Mathematics - 2015-11-24 (page 3 of 6)

Clarinetist

04:00

@TedShifrin Reminder that my program's online... I might consider getting access to UW-Madison's library, but I don't recall it being cheap

Ted Shifrin

Interesting. I wonder what the definition of degree is.

Oh right @Clarinet.

Ben Lim

Ok, I think one example would be the map from M_{1,1} to the j-line. I'm told that has half a degree.

Ted Shifrin

I'm too classical. I just can't make sense of this.

Presumably it can only happen in some finite characteristic setting.

Ben Lim

No it happens over $\Bbb{C}$.

Ted Shifrin

Oh.

Ben Lim

04:03

So let's put it this way. We have a map f from the lambda line $X$ to the j-line $Y$ that factors through M_{1,1}.

The map from $X \to Y$ has degree 6.

Ted Shifrin

Ah, so it's a formal definition with multiplicativity of degree?

Ben Lim

I dunno probably yea. But it's got something to do with groupoid cardinality and automorphisms that I don't fully understand.

Anyway, I was going to say the map from $X \to M_{1,1}$ has degree 12, so in any reasonable sense the map from M_{1,1} to Y must have degree 1/2

Ted Shifrin

Right, that's what I meant about formal multiplicativity.

Ben Lim

yea.

Clarinetist

So apparently there's a connection between experimental design and Galois fields mind blown

Ben Lim

04:05

@anon You should apply here.

Ted Shifrin

Doesn't make sense to me, though.

Ben Lim

@TedShifrin Anybody at your school applying to here for grad school?

Ted Shifrin

Yes, @Clarinet. Look at Lidl/Pilz Applied Abstract Algebra. All sorts of such stuff.

I'm gone, @BenLim. I've retired. I am not sure. Our super-star student asked me to write only a few letters (for schools that wanted 4 letters), so I don't know if Stanford is on her list.

Ben Lim

oh ok.

Ted Shifrin

So these aren't functions in the usual sense, @BenLim. Preimages won't work if they are.

Ben Lim

04:09

huh what is not a function?

Ted Shifrin

the mapping between stacks

Ben Lim

ah yea. For a map from a scheme to a stack you can make sense of that by Yoneda's Lemma.

But atm I don't know how to say what a map from a stack to a scheme is.

Sajindia

What do you guys think of my profile picture?

Ted Shifrin

right, but set theory for functions fails for this degree 1/2 stuff to happen.

Ben Lim

yea.

i think the point is that M_{1,1}(T) for a scheme T is not a set but a groupoid.

Ted Shifrin

04:11

Groupoids aren't sets?

Ben Lim

in a set there's only one possible arrow from one element to another right?

(if there is even one)

but in a groupoid there is possilby more than one arrow.

Ted Shifrin

ah, so that explains fractional degree, then.

Ben Lim

yea.

Ted Shifrin

If there are two arrows going out, then you see degree 1/2.

Sajindia

:(

Ben Lim

04:13

Yea I think the 1/2 thingy is related to the fact that a "generic" elliptic curve has automorphism group of size 2.

Ted Shifrin

Right, the hyperelliptic involution.

Ben Lim

yea

the presence of automorphisms presents the functor from being representable.

Ted Shifrin

I'm glad I'm too old to learn this stuff :D

Ben Lim

Because the naive functor M_{1,1} : Sch \to Set is not an fpqc sheaf.

@TedShifrin Don't worry, as I get older I'm starting to think less and less about the formalism.

Ted Shifrin

gulp

Ben Lim

04:15

?

Ted Shifrin

the gulp was at the first sentence ... a functor not being a fpqc sheaf :P

Ben Lim

Ok here's a translation: There are non-isomorphic elliptic curves over Q that become isomorphic over Q(\sqrt{2}). Is that better? The fpqc part is just that the map Spec Q(\sqrt{2}) \to Spec Q is fpqc.

Ted Shifrin

OK, that's nominally better. Thanks :)

Ben Lim

@TedShifrin See all these complicated words just mean something concrete

Mike Miller

@Ted: Maps of degree 1/2 should exist for the same reason you might have non-integer Euler characteristic for an orbifold, I think. I don't know if people think about degrees of maps between orbifolds.

