Talk:Minkowski's question mark function

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Just a comment, this has to be the worst example of bad choice of notation...what was he thinking?? Revolver 07:06, 21 May 2005 (UTC)

Dates back to the dawn of the discovery of pathological functions, at the turn of the 20th century. Maybe it was felt to be very confusing? And maybe he had a twisted sense of humour. linas 00:36, 22 May 2005 (UTC)

Why is ? any crazier than ! (asks one who shorthands sin and cos with $ and ¢)? Kwantus 2005 June 29 17:52 (UTC)

Worse yet, it conflicts with a much more useful notation, d?k for the binomial coefficient d!/(d−k)!k!, generalizing to ?〈i₀,…,i_n〉 for the multinomial coefficient of a multi-index. Without regard for degree we can write

$? I {\mathbf x}^I$

as the general term of a multinomial expansion, where the multi-index exponent, as usual, means

$x_0^{i_0} x_1^{i_1} \cdots x_n^{i_n}$ ,

and

$|I| = i_0 + i_1 + \cdots + i_n$

equals the degree. Ah well; there's little chance of confusion. --KSmrq^T 20:53, 2005 August 31 (UTC)

[edit] Question

The fractal and self-similar nature of the function is unclear. Exactly how does the modular group describe the self-similarity? AxelBoldt 03:52, 29 March 2006 (UTC)

Yes, its a sloppy statement; it is only the "period-doubling monoid" inside SL(2,Z) (and not the modular group PSL(2,Z)) that applies to period-doubling fractals. (Different people seem to call this monoid different names). Consider the operators R and S:

[S ?](x) = ?(x / 1 + x) = ?(x) / 2

Note that the above is a self-symmetry for x in [0,1]: the question mark on the interval [0,1/2] is a half-size of the whole thing. Let R be a reflection:

[R ?](x) = ?(1 - x) = 1 - ?(x)

Then R and S generate the monoid: that is, any string of the form

S m R S n R S p ...

for positive integers m,n,p ... is a self-symmetry of the question mark. The requirement that m,n,p be poistive is what makes it a monoid,nt a group. linas 02:31, 30 March 2006 (UTC)

I fixed the article. The relationship to SL(2,Z)/PSL(2,Z) etc. is not hard but has some subtle confusions.linas 03:12, 30 March 2006 (UTC)

[edit] question

Is this function really absolutely continuous?

[edit] question/correction? =

When reading the page 84 of my copy of John Conway's "On Numbers and Games" (2nd edition, 2001, A K Peters, Ltd) I see at the top:

+---------------+
|               |
| (1+sqrt(5))/2 |  =  5/3
|               |
+---------------+

The function here called [x] is traditionally called "Minkowski's Question-Mark Function," and has interesting analytic properties.

(Where [x] is my ascii rendition of "x in a box"). So Conway _does not_ indicate the inverse of Minkowski's ? with a box as is claimed in the article, but the Minkowski's ?-mark function itself!

BTW, I created a new index entry to Sloane's OEIS for related sequences: http://www.research.att.com/~njas/sequences/Sindx_Me.html#MinkowskiQ You may add it to the external links section.

Yours, Antti Karttunen, his-firstname.his-surname@gmail.com

PS. You (Linas) erroneously claim on your "Wacky Thoughts" page that Minkowski's Question mark-function maps algebraic numbers to rationals, whereas it's only the "quadratic surd" -subset of them. (Like correctly explained on this page.)

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Talk:Minkowski's question mark function

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