Mathematics - 2015-07-06 (page 2 of 4)

for some, right. so you want some pair (x_0, x_1) such that $f : t \mapsto x_0t + x_1(1-t)$ doesn't lie in $\Bbb R^2-\{0, 0\}$, i.e., crosses the origin once. you have to find that pair.

recall that $f$ is the straightline joining $x_0$ and $x_1$. now think about this.

Remember me

12:12 PM

@BalarkaSen did you think about the kuratowski set I asked you about....

Balarka Sen

nope, I decided I didn't want to think about it :)

besides, I have a lot to get done.

Remember me

Oh... :)

Chris's sis the artist

@Hippalectryon maybe we should ask for some opinions on main?

Remember me

I think I will post that as an question....

Balarka Sen

sure, go on.

let me know when you have done this path connectedness problem.

(to get you interested : path connectedness is an important criterion of topological spaces that's used in the construction of fundamental groups)

Remember me

12:16 PM

Okay i think in few minutes or so... Also I got an idea for the answer for it...
I have to just find the equation of a curve@BalarkaSen

I knew it was coming @BalarkaSen :p

Balarka Sen

No, I didn't tell you to find the equation of the curve. I just asked what makes $f : t \mapsto x_0t + x_1(1 - t)$ fail in $\Bbb R^2 - (0, 0)$.

Chris's sis the artist

@Hippalectryon or better drop it, I feel I only receive horrible answers. :-)

Balarka Sen

If you can figure this out, finding the equation is no problem. You can take that as an exercise if you want afterwards, but I don't want you to construct an equation. Too tedious.

Hippalectryon

@Chris'ssistheartist Well you can always try after a few days if you still don't have anything satisfying enough

Remember me

Okhay...

Chris's sis the artist

12:18 PM

@Hippalectryon I have one line proof but I wanted to see the way people think.

@Hippalectryon send an e-mail to all those tough professors and ask them all to find a solutin on half a line. :-)

@Hippalectryon I'm not kidding.

Hippalectryon

@Chris'ssistheartist Ugh I don't know many tough professors, and the few I know would probably see it as a waste of their time

Chris's sis the artist

In my book what I used will be called a star ... (something)

Hippalectryon

They're into more mainstream maths

Chris's sis the artist

OK

Soham Chowdhury

@BalarkaSen ever the thoughtful teacher :P

Balarka Sen

12:31 PM

hahaha

Soham Chowdhury

@BalarkaSen and this is straight homotopy-ish. you're preparing him?

Balarka Sen

I know you guys will never do any of my problem if I don't give reference to enough motivations.

Soham Chowdhury

I sort of like Hatcher's notes. I think I'll have a good time with his text later.

Chris's sis the artist

@Hippalectryon Sometimes I talk as if I was arrogant, but I'm just very nice. :-)

Balarka Sen

And for you people, they seem to come easy enough. "motivations" = "big names I don't understand"

Soham Chowdhury

12:32 PM

or "things I want to study"?

Balarka Sen

nah.

Soham Chowdhury

but I don't want to get into this argument again.

Balarka Sen

now you want to study algebraic number theory instead of algebraic topology.

yeah, me neither.

Soham Chowdhury

no, not really.

not right now.

Hippalectryon

@Chris'ssistheartist :D

Balarka Sen

12:33 PM

@SohamChowdhury no, no. those're just basic things about paths.

Soham Chowdhury

but I will, of course. just like you'll study algebraic geometry.

Balarka Sen

homotopies are not point-set topology.

Chris's sis the artist

@Hippalectryon :D

Balarka Sen

@Soham algebraic topology and algebraic number theory don't go very well, keep that in mind. well, not unless you want to be an arithmetic geometer.

Soham Chowdhury

arithmetic topology is a thing too, I've heard.

Balarka Sen

12:35 PM

By "I will study algebraic geometry", I don't mean "algebraic geometry is so cool". I just mean I will study it, next year.

Soham Chowdhury

but enough fantasizing. worst case, I end up studying algebraic geometry. all that cohomology stuff is in there, too, right?

