Wikipedia:Reference desk/Archives/Mathematics/2010 May 31

= May 31 =

Notational Inconsistencies in Vectors
I'm confused about the use of $$|$$ in notations for the length of a vector. According to the Dot product article:


 * $$|\mathbf{a}| = \sqrt{\mathbf{a} \cdot \mathbf{a}}$$

However, the Norm_(mathematics) article writes:


 * $$\|\mathbf{x}\| := \sqrt{x_1^2 + \cdots + x_n^2}$$

So my question is in a couple of parts:


 * 1) Is there a difference between ||a|| and |a|? Is one using the wrong notation?
 * 2) If not, what's the difference between the two?


 * The dot product article says: a.a is the square of the length of a (which is written as ||a|| on the Norm article).
 * This implies that:


 * $$|\mathbf{a}| = \sqrt{\|\mathbf{a}\|^2}$$

which would imply that ||a|| and |a| are the same.

Thanks in advance --Philipwhiuk (talk) 12:22, 31 May 2010 (UTC)
 * There's no real difference between the two, it's just a different notation for the same thing. Some people reserve |x| for (real or complex) absolute value and denote more general norms with ||x||, some don't bother and use the simpler |x| for all norms.—Emil J. 12:36, 31 May 2010 (UTC)
 * Many thanks for your quick reply EmilJ. --Philipwhiuk (talk) 12:39, 31 May 2010 (UTC)


 * ... and, of course, $$|\mathbf{a}| = \sqrt{\mathbf{a} \cdot \mathbf{a}} = \sqrt{a_x^2 + a_y^2 + a_z^2}$$ for a 3-D vector $$\mathbf{a}$$.   D b f i r s   16:38, 31 May 2010 (UTC)

Percentage increase
if I have four things, and I add another thing, is that usually called a 25% increase in things (a percentage of the original amount) or a 20% increase (a percentage of the final amount)? 81.131.69.251 (talk) 18:47, 31 May 2010 (UTC)
 * The former, i.e. a 25% increase. —Qwfp (talk) 18:51, 31 May 2010 (UTC)
 * Ta. 81.131.69.251 (talk) 18:53, 31 May 2010 (UTC)

Please help with a couple questions I've been stuck on for weeks! Curvature, projective modules & metric spaces
Hey everyone, I'm revising for my exams (hooray) at the moment, and having arduously completed about 10 years of past papers, I've been left with about 4 or 5 questions which I just can not for the life of me finish. I know you're meant to post all your working etc. here but as I said, these are the only questions left I can't get my head around, so there's very little working I've managed to achieve or else I would probably have been able to finish the problem with enough perseverance.

I appreciate I'm asking a for lot of help, so if you just have even 1 suggestion for any of these problems which could be of use then please do shout, the more of these I can get sorted before tomorrow the better, and of course I don't expect you to have to go out of your way to pick up my slack when I can't figure something out! :)


 * 1. (Final part of a long question) Show that a finitely generated projective module over a principal ideal domain is free.       I've shown that every free module over an arbitrary ring is projective, where here I'm using the definition that an R-module p is projective if whenever we have module homomorphisms $$f: M \to N$$ and $$g: P \to N$$, with f surjective, there exists a homomorphism $$h: P -> M$$ with $$f \circ h = g$$.


 * 2. Let (X,d) be a metric space with at least 2 points. If $$f: X \to \mathbb{R}$$ is a function, we define Lip(f) $$Lip(f)\,=\,sup_{x\neq y}\frac{|f(x)-f(y)|}{d(x,y)} + sup_z |f(z)|$$, provided this supremum is defined. Now with $$X=\mathbb{R}$$, suppose $$(f_i)_{i=1}^{\infty}$$ is a sequence of functions with $$Lip(f_i) \leq 1$$ and with the property $$f_i(q)$$ converges as i tends to infinity, for every rational q. Show the $$f_i$$ converge pointwise to a function f satisfying $$Lip(f) \leq 1$$. Suppose also $$(f_i)_{i=1}^{\infty}$$ are any functions with $$Lip(f_i) \leq 1$$. Show there is a subsequence $$f_{i_1},\,f_{i_2},...$$ which converges pointwise to a function f with $$Lip(f_i) \leq 1$$.      Where to start? I showed that Lip(X) is a vector space over $$\mathbb{R}$$ and that Lip is a norm on it, and then from then on I have no clue where to go. I tried to work out how to treat the irrational numbers in the first part and I presume it utilizes the density of the rationals, but I haven't been able to get anywhere at all from there - it was a hard year for Analysis and this was definitely the worst question, completely off the course.