When someone says stack to me I always think orbifold and what they say sometimes makes sense.

Ted Shifrin

04:17

Oh, but this is a very different situation, I think, @MikeM. You actually are in a setting where you don't have functions in the traditional sense.

Ben Lim

@MikeMiller There's a post by qiaochu on this I think, fractional euler characteristic and related ideas.

Ted Shifrin

I remember thinking about the orbifold example with intersection homology many moons ago.

I guess a vague analogy is trying to give a "morphism" from $\Bbb Z/2$ to $\Bbb Z/4$. :P

Anyhow, I'll leave the math to you young'uns.

I'm tantalized by the wiki page that says a group action on a set corresponds to a groupoid. Since I love group actions so much. I'll have to mull that over sometime.

Wow, Alan Weinstein even wrote an article on it.

Mike Miller

@Ted: I have come to like orbifolds quite a bit after first hating them quite a bit. I dunno why.

Ted Shifrin

Well, most of them are coming from the hot-shot undergraduate programs. For the most part, undergraduate education in this country is very undemanding.

Ben Lim

@TedShifrin True the people in my year are from like MIT, UChicago and Princeton.

Ted Shifrin

04:23

I never hated them, @MikeM. Also thinking about symmetric products of things of higher dimension than curves leads to such quotients.

Right, @BenLim, so they already took grad courses at such places.

Julian Rachman

Hey. Does anyone know how to number a diagram in LaTeX like

<diagram> (1)

Ben Lim

@JulianRachman Put your diagram in the begin equation environment.

Ted Shifrin

Well, most of the US is way different, too, @BenLim. Most of UGA's best students got crucified by me their freshman year and that got them sweating, but most of the rest of their courses weren't so tough. Now that I'm gone, we'll see ...

Ben Lim

@TedShifrin hahahahahahahah

Julian Rachman

@Ben Ok and?

Ted Shifrin

04:25

The figure environment normally numbers for you, @Julian.

Ben Lim

that auomatically numbers it.

Julian Rachman

not tikz-cd

Ben Lim

begin equation followed by tikzcd

Julian Rachman

ok

Ted Shifrin

MichaelAlbanese hadn't commented to me about feeling underprepared at StonyBrook, but he actually did a masters in Australia, I think.

I'm sure there are some American students who are overwhelmed, @BenLim. Maybe not the ones taking second- and third-year grad courses like you :P

Ben Lim

04:27

@TedShifrin I'm taking the first year graduate algebra, analysis classes and math216 which is ravi's algebraic geometry. nothing crazy.

Ted Shifrin

When I was at Berkeley, approximately 60-75% of the grad students dropped out the first few years.

Ben Lim

wow.

Ted Shifrin

This stacks stuff is not first year stuff.

Ben Lim

@TedShifrin Well I spent a lot of time in undergrad understanding basic stuff in moduli theory.

Ted Shifrin

Yeah, that's not standard undergrad.

Or even first-year grad.

Ben Lim

04:29

@TedShifrin ok true.

Mike Miller

@Ted: That's special to Berkeley, whose admissions process may very well be called unethical.

Ted Shifrin

Just calm down, @BenLim. You'll do fine. Most grad students most places go through various stages of freaking out periodically.

Ben Lim

@MikeMiller Why is it unethical?

Ted Shifrin

It's changed some, @MikeM, but I tend to agree with you.

When I was there the grad program was 400 ... now it's something like 200-250?

They admitted many students who had no prayer of being successful, @BenLim.

Some of us (successful) students ranted about it, indeed.

REUs are overblown, seriously.

There's also the whole question of financial support.

Ben Lim

@MikeMiller wow ok. You can delete your comment.

Ted Shifrin

04:32

In all my years at UGA, I very rarely encouraged students to apply to Berkeley. Some went to MIT, Penn, Chicago, ...

I hate all this deletion. I refuse to delete.

Do you actually understand Grassmannians geometrically, @BenLim? :D As homogeneous spaces, as manifolds, etc.?

Mike Miller

@Ted: Some of us are hoping to have careers and don't like gossip attached to our name. You know I delete very little I say.