Balarka Sen

yes, @Soham. more like a topic, but I am not knowledgeable enough.

of different tastes.

Tobias Kildetoft

@SohamChowdhury Yes, there is also plenty of cohomology in algebraic geometry

Soham Chowdhury

yes.

Balarka Sen

like I said, you don't want to do cohomology from alg. top. and cohomology from alg. geo. both unless you want to do algebraic/arithmetic geometry.

Tobias Kildetoft

12:37 PM

@BalarkaSen It is the same basic idea in both cases

Soham Chowdhury

like I can study all those things at the same time. I'm too dumb.

Balarka Sen

surely singular cohomology and etale cohomology are vastly different?

I mean, both are a cohomology theory, sure.

Soham Chowdhury

anyway, however much you may think it's just an infatuation, Balarka, I genuinely am interested in topology right now. and I have a bunch of things I want to study after that. haven't really decided, and I don't even think I should think that far ahead.

there's just so much cool stuff to explore. it honestly gives me chills sometimes. :)

Tobias Kildetoft

@BalarkaSen Most cohomology boils down do picking a resolution and applying a functor (and then taking (co)homology of the resulting complex)

Balarka Sen

@SohamChowdhury I know you're interested in topology, because you're studying topology.

Soham Chowdhury

12:39 PM

yes, but it's not just a passing fancy

Balarka Sen

But you're not interested in algebraic topology :P Just see, I told you about p-adics, and you jumped up saying I'd like to do algebraic number theory instead.

@SohamChowdhury Yes, it's not.

Soham Chowdhury

@BalarkaSen tsk, tsk not "instead".

"as well, later".

Balarka Sen

shakes head

:P

Soham Chowdhury

I want to learn about -- don't kill me -- higher-dimensional holes.

runs for his life

Balarka Sen

Just do what you're studying right now.

Soham Chowdhury

12:41 PM

yeah.

Balarka Sen

equivalently, what you like.

Soham Chowdhury

@BalarkaSen and, for the last time, I am. you'll see. ;)

ciao, I've got work to do.

Remember me

@BalarkaSen got the point if..
t=-x_0/(x_1-x_0)

Then it goes through 0,0

Hello@TobiasKildetoft

Balarka Sen

that doesn't make sense. your t must be inside [0, 1]

@SohamChowdhury yeah, now that you're with SB, I guess you will.

Tobias Kildetoft

@Rememberme Hi

Balarka Sen

12:43 PM

@TobiasKildetoft I don't understand resolutions well, but I'll believe you.

Remember me

Okay .....

Balarka Sen

for the second time, you're mixing up quantifiers.

Tobias Kildetoft

@BalarkaSen But then, I mostly do cohomology in some sort of algebraic context

Balarka Sen

you have to find an $(x_0, x_1)$, not a $t$.

yeah, @Tobias.

Tobias Kildetoft

@BalarkaSen But then, for example the proof that De Rham and Cech cohomology coincide (whe they both apply) is pretty much just algebra (via spectral sequences)

Balarka Sen

12:49 PM

ah?

interesting.

Remember me

Hmm

Balarka Sen

@Tobias did I tell you about the recent problem I have been thinking about? (I am not sure if I should : I have been bragging about my small discoveries ever since I started working on it, haha)

@BalarkaSen No

wanna hear?

sure

12:55 PM

ok, so we know that galois theory and covering spaces are of strikingly similar nature. that is, $\pi_1(X, x_0)$, defined as homotopy classes of loops based at $x_0$, can also be thought of as deck transformation group of the universal cover $p : (\widetilde{X}, \tilde{x_0}) \to (X, x_0)$. this is very similar to the definition of $\mathsf{Gal}(\bar{k}/k)$, where $\bar k$ is the algebraic closure of $k$. there has been lots of theories unifying these, afaik.

now, if the two theories are similar, there should be an analogy for the homotopy-classes-of-loops definition of $\pi_1X$ in Galois theory too. so the punchline of the problem is : can we "see" loops and homotopies in $\mathsf{Gal}(\bar k/k)$?

