I can recall some of them off the top of my head.

Well, let's forget the abelian ones.

@rubito See the 3rd message from the top in the "starred" list at right. There's a link for "LaTeX support for chat" that allows you to render LaTeX code on any webpage. (That way, if you come in here later, you can see all the math in its rendered glory.)

Yeah, sure.

Since $\varphi(8)=4$, we can extend cyclic groups of order $8$ by cyclic groups of order $2$.

That gives the usual $a\mapsto a^{-1}$ procedure.

Yes, ok.

In fact the quaternions are this and then taking a quotient.

Well, there are a bunch.

@PedroTamaroff Yes, and the generalized quaternions are just semidirect freaks.

Not going to think about it now. =P

@BalarkaSen With quotients!

@PedroTamaroff True.

Have you ever studied wreath products, @Pedro?

@BalarkaSen No, just heard of them. They are used to construct Sylows of $S_p$, for example.

God, I just realized by that, I even forgot galois theory over quintics!

I don't recall the proof of the fact that quintics are solvable iff galois group is contained in $F_{20}$

Takes some fair bit of Sylow, I guess, as far as I can recall.

@PedroTamaroff $A_5 \cong \text{PSL}_2(\Bbb F_5)$

@BalarkaSen Look at that.

How are you going to prove that one?

It suffices you show the RHS is simple. =)

@PedroTamaroff Such an algebraic perspective.

@BalarkaSen Well pardon me, Mr.

Yuck, I hate that derivation.

What derivation?

@PedroTamaroff Through the fact that both are of order 60 and simple.

@BalarkaSen Why so? It's a simple approach (PUN INTENDED). Also, Galois worked with those, he has a big result on their simplicity. Why not honour Galois? =D

@PedroTamaroff Well, actually you get more arithmetic information of these if you think geometrically.

@BalarkaSen OK...?

Enlighten me.

Would someone mind taking a look at this math.stackexchange.com/questions/782387/dedekind-sum-problem

@PedroTamaroff Consider $A_5$ as the rotational symmetry of an icosahedron. Think of the dual of icosahedron, a dodecahedron. And pair opposite faces to get 6 pairs.

@BalarkaSen Sorry, dual of a Platonic solid...?

Note that $\text{PSL}(2, 5)$ acts over $\Bbb P^1(\Bbb F_5)$, i.e., six points by preserving the collineations.

@PedroTamaroff Yes, dualization of a Platonic solid.

Sorry, I am not following all that geometry.

@PedroTamaroff That's sad =(

It's a beautiful piece of geometry.

@BalarkaSen Sad? How so...? I just never studied it.

The only geometry I like, actually.

Anyone know an example of an exact sequence 0-> F -> G -> H-> 0 of sheaves of abelian groups where G is flabby but F,H are not?

@sdf Flabby?

LOL just googled that.

yup flabby or flasque

Mathematicians are running out of words!

@PedroTamaroff That's Agro-mathematics, sheaves, stalks, germs, ...

@sdf I suspect flasque is the original French word.

@DanielFischer =D

Nono, it makes sense though, because it means that it can be flattened out on the topological space, at least in a sense.

@sdf Flabby was "every section can be extended to a global section", iirc?

@sdf Hehe, OK. I shouldn't even meddle with that über math.

Dratted internet connection.

@DanielFischer yes that's basically it

@sdf And the sequence is exact means it's exact on each stalk, doesn't it?

@DanielFischer Correct.

My lecturer like to ask for counterexamples in exams so i am trying to make a list, it is taking so long though.

Hmmm. Nothing springs to mind, sorry.

@PedroTamaroff I simple say that by considering all that ugly arguments (one is what you showed, another with the fact that $S_5$ acts over it's 6 sylow 5-subgroups) one usually gives.

In the comments here the asker wants to read more about the polynomial resultant than is mentioned in Wikipedia and Mathworld. If anyone knows e.g. a dedicated article, please comment :-)

@ccorn I know of a book that gives good description on resultants and discriminants, I just can't recall the name. let me see for a moment.

@ccorn Lang's Algebra?