Ben Lim

@TedShifrin I have maybe thought about this perspective like once in my life?

@MikeMiller Word.

Ted Shifrin

Fair enough, @MikeM. Am I expected to follow suit? I have no career :P

Well, @BenLim, I would encourage you to learn that perspective. Seriously.

Mike Miller

Nope... just don't want something I said showing up on google.

Ted Shifrin

Yes, I get it.

Ben Lim

04:34

@TedShifrin Fair enough.

Ted Shifrin

And I haven't said anything here that I haven't said to various students and in public.

PVAL

@TedShifrin I'm so sick of having to give any credit to wrong answers.

Ted Shifrin

@PVAL, I don't know why you kvetch to me. I gave (generous) partial credit for correct methodology, even if the answer was flawed.

Ben Lim

@TedShifrin This is something I like about Ravi's style. He doesn't want to mention unnecessary machine if you don't need it. Rather, he'd rather his students work things out with as little machinery as possible and see how far they go. When the time comes to introduce the big weapons, then he'll happily ask you to read about it.

Ted Shifrin

Yes, @BenLim, and working out lots of concrete examples is excellent, too.

You're teaching ODEs, @PVAL? I guess I would want some context for your kvetch :)

Ben Lim

04:37

@TedShifrin yea. If you read about this isom scheme in general it's like super complicated. I was extremely happy to be able to write down equations for it:D

Ted Shifrin

Yes, yes, that's excellent that you did that, @BenLim.

Even if you can do everything super-abstractly another way.

Ben Lim

@TedShifrin I remember reading oh Isom sits inside of the hilbert scheme therefore its representable blabla. And this hilbert scheme thingy is very complicated.

PVAL

Students can't find the eigenvectors of a 3by3 matrix in order to solve a system of ODES, I don't understand what "correct methodology"means in this context. You can either do it or you can't.

Ted Shifrin

Ah, well, I admit that was a topic I had trouble giving partial credit for, especially when they messed up the characteristic polynomial. But I usually had a rubric breaking the problem down and awarding partial credit.

PVAL

I just wonder if theres a field where its useful to sort of, kind of, not really be able to find the eigenvectors of the matrix.

Ted Shifrin

04:40

In real life they use a computer to find those.

PVAL

Well I am grading 120 quizzes a week plus trying to actually teach, so it certainly is a rush job.

Ted Shifrin

I made my students do that stuff by hand on exams, but let's not kid ourselves about what they have to do in a job or in the real world.

Yeah, I hate what you're having to do.

I think you should make up clever quizzes with fewer places for arithmetic errors.

PVAL

I don't write them.

Ted Shifrin

Oh.

PVAL

The quizzes are all arithmetic

Ted Shifrin

04:42

Yeah, but I know from years of teaching that stuff that students always mess up that arithmetic.

And even though I told them 1000 times that if $A-\lambda I$ failed to have a nontrivial nullspace, their $\lambda$ couldn't be an eigenvalue, under time pressure on tests, I still saw such things.

I really do wonder why we're teaching 19th century style during the 21st century.

We should test concepts by making the arithmetic as trivial as possible on exams.

@Stan: I answered above.

PVAL

I think one time the students had to prove something (the sum of a soln to a homogeneous equation and a sol. to a non-homog is a soln to a non-homog equation) . I proved a similar thing to help them (that the sum of two homog soln's was again a homog solution). Many students complained about the 0 they got after writing down exactly what I said!!

Ted Shifrin

You'd like to think that UT students have progressed past memorization/regurgitation.

PVAL

There

Ted Shifrin

But reading is difficult. Understanding is more so.

PVAL

There's some fantastic undergrads here. Many land in the top 3 schools for PhD programs. There's about 40k others though.

Ted Shifrin

04:46

Yes. Remember that even at MIT, Princeton, UT, etc., only a very small number of math majors go on to Ph.D. work.

Ben Lim

@PVAL and where is this?

PVAL

at UT

austin

Ted Shifrin

Texas, @BenLim.

Ben Lim

ah ok.

Ted Shifrin

UT also = U Tennessee :)

PVAL

04:48

I think I told you the book was something it wasnt.