I have found a few analogies, but I am still working on this as of now. It seems to be a fun problem.

Remember me

Hmm

Tobias Kildetoft

@BalarkaSen Interesting

Balarka Sen

yeah, but I don't have enough machinery to "prove" anything yet. just searching for the correct analogs.

barznjy

let n = 1,2,.....

It is obvious that the binomial coefficient of $\binom{n}{k}$ is $\frac{n!}{(n-k)!k!}$.

what is the binomial coefficient of $\binom{-n}{k}$

Semiclassical

1:12 PM

@barznjy: an alternative way of writing your definition of the binomial coeffficients is as $$\binom{n}{k}=\frac{n!}{(n-k)!k!}=\frac{n(n-1)\cdots(n-k+1)}{k(k-1)\cdots 1},$$ and that second version of it works for any number $n$ (positive integer or not)

barznjy

@Semiclassical Matlab cannot compute factorial(negative integer)

Semiclassical

the numerator i wrote is not the factorial of a negative integer

barznjy

@Semiclassical do you think it is possible to write a Matlab code for this ?

Semiclassical

certainly. you can probably do it from scratch without much trouble by creating an array of the factors and taking their product. alternatively, there's probably a matlab implementation of the pochhammer symbol (which is what the numerator amounts to)

though, it looks like matlab already can handle binomial coefficients with negative arguments: NChooseK

actually, nvm. that's for the symbolic toolkit

Soham Chowdhury

2:09 PM

and then, of course, replace factorials with $\Gamma$ to generalize to arbitrary complex $n,k$?

Rajesh D

Hi @all

small request, could you check whether my notation is readable in this answer mathoverflow.net/a/210787/14414

Soham Chowdhury

@Balarka, do these topologies have any particular names?

Tobias Kildetoft

@SohamChowdhury As mentioned in the paragraph, they give the same topology

Soham Chowdhury

oh, sorry. I meant "metrics", not "topologies".

Rajesh D

I have one general question, the mathematics courses on coursera are generally too applied and they dont really teach pure math at all. Why is that? for example, they dont teach differential geometry, topology, etc.,

Soham Chowdhury

2:21 PM

@RajeshD because most people don't care about non-applied things?

and the ones who do come and waste time in this kind of chatroom :P

Random Variable

@DanielFischer Is it still too hot to think?

Balarka Sen

@SohamChowdhury $d(x, y) = |x - y|$ is just the archimedian metric.

Soham Chowdhury

the others.

Balarka Sen

the max metric, I think, is called $\ell_\infty$ metric

Soham Chowdhury

I think I saw Alex talking about those once.

Rajesh D

2:24 PM

okay....well, but the puremath is not really non applied , its just not apparent that they can be applied, thats the interesting thing about them

Soham Chowdhury

what are you doing right now?

Rajesh D

@SohamChowdhury

Balarka Sen

@SohamChowdhury trying to get some schoolwork done, took a break since I am talking to prof about something.

Soham Chowdhury

well, but few people study pure mathematics with the sole intention of applying it.

@BalarkaSen who's your prof?

Balarka Sen

you know the guy.

Soham Chowdhury

2:25 PM

Mj?

yes.

ooh, cool.

I never said 'sole'

@RandomVariable No. But I'm busy for a few more minutes.

Soham Chowdhury

@RajeshD yes, I understand.

Semiclassical

2:26 PM

@SohamChowdhury that's true for a lot of science in general, come to think of it

Soham Chowdhury

but not many people understand that the joy of pure math is just that -- "OMG this is so beautiful".

leastways, I'd think so.

anyway, I don't think my philosophising is productive at all, so I'll zip it.

Balarka Sen

@Soham did you think about the short exact sequence problem much?

Soham Chowdhury

@BalarkaSen will, tomorrow in the bus.

Semiclassical

@soham it's not quite what you're getting at, but you might find this essay an interesting read

Balarka Sen

i recall you already made a few guesses, but you didn't give me the maps.

