@MartinSleziak ok, fair enough :P

I had forgotten what the question is.

lol. that was not so hard.

BTW my guess would be: If $F\subseteq G$ are an infinite fields and a infinite subfield, then for each cardinal $\varkappa$ such that $|F|\le\varkappa\le|G|$ there is a subfield with cardinality $\varkappa$.

sigh.

@AlexWertheim!

Hello @BalarkaSen. :)

How goes it?

How's life?

I'm well :)

Glad to hear it. Life is good.

What have you been thinking about?

I presume you've become a geometer/topologist under the influence of people in U of Berkeley?

Lol, I'm at UCLA, my friend. But if you mean to ask whether Mike has converted me yet, the answer is no. :)

Ah, alright. OK, good to hear.

:P

I don't anticipate I'll be doing geometry or topology any time soon. Still feeling pretty firmly in the number theory camp.

nice!

Let me know when you learn Arakelov theory.

Haven't been thinking about too much that's interesting. Been doing a bit of reading about elliptic curves and modular forms.

Lol, I'll be sure to let you know. What have you been thinking about?

that's nice. I don't know anything about either of those two.

I? I have studied Hatcher chapter 3. Studying some differential geometry and multivariable calculus, and discovering that these are not boring at all. Lots of topological ideas/food for thought hiding in calculus and diffgeo.

Lol, were you once of the opinion that they were boring? ;)

That's good. Always valuable to broaden one's horizons, I've found.

Yes, I was. I am growing up :)

@AlexWertheim That's true.

Oh, by the way, I think I have some number theoretic thing to ask you.

Have you heard/studied anything about this ergodic number theory stuff?

NIGHT, @AlexClark :P

Neat. Feel free to ask, though I'll likely be unable to answer.

Not much, I'm afraid. I believe Tao does some work in that area, no?

I am not saying night anymore to you.

Can I ask what it means for a sequence to converge pointwise?

@AlexWertheim I presume. I was in an ergodic NT seminar a few months ago, and they have apparently proved fantastic number theoretic facts about quadratic forms using real analysis on Lie groups.

@Khallil: a sequence of what? Functions, I presume?

Wasn't specific enough. Yep, I mean a sequence of functions, @AlexWertheim.

A sequence of (real-valued) functions $\{f_{n}\}$ on a domain $D$ converge pointwise if for each $x \in D$, $\{f_{n}(x)\}_{n=1}^{\infty}$ converges (as a sequence of real numbers!).

Wikipedia: Pointwise convergence. (And you'll find there exactly what Alex said.)

@AlexWertheim Have you heard of the Oppenheim's conjecture?

So if I had a sequence $\{ f_n \}$ converging pointwise on a domain $D$ to a function $f$, then I'd have that $\forall x \in D$, $\{ f_n (x) \}_{n \geq 1}$ converges to $f(x)$, @AlexWertheim?

@BalarkaSen: I hadn't before this moment. Looking it up now.

That's right, @Khallil.

It says that if $Q$ is an indefinite irrational quadratic form of at least $n \geq 5$ variables, then $Q(\Bbb Z^n)$ is dense in $\Bbb R^n$.

How about for series converging pointwise, @AlexWertheim?

Actually, Davenport asked this for $n \geq 3$, I think.

I have a problem extending that definition to a series. Do I just write them as functions of the one variable they're in terms of, @AlexWertheim?

@Khallil: how do you define the convergence of a series usually?

@BalarkaSen: neat, sounds like a very strong result.

An infinite series converges if it's sequence of partial sums converges, @AlexWertheim.

People made some progress using classical analytic number theory like circle method, etc. But the problem was finally solved by Ratner using dynamics on Lie groups. Pretty wacky, eh?

@Khallil: ok. How could you make a sequence of partial sums out of what you're given? Think about what the word pointwise meant before, and see if you can come up with something similar for pointwise convergence of a series. If you're still stuck, I'll help.

That is wacky. Those sorts of results always feel like strange magic to me. :)

@Jasper how are you doing?