Ted Shifrin

I don't berember.

PVAL

It's actually Braun. I'm still waiting on my copy!

Ted Shifrin

Oh, I don't know that book, but I've heard of it.

Great. End of semester and you're expected to TA? Ugh.

PVAL

I cant imagine having heard good things.

Well .djvu is a hell of a format.

Stan Shunpike

@Ted $y \rightarrow 0$ and $x \rightarrow \infty$

Ted Shifrin

04:50

Right, @Stan. So you are not in general guaranteed a max. When there is one, LM will find it (plus extraneous points, possibly).

PVAL

I think it has been available since like week 5 or 6, but after 6 weeks of teaching without a hard copy whatever.

I got 1 day notice on my assignment as well!

Ted Shifrin

@Stan: This is why I make such a fuss about compactness and the max value theorem in the book.

Sounds like they don't do a great job of making their grad students feel loved, @PVAL.

Stan Shunpike

@TedShifrin yeah I think I am starting to realize why

Ted Shifrin

When I was associate head, the grad director and I tried to make TA assignments way earlier.

Stan Shunpike

Ohhhhhhhh

Ted Shifrin

04:52

Oh, @Stan, it looks like I'll be in Chicago mid-June.

Stan Shunpike

facepalm

Sajindia

@TedShifrin what do you think of my profile picture?

Ted Shifrin

Whom are you facepalming this time?

Stan Shunpike

Nice! I can't wait :D

Ted Shifrin

Who is it, @Sajindia?

Stan Shunpike

04:52

Me! Lol the max value theorem. Now I get what it's for.

Ted Shifrin

Duh @Stan :D

Sajindia

@TedShifrin me

Ted Shifrin

Elegant outfit, @Sajindia.

Sajindia

thanks :D

Ted Shifrin

Not to mention hair dye.

Stan Shunpike

04:54

@Ted Too much time around economists. they have the worst respect for mathematical rigor.

For instance

Ted Shifrin

Well, don't blame me, @Stan. I was trying to train you properly before they polluted you :D

Sajindia

yes :D I like to dye my hair

I make youtube tutorials

Stan Shunpike

I learned about this thing called the Coase Theorem. It isn't even a mathematical claim! There's no proof. Nothing! Yet they call it a theorem! What nonsense

Sajindia

of make up

Ted Shifrin

Cool, @Sajindia.

I was pronouncing that Co-ass, @Stan :D

Stan Shunpike

04:56

LOL

How is life in Cali?

Ted Shifrin

I miss my GA chiropractor. My back/neck/shoulders are in pain most of the time, despite visits to the chiropractor and masseuse. Frustrated not to be tutoring the way I wanted, but I am leaving so I won't rant about that.

Talk to you soon, @Stan.

Stan Shunpike

K ttyl! @Ted

Mike Miller

@BenLim: This reminds me of a talk Okounkov gave. The only person who could understand his talk would have had a good knowledge of both Hilbert schemes and the Seiberg-Witten equations. There are at most three people in my department that's true of, and I don't know if it is for all of them.

Sajindia

@TedShifrin here's my youtube channel if you want to check it out[youtube.com/watch?v=7NRm7lW24uk]

morphic

What is this makeup

Too bad no subtitles

Sajindia

05:06

sorry :( @morphic

morphic

To be fair I took Spanish class for like 5 years so I should know a little but I don't even know the alphabet lol

Sajindia

the alphabet is the same but with ñ

PVAL

@MikeMiller Shende gave a talk on "Constructable Sheaves and Contact Structures". I can't imagine anyone here understood much of it.

Mike Miller

Sivek gave what is presumably the same talk here but engaged with the audience and spent most of the time explaining what a constructiible sheaf was. I forgot most of it but I remember thinking I understood it then.

Of course it was a topology group seminar as opposed to a colloquium...

morphic

What is the difference between a seminar and a colloquium

Mike Miller

05:13

A colloquium is intended for a general audience of math faculty and grad students. It should (but in practice needn't be) accessible to any attending grad student.

morphic

Oh

Mike Miller

A seminar is a talk or a series of talks given to/by experts in the field. They are usually much more technical.