Rajesh D

2:28 PM

Any ideas how to proceed from here on? : mathoverflow.net/a/210787/14414

Soham Chowdhury

I've read that already, @Semi.

Semiclassical

ah

Soham Chowdhury

@BalarkaSen don't recall. $V_4$ and all?

I remember I did guess, but not what the guesses were.

Balarka Sen

oh?

i told you not to google.

Random Variable

@DanielFischer OK. I'll wait. I had a question about something I asked you last year.

Soham Chowdhury

2:29 PM

@BalarkaSen eh, I didn't.

never did.

Balarka Sen

oh, I misread as "I remember I did guess, but not my guesses"

Soham Chowdhury

oh haha

Balarka Sen

ok, then recall and write down the maps in $\Bbb Z_2 \to G \to \Bbb Z_2$

Soham Chowdhury

hmm.

Daniel Fischer

@RandomVariable Okay, I'm no longer busy for the moment.

Soham Chowdhury

2:30 PM

just remind me. what is the SES condition, @Balarka?

Balarka Sen

fact that might interest you : your guesses were right, and those are all of them.

Soham Chowdhury

@BalarkaSen you told me that, I think. $V_4$ and $C_4$?

Balarka Sen

@SohamChowdhury $A \to^f B \to^g C$ is a short exact sequence if $\ker g = \im f$, where $f$ is injective and $g$ is surjective

Soham Chowdhury

oh, did I tell you I now have awesome intuition for quotienting? (compared to before)

Balarka Sen

equivalently, $B/A \cong C$

ok, @Soham?

Soham Chowdhury

2:32 PM

right.

then definitely Klein and cyclic of order 4.

I'll find maps.

Balarka Sen

what's your intuition?

Soham Chowdhury

timidly Lagrange's theorem?

Balarka Sen

no, I mean what's your intuition for quotienting?

Soham Chowdhury

huh?

I just "feel" quotients better now compared to before.

Balarka Sen

oh. well, i think of intuition for something as a model which you can use to simplify the thing you want to understand

Soham Chowdhury

2:35 PM

(i.e. I get why you guys talk about "quotienting stuff out")

@BalarkaSen ah, that's later.

Balarka Sen

ok, cool

artin gives a lot of intuition for quotients, iirc

Soham Chowdhury

first grab the eel and pull it out of the water, then you can bang it on the head, then skin it, then . . . left as an exercise to the reader

@BalarkaSen I read Artin on the bus.

oddly, Aluffi did it better. dunno why.

Balarka Sen

@SohamChowdhury grothendieck-soham method for proving something?

2

Soham Chowdhury

when I understood the correspondence thm, everything clicked.

@BalarkaSen haha

$\hskip-0.5in \huge{\uparrow}$the rising fisherman

Balarka Sen

@SohamChowdhury yep, lattices are cool.

Random Variable

2:37 PM

@DanielFischer Does the argument here necessarily break down if $\sum a_{n}$ does not converge absolutely? It might no longer be the case that we have uniform convergence on the entire unit circle.

Balarka Sen

haha, @Soham, that one's even better

Soham Chowdhury

now with an arrow! B|

Balarka Sen

i am not starring it anymore.

:P

Soham Chowdhury

because these kids mess with the $\LaTeX$?

Balarka Sen

because the chat'd get messed up

Soham Chowdhury

2:39 PM

ah, I see.

Hatcher is very good at conveying intuition imo.

Balarka Sen

yes.

yes indeed.

ok, I am gonna leave. make sure you figure out the maps.

Soham Chowdhury

sure.

ciao.

Daniel Fischer

@RandomVariable I don't think it necessarily breaks down. But with only conditional convergence, if the result still holds, it will be hairier to prove.

Lucio D

Hi @DanielFischer How's it going

Daniel Fischer

@LucioD Fair enough. The temperature is back to life-supporting levels.

Lucio D

2:45 PM

@DanielFischer Where are you from?

Daniel Fischer

Europe.

Lucio D

@DanielFischer Oh yeah, I heard there are a number of heat waves.

Daniel Fischer

We typically have one per year. And that's more or less all the summer we get, the rest of the year is un-hot and rainy.