@Jasper :-)))

@Jasper aaaaaaaaaa, I see! :D Is it a good movie? :-)

Specifically, they used the following result (from my notes) : $G$ be a Lie group, $\Gamma$ be a lattice of $G$. $H < G$ be a subgroup generated by unipotent one-parameter subgroups. Then image of any $Hx$-orbit in $G\setminus \Gamma$ is algebraic (there is a closed subgroup $L$ of $G$ such that $H \leq L \leq G$ and the closure of $Hx$ is $Lx$).

@Jasper lol

@AlexWertheim It's pretty amazing in the sense that I don't even understand how these two relate :P

@BalarkaSen I guess that's the creativity of math for you. :) Very nice stuff.

Ratner proved the above using ergodic techniques, and apparently Raghunathan had already proved that the above implies the Oppenheim-Davenport for $G = SL_3(\Bbb R)$.

(Or so my notes claim)

@AlexWertheim I don't understand a thing here except the statement of the problems, so presumably I won't try to understand much about this in future. But you might have a go. Who knows, maybe you will use ergodic theory later on to prove some very disconnected thing like this :)

Lol, that would certainly be nice. :) I would like to do some reading about ergodic theory at some point, powerful stuff. I've got to clear some stuff on the list before it, though =P

Anyway, time to go do some work. Nice chatting with you, @Balarka. See you all later

Right. Well, good luck to you!

Have fun with grad school :)

Hey everyone, I've got another algebra question I'd like to run by you. This has a somewhat different flavor than my last one.

Suppose you are handed the multiplication table of something that is claimed to be a group. You would like to verify that it really is a group.

Checking for the existence of an identity is easy, you just see if there's a row that says "1, 2, 3, 4, ..."

And if I understand correctly, checking for the existence of inverses is easy too. You just verify that the table is a Latin square.

But is there fast way to check for associativity?

associativity is horrendous

I don't think there is.

@DavidZhang Not as far as I know.

The naiive algorithm of simply testing every 3-tuple is clearly O(N^3).

Hmm, so it's not possible to do better than that?

Post on the main: Is there an easy way to see associativity or non-associativity from an operation's table?

It seems to me that associativity implies at least a bit of structure, and it might be possible to test for that structure in, say O(N^2 log N) time

@Martin Would that really take less than O(n^3) time?

OK, I guess something O(n log n).

If you are supposed to do this by hand, you can often guess: "Look, this is the table as this well/know group." (If the table is not too big.)

@MartinSleziak That is true, but I am interested in the time complexity of this problem.

oh, no, it's actually O(n^2 log n). The quadratic term comes from constructing the two Cayley table, and the log term for comparing them.

Hmm.

Then the answers posted there might provide you some good reasources.

Hey, my guessed time complexity was right! Thanks.

Well, it's what I believe, and it's been a long time since I worked through Crandall & Pomerance.

So you'd better check this through by someone else.

OK, I need to head to bed.

I finally managed to get through Garfinkle's 3 papers in order to understand how to determine the left KL cells in type $B$. Men, that was a slog.

anyone know about matrix norms? math.stackexchange.com/questions/1517103/…

ohhhh finally i found it !

Hey is $x(t) + \cos(x(t)) = t$ invertible

Trying to solve for $x$ but it proves quite prickly

@Slereah It is not a function

So no solution?

so invertible does not make sense

Ah well, bad terminology

Just want to find what $x(t)$ is

@DavidZhang I see that you are also asking about Axiom of Choice. You should be more careful in statement of your problem, if you want to ask it in ZF. Namely, you should clarify what you mean by infinite.

Dedekind-infinite is equivalent to "has a countably infinite subset".

But in ZF this is not equialent to Tarski's/Kuratowski's definition of infinite set: en.wikipedia.org/wiki/…

@MartinSleziak Hmm, that is an interesting point. Is "S is infinite if S is not bijective with [1,...,n] for any n" a reasonable enough definition?

By the way, for those who were involved with that previous discussion, here is the new AC-less question: math.stackexchange.com/questions/1517835/…

That would be the "usual" definition. (Sometimes called Tarski-finite, Kuratwoski/finite, IIRC.)