@ccorn Victor V. Prasolov, Polynomials

It's a nice book, cover's one of Hilbert's problems. (The 17th I think)

It pretty much gives a survey on almost everything of elementary theory of equations.

@BalarkaSen @PedroTamaroff Those are great refs. If you would be so kind to fill them in there; otherwise I'd do that and quote you as epi-sources...

@PedroTamaroff But you should meddle with Gruber math.

@BalarkaSen +1 that book is very nice, its got lots of nice stuff about invariant theory too.

@sdf Yiss.

@BalarkaSen @PedroTamaroff Thanks to you both.

OK, I think I need to go and get some sleep right now. Bye, everyone!

If I have $A\in \mathcal L(\mathbb C^2)$ and am given $^XA$ and basis $X$, to find $^X(A^*)$ I just transpose and conjugate $^XA$, right?

Anyone have time to look at this math.stackexchange.com/questions/782387/dedekind-sum-problem. If someone could solve this it would be very helpful.

Sorry changed title math.stackexchange.com/questions/782387/…

If anyone can glimpse in here and help me I'll be very greatful, math.stackexchange.com/questions/782380/…

@robjohn an fyi: math.stackexchange.com/review/suggested-edits/205254 (the editor claims that the question is from an active exam...)

@anorton: No more takehome exams and probably graded homework is on the way out. Glad I'm retiring in a year ... Also glad after the grades I assigned in senior diff geo ...

@TedShifrin AGH! Catalan numbers. =)

Evening, @Ted.

Not my mug of tea, @Pedro. And I don't need them to teach probability ... :)

Hi. @DanielF

@TedShifrin :( That is sad. As someone who does the work themselves for graded homework and exams, it makes me upset that at those people who are ruining it for the rest of us. :|

Is there any way to find the value of this integral? $$ \int_{0}^{\frac{\pi}{2}} \dfrac{\ln(\tan \theta)}{\sec^2 \theta} \text{ d}\theta $$ I got the question from a maths thread and it's stumped me for a while!

Yeah, @anorton, me too. You want to retire, too? :)

@anorton Probably trolling.

I doubt it, @Pedro. I bet it's the prof.

@TedShifrin What are we betting?

@TedShifrin haha. I'll stick it out 3 more years at least--then I'll have a degree.

I wouldn't bet on it being the prof, given that they claim to be a TA. :P

@Shisui: Did you try an obvious substitution?

Dinner in August @Pedro

@TedShifrin I was thinking of $ u = \tan \theta $.

Oh, where was that about the TA?

@TedShifrin Not bad! I know a nice place in NJ, though... HEHEHE.

@TedShifrin The "TA" wrote "answers" to all three of the OP's questions.

Ok @Shisui ... So?

@PedroTamaroff Even though it may be trolling, our (apparently) agreed-upon policy is to lock/close quickly, and confirm the identity of the claimed TA/prof: meta.math.stackexchange.com/questions/9021/…

@TedShifrin I ended up with $$ \int_{0}^{\infty} \dfrac{ \ln(u)}{(1+u^2)^2} \text{ d}u $$

Ah @anorton ... I see. I might owe @Pedro a dinner. Guess he'll have to come collect it.

However, that was my original problem to start off with but in terms of $x$ haha!

No, @Shisui, rookie error. Where's your $du$?

@anorton: I want you to police for me next year. :) Actually, I've decided to count homework only 10% for my probability course because of this ....

@TedShifrin I don't see where my mistake is $$ u = \tan \theta \implies \text{d}u = \sec^2 \theta \text{ d}\theta $$ $$ \begin{aligned} \int_{0}^{\frac{\pi}{2}} \dfrac{\ln(\tan \theta)}{\sec^2 \theta} \text{ d}\theta & = \int_{0}^{\frac{\pi}{2}} \dfrac{\ln(\tan \theta)}{(\sec^2 \theta)^2} \sec^2 \theta \text{ d}\theta \\ & \overset{u = \tan \theta}= \int_{0}^{\infty} \dfrac{\ln(u)}{(1+u^2)^2} \text{ d}u \end{aligned} $$

P.S. @Pedro: What about our tennis debacle?