@RandomVariable If the limit function is continuous, I think we can modify the argument to use the Fejér kernel instead of the Dirichlet kernel. Then we have a uniformly convergent sequence of trigonometric polynomials, and everything should work. For sufficiently ugly $\sum a_n \sin (nx)$, I can imagine the result becoming false.

Lucio D

I prefer un-hotness and rain myself.

Chris's sis the artist

@r9m @Hippalectryon

Lucio D

2:59 PM

@DanielFischer I thought you were American?

Hippalectryon

?

Chris's sis the artist

@Hippalectryon $$\int_0^{\pi/2} \text{Li}_3(\phi \tan (x)) \, dx$$

Daniel Fischer

@LucioD Eight degrees and drizzle become pretty boring (and uncomfortable) after a while.

Hippalectryon

@Chris'ssistheartist $\phi$ ?

Chris's sis the artist

@Hippalectryon golden ratio

Hippalectryon

3:01 PM

Oh ok

Daniel Fischer

@LucioD Where did that idea arise from? I always spell neighbourhood, behaviour, colour, centre etc.

r9m

@DanielFischer can you help here .. the OP is asking if we can use Lindeloff Theorem here .. but I'm not sure why we'd need that here :|

Chris's sis the artist

Upvote

0

Q: Calculating $\int_0^{\pi/2} \text{Li}_3(\phi \tan (x)) \, dx$

Differentiation under the integral sign is one thought. What else would you try? $$\int_0^{\pi/2} \text{Li}_3(\phi \tan (x)) \, dx$$ where $\phi$ is golden ratio.

r9m

@Chris'ssistheartist kore wa nan desu ka?

Chris's sis the artist

@r9m lol, what is that? :-)

r9m

3:04 PM

@Chris'ssistheartist okay! (+1)

Lucio D

@DanielFischer Might have confused you with Ted.

Daniel Fischer

@r9m Hmm. I'd think proving that the function doesn't grow too fast on $\mathbb{R}$ isn't easier than proving it's bounded.

r9m

@Chris'ssistheartist what could it be? (japanese :P)

Chris's sis the artist

@r9m hehe, no chance for me to recognize it. :-)

r9m

@Chris'ssistheartist could have just googled it :-) I pick up random japanese phrases as I watch jap anime :)

@DanielFischer I see ..

Chris's sis the artist

3:06 PM

@r9m OK :-)

Random Variable

@DanielFischer I wanted to adapt your argument to show that $$\int_{0}^{\infty} \frac{\sin^{3}x}{x}\sum_{n=0}^{\infty} a_{n} e^{inx} \, dx = \sum_{n=0}^{\infty} a_{n} \int_{0}^{\infty} \frac{\sin^{3} x}{x} e^{inx} \, dx $$ where $a_{n}$ are the coefficients of the Maclaurin series for $\exp \left(\frac{z-1}{z+1} \right)$ (which I think converges conditionally at $z=1$). Things work out nicely if I assume that the switch is valid.

Soham Chowdhury

@Balarka: for $G=\Bbb Z_4$, I think $f:[x]_2\mapsto[2x]_4$ and $g:[x]_4\mapsto[x]_2$ should work.

it's great fun playing with SESes.

Chris's sis the artist

@Hippalectryon

Hippalectryon

Polygamma 3 O_o that's unexpected where does it come from ? @Chris'ssistheartist

Chris's sis the artist

@Hippalectryon Can't tell, sorry. Wait for my book. :-))))

Hippalectryon

3:12 PM

:P ok

Daniel Fischer

@RandomVariable If things work out nicely, that's usually an indication that the switch is valid. A justification may still be difficult. Gut feeling is that it's okay, but I don't trust gut feelings before I have a proof.

iluso

Hi ! Any idea ?
http://math.stackexchange.com/questions/1315109/distributing-groups-of-objects-into-boxes

Voyska

This might be stupid, but: Is there a function that hasn't limits everywhere? I mean, some non trivial function. I guess that the void function would play that role.