PVAL

It was a geometry seminar. There wasn't a topologist in the audience. He drew something like and x with a circle in it and said something that was very strange, and at the end said this was a "Legendrian isotopy". It's strange when someone does things are incomprehensible to me and at the end says its something that I feel I understand quite not terribly.

I don't think it was a bad talk.

I'm not trying to insult him as a a speaker.

Mike Miller

Fair enough.

PVAL

If Bob was there, he might have understood it.

I'd have liked to ask him if he understood it as well.

Josh

06:09

Hi!

@TedShifrin @MikeMiller @BalarkaSen You three really make the room go, huh

Silent

06:24

@anon, how to show that a linear combination $cv+dw$, with $c,d \ne 0$, of two eigenvectors corresponding to different eigenvalues is never an eigenvector?

Kened Yassine

06:41

Hi,

-1

Q: Need to understand this calculus

Could you please explain me this Calculus: \begin{align*} Q(X>x) &= \int_{x}^{\infty } f(y)dy \\ &= \frac{1}{2\pi} \int_{-\infty}^{+\infty} \varphi_X(u)\int_{y>x}^{+\infty}e^{-iuy}dy du \\&= \frac{1}{2\pi} \int_{-\infty}^{+\infty} \varphi_X(u)\frac{e^{-iux}}{iu}du \\ &= \frac{1}{2}+ \frac{1}{\...

Enjoys Math

08:12

? math.stackexchange.com/questions/1544062/…

Can a non-finitely generated semigroup be a subsemigroup of a finitely-generated one?

刘哲诚

09:51

is there any difference between the implicit/parametric equation of an upright cone vs one where the base is @ z=0?

user174558

10:13

@刘哲诚 Liu Zhe Cheng, what is an upright cone?

刘哲诚

where the apex is at the origin

like an upside down cone

google.com/…

user174558

A cone is a cone, and its equations change if you want to put it in a different position in space. That is all.

刘哲诚

can you tell me what the parametric equation for that would be

user174558

You should work it out yourself.

刘哲诚

mathworld.wolfram.com/Cone.html

what would h/u be in this case?

user174558

10:26

@Chris'ssistheartist It's amazing how some people keep saying bad things about you that are not justified and fail to see their own faults and then react when others point it out. I hope you will return to this chat!

Kasper

10:36

0

Q: Searching for prezi like software that supports LaTeX natively

I've been making some mathematical presentation in prezi, for example: https://prezi.com/aexypzd25tmd/spectral-theorem-from-proofs-of-the-book/ This is a prezi that I presented today, but it took me a hell lot of work, to get LaTeX working in prezi. I was wondering if there are any prezi alterna...

presentation

user174558

@Kasper It is taking forever to load here. Are you aware of the beamer package in TeX? It is one of the standard packages used for creating projector presentations and handouts.

Kasper

@JasperLoy oh that is bad... stupid slow prezi :P

yeah, I'm aware, but I kind of like some features of prezi. the zooming out, and in features make it nice to kind of give a big picture overview of the whole proof

user174558

@Kasper Hard to have the best of all worlds in one package. If I were you, I would just use TeX and nothing else.

Kasper

I would advise the same to people I think (after finding out how much time getting TeX working in prezi takes)

user174558

@Kasper Yes, better to eat a pretzel instead.

Kasper

10:48

but I gave this prezi today, and the public said that they really like the structured way you can lay out a proof in prezi

so, I'm now thinking about maybe making some prezi like software myself for this, that support mathjax, (I am a programmer)

but I was wondering, maybe someone else already tried the same thing (which would save me a lot of time :P )

user174558

@Kasper Good luck! I like to give presentations using chalk and board only.

Kasper

thanks :)

my goal is to get a job where I can work full time on software for mathematics (not for kids, but university level mathematicians)

user174558

11:03

@Kasper I hope you achieve your goal. But I would think I only need TeX, though I am not a mathematician.

Kasper

there is a company in Amsterdam that is doing this, so I hope they hire me

user174558

I am looking for free software for geometrical figures.