@TedShifrin Well, that can happen before dinner.

I apologize, @Shisui: You were right, and I was a tired dope.

@TedShifrin I'd love to, but I don't think I'd have enough time. :p

@Pedro One other reason I don't really think they're trolling is because there really is a Dr. Pollak who is known for combinatorics.

@TedShifrin The question is really getting to me too, haha!

@Shisui: We need complex variables to do this.

It's a standard sort of contour integral with residues.

@TedShifrin That makes a lot of sense. The thread I was looking at had a bunch of integrals that can be done via contour integration.

I need to have a look at it in the near future :)

Can people please take a look at this for me (and/or upvote) math.stackexchange.com/questions/782387/… Thanks.

You need to exploit the "multivalued" nature of $\log$.

G'night guys :)

'night

"Night" sounds good to me :)

@anorton Then someone is in for a spanking!

Off to dinner... be back later.

Hi @Ted

@pedro well that's just inappropriate

@anorton Thanks. There is not much we can do without some sort of corroboration that is indeed an exam question and not simply someone claiming so. Thanks for the heads up.

@Mike ORLY?

Hi @rob

@ಠ_ಠ hello

@anorton I see that it has already been deleted.

I got back my calc and probability exams today and I failed both

Greetings

@Chris'ssis hey there

@robjohn Hi :-)

I've just created this integral ... $$\int_0^{\infty} \frac{\log(\cosh(x))}{x} e^{-x} \ dx$$

It would be interesting to find a way to shoot it down in a very few steps.

@Chris'ssis I will look into it now.

@robjohn OK. Thanks. :-)

hmmm, I also need to find some identities with polygamma ...

@robjohn

Can you help me to count, rob? =)

Can people please take a look at this for me (and/or upvote) math.stackexchange.com/questions/782387/… Thanks.

@ruadath This is like the fourth time you post that here.

Be patient. Stop spamming...

I like putting Spam in ramen

@Pedro I did an exercise not all too different from the one I tried to help you with yesterday: Prove that if $X$ is infinite, compact, Hausdorff, then $\operatorname{Spec}(C(X))$ is non-Noetherian.

@KarlKronenfeld O_O

@PedroTamaroff This one doesn't require any cool analysis tricks though (of course).

@KarlKronenfeld Well, let me think about what a prime ideal of $C(X)$ looks like.

Also, if $X$ is compact you might as well use $X=K$ =D

Notation...

Wait, what does it mean for a topological space to be Noetherian?

OK sorry

@PedroTamaroff No infinite descending chains of closed sets.

@KarlKronenfeld Ah, OK.

So, what do prime ideals is $C(X)$ look like?

Hint, focus on the maximal ones.

@KarlKronenfeld I am quite clueless, Karl.

Or, whatever-your-name-is.

@PedroTamaroff You end up identifying $K$ with the max-spectrum.

The set of all continuous functions which are zero at $x$ forms a maximal ideal.

(And vice versa)

Then, topologically, the max-spectrum is finer than the original topology on $Y$.

$\mathfrak m_x=\{f\in C(X):f(x)=0\}$.

@KarlKronenfeld How do I show that is maximal?

Hi mr eyeglasses

Hi @Ted

I got my failed calculus and probability exams back today

Well I got stuck at Catalan numbers but not a counting involutions in $S_n$.

@PedroTamaroff You have a natural evaluation homomorphism. (Reminiscent of polynomial rings!)

What does failed mean, mr eyeglasses?

I got a 70 on calc and 80 on probability

@KarlKronenfeld Wait...

This is the first time I got below 90s before

@PedroTamaroff No, no. Fix $x\in X$, and define $ev_x\colon C(X)\to\mathbb R$.

Those are not fails. At worst, low C and low B. Chill!

@KarlKronenfeld I thought that $C(X)$ meant $C(X,X)$!!!

oh

I was going nuts, Karl.

Literally.

likes nutsy @Pedro

If I get 100 on both finals (unlikely) I can still pull 'A's in both classes. Just not A+ like I wanted

Mr eyeglasses, obsessing over grades is counterproductive. I wish the D and F students in my class cared more...