Chris's sis the artist

@Hippalectryon It's a long time since I managed to created bridges bewteen the polylogarithm area and polygamma area.

Soham Chowdhury

@Chris'ssistheartist here's Balarka creating bridges between Galois theory and covering spaces, and you building these. everyone's building bridges :P

Chris's sis the artist

3:21 PM

@SohamChowdhury :D

r9m

Too many Bob The Builders around here ... :P

Daniel Fischer

@r9m I just scrolled down and saw a deleted answer mentioning Lindelöf's theorem there. The OP of the question may have gotten the idea from that answer.

(And if you stop slacking and get your 10k rep, you can see deleted answers yourself.)

r9m

@DanielFischer slacking?! :P You sound like my professors :P ..

Daniel Fischer

XD

Mary Star

Hello!! Is someone of you familiar with multipoint evaluation?

In my book there is the following part about Multipoint Evaluation:


The forward transform of evaluation-interpolation is multipoint polynomial evaluation over a field $F$. We shall focus our attention on the following

Problem $P_N$ of "size" $N$: Evaluate a polynomial $a(x)=\sum_{i=0}^{N-1}a_ix^i$ of "length" $N$ (length=degree+1) at each of a set $E_N=\{a_k\}_{k=0}^{N-1}$ of $N$ distinct points $a_k \in F$ (the "evaluation points").

The solution of $P_N$ is the collection of polynomial values $A_k=a(a_k)$ ($k=0, \dots , N-1$).

@DanielFischer do you have an idea?

Daniel Fischer

3:29 PM

Never heard of multipoint evaluation.

Mary Star

Ok... No problem... @DanielFischer

Soham Chowdhury

@Balarka, the same maps work for $G=V_4$, if you relabel as follows: $(0,1,2,3)\mapsto(1,a,b,c)$.

I've checked the maps by hand. I want my prize.

:P
(also, this was a valuable exercise; thanks)

Lucio D

@DanielFischer Not sure if you remember our discussion regarding upper semi-continuity. I found another definition: $f$ is upper semi-continuous at $x_{0}$ iff for every sequence $x_{n} \rightarrow x$ it follows that $\limsup\limits_{n \to \infty}f(x_{n}) \leq f(x_{0})$. I am using the definition of upper semicontinuity from wiki.

Which is $f$ is upper semi-continuous at $x_{0}$ iff $\limsup\limits_{x \to x_{0}}f(x) \leq f(x_{0})$. Can you see how it can easily be shown that these are equivalent?

Daniel Fischer

@LucioD They aren't. Only for nice enough spaces. Then it's the same proof as that $f$ is continuous at $x_0$ if and only if for every sequence $x_n \to x_0$ you have $f(x_n) \to f(x_0)$.

Soham Chowdhury

just to clarify, @Balarka:
for $\Bbb Z/4\Bbb Z$, the maps are $f:(0,1)\mapsto(0,2)$, $g:(0,1,2,3)\mapsto(0,1,0,1)$.
for $V_4$, the maps are $f:(0,1)\mapsto(1,b)$, $g:(1,a,b,c)\mapsto(0,1,0,1)$.
(the homs map corresponding elements of each seq.)
and since there are no other iso classes of order-4 groups, done.

Lucio D

3:33 PM

@DanielFischer First countable spaces?

Daniel Fischer

@LucioD Yes. More generally, sequential spaces.

But ordinary people don't need to care about sequential spaces that aren't first countable.

Lucio D

@DanielFischer What do you care about?

Daniel Fischer

Food. Tea. Cats.

Random Variable

@DanielFischer Is it obvious whether or not the Maclaurin series for $\exp \left(\frac{z-1}{z+1} \right)$ converges on the unit circle (excluding $z=-1$)? I wasn't able to figure out the general term. I came to the conclusion that it probably does using the first 20 terms of the series.

Lucio D

@DanielFischer Tea? Are you British?

Daniel Fischer

3:38 PM

@RandomVariable No idea. But twenty terms is not much to base a hunch on when infinity is involved.

@LucioD Unfortunately, no.

skill patrol

So you would like to be?

