Recent content by hatsoff

Well there are a couple of things to keep in mind, here. The first is this: would the (presumably small) part of number theory which is useful for cryptography have been developed anyway, once crypographers understood how it might be useful? Sure, we would have had to wait longer for the...

But that's not true. We lose a lot. Oogles of money is funneled into promoting all this useless math, e.g. research grants and conference budgets. Research professors are paid six-digit figures and only teach a few classes per year because the bulk of their time is spent writing pointless...

It sounds like you aren't really disagreeing with my first point about a lot of math *currently* being useless. If I understand you correctly, you're saying "okay, we don't have any use for it right now now but we might find some use for it in the future." Is that right? But I think that's a...

Sure. The constructions of hereditarily indecomposable spaces. Classifying closed ideals in operator algebras. Pretty much anything in number theory (other than the cryptography stuff). Etc.

I've published a dozen or so research level math papers (most with co-authors), and every single one of them is utterly pointless. I do it because that's what research mathematicians have to do in order to get and keep a professorship. Publish or perish, as the old saying goes. I believe a lot...

Thanks for the response. My first thought is to let \varphi(m,n)(x_1,\cdots,x_{m+n})=f(m,n)(x_1,\cdots,x_m)+g(m,n)(x_{m+1},\cdots,x_{m+n}) so that each φ(m,n) is (m+n)-multilinear, and then apply the alternation mapping A to get the antisymmetric multilinear map h(m,n)=A(φ(m,n)), that is...

Homework Statement Let M and N be orientable m- and n-manifolds, respectively. Prove that their product is an orientable (m+n)-manifold. Homework Equations An m-manifold M is orientable iff it has a nowhere vanishing m-form. The Attempt at a Solution I assume I would take nowhere...

Homework Statement Consider two long coaxial metal cylindrical tubes, with radii a and b and length L. (You may assume a,b<<L. Also a<b.) Suppose the inner cylinder is given a charge +Q and the outer cylinder a charge -Q. Using Gauss' Law, compute the electric field for all r between a and...

OH, I see now! That was silly. Thanks.

Homework Statement A hollow spherical shell carries charge density \rho=k/r^2 in the region a\leq r\leq b. Use Gauss' Law in integral form to find the electric field in three regions: (i) r<a, (ii) a<r<b, (iii) r>b. Homework Equations Gauss' Law in integral form...

Thanks, I appreciate the link, and indeed that is a very nice proof of the theorem in question. However, I'm looking to finish the particular approach I was given. Basically, I have to show, using the fact that for all h\in L_q we have \lVert (\delta * f)h\rVert_1\leq\lVert\delta\rVert_1...

Homework Statement Prove the following: If \delta\in L_1(\mathbb{R}^n) and f\in L_p(\mathbb{R}^n) then the convolution \delta * f\in L_p(\mathbb{R}^n) with \lVert \delta * f\rVert_p\leq\lVert\delta\rVert_1\lVert f\rVert_p. Homework Equations We use the natural isometry (or isometric...

Latex is definitely the way to go, but it's a little strange at first. For Windows XP/Vista/7, you need three pieces of software which will work together: (1) Sumatra PDF Viewer (2) MikTex: Use the "'Basic MiKTeX 2.8' Installer" found here. (3)...

Well, that looks fine to me. But the other way to do it, if you prefer to avoid ... type notation, is to choose x\in A\cap\bigcup_{n}^{i=1}B_{i}, and say that therefore x\in A and x\in B_i for some i\in\mathbb{N}. Therefore x\in A\cap B_i\subseteq\bigcup_{n}^{i=1}\left(A \cap\right B_{i})...

(a) If L is linear, then L(u+v)=Lu+Lv. So if we have Lu=0, then Ly=L(u+v)=Lu+Lv=0+Lv=Lv=f. (b) Since L is linear, then there is a zero vector u with Lu=0 and 0+y=y. Choose u=0 and v=y. Then u+v=0+y=y.