/sci/ - Science & Math

>>7747005
It just some rant, if something seems fuzzy then because it is.

>>7746997
Okay, so only should, I think, again see it under the light that we pass from something discrete to something smooth.

So one starting point is the Bernoulli polynomials [math]B_k(x)[/math]. They are characterized on one side by behaving like monomial building blocks:

[math] B_k’(x)=k\,B_{k-1} [/math]

(several function do this, especially, of course, monomials functions x^k)
and made unique in that picking up smooth information about their coefficients is leading to the most simple algebraic building blocks:

[math] \int_x^{x+1} B_k(y)\, dy=x^n [/math]

The first few are shown in pic related and they can be compactly defined as coefficients of the z-expansion of the function [math] \frac {z} {{e^z} -1} {e} ^{x\,z} [/math]

[math] \frac {z} {{e^z} -1} {e} ^{x\,z} = 1 + \frac{1} {1!} (x- \frac {1} {2} ) z + \frac {1} {2!} (x^2-x + \frac{1} {6} ) z^2 [/math]
[math]+ \frac{1} {3!} (x^3 - \frac{3 x^2} {2} + \frac{x} {2} + 0) z^3 + O (z^4) [/math]

[math]= 1 + \sum_{k=1}^\infty \frac{1} {k!} B_k(x) \,z^k [/math]

I noted as similar generic expansion in (>>7744515). If f is a series with coefficients a_k, and a_0 not zero, then

[math] \frac { f(0) } { f(f(0)\,z) } = \frac { 1 } { \sum_ { k=0 }^\infty \frac { a_k } { a_0 } (a_0\,z)^k } =1-a_1\,z+(a_1\,a_1-a_0\,a_2)\,z^2-(a_1\,a_1\,a_1-2\,a_0\,a_1\,a_2+a_0\,a_0\,a_3)\,z^3+ O (z^3) [/math]

where we compare something of zeroth order f(0) to the full thing f(z).
(the above expression is normalized, f(f(0)·z), so that if f(0) isn’t 1 it factors out)

The particular number pops up when [math]a_n = \frac {1} {(n+1)!}[/math], so that [math] a_1\,a_1-a_0\,a_2 = \frac {1} {2!2!} - \frac {1} {3!} = \frac {1} {12} [/math].

If you take a look at the Todd class pic above (>>7746983), you see that this is exactly about the Bernoullis

(cont.)

>>7747005
It just some rant, if something seems fuzzy then because it is.

>>7746997
Okay, so only should, I think, again see it under the light that we pass from something discrete to something smooth.

So one starting point is the Bernoulli polynomials [math]B_k(x)[/math]. They are characterized on one side by behaving like monomial building blocks:

[math] B_k’(x)=k\,B_{k-1} [/math]

(several function do this, especially, of course, monomials functions x^k)
and made unique in that picking up smooth information about their coefficients is leading to the most simple algebraic building blocks:

[math] \int_x^{x+1} B_k(y)\, dy=x^n [/math]

The first few are shown in pic related and they can be compactly defined as coefficients of the z-expansion of the function [math] \frac {z} {{e^z} -1} {e} ^{x\,z} [math]

[math] \frac {z} {{e^z} -1} {e} ^{x\,z} = 1 + \frac{1} {1!} (x-\frac{1} {2} ) z + \frac{1} {2!} (x^2-x + \frac{1} {6} ) z^2 + \frac{1} {3!} (x^3-\frac{3 x^2} {2} + \frac{x} {2} + 0) z^3 + O (z^4) = 1 + \sum_{k=1} ^\infty \frac{1} {k!} B_k(x) \,z^k [/math]

I noted as similar generic expansion in (>>7744515). If f is a series with coefficients a_k, and a_0 not zero, then

[math] \frac { f(0) } { f(f(0)\,z) } = \frac { 1 } { \sum_ { k=0 }^\infty \frac { a_k } { a_0 } (a_0\,z)^k } =1-a_1\,z+(a_1\,a_1-a_0\,a_2)\,z^2-(a_1\,a_1\,a_1-2\,a_0\,a_1\,a_2+a_0\,a_0\,a_3)\,z^3+ O (z^3) [/math]

where we compare something of zeroth order f(0) to the full thing f(z).
(the above expression is normalized, f(f(0)·z), so that if f(0) isn’t 1 it factors out)

The particular number pops up when [math]a_n = \frac {1} {(n+1)!}[/math], so that [math] a_1\,a_1-a_0\,a_2 = \frac {1} {2!2!} - \frac {1} {3!} = \frac {1} {12} [/math].

If you take a look at the Todd class pic above (>>7746983), you see that this is exactly about the Bernoullis

(cont.)