Wikipedia:Reference desk/Archives/Mathematics/2009 December 20

= December 20 =

hyperbolic tangent derivative
How is this expressed in Excel? Most references show it as 1-tanh2(x). In Excel is this 1-tanh(x)2? 71.100.0.206 (talk) 01:55, 20 December 2009 (UTC)
 * Why did you revert someone else's inquiry? Did you not know that this is prohibited? -- PS T  03:19, 20 December 2009 (UTC)
 * Presumably it was an accident... in any case, thanks for finding and fixing it. Eric.  131.215.159.171 (talk) 12:59, 20 December 2009 (UTC)


 * Yes. The notation f2(...) for f(...)2 is especially common when f is a (broadly) trigonometric function. This notation conflicts with the (also common) use of f&minus;1 for an inverse function, but that cannot be helped. –Henning Makholm (talk) 02:12, 20 December 2009 (UTC)
 * Indeed, just ensure you never combine the notations as sin-2x - you will cause people's heads to explode! --Tango (talk) 13:05, 20 December 2009 (UTC)
 * I agree, but if pressed, I would probably interpret that as $$(\arcsin x)^2$$. Likewise, in an unofficial context, I might use $$A^{-T}$$ to mean A inverse transposed. -- Meni Rosenfeld (talk) 13:29, 20 December 2009 (UTC)
 * I've used that one in papers.195.128.250.121 (talk) 00:12, 21 December 2009 (UTC)
 * But it could easily mean 1/sin2x. --Tango (talk) 17:39, 20 December 2009 (UTC)
 * Or arcsin(arcsin(x)). –Henning Makholm (talk) 22:18, 20 December 2009 (UTC)
 * Help! My head! No more! :) Dmcq (talk) 00:46, 21 December 2009 (UTC)
 * So soon? And we didn't even talk about how superscripts denote higher-order derivatives, with the possible extension of using negatives for antiderivatives... -- Meni Rosenfeld (talk) 16:00, 21 December 2009 (UTC)
 * I think all sane people use f(n) rather than fn for higher derivatives. There's always the insane authors to worry about, though. Algebraist 16:20, 21 December 2009 (UTC)

Local class field theory
Suppose $$L / E / K$$ is a tower of finite extensions of p-adic fields, pairwise Galois, with L/E abelian. Let $$ H = N_E^L(L^\times)$$. Let $$G = Gal(L / K), A = Gal(L / E), B = Gal(E / K)$$. One can easily see that every element of G fixes E and H (not pointwise). Therefore, we have an action of G on the quotient group $$E^\times / H$$, that is, a homomorphism
 * $$G \to Aut(E^\times / H)$$.

The Artin map gives us an isomorphism of $$E^\times / H$$ with A, so in fact what we have is a homomorphism $$\alpha : G \to Aut(A)$$. One sees that A is in the kernel of this homomorphism.

Now, A is a subgroup of G. As such, conjugation gives us a homomorphism $$\beta : G \to Aut(A)$$. As A is abelian, it is in the kernel of this homomorphism. I claim: (1) these two homomorphisms $$\alpha$$ and $$\beta$$ are equal.

With some work I was able to show that if B is cyclic, then G is the direct product of A and B (that is, $$\beta$$ is trivial) if and only if $$\alpha$$ is trivial. In fact I could show this holds whenever B is a product of cyclic groups of pairwise relatively prime orders. (This requires Hilbert's Theorem 90.) I claim:  (2) if B is abelian, then G is the direct product of A and B if and only if $$\alpha$$ is trivial. One of the directions -- G is a direct product implies $$\alpha$$ is trivial -- is easy.

Finally, I am (3) looking for conditions under which $$A \to G \to B$$ splits, i.e., conditions under which G is a semi-direct product of A and B.