Lucio D

@DanielFischer It is the implication that does always hold that I am interested in proving. If $\limsup\limits_{x \to x_{0}}f(x) \leq f(x_{0})$ then $\limsup\limits_{n \to \infty}f(x_{n}) \leq f(x)$ whenever $x_{n} \rightarrow x$. Can this easily be shown?

Daniel Fischer

@LucioD Depends on what you call "easily", but basically yes.

Lucio D

@DanielFischer Can you give me a hint of how to do this

Daniel Fischer

Given $\varepsilon > 0$, there is a neighbourhood $U_\varepsilon$ such that $f(x) \leqslant f(x_0) + \varepsilon$ for all $x\in U_\varepsilon$.

Since $x_n \to x_0$, there is an $n_\varepsilon$ with $x_n \in U_\varepsilon$ for $n \geqslant n_\varepsilon$.

Hence $\limsup\limits_{n\to\infty} f(x_n) \leqslant f(x_0) + \varepsilon$.

Balarka Sen

3:52 PM

@SohamChowdhury yes, that is correct.

@SohamChowdhury i don't understand those notations. what are $0, 1, 2, 3$ and $1, a, b, c$?

oh, I see, integers mod 2 and those letters are elts of $V_4$.

Random Variable

@DanielFischer I know. I was hoping there was some theorem about convergence on the boundary that I didn't know about.

Balarka Sen

I haven't checked if your second maps are correct, @Soham. the thing with $V_4$ is apparent once you identify it with $\Bbb Z/2\times \Bbb Z/2$ (indeed, these are isomorphic). $\Bbb Z/2 \to \Bbb Z/2 \times \Bbb Z/2 \to \Bbb Z/2$ is defined by letting the first map to be inclusion into the first factor, and the second map to be projection onto second coordinate.

Lucio D

@DanielFischer Oh okay, so you are not using the definition of $\limsup\limits_{x \to x_{0}}f(x)$ directly, but instead using the implication that $f$ is semiconinuious at $x_{0}$ iff $\limsup\limits_{x \to x_{0}}f(x) \leq f(x_{0})$?

Daniel Fischer

@RandomVariable Boundary behaviour can be bad. Is there any known semi-explicit formula for the coefficients?

Balarka Sen

@Soham ps : no point to use the fact that there are no groups of order 4 to conclude these are all, because I am going to use that these are the only exact sequence to conclude the there are no groups of order 4 :)

remind me later, I'll tell you about the homological algebra lemma/

Daniel Fischer

4:00 PM

@LucioD Yes. You can of course leave that out to see $\limsup\limits_{n\to \infty} f(x_n) \leqslant \limsup\limits_{x\to x_0} f(x)$ whenever $x_n \to x_0$. [Where you either allow or disallow $x = x_0$ resp. $x_n = x_0$ for both sides of the inequality.]

Random Variable

@DanielFischer I don't know of any formulas.

Daniel Fischer

@RandomVariable In that case, urgh. No idea how it would behave on the boundary.

Lucio D

@DanielFischer I was trying to prove directly from definition of $limsup\limits_{x \to x_{0}}f(x)$ but the definition is quite irritating (at least the one from wiki), so I was having some difficulty. I am talking about the definition $\limsup\limits_{x \to x_{0}}f(x) = \lim\limits_{\epsilon \to 0}(\{f(x): x \in B(x_{0},\epsilon)\setminus\{x_{0}\}\})$.

Balarka Sen

@anon you're around?

if not, here goes : $i_1 : L \hookrightarrow \bar{k}$ and $i_2 : L \hookrightarrow \bar{k}$ be two embeddings. then is it true that there is a $g$ in $\mathsf{Gal}(\bar{k}/k)$ such that $i_1 = g \circ i_2$?

I guess there is.