@KarlKronenfeld Look at what you did to me.

When I saw my calc grade I felt sick so I went to the toilet and I threw up.

@PedroTamaroff aw

Well, mr eyeglasses, figure out what you didn't understand, and fix it.

Cashews are yummy, but fattening @Pedro

@TedShifrin Really?

Dang it, I have been watching my weight lately.

Put down 9 pounds or so.

Watching it grow? Me too ...

@TedShifrin OH, YOU!

Volley to the deep backhand corner :)

@TedShifrin I have to get back to playing tennis.

Bah why does one require dominated convergence

Only classes are making me rusty.

@N3buchadnezzar DCT is awesome.

Love it.

Caress it.

I just wrote $\sum\limits_{n\geqslant 0}x_nx^n$. My professor will hate me. =D

@PedroTamaroff why? :p

$x_nx^n$.

who cares!'

This is kinda an off-the-wall question, but does anyone here know of good, free texts on various math topics? E.g. Generating Functionology is freely available by the author, and is a well-known text. So is Analytic Combinatorics by Sedgewick and Flajolet. Is there a list of well-known texts that are free? :-) (And I mean "free" as in "free via legal means, per US law code," in case someone was going to ask...)

(And I'm back from dinner, but that's kinda obvious.)

@anorton Abott's text on analysis? Not sure if 100% free though.

@anorton I think there is a list on site

@anorton You are you so caught up in free text btw=

@N3buchadnezzar Huh?

@PedroTamaroff I'll look for that one... I should learn analysis.

I mean proper books are just as easy to get a hold on

@N3buchadnezzar Oh! Ok--I wasn't parsing "you are you" correctly. I think I get your drift. :) But, yeah--I know I can buy "proper" books, and I'll do so if I'm serious about learning a topic, but sometimes I just want a free pdf to scroll through.

Well you can find them pdf's anyway ;)

I buy books for topics I intend to learn, but as you - I just casually stroll through them for now.

@anorton PMA is avaible

@N3buchadnezzar Eh... as someone who thinks that I might write a book someday, I wouldn't want others just picking a free copy off the net. :P What is PMA?

Well I do buy them! So I think it's fair...

@anorton Tsk tsk! =D

The closest thing I can match to that acronym is "Principles of Mathematical Analysis," but I don't think I've heard of that...

wow really

@PedroTamaroff Haha, yeah. Believe it or not, all my "official" classes have consisted of the engineering calculus sequence through diff eq, a linear algebra course, and an intro to proofs course. I've just been scraping other stuff together from various free sources...

I've never had analysis, or number theory, complex analysis, etc.

I was working on abstract algebra, then my "real school work" overtook me.

@ಠ_ಠ Something tells me I asked a dumb question...

Gonna read this one soon amazon.com/Course-Real-Analysis-John-McDonald/dp/0127428305/…

@anorton Its a classic

@N3buchadnezzar Huh! Would you look at that--I just picked three math-y words, strung them together, and it came out right! :P

Cool.

You're referring to this one, right? amazon.com/…

(Rudin?)

@anorton Yeah, although you can find it much cheaper elsewhere.

ok

Picked up my copy for 9.99 or something

I'll put that on my list to buy.

Well...

Use it as secondary source, it is one of the finest books once you are familiar with the topics. As a first introduction it can be very terse.

I like terse, so we'll see. My favorite problems/books I've spent 30 min-1hr just reading a page, and trying to wrap my brain around "why is this right?"

As a reference and as a way to look up proofs it is exellent. My only criticism is that rudin always provides a single proof, and seemingly always the shortest he can find. :p

That does seem like it could get annoying, if it's always the shortest proof.

I mean the proofs are complete, they are just somewhat short :p

ok

@Chris'ssis Seems I missed a lot of fun stuff ?!! :D

@anorton Here is an example. i.sstatic.net/v4c35.png This is a proof of the Bohr-mullerup theorem.

Took me some time to fully understand that one.

::opens image, and reads::

Not entirely sure how that implies that $f$ is uniquely determined at first glance, but I certainly understand what you mean by terse. :)