Talk:Blancmange curve

Hurst parameter

I don't think that Hurst parameter should redirect to this page. I was going to add a link from the page on fractional Brownian motion, but this page is no more about the Hurst parameter than that page is.

Would someone like to write a real Hurst parameter page?

LachlanA 23:34, 19 June 2006 (UTC)

Fractal dimension

Does anyone happen to know what the fractal dimension of this function is? It's not listed anywhere on the Internet, but a quick box-counting estimate gives 1.03037, which suggests it could well actually be 1, at least for w=1/2. 82.12.108.65 13:25, 10 February 2007 (UTC)

Differentiability

I reduced the section "differentiability", that contained wrong and unclear statements. However it may be expanded again hopefully. A more complete exposition should give the modulus of continuity of $T_{w}$ for w > 1/2 ( Hölder, I guess) and if possible, the exact value of the total variation of $T_{w}$ on the interval [0,1]). Also, a more precise statement about the differentiability for w ≥ 1/2 is still lacking. pma 16:28, 28 December 2018 (UTC)

A useful reference should be: Nowhere Differentiable Functions: The Monsters of Analysis , by Marek Jarnicki and Peter Pflug. pma 09:58, 18 January 2019 (UTC)

Unclear passage

The section Integrating the Blancmange curve contains this passage:

"This integral is also self-similar on the unit interval, under an action of the dyadic monoid described in the section Self similarity. Here, the representation is 4-dimensional, having the basis $\{1,x,x^{2},I(x)\}$. Re-writing the above to make the action of g more clear: on the unit interval, one has

$[g\cdot I_{w}](x)=I_{w}\left({\frac {x}{2}}\right)={\frac {x^{2}}{8}}+{\frac {w}{2}}I_{w}(x)$."

Previously that article was discussing "representations" of the Blancmange curve or its integral via various infinite series.

But here the word "representation" is being used in a different way that I don't understand, referring to the assertion that the representation is "4-dimensional" and has a given "basis" These words suggest that some vector space is being considered here. If so, it it necessary to state what vector space is being referred to.

Or if that is wrong, SOME much, much, much clearer explanation of what "dimension" and "basis" are about would be necessary for this passage to be comprehensible. 2601:200:C000:1A0:211A:BDC0:4259:59DA (talk) 14:28, 12 July 2022 (UTC)

Given some space X of "things" having some symmetry, one generally identifies that symmetry as a symmetry group. Elements g of the group G act on X as $g:x\mapsto gx$. Such actions can be given a group representation R in n-dimensional space, whenever one can find an integer n and a collection of n x n matrices M such that, for each group element g in G, one has a matrix $M_{g}$ such that there is a commuting diagram $R(gx)=M_{g}R(x)$. It is "commuting" in that the order of R and g are exchanged with one-another. In this representation, each element x in X corresponds to some vector R(x) in that n-dimensional space. In this article, its not actually a group, but a monoid, but the same ideas apply.

A basis for an n-dimensional vector space is commonly written as $\{e_{1},e_{2},...,e_{n}\}$. The basis here is $\{1,x,x^{2},I(x)\}$, that is, $\{e_{1},e_{2},e_{3},e_{4}\}\mapsto \{1,x,x^{2},I(x)\}$. That is, given ANY 4-D vector (a,b,c,d) one has the corresponding vector representation $a+bx+cx^{2}+dI_{w}(x)$ and this vector transforms under g as the matrix given in the article, which is $g:(a,b,c,d)\mapsto (a,b/2,c/4+d/8,wd/2)$. This is just a linear equation. Does this answer your question? I will try to edit the article to clarify this. 67.198.37.16 (talk) 04:16, 27 December 2023 (UTC)