 * 3. Deduce (from the fact that the Gaussian curvature K of a surface locally given by the graph of an infinitely differentiable function $$f: U \to \mathbb{R}$$, where U is an open subset of $$\mathbb{R}^2$$, is equal to $$K = \frac{f_{xx}f_{yy}-f_{xy}^2}{(1+f_x^2+f_y^2)^2}$$ - this was the first part of the question, so this is assuming I calculated that right!) that if $$\Sigma \subset \mathbb{R}^3$$ is a compact surface without boundary, its curvature is not everywhere negative. Give with justification a compact surface in $$\mathbb{R}^3$$ without boundary whose Gaussian curvature must change sign.    I can see that the curvature is negative if and only if the second fundamental form is negative, but I'm not sure if they're looking for a geometrical argument here or an algebraic one - personally I'd prefer the latter if it really is a 'deduce' and not a 'show'. In addition, perhaps I'm getting confused, but I thought a compact surface had to be closed and bounded, in which case surely it necessarily has a boundary? I don't think I can do the last part without understanding what they mean by 'without boundary' - could we just for example attach a hemisphere to part of a hyperboloid (in a sortof 'icecream cone' fashion) to get a surface with both positive and negative curvature, or is there more to it than that?

As above, I'm sorry this post is so long and wordy, but I've spent weeks and weeks doing past papers and I've whittled it down to just these questions to ask for help on, so if you can suggest anything in any case which might be of use in any of the 3 questions please don't hesitate, even if you only have time for a quick comment. I would say that questions 2 and 3 are probably the most pressing, since I was able to tackle the majority of Q1 without any problems. Thankyou so so much in advance! :-) Otherlobby17 (talk) 18:49, 31 May 2010 (UTC)

I think I have some suggestions for part 2 of question 2 Suppose also $$(f_i)_{i=1}^{\infty}$$ are any functions with $$Lip(f_i) \leq 1$$. Show there is a subsequence $$f_{i_1},\,f_{i_2},...$$ which converges pointwise to a function f with $$Lip(f_i) \leq 1$$. We just need to find a subsequence that converges for all rationals then use the first part. All the fi must have values within [-1,1] since their Lip is less than 1. By the Tychonoff theorem, [-1,1]^(omega) is compact; it's also first countable so it's sequentially compact. Now enumerate the rationals in the real numbers as q1,q2,q3... And write out the q1,q2,q3... horizontally and the f1,f2,f3... vertically like a multiplication table, and then evaluate each fi at qj.

Then this is a sequence of sequences in [-1,1]^(omega) (the rows are the elements of [-1,1]^(omega)), and so has a convergent subsequence. This subsequence is the one you want. I might be wrong I’m only a student. I think user:pma has expertise in this area Money is tight (talk) 04:16, 1 June 2010 (UTC)

For 1), you probably know the structure theorem for finitely generated modules over a PID. It shows that any finitely-generated torsion-free module over a PID is free. Now, over any ring, a projective module is a direct factor of some free module. (Find a surjection from a free module onto it and then split it.) Since a submodule of a free (and hence torsion-free) module is torsion-free, we're done.

Actually, though, you don't need finitely-generated. That's because there's a theorem (using Zorn's lemma) that says that a submodule of a free module over a PID is free. 70.51.11.224 (talk) 05:16, 6 June 2010 (UTC)

For 2), take a real number r and show that f_i(r) is a Cauchy sequence. Take some q close to r. For large i, the terms f_i(q) will be within a of each other. Then by the Lipschitz condition on f_i, the terms f_i(r) will be within 2|r-q| + a of each other. You can make this smaller than any epsilon you pick, by a suitable choice of q and then a. So f_i(r) is Cauchy. I imagine the fact that the limit function has norm <= 1 can be proved by some passage to the limit, but I haven't looked at it in detail.

For the second part of 2, the point is to find a subsequence satisfying the above condition. Enumerate the rational numbers q_1, q_2, etc. Since the sequence f_i(q_1) is bounded by 1, you can pick a convergent subsequence. Within that subsequence of the f_i, pick a subsequence such that f_i(q_2) converges. And so on. Now you have a sequence of nested subsequences. Extract a single subsequence from that by a diagonal construction. This subsequence will be such that f_i(q) converges for all rational q. 70.51.11.224 (talk) 05:35, 6 June 2010 (UTC)

By the way, it looks like the last part of 2 can also be deduced from the Banach-Alaoglu theorem. 70.51.11.224 (talk) 05:40, 6 June 2010 (UTC)