Kasper

@Kasper I'm thinking about things like a TeX editor that integrates with mathematica. So that you can select your formula in your editor, and it may calculates it

user174558

TeX is too complicated for geometry.

Kasper

yeah, geometry is also something interesting

user174558

11:07

It takes a lot of code to do a simple figure, especially 3D.

user174558

The most powerful 3D package in TeX is pst-solides3d.

user174558

With it you can create all kinds of 3D graphics.

user174558

Take a look at pstricks.blogspot.com!

Kasper

so basically you would want a WYSIWYG geometry kind of editor built in a TeX editor?

user174558

Well, it's OK if it is not part of TeX. You can import the graphics as external graphics in jpg.

user174558

11:10

But as a software developer, you need to cater to your market, so you need to find out who your market is first!

Kasper

that is true, but this company I want to work for, has already market, many dutch university pays them so that they can make interactive university calculus courses

I hope they are also intersted in things like these :)

I have been experimenting with some kind of mathjax editor like this:
http://kasperpeulen.github.io/PressAndHold/

user174558

Calculus books in America are way too expensive! Stewart costs over 200 USD on Amazon.

user174558

He is dead. Long live Stewart.

Kasper

yeah, that is insane

Mathematicians should unite and make an open source free calculus course of the same quality.

user174558

Apostol Calculus I costs over 200 USD and Apostol Calculus II costs over 200 USD.

user174558

11:14

That's over 400 USD for calculus. Total nuts.

user174558

@Kasper There are a number of good texts available already, but they might not be dumbed down enough for Americans.

Kasper

hahaha ;)

user174558

So you need a cheap calculus text that is seriously dumbed down.

user174558

And then it will sell like hotcakes, or pancakes if you like.

user174558

Each new edition of these calculus books has more colourful pictures than the previous one.

Kasper

11:19

I've been thinking about, if you would make a interactive ebook. And let the user dumb it down themself how they want.

I picture this like, you began with a short but sufficient explanation, that would satisfy most people that use it more as reference

and then you can click on the explanation, to let it expand with more intuition, more visuals, and click again, until it as dumb as you can think of :P

user174558

So you are paying for the coloured ink.

Chris's sis the artist

@JasperLoy JASPER!!!

user174558

@Chris'ssistheartist Hi Chris.

Chris's sis the artist

@JasperLoy Hi. Well, this is life sometimes. :-)

user174558

@Chris'ssistheartist I think lots of things get misread in a chat room, considering there is no body language or tone.

Chris's sis the artist

11:21

@JasperLoy Possibly.

user174558

But smart people should be able to interpret a line in over 9000 ways before anything.

user174558

@Kasper As an example, Canuto and Tabacco's Mathematical Analysis I and II is way more affordable and already sufficiently dumbed down. I think Americans should switch to something like that.

Chris's sis the artist

11:35

@JasperLoy All stuff told by Ted is a piece of cake compared with the unstar message yesterday that means taking (abusively) the freedom of expression (on chat).

Tobias Kildetoft

@Chris'ssistheartist There is no freedom of expression on chat

Chris's sis the artist

Just for record: I ask that moderator to resign (in a hurry if possible).

@TobiasKildetoft Yes, it is within the rules.

Anyway, I have research to do.

BBL

Tobias Kildetoft

@Chris'ssistheartist what rules?

Chris's sis the artist

@TobiasKildetoft See chat guidelines

Tobias Kildetoft

@Chris'ssistheartist Nothing there about freedom of expression

Chris's sis the artist

11:40

@TobiasKildetoft Well, it's simple. As long as you don't break the rules you should have the same privilege like other users to express yourself on chat.

Tobias Kildetoft

@Chris'ssistheartist That is not at all the same as having freedom of expression

Chris's sis the artist

@TobiasKildetoft Well, I repeat, as long as you don't break the rules you should benefit of the freedom of expression. Or wait a second. You failed contradicting me with math and try in another area?

@TobiasKildetoft Seriously, speaking do you teach students? If I were a student I wouldn't allow you to talk to me.

Tobias Kildetoft

@Chris'ssistheartist I am not sure I ever tried contradicting you with math (how would you even do that?)