So: I have been unable to find any proof for claims (1) and (2). (I don't know if they are true or not, I conjectured them.) In particular, with (1), I don't have a sufficiently explicit form for the Artin map to even attempt a proof. As for (3), I don't really have any ideas. Would anybody be able to give any guidance with these 3 points? I'd like hints -- very small hints -- just to point me in the right direction. Thanks. Eric. 131.215.159.171 (talk) 02:40, 20 December 2009 (UTC)

New Identity?
Hi everyone, I found this identity(which I cannot think of any practical use):$$B^{\sqrt{\frac{2x^2 (B - ln(C))}{ln(B)}}} = \frac{B^{Cx}}{Cln(B)}$$. Anyway, I haven't checked it so could anyone help me validate it? P.S. There is a possibility of a constant appearing which has to be added to one side. Thanks! The Successor of Physics  11:31, 20 December 2009 (UTC)


 * If it's an identity then it should be true for any valid values of x, B and C (i.e. as long as the term in the square root is not negative and the terms inside the logs are positive), yes ? So let's try:
 * $$x=B=C=2$$
 * $$B-\ln(C) = 2-\ln(2) = 1.306852819$$
 * $$\frac{2x^2 (B-\ln(C))}{\ln(B)} = \frac{8 \times 1.306852819}{\ln(2)} = 15.08312065$$
 * $$B^{\sqrt{\frac{2x^2 (B-\ln(C))}{\ln(B)}}} = 2^{\sqrt{15.08312065}} = 14.76080351$$
 * $$\frac{B^{Cx}}{C \ln(B)} = \frac{2^{4}}{2 \ln(2)} = 11.54156033$$
 * But 14.76... is not equal to 11.54... Have I made a mistake ? Gandalf61 (talk) 12:27, 20 December 2009 (UTC)
 * $$B^{\sqrt{\frac{2x^2 (B-\ln(C))}{\ln(B)}}} = 2^{\sqrt{15.08312065}} = 14.76080351$$
 * $$\frac{B^{Cx}}{C \ln(B)} = \frac{2^{4}}{2 \ln(2)} = 11.54156033$$
 * But 14.76... is not equal to 11.54... Have I made a mistake ? Gandalf61 (talk) 12:27, 20 December 2009 (UTC)
 * $$\frac{B^{Cx}}{C \ln(B)} = \frac{2^{4}}{2 \ln(2)} = 11.54156033$$
 * But 14.76... is not equal to 11.54... Have I made a mistake ? Gandalf61 (talk) 12:27, 20 December 2009 (UTC)


 * If you try x = 0, then the left side goes to one but the right side still depends on B and C. Eric.  131.215.159.171 (talk) 12:52, 20 December 2009 (UTC)
 * Look at the PS(It's obvious this was derived from an antiderivative)! The Successor of Physics  12:56, 20 December 2009 (UTC)
 * Sorry, they're not equal up to an additive constant either. Try $$x=3$$ and the difference between the terms will be different than with $$x=2$$.
 * Anyway, any identity looking like what you've written above can probably also be reached with basic algebra. -- Meni Rosenfeld (talk) 13:21, 20 December 2009 (UTC)

They can't be equal when x is negative. Michael Hardy (talk) 17:21, 20 December 2009 (UTC)

Do you have any reason to believe that this "identity" holds? I don't see any. It kinda seems like nonsense. For example if you let x = C = 1, then you end up with $$B^{\sqrt{\frac{2B}{ln(B)}}} = \frac{B}{ln(B)}.$$ Taking the ln of both sides, $$\sqrt{2B\ln B} = \ln B - \ln\ln B,$$ which clearly doesn't hold since √B grows much faster than lnB. Rckrone (talk) 19:21, 20 December 2009 (UTC)

Unless you mean that for a specific C and B it holds for all positive x, in which case you can find such B and C by solving for ClnB = 1 and C = sqrt(2(B-lnc)/lnB). In terms of this additive constant you mentioned, I'm not sure where you mean it should be, but with enough arbitrary constants in the right places I'm sure you could make things work out for more general B and/or C. Rckrone (talk) 19:37, 20 December 2009 (UTC)
 * Hey, Thanks, Rckrone(I'm an idiot)! The Successor of Physics  05:54, 21 December 2009 (UTC)

