Wikipedia:Reference desk/Archives/Mathematics/2012 July 1

= July 1 =

Which font is used in equations
Hello,

My name is Steven and i have an question regarding what font is used when math/physic equations are used in pages, i would really like to know, please and thank you for your time.

Sincerely, StevenCelusCane (talk) 06:15, 1 July 2012 (UTC)


 * Wikipedia's math mode uses LaTeX; see WP:MATH. As far as I know, it uses the Computer Modern fonts that are the default for LaTeX. Looie496 (talk) 06:37, 1 July 2012 (UTC)


 * A new way of displaying the maths will very probably be brought in soon using MathJax using the implementation of the STIX fonts which they describe as being Times compatible - but then again Computer Modern is pretty much Times compatible. Dmcq (talk) 09:57, 1 July 2012 (UTC)


 * Computer Modern resembles Times to the casual observer, but it's not "compatible" in the sense of matching spacing. —Tamfang (talk) 22:08, 1 July 2012 (UTC)


 * Times and Computer Modern are also not compatible in that using both in the same text looks ugly. The italic letters kvwxyz have completely different forms, the bar of the italic letter t have a different style, and the upper left serifs of the italic letters ijmnpruvy also differ noticably.  The roman variant of these fonts are much more similar in style: the easiest letters to distinguish (not counting Q) is e: e has its bar in the middle in Computer Modern, but above the middle in Times.  There are of course widely used Times-based fonts for mathematics too.  &#x2013; b_jonas 23:20, 11 July 2012 (UTC)

Probability question
So, say there were $$n$$ runners in a race (including yourself), and you know you came $$k^{\text {th}}$$ overall (so there were $$k - 1$$ people in front of you). Now, suppose that $$j$$ of those $$n$$ people, where $$1 < j < n$$, form some subset of the runners that you are particularly interested in (all your friends who ran or whatever); and say you count yourself a member of $$j$$ for present purposes. If the place each runner comes in the race is uniformly distributed, how do you work out the probabilities that you came 1st, 2nd, etc. out of $$j$$? I tried working it out for the case where $$j - 1 \geq k - 1$$ and you come $$k^{\text {th}}$$ out of $$j$$ (so, all the people who came in ahead of you were, in fact, your friends), and I tried: $$P \left ( \text {all those who finished ahead of you were members of } j \right ) = \frac{j - 1}{n - 1} \cdot \frac {j - 2}{n - 2} \cdots \frac{j - \left ( k - 1 \right )}{n - \left ( k - 1 \right )}$$, but even there I'm not sure, because do I need a Combination or a Permutation in there somewhere? And what about the case when only $$\frac{i}{k - 1}$$ of those who finished ahead of you were members of $$j$$, where $$i < k - 1$$? 131.111.7.71 (talk) 12:32, 1 July 2012 (UTC)


 * You have j&minus;1 friends among n&minus;1 runners. The probability that m&minus;1 friends are among the k&minus;1 runners in front of you, and the remaining j&minus;m friends are among the n&minus;k runners behind you, is the hypergeometric distribution
 * $$P(m)=\frac{\binom{k-1}{m-1}\binom{n-k}{j-m}}{\binom{n-1}{j-1}}$$
 * Bo Jacoby (talk) 20:30, 1 July 2012 (UTC).

"Best" twice differentiable upper bound approximations of functions for optimization?
I wanted good twice differentiable approximations of the absolute value and hinge loss (i.e. max(0, 1-x)) functions for the purpose of minimizing functions composited of such expressions with gradient descent and Newton-like methods, and instead of just taking any which one (there are some proposed approximations in the literature), I sought some principles for doing so (apologies for the long-winded prelude to questions..).

The composite function is multidimensional and a well-conditioned Hessian is desirable, so requiring |f″(x)| ≤ a for some a seemed natural. Also, both g1(x) = |x| and g2(x) = max(0, 1-x) are used to measure "badness", so requiring f(x) ≥ g(x) seemed sensible. Finally, to measure the error of approximation, the area between the curves is at least a visually appealing choice (for lack of really well-motivated criteria..). Comments on these quasi-arbitrary choices are welcome, but let's proceed to the consequences.

To be in the epigraph of f, a point must be in the epigraph of a parabola with second derivative a contained in the epigraph of g, so the boundary of the union of such parabola epigraphs is a candidate for f (think of running a parabola-point pen over g and seeing where it fits). If the resulting f satisfies |f″(x)| ≤ a, we're done. This actually solves the problem for both g1 and g2, but leaves me more curious than before about the general case!

Surely this type of thing cannot be new. Is there anything interesting written about such/similar things on Wikipedia (or elsewhere)? Is there some kind of practical solution method for the general case? Minimizing functionals is the ken of variational calculus, but the constraints of the problem seem to make it inapplicable..? Is there some way to characterize the solutions, e.g. is f always piecewise quadratic (with f″(x) = ±a?) in regions where f(x) ≠ g(x)? (this from the intuition that either the "pen" freely follows the "paper" or can't quite probe a "hole", leaving a mark of the quadratic tip; I think even something like g3(x) = -rect(x) will result in such an f, but it's more complicated because of overshoot akin to Gibbs phenomenon) If so, is there a practical way to find the piece endpoints? Breakdown points of the curve drawn by the "pen" look like a natural starting point, but they won't suffice even for g3. -- Coffee2theorems (talk) 13:49, 1 July 2012 (UTC)
 * Here's one small point: it's not the case that the boundary of the union of all a-parabolas in the epigraph of any g has a second derivative bounded independently of g. Consider $$g(x)=e^{-kx^2}$$ (rather silly for a cost function, but maybe add $$x^2$$ to it to say "you must be near but not too near"): as k is made larger, the union of parabolas approaches the entire upper half plane except for a sharp-topped "tent" over the spike of g.  This is related to your "Gibbs phenomenon", but here the ringing has to be in x, not y.  --Tardis (talk) 05:48, 11 July 2012 (UTC)