line number with length equal to the pole's residue.

Possibly Taylor series require numbers with a different dimension outside of his radius of convergence, to be able to calculate pass singularities.
I have the result, but I do not yet know how to get it.
Reply
#12
For the " tommy octonion " we have the following Jacobi matrix.
By that I mean the Jacobi matrix for taking f(X) = X^2 in that number system.

see pictures.

The inverse of that matrix corresponds to taking the square root whenever the matrix is invertible (the determinant of the original Jacobi is not zero).
( see : https://en.wikipedia.org/wiki/Inverse_function_theorem )

regards

tommy1729
Tom Marcel Raes


Attached Files Image(s)
       
Reply
#13
Off topic but I like that you use a day planner/agenda to write your math in, lol.
Reply
#14
My friend posted the question at Mathoverflow :

https://mathoverflow.net/questions/38711...under-sqrt

So it is more formal now.

Some of you are probably on MO.

regards

tommy1729
Reply


Possibly Related Threads...
Thread Author Replies Views Last Post
  Tommy's Gaussian method. tommy1729 20 1,540 08/19/2021, 09:40 PM
Last Post: tommy1729
  tommy's simple solution ln^[n](2sinh^[n+x](z)) tommy1729 1 4,751 01/17/2017, 07:21 AM
Last Post: sheldonison
  Tommy's matrix method for superlogarithm. tommy1729 0 3,137 05/07/2016, 12:28 PM
Last Post: tommy1729
  Dangerous limits ... Tommy's limit paradox tommy1729 0 3,280 11/27/2015, 12:36 AM
Last Post: tommy1729
  Tommy's Gamma trick ? tommy1729 7 11,616 11/07/2015, 01:02 PM
Last Post: tommy1729
  Tommy triangles tommy1729 1 3,821 11/04/2015, 01:17 PM
Last Post: tommy1729
  Tommy-Gottfried divisions. tommy1729 0 2,908 10/09/2015, 07:39 AM
Last Post: tommy1729
  Tommy's hyperlog tommy1729 0 3,041 06/11/2015, 08:23 AM
Last Post: tommy1729
Sad Tommy-Mandelbrot function tommy1729 0 3,395 04/21/2015, 01:02 PM
Last Post: tommy1729
  tommy equation tommy1729 3 7,039 03/18/2015, 08:52 AM
Last Post: sheldonison



Users browsing this thread: 1 Guest(s)