Rouché's theorem
Rouché's theorem is so called, because ... ? Was there a mathematician named Rouché? --Andreas Rejbrand (talk) 15:30, 20 December 2009 (UTC)
 * Apparently so. Why it's not in the article... - Jarry1250 [Humorous? Discuss.] 15:46, 20 December 2009 (UTC)
 * Thank you. --Andreas Rejbrand (talk) 16:08, 20 December 2009 (UTC)
 * It is almost certainly named after Rouché, but that doesn't mean he had anything to do with coming up with it! Mathematical theorems are often named after someone that popularised the theorem rather than first proved it. Sometimes they are named after someone that conjectured it, rather than proved it, as well - eg. Wile's theorem. --Tango (talk) 16:13, 20 December 2009 (UTC)


 * You mean Wiles' theorem (not that that exists any way).- Jarry1250 [Humorous? Discuss.] 21:55, 20 December 2009 (UTC)
 * Obligatory link to Stigler's law of eponymy (due to Merton). Algebraist 20:38, 20 December 2009 (UTC)


 * Wow, in this case it was proved by the person it's named after, see Theory of complex functions By Reinhold Remmert p. 392.--RDBury (talk) 06:31, 21 December 2009 (UTC)

Matrix subgroups
Let MatnR denote the space of n&times;n matrices with real entries, and let SymnR denote the space of n&times;n symmetric matrices with real entries. Given M &isin; SymnR let GM denote the set of X &isin; MatnR such that XTMX = M; where XT denotes the transpose of X. I started thinking about the GM by thinking about metrics and isometries. For example, the orthogonal transformations are the special case when M is the n&times;n identity matrix. The matrices X &isin; GM are the linear transformations which preserve the two-form (u,v) ↦ uTMv. It's easy to show that each GM forms a group under matrix multiplication. Also, the GM seem like they would quite like to be a vector spaces: for example, if X,Y &isin; GM and &lambda;, &mu; &isin; R then &lambda;X + &mu;Y &isin; GM. The only problem is that the zero matrix does not belong to GM unless M is itself the zero matrix, and that is a very dull example. There is some linear structure here too. If X &isin; GM and X &isin; GM then X &isin; G&lambda;M+&mu;N for constants &lambda;, &mu; &isin; R. I'm more of a recreational algebraist; so go easy on me. Please try to Wikify as much as possible. Dr Dec (Talk)  22:14, 20 December 2009 (UTC)
 * I was wondering what kind of algebraic structure the GM have. They seem like they would like to be rings but with multiplication and addition reversed; i.e. multiplication taking the role of addition (e.g. there is a multiplicative identity, but no additive identity) and vice versa. But that doesn't quite work either because they're not abelian with respect to multiplication.
 * I was wondering how the GM fit together in the whole space of matrices.
 * These are the indefinite orthogonal groups, of which I know nothing, but the article may be of use. Note that your statement "if X,Y &isin; GM and &lambda;, &mu; &isin; R then &lambda;X + &mu;Y &isin; GM." is not true, and these groups aren't much like linear spaces. They are manifolds, though, and hence Lie groups. Algebraist 23:00, 20 December 2009 (UTC)
 * Yeah, I wrote the wrong thing; thanks. What I meant to say was that if X &isin; GM and X &isin; GN then X &isin; G&lambda;M+&mu;N.
 * In that case you're just saying that the set of symmetric bilinear forms preserved by a matrix X forms a vector space, which is both true and unsurprising. Algebraist 16:17, 21 December 2009 (UTC)
 * If you want ring-like structures along these lines, you could look at the associated Lie algebras, which consist of matrices X such that XTM = −MX. Algebraist 23:41, 20 December 2009 (UTC)
 * I would also point out that a structure is not even slightly ring-like unless its two rules of composition are connected by the distributive law in the right direction. If distributivity is lacking, then it matters not at all whether each composition viewed separately has properties one would expect of a ring addition/multiplication –Henning Makholm (talk) 23:28, 20 December 2009 (UTC)