Daniel Fischer

@LucioD Note that by monotonicity, we can write that as $\inf\limits_{\varepsilon > 0}\dotsc$. Since the tail of every sequence $x_n \to x_0$ which doesn't attain the value $x_0$ lies in $B(x_0,\varepsilon) \setminus\{x_0\}$, we have $\limsup\limits_{n\to\infty} f(x_n) \leqslant \sup \{f(x) : x \in B(x_0,\varepsilon)\setminus \{x_0\}\}$. That for every $\varepsilon > 0$.

Balarka Sen

4:13 PM

oh, and by embeddings, I mean embeddings that fix $k$ lying below. otherwise there'll be wild things for which that doesn't hold, I guess

Random Variable

4:26 PM

@DanielFischer This all relates to a very general integral formula that I derived yesterday that I can justify in some cases but not in others. I wanted to use the formula to answer a question on M.SE. Unfortunately, that question involves one of the cases I can't completely justify. Should I post an answer with a caveat, or should I not post anything?

Daniel Fischer

@RandomVariable I'd say post with a caveat, quite possible that somebody can justify it completely.

r9m

@Chris'ssistheartist why did you delete the question?

skill patrol

Pride?

Or perhaps saving it for the book :-)

Lucio D

@DanielFischer That does make sense except I don't understand the reasoning as to why monotonicity allow you to rewrite the $\lim\limits_{\epsilon \to 0}...$ as $\lim\limits_{\epsilon > 0}...$?

r9m

@skillpatrol I saw there were 2 comments when I came back ... but it was already deleted by that time, so I didn't get to see what those comments were ..

Daniel Fischer

4:39 PM

@LucioD $s(\varepsilon) := \sup \{ f(x) : x \in B(x_0,\varepsilon) \setminus \{x_0\}\}$ is a monotonically increasing function of $\varepsilon$ (on $(0,+\infty)$). For a monotonically increasing function $m \colon (0,+\infty) \to [-\infty,+\infty]$ you have $\lim\limits_{\varepsilon \to 0} m(\varepsilon) = \inf\limits_{\varepsilon > 0} m (\varepsilon)$.

skill patrol

@r9m ok

Lucio D

@DanielFischer I see, thanks.

r9m

@DanielFischer This is my idea of gaining reputation while completely slacking off :P I guess it was cheap but couldn't help it ... :P

Chris's sis the artist

@r9m I calculated it and it wasn't as I expected to be, referring to the answer. The question might have put too much pressure on people.

r9m

@Chris'ssistheartist I see!!!! I thought some one might have posted a duplicate link or sth like that .. so I got worried ..

Lucio D

4:49 PM

@DanielFischer You following Wimbledon at all?

Chris's sis the artist

@r9m No, I just deleted it.

Daniel Fischer

@LucioD No, it's following me. I'm farther to the east. As long as Federer or Murray win it, everything's fine.

r9m

@Chris'ssistheartist I see! okay :-) If it's very difficult I don't wanna try it .. is it doable by my level of knowledge?

Remember me

Oh you following Wimbledon nice....@DanielFischer

Where has @AlexClark been .... I haven't met him since days :(

Lucio D

@DanielFischer Both of them are playing well. Hoping Federer does well, don't see him having a great chance of winning slams after this season. The younger players are becoming much more competitive.

Remember me

4:55 PM

Huge shocker .....
Nadal was defeated @LucioD

Chris's sis the artist

@r9m I couldn't answer for you, you know better. :-) Well, some problems are based upon research, and I doubt one can easily do them from a first try, but I might be wrong, of course.

Lucio D

@Rememberme Yeah it was. His game still a bit off after his injury. But I think he can still win some slams. But not sure if he has the longevity like Federer.

mixedmath

hi all

As we do from time to time, the Community Math Blog [this one] is looking for posts and authors interested in writing posts! If you have something that you think would interest the community (like a tutorial, exposition, an interesting set of problems, bits of news, et.) and you're interested in writing something about it well, then I bet the blog is looking for you

5

r9m

@Chris'ssistheartist :-) I wanted to know if the tools you used were highly specialized :)

Chris's sis the artist

@r9m I can only tell you they are very clever. :-)

Remember me

4:59 PM

@r9m you can write up your blog on mse blog...

Transcript for

$Mathematics$ Mathematics