what is a good website for plotting this graph?

I think google has osmething which can do this for you

If you use spreadsheets I think...

Alternatively, you could use Sage to graph it for you.

ok thanks i'll have a look

Man it feels good to be back after a while

Anyways...

Bye until June 10th when my exams are over! :)

wow, google actually does do it. tinyurl.com/27nk2rps

fixed the URL

@robjohn How would you argue that $ \lim_{s \to 0^{+}}\sum_{n=1}^{\infty} \frac{e^{in}}{n^{s}}$ goes to $\operatorname{Li}_{0}(e^{i})$? Would you need to show that $\operatorname{Li}_{v}(e^{i})$ is continuous at $v=0$?

@RandomVariable how can one confirm continuity for a function not defined at the point in question?

@robjohn The function $\operatorname{Li}_{s} (e^{i})$ is defined at $s=0$, but not by that series.

(I didn't mean to use $v$.)

@RandomVariable You pretty much need to do what was done in the answers to the question.

@robjohn Which makes the answer I posted pointless.

@RandomVariable well, if the work has been done for the function already, then it wouldn't be pointless. $\mathrm{Li}_0(x)=\frac x{1-x}$ is gotten by analytic continuation for $|x|\gt1$.

showing the limit is the point of Abel's Theorem.

but that is in $x$... I am not sure how else to show it is continuous in $s$

The idea behind nowhere dense sets: We take a closed set S in a complete metric space X and if S has empty interior then S is no-where dense (intuitively, every open ball in X contains a smaller ball which has no element of S). So how to define it for any set T (not necessarily closed)? We consider cl (T) (the smallest closed set containing T) and define T to be nowhere dense if int cl(T) is empty. Intuition works again. Straight lines in $\mathbb R^2$ are no-where dense for example.

It follows that $\mathbb R^2$ can't be a countable union of straight lines.

@leslietownes: The video I shared yesterday is so amazing :-).

@RandomVariable: Showing that $\operatorname{Li}_s\left(e^i\right)$ is continuous at $s=0$ is the key, as you asked at the beginning. If you can show that (which the other answers do, essentially) or quote that result from somewhere, your answer is fine. The Wikipedia article doesn't seem to say that (though I am sure it must be true).

If $\epsilon\to 0+$ then $\int_{-\infty}^\infty e^{-x^2}(e^{-\sqrt{4k\epsilon}x-\epsilon} -1) dx\to 0$?

@Koro this definition is for any topological space

@Jakobian I was wondering why no-where dense S is defined in X as int (cl S) is empty. It can be proven using the definition that every non empty open set in X contains an element from S but I wanted to know the intuition behind it.

The video explained that very nicely :).

For any real valued function defined on R, there exists an M>0 and a non empty open set U in R such that for any u in U, there exists a sequence $x_n$ in R such that $x_n\to u$ and that $|f(x_n)|\le M$.

Can anyone please help me with how to prove this? Thanks.

Does anyone know how letters of recommendation work on mathprograms.org?

Hello @robjohn :).

How to write latex of negation of $\to $?

that is, negation of $x_n\to x$.

"$x_n$ does not converge to $x$".

$\nrightarrow$

$\ddot\smile$

@Koro There is no single symbol that says that, though you might use $\not\to$ and add a definition.

when defining a symbol i would be very explicit about whether you are assuming that x_n is convergent (even if it is clear that you aren't). the epsilon-N statement doesn't naturally require splitting into these cases, but a human being reading it might need that nudge.

i remember a lot of analysis students negating "x_n converges to x" as "x_n converges to something that isn't x," whether explicitly or in their thinking.

Indeed. See this answer, for example.

@Koro PLEASE do not write mathematics in symbolic sentences. I know you love it, but mathematicians do not. And make the title something intelligible, not a pile of mess.

Well, at least Ted doesn't ;-)

for one thing, long strings of symbolic characters in between a single set of $ tend not to be spaced appropriately. there is probably some logician package for this. i don't recommend finding it or using it.

@robjohn We were taught this by a very stern Jim Munkres as we began point set back in 1972.

I do agree that it is bad for a title. I also think that the logical statement can be useful, when used properly.

my instructors were similar. most of them, anyway.

but when English and logical are mixed, it becomes ugly

what really set my advisor off was using acronyms.

DUA (Don't use acronyms)

i never used them, but he would stop in the middle of discussing something to rant about someone else using them in a paper.

If you want to use formal logic in the privacy of your bedroom, do so. But it has virtually no place in public published mathematics.

@TedShifrin I did try to avoid symbols. Please understand that without symbols, the title word limit was exceeding so I put symbols in the title and full language in the post.

dump the title.

Reread the purpose of a title here.

mixing english and logical is a bad idea. my least favorite might be using a symbol for english words that even logicians don't use. e.g. that weird epsilon-like thing for 'such that.'

@leslietownes what is DOD? Any guess?

koro: what context? i haven't seen it in the math that i studied.

DOA

Department of design

:D

I've see it usually for Department of Defense

in the usa it would more likely be taken to mean department of defense.

we have a big one and spend a lot of money on it. we're very proud of the money we pour into it.

@robjohn I tried shortening the title to bring it within the allowed word limits but then using symbols seemed like the last thing to do.

@Koro You don't need to state the whole problem in the title.

I know that sometimes coming up with a good title requires some creativity, but that is a good skill to practice.

I'm confused. Sometimes I see comments like 'your title should state the problem'.

sometimes, it's some other thing.

To avoid trouble, I tend to include most of the problem statement in the title.

a title should always be more specific than something like "problem in real analysis" but does not need to be the whole thing. i think most people are unlikely to parse the symbolic stuff.

at least, i am :D

In particular, the actual question needs to be in the question, NOT in the title.

The title should convey the subject matter in broad terms, but not so broad as to say nothing.

I received an answer now to my question. :).

If $(\rho, V)$ is a finite dimensional complex linear representation of $sl_{3,\mathbb{C}}$ and $W$ is a subrepresentation containing all of the heighest weights of $V$ then must $W$ equal $V$?

@robjohn For fixed $z \ne 1$, $\operatorname{Li}_{s}(z)$ is apparently an entire function in $s$. That's at least what it seems to say here.

@RandomVariable I have no doubt it is. I am sure that $\operatorname{Li}_0\left(e^i\right)$ is a nice point. However, the pole at $\operatorname{Li}_0(1)$ is not mentioned at all on that page. Wolfram is not always good about that sort of thing.

It would be nice to see how they would justify it.

@leslietownes Here's a cape made from golden orb weaver silk:

looks expensive!

@leslietownes Indeed. I don't think it's for sale. It took several years to make.

Spider body fur can be extremely silky, far softer than any animal fur I've touched, including kangaroo and baby alpaca.

Some spiders make a special milk web for their babies. But some make a liquid milk, which is deposited around the nursery web, or in some cases the babies suckle it from the mother.

Many spiders communicate by touch, especially during courtship, but also when negotiating territorial boundaries. A common action involves gently stroking a foreleg of the other spider.

Munchkin’s next outfit!

If you're careful, you can "shake hands" with a spider by stroking their leg with a thin stick, eg a bamboo skewer. But I've only tried it with Daddy Long-legs spiders.

one time at our old house i came home to find a brown widow had made a pretty cool web on our doorstep. it's a shame munchkin wasn't older, she would have loved it.

another time one of those big orb weavers did a masterpiece right in front of our front door. the web did not include inspirational messages.

Golden orb weaver webs can look beautiful when the sun hits them at the right angle. But they can sure be sticky when you accidentally walk into them. :)

my brother had a brown recluse in his backyard when he lived in Sydney