Wikipedia:Reference desk/Archives/Mathematics/2011 April 4

= April 4 =

Notation
In this paper, in Theorem 1, what does p(G) mean? —Preceding unsigned comment added by 210.212.167.113 (talk) 06:14, 4 April 2011 (UTC)
 * It is defined in the paragraph just before that theorem:
 * let p(G) denote the number of points of G.
 * &#x2013; b_jonas 07:27, 4 April 2011 (UTC)

Trivial question has me defeated
How do you solve $$2^x+2^{4-x}=17$$? I saw this on a "homework help" site and thought I would be able to do it easily. I could probably have done this when I was at school but I don't know where to begin. I have plotted the function and got the answers (0 and 4), but how do you solve it algebraically? I have thought of logarithms but I don't know to apply them. -- Q Chris (talk) 08:44, 4 April 2011 (UTC)


 * Try starting with the substitution y = 2x Dmcq (talk) 08:53, 4 April 2011 (UTC)


 * OK I get $$y+2^4*2^{-x} = 17$$
 * which gives: $$y +{2^4}/y = 17$$
 * multiply by y (and evaluate $$2^4$$ as 16:) $$y^2 +16 = 17y$$ or $$y^2 -17y + 16=0$$
 * $$(y-1)(y-16) = 0$$
 * which gives $$y=16$$ and $$y=1$$
 * substitute back in y = 2x
 * $$2^x=16$$ gives 4 and $$2^x=1$$ gives 0
 * Thanks Dmcq great stuff, your clue got me there! - Q Chris (talk) 09:27, 4 April 2011 (UTC)
 * Thanks Dmcq great stuff, your clue got me there! - Q Chris (talk) 09:27, 4 April 2011 (UTC)

sets of numbers with bounded subsets
Hi. Is there a proper name for numbers $$p_i$$, with $$p_i\in\mathbb{R}^n$$ for $$i=1,\ldots,k$$, with the modulus of the sum of any subset of the $$p_i$$ less than or equal to one? That is,



A\subseteq\{1,2,\ldots,k\}\longrightarrow\left|\sum_{i\in A}p_i\right|\leq 1$$

(I'm not sure whether $$\left|\sum_{i=1}^kp_i\right|$$ should be one or less than one).

Is this system known by any standard name? How about if $$ p_i\in\mathbb{C}^n$$? Robinh (talk) 10:04, 4 April 2011 (UTC)

Stalk
This is a really simple question I'm just a bit confused... The usual definition of Stalk at a point x defines it as a quotient set of the disjoint union of all the F(U) with x in U, and then this set is given the usual group structure. The stalk is the direct limit of all the F(U) as sets, but is it the direct limit of all the F(U) as groups? Money is tight (talk) 10:56, 4 April 2011 (UTC)


 * Yes, because it satisfies the universal property in the category of groups. It's universal in sets, so you only need to convince yourself that the universal mapping is a group homomorphism if all the relevant mappings from the F(U) are also assumed to be group homomorphisms.  You're right though that this isn't the construction of the direct limit that one would use in a category theory setting (where the disjoint union would get replaced with a coproduct in the category of groups), but the two things must agree in the end by universality.   Sławomir Biały  (talk) 11:23, 4 April 2011 (UTC)

Latex resize align*
I want to resize an align* in latex to be a single column width. Everything I've tried throws tons of errors. The computer science people I've asked have no clue (most suggest using resizebox, which doesn't work and others suggest minipage, which only resizes the caption). Any mathematics people know how to do this? -- k a i n a w &trade; 15:05, 4 April 2011 (UTC)

Should have noted that I am currently using \tiny to make the equations smaller, but I'd prefer them to be \columnwidth instead of just \tiny. -- k a i n a w &trade; 15:15, 4 April 2011 (UTC)


 * What are you using for multiple columns? It can be done directly via an environment such as \begin{multicols}{2}, but there are several packages that do it. Minipage should work for the entire contents of align* (unless something else is creating a conflict). I would start by trying different methods for multiple columns, some of them have very limited interoperability. SemanticMantis (talk) 13:53, 5 April 2011 (UTC)