Okay, in that case, it is the definition I want to go with. I will edit my question accordingly.

If you replace "infinite" by "Dedekind infinite", you at least know that $K$ has a countable subset.

@MartinSleziak Done. And yes, I suppose that would make the problem somewhat easier.

any one intersested in combinatorics plz check if this answer is correct

spelling/calculation

@Slereah It's invertible, but you can't write down the inverse.

(probably)

Well bollocks

Hm, what to do

@AntonioVargas which part is invertible?

What would be a nice function that is 1) always positive 2) periodic 3) globally smaller when < 0

@Tobias, the map $x \mapsto x + \cos x$

And that isn't $1 + \frac{1}{2} \sin(x)$ or some variant

Since this is what gave me this equation

I tried the bump function but it doesn't seem to be integrable

@AntonioVargas But it is $x(t)$

@Tobias, it's just notation. The solution to the equation $x + \cos x = t$ depends on $t$, so you might as well call it $x = x(t)$.

@AntonioVargas I thought he was solving for the function $x$

Yup, the function $x$, which depends on $t$

is the solution $x$ to the equation $x + \cos x = t$

@AntonioVargas Ohh, I get it now. Man, I am slow tonight

;)

Is there a natural coordinate system for a surface with a handle?

Or does it require several coordinate charts

(That is $\mathbb{R}^2 \# T^2$)

What do the $\#$ and $T^2$ mean, @Slereah?

I'm given a series and want to show that it doesn't uniformly converge so I decided to find a closed form expression, say $f_n(x)$. However, I don't know what the function it would converge to in the definition, say $f$, is so I'm just stuck with considering $|f_n(x) - f(x)|$. Do you know how I could proceed?

@Khallil Connected sum and the torus

@Khallil It is rather difficult to give a general advice without knowing $f_n$. But maybe you could have a look at some posts tagged uniform-convergence (maybe add some reasonable search terms to the tag.) You might find them solutions of some exercises of similar nature.

This might help you get started.

Or you could use Cauchy criterion.

If $Y \subset X$, can a set be closed in $X$ but not in $Y$?

Or another way to prove uniform convergence is M-test.

@Simeon The set you are talking is a subset of $Y$?

I.e., you're asking this question for some $A\subseteq Y$, where $Y$ has the subspace topology?

By definition, $A$ is closed in the subspace topology if and only if $A\cap Y$ is closed in $X$.

If $A\subseteq Y$, then $A\cap Y=A$.

@TobiasKildetoft Are you familiar with the simplex method?

@evinda No

hi - off topic. Why can't we describe the elements of a sigma-algebra? Is there an element of some sigma-algebra which cannot be generated by the generators of the sigma-algebra by using countable set operations?

Doesn't seem off-topic to me. (I don't know the answer.)

Hey guys. I'm looking at the claim that any point on a compact Riemannian manifold has a neighborhood which is a ball of radius $\rho_0>0$ which is also a convex normal neighborhood. In some sense, $\rho_0$ is the "lower bound" for the size of a normal neighborhood.

I already know that every point $p\in M$ has a normal neighborhood with "size" $\rho >0$.

I also know that if I can cover $M$ with such neighborhoods, I can just take $\rho_0$ as the smallest one.

And because $M$ is compact, if I can find an open cover, I can extract a finite subcover and get $\rho_0$ from it.

So how do I show that the normal neighborhoods form an open cover?

My text says: "...$\exp_p$ is smooth in $p$. If thus $\exp_p$ is injective and of maximal rank on a closed ball with radius $\rho$ in $T_pM$, there exists an nbd $U$ of $p$ s.t. $\forall q\in U$, $\exp_q$ is injective and of maximal rank on the closed ball with radius $\rho$ in $T_qM$."

I don't see why that is true or how that helps to prove the theorem.

hi guys

@TheSubstitute I am not entirely sure what exactly is meant by describe the elements. But maybe this answer might be useful for you. Or maybe you want the description of $\sigma$-algebra generated by given set which is based on transfinite induction - you can probably find more about it, but it is mentioned in this answer.

Time for me to get some sleep....