ATON is a hopefully evolving classification theory. It aspires to unify knowledge around numbers and prefers naive methods. Some of the older posts are wrong but I'll keep them for the sake of continuity.

Sunday, November 12, 2006

Mirror Set, rev-2

After discovering the multifactorials, or Barnes G-functions, a revision is necessary since G-functions are more general. Mirror Set maps pairs of naturals to rationals;





Where m,n are natural, G(m,n)=m!n is multifactorial of m to depth n. Revising and broadening Mirror Set previously defined as M(m)=M(m,2). This form is more general and combines G-functions and Zeta function. It has a ying-yang look and variables do not commute, G(m,n)=/G(n,m). Here are some small values;

M(1,1-4) = (1, 1/2, 1/3, 1/4)
M(2,1-4) = (2, 1/2, 2/9, 1/8)
M(3,1-4) = (6, 3/2, 8/9, 3/4)
M(4,1-4) = (24, 18, 4^4/3, 6^4)

Previously I took M(3,2) as the center. Now M(2,2) =2/4=1/2 takes the center, standing for time/space. {2,2/9}={light,gravity}, why 2/9 for gravity? (3/2,1/2}={strong,weak} is reasonable.

Applying continuity by setting m-->z and summing gives a harmonically weighed G-function. For G=1 it is the Zeta function;






Assuming continuity holds, G may be generalised to a function of 2 complex variables;






Where z!w stands for the multifactorial to depth w, G(z,w). The form of M seems to imply the extension of G but it remains to be seen if it can be made continuous. I think G works like a local correlation function.

Thus naturals take the center stage of ATON, instead of sets. Maybe not surprising. Don't they (mathematicians) consider number theory as the purest?

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