So consider $\mathbb{S}^1\times \mathbb{R}$. Since there exists a smooth covering map $\pi: \mathbb{R}^2\rightarrow \mathbb{S}^1\times \mathbb{R}$, $\mathbb{S}^1\times \mathbb{R}$ admits a Riemanian metric making it into a complete Riemannian manifold under $\pi^{*}g$, where $g$ is euclidean metric. here, $\pi$ is also a local isometry, and so this means under this metric $\mathbb{S}^1\times \mathbb{R}$ has constant sectional curvature 0, right?
My notes say the following: There exists a smooth covering map $\pi: \mathbb{R}^2\rightarrow \mathbb{S}^1\times \mathbb{R}$. Therefore, $\mathbb{S}^1\times \mathbb{R}$ has a metric for which $\pi$ is a local isometry and under such a metric, $\mathbb{S}^1\times \mathbb{R}$ is complete
You have a covering space, so interpret my question in your context and think. You don’t need any of this covering space talk here, but your prof did it for a reason, I suppose.
How does $\Bbb R\times \Bbb R$ naturally cover $S^1\times\Bbb R$?
Can I modify the product metric on the cylinder to give me a riemannian metric which preserves completeness but now the cylinder has sectional curvature atleast 1?
ARG! We are leaving for Tucson tomorrow morning, and won't be back for more than a week, yet the small child decided that no, leftovers aren't good enough (despite the fact that they are all going to have gone off by the time I get home), so she pulled out a pint of beans from the freezer and refried them.
I don't know if I should be upset that she is wasting food, or impressed that she refried her own damn beans.
If $u(x,t) $ is a C^2 function satisfying the following conditions : $u_{tt} - c^2 u_{xx} + ku = 0$, $u(x,0) = 0$ for all $-\infty<x<\infty$, $\lim_{|x|\to\infty} u_t(x,0) = 0$ and $\lim_{|x|\to\infty}u(x,t) = 0$ for all $t>0$ then what is the sign of $E(t)- E(0)$ for $t>0$ where $E(t) = \frac{1}{2}\int_{-\infty}^\infty u^2_t(x,t)+c^2u_x^2(x,t)dx$ ?
$E(t) - E(0) = \frac{1}{2}\int_{-\infty}^\infty u_t^2(x,t) - u_t^2(x,0) + c^2 (u_x^2(x,t) - u_x^2(x,0)) dx$. I think the second condition implies $u_x(x,0) = 0$ and last one implies $\lim_{|x|\infty} u_x(x,t) = 0$. But I don't know how to use the given pde.
It's deciding the sign of $E(t) - E(0)$ so I think differentiating w.r.t. $t$ is not a great idea. It only decides if it's increasing or decreasing.
I want to write some learning software for math (Django website) in my spare time
Right now got a construction job
Kind of like "learn proof of X" content, but users can add in more proofs, expand certain points of proof and so on... The bottleneck is not getting the computer to understand math, it's helping the user learn math quicker
For example, I want to show that the standard complex satisfies $d^2 = 0$, but am iffy about expanding that by hand. The proof page will show the different steps in expansion and allow different ways of viewing the formulae
Support for commutative diagrams I have experience with on a Django site using Quiver CD editor as a frontend. But I need to write support for parsing & rearranging formulas
The user should be able to enter in regular mathjax and have the system parse it and auto-create some parts of the content
You need hands-on experience with playing with calculations and concrete examples. You also need to understand the hypotheses of the theorems you find.
I meant it when I said to try to find a metric of curvature $\ge 1$ on the plane.
Exercise: If $F$ is a continuous distribution function on $(\mathbb R, \mathscr B, \mu_{\mathcal L})$ with distribution $\mu_F$, use Fubini's theorem to show that $\int_{\mathbb R} F(x) \, d\mu_F(x) = \frac{1}{2}$, and show that if $X_1, X_2$ are i.i.d random variables with common distribution $F$, then $P(\{X_1 \leq X_2 \}) = 1/2$ and $\text E(F(X_1)) = 1/2$.
I am unsure of what to do but I will try to type some thoughts.
First of all, this seems vaguely connected with the probability integral transform, although that observation may not be permitted to help solve this problem.
Second, the integral and the expectation look very similar - I am not sure if they are the same thing, though.
Third thought: Fubini's theorem says (roughly) that an integral over a product space can be computed as an iterated sequence of integrals over the spaces that make up the product. I don't see a product space here.
Might have more thoughts coming; not sure.
A couple definitions might be worth writing down.
A distribution function is a montone increasing, right continuous function $F \colon \mathbb R \to \mathbb R$.
If $F \colon \mathbb R \to \mathbb R$ is a distribution function, there is a unique Borel measure $\mu_F$ on $\mathscr B$ such that $\mu_F((a, b]) = \hat F(b) - \hat F(a)$ for all $a, b \in \hat{\mathbb R}$, $a < b$.
($\hat F$ is the extension of $F$, to the extended reals I think.)
(I hope this doesn't come across as spamming this chat room, sorry.)
I probably need to think more about what that integral means, in terms of defintions.
Maybe the product space is the product of $(\mathbb R, \mathscr B, \mu_{\mathcal L})$ and $(\mathbb R, \mathscr B, \mu_F)$. Not sure; need to think about it.
This link seems relevant, but Graham Kemp's answer uses Riemann integration, and seems to assume that $F$ is differentiable, which seems unwarranted in my case.
Maybe I need to write an integral over a product space, use Fubini's Theorem to turn that into two individual integrals, with an integral with respect to Lebesgue measure on the inside, so I can evaluate that somehow, and then get the result Dunno. Just riffing on another exercise I completed that went sort of like that.
I feel bad about posting so many messages here. Maybe I'll go write up a question and post it on the main site.
Showing that $\sum_{n=1}^\infty\frac {1}{n^2a_n}$ diverges if $\sum a_n,a_n\ge 0$ converges.
Since $\sum a_n$ converges, there exists $N$ such that for any $m\gt N$, the following holds:
$$a_{m+1}+a_{m+2}+\cdots+a_{2m}<\frac 12$$
By AM$\ge$ HM, $$\frac{\sum_{n=m+1}^{2m}a_n}{m}\ge \frac m{\sum_{n...
@robjohn I have been thinking about that since someone posted an answer (now deleted) using this idea. And then someone posted that query in the comments to my above linked post.
Re your question: First of all, you are not allowing constants (let alone more complicated things). But make $a_n = 1/n^2$ unless $n=k^3$ for some $k$. And then what?
> In your Easter bonnet, with all the frills upon it, You'll be the grandest lady in the Easter Parade. I'll be all in clover and when they look you over, I'll be the proudest fellow in the Easter Parade.…
It's a sweet song, but it's also pretty corny. I don't mind some Irving Berlin stuff, but for music from that era I prefer Jerome Kern and Dorothy Fields.
