Wikipedia:Reference desk/Archives/Mathematics/2012 December 20

= December 20 =

What the heck does this mean?
Apparently, a(n+1) is the number of compositions n=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 7*p(k+1) Sagittarian Milky Way (talk) 06:23, 20 December 2012 (UTC)

Here are constants for this:
 * n  a(n)
 * 1 1
 * 2 1
 * 3 1
 * 4 2
 * 5 4
 * 6 8
 * 7 16
 * 8 32
 * 9 64
 * 10 127
 * 11 253
 * 12 505
 * 13 1008
 * 14 2012
 * 15 4016
 * 16 8016
 * 17 16000
 * 18 31936
 * 19 63744
 * 20 127234
 * 21 253961
 * 22 506910
 * 23 1011800
 * 24 2019568
 * 25 4031088
 * 26 8046112
 * 27 16060160
 * 28 32056322
 * 29 63984903
 * 30 127714833
 * 31 254920736
 * 32 508825640
 * 33 1015623664
 * 34 2027200176

Anisotropic quadratic form on 2-dimensional space over finite field of odd order: how many "square" points?
Hello,

suppose $$Q$$ is an anisotropic quadratic form on a 2-dimensional vector space over a finite field of odd order. Let us denote 1-dimensional subspaces as points. $$Q$$ will assume either square values on all non-zero vectors in a point, or non-square values on all such vectors. On how many points does it assume square values?

I know the answer is exactly half: (q+1)/2, but what is the most elegant way to see this? I mean: is a coordinate-free proof possible?

Many thanks, Evilbu (talk) 09:56, 20 December 2012 (UTC)

Solving a function containing xe^x
I'm trying to solve an equation of this form for x: $$ae^x+bxe^x+cx+d=0$$ I already know that the xe^x term makes this pretty much impossible to do in closed form, because the inverse of that is the Lambert W function. Is there a good numerical approach outside of Newton's method? Is there some way to quickly approximate a solution for the initial guess? The solution is going to be evaluated very quickly in a control loop on an embedded system, so I'm trying to simplify and optimize things as much as possible. I doubt the constants are related in a useful way to take advantadge of any special cases of the form, but let me know if you want me to post the (rather ugly) formulas for them. 209.131.76.183 (talk) 21:05, 20 December 2012 (UTC)
 * Expand the exponential functions
 * $$ae^x+bxe^x+cx+d=a+d+(a+b+c)x+\sum_{k=2}^\infty(a+kb)\frac{x^k}{k!}$$
 * Truncating after the linear term
 * $$a+d+(a+b+c)x=0$$
 * gives the approximate solution
 * $$x=-\frac{a+d}{a+b+c}$$
 * which is used as one initial guess for the next equation
 * $$a+d+(a+b+c)x+(a+2b)\frac{x^2}{2}=0$$
 * and so on. Solve the polynomial equations numerically (See Durand-Kerner method). Bo Jacoby (talk) 21:48, 20 December 2012 (UTC).


 * Thanks! That looks great! My only question is how $$(a+kb)$$ got into the sum. Shouldn't it be $$(a+xb)$$? It looks like the same method will work, I just want to make sure I'm not missing something. 209.131.76.183 (talk) 12:58, 21 December 2012 (UTC)
 * Note that
 * $$xe^x =x\sum_{k=0}^\infty \frac{x^k}{k!} =\sum_{k=0}^\infty (k+1)\frac{x^{k+1}}{(k+1)!} =\sum_{k=1}^\infty k\frac{x^k}{k!} =x+\sum_{k=2}^\infty k\frac{x^k}{k!}$$
 * where the intermediate index k+1  is renamed to  k . Bo Jacoby (talk) 17:08, 21 December 2012 (UTC).

Riemann zeta function of the empty set
I recently ordered a book about the Riemann Zeta Function from Cambridge and it talks of finding the Riemann Zeta Function of the empty set. However it does not go into detail, how would you do this? And what would the Riemann Zeta function of the empty set? — Preceding unsigned comment added by 86.140.200.90 (talk) 21:06, 20 December 2012 (UTC)
 * For any function f and set A, you can find f(A), which is described at image (mathematics). When A is the empty set, the image is always the empty set. So in this sense the Riemann Zeta function of the empty set is the empty set. This is 100% trivial, so maybe something more interesting was meant in your book. Staecker (talk) 12:52, 21 December 2012 (UTC)