Wikipedia:Reference desk/Archives/Mathematics/2007 January 20

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January 20

Arcsin sqrt(x) transformation

My officemate had a reviewer on a paper write "data should be transformed by arcsin sqrt(x)." I recall (I think) that this is a normalizing transformation for some kind of data set (binomial/n, I think), but neither she nor I can find a reference on it. Any ideas? --TeaDrinker 00:30, 20 January 2007 (UTC)

Google gave an immediate reference: www.tina-vision.net/tina-knoppix/tina-memo/2002-007.pdf

The point seems to be that if the data are binomially-distributed, the transformation gives a variance independent of the mean. I question, however, the arrogance of the reviewer in saying what should be done, without explanation or even considering that another approach could be valid.86.132.163.126 12:25, 27 January 2007 (UTC)

Thanks for the replies! That was what I was looking for. --TeaDrinker 03:45, 6 February 2007 (UTC)

Rounding to a rational number

This might be better at the computing desk, but it's quite mathematical: is there a good algorithm for discovering the fraction (with an upper bound on the denominator) closest to a given number? More specifically, given a real number r (without loss of generality, we can suppose it to be on [0,1)) and a positive integer d, I want to find the coprime a and b, $b\leq d$, that minimizes $\left|{\frac {a}{b}}-r\right|$. Two simplifications are possible: I'm okay with r in fact being a rational already (with the denominator known), and the coprime property is optional (because reducing a fraction is trivial). It's obviously possible to test all denominators up to d and see which allows us to come closest; it's in fact possible to test only denominators greater than $d/2$ since any ${\frac {a}{b}},\;b\leq {\frac {d}{2}}$ can be rewritten as ${\frac {2a}{2b}},\;b\leq d$ (abandoning the coprime property like I said). I'm interested in an algorithm with better than $O(d)$ running time, however. Any suggestions (or even proofs of impossibility!) would be appreciated; thanks. --Tardis 04:01, 20 January 2007 (UTC)

The name for this general topic is Diophantine approximation. A common approach involves continued fractions. —David Eppstein 04:18, 20 January 2007 (UTC)

In case you want more details, the section you need to read is Continued_fraction#Best_rational_approximations. This gives explicit instructions for determining the best rational approximations to a real r ("best" in your sense, i.e. closer to r than any other rational number with a smaller denominator) from the expansion of r as a continued fraction. A sequence of best rational approximations for increasing values of the denominator d can be found by applying a fairly simple algorithm - no searches are required. Gandalf61 12:48, 20 January 2007 (UTC)

I think I'm confused. Since the rationals are dense in the reals, given any real number r, there exists a sequence of rationals that approach r; i.e., it is impossible to "minimize" that quantity you have written above. For any $\epsilon >0$ there exists a rational number $a/b$ so that $|a/b-r|<\epsilon$. Are you interested in finding such sequences, or have I totally missed the piont? –King Bee ^{(T • C)} 16:16, 20 January 2007 (UTC)

The key point is the constraint $b\leq d$. In other words the denominator of the rational number must not exceed some upper bound d. For any given positive integer d there is a best (i.e. closest) approximation to r among all the rational numbers with denominators less than or equal to d. Gandalf61 16:28, 20 January 2007 (UTC)

Ahh, I understand perfectly now. I should read more closely next time. =) –King Bee ^{(T • C)} 16:29, 20 January 2007 (UTC)

The continued fraction that converges most slowly is all 1s. Thus its value, f, satisfies f = 1+¹⁄_f. Rearranging to f²−f−1 = 0, the root we want turns out to be the golden ratio, approximately 1.618. The denominators of its successive approximants are the Fibonacci numbers, which grow in (roughly) geometric progression. So even in this worst case we'll do better than O(d), and usually we'll do much better.

When implementing the continued fraction approach in floating point, it is a Bad Idea to repeatedly take reciprocals. Instead, treat the floating point number as an exact rational number whose (unreduced) denominator is a power of two, and use the GCD algorithm to generate continued fraction terms one at a time. The first step or two can be tricky to handle robustly, but after that the numerator and denominator fit comfortably in integer variables. Safe testing for termination must also be handled carefully, especially when the deciding term is even. --KSmrq^T 06:35, 21 January 2007 (UTC)

mathematics

we dont know, how to get the definitions and references of some mathematical terms. —The preceding unsigned comment was added by 203.101.67.2 (talk) 10:09, 20 January 2007 (UTC).

You can try this website. http://mathworld.wolfram.com/ Zeno333 10:20, 20 January 2007 (UTC)

Wikipedia is very good for that, it isn't as technical as mathworld. Mathworld is slightly more comprehensive, but not by much. Oskar 18:28, 20 January 2007 (UTC)

Also check out Math Forum's site! --JDitto 03:26, 25 January 2007 (UTC)

Name of a line

what is a line that divides some figures into mirror-image halves.

The Reflection symmetry article calls it an "axis of symmetry", thinking back to school, we called it a "line of symmetry" - presumably the former is the more technically correct answer. For a Three dimensional object, its a "plane of symmetry".

Hope this helps - cheers, Davidprior 20:00, 20 January 2007 (UTC)