Yes, I teach students

Chris's sis the artist

@TobiasKildetoft Well, yes you tried to say that my math stuff is random stuff but you have no idea of what you say.

Tobias Kildetoft

@Chris'ssistheartist Well, fortunately, my students don't think they are the next Ramanujan or make other strange claims. So I don't need to point out when they do.

Chris's sis the artist

11:45

@TobiasKildetoft You, and also Ted, thank to God you don't have in your classes students like me.

Tobias Kildetoft

@Chris'ssistheartist Yep, very much so

Chris's sis the artist

Back, by freedom of expression in chat I meant that you can say all you want to say as long as you don't break the rules, expressing ideas, opinions, and so on, you express yourself freely.

Chantry Cargill

As entertaining as this is to read, I don't see it going anywhere positive.

Tobias Kildetoft

@ChantryCargill You're right, sorry

Chris's sis the artist

I don't see how using moderators powers to restrict my freedom of expression is anything good, other than making fun (of a user).

I thought that mod powers should be used wisely, but I might know nothing about it, and find now the things are totally different from what I thought.

OK, it's enough for me.

Paradox 101

11:51

Can anyone explain how do I find the limit of the integral of a sequence of function?

Just the bare essentials through which i can fnd a solution?

Chris's sis the artist

@TobiasKildetoft So, if a student comes to you and tells about his/her dreams in mathematics, you might possibly tell him/her "you make strange claims" as you told me above?

Tobias Kildetoft

@Chris'ssistheartist Not the first time, no. But if they keep on the way you do, at some point I probably will, because otherwise that person will be completely impossible to teach anything.

Chris's sis the artist

@TobiasKildetoft never play with people's dream. The fact that some dreams seem to you strange means nothing, and you shouldn't discourage anyone especially for the fact that you said you teach students.

@TobiasKildetoft A gifted student with a weak personality might believe what you say and live in math mediocrity.

Think well of what you tell them.

BBL (I have to finish some proposals)

Chantry Cargill

@Chris'ssistheartist See the answer here (might be something you're looking for): meta.math.stackexchange.com/a/22049/109879

Chris's sis the artist

@ChantryCargill thanks

One more thing and I'm done (a quote I like very much)

"Don't let someone else's opinion of you become your reality." - Les Brown

BBL

Private Pansy

12:04

So. My friends were working through some past year's exam papers for the upcoming exam, and came across a problem which seems unintelligible.

Not asking for a solution or anything, just that we can't even parse what the question is trying to ask

Huy

I assume you're asked to write down a "list" of all equivalence classes?

Tobias Kildetoft

@PrivatePansy And there is nothing more? That does seem very strange

Chantry Cargill

Maybe it means the set of all ordered pairs subtract each equivalence class defined by the relation?

Private Pansy

Hmmm. I found the question. I think my friend might have missed some context. Let me read the first 6 parts

Okay. I think I got it. Well, the first part at least. Still not sure how the classes can be used to represent irrational numbers

The question redefined the congruence sign to be this relation

Tobias Kildetoft

12:33

@PrivatePansy Ahh, then it makes much more sense

Paradox 101

13:05

If we have limits of an integral of a sequence of functions, should we first expand sequence or directly find the integral?

Huy

14:25

@Paradox101: What do you mean by "expand sequence"? If you mean "change the order of limit and integral", then NO, don't do that. In general, that is not allowed.

shcolf

14:38

hello

Anyone is familiar with Schreier trees?

L33ter

14:51

Hey @anon here?

Let $\alpha$ be a path in X from $x_0$ to $x_1$. we define $\hat{\alpha} : \pi_1(X,x_0) \rightarrow \pi_1(X,x_1)$ by the equation $\hat{\alpha}([f]) = [\bar{\alpha}] . [f] . [\alpha]$

what is here the alpha bar

they didn't define it

@Huy you know some algebraic top right ?

maybe can you tell me what is alpha bar ?

I guess it should be a path from x1 to x0

maybe

I see

Huy

ah, the bar

that's probably inverse

iirc you apply conjugation

does it make sense in the context?

L33ter

I am just thinking of why it will have same start and end point at $x_1$ just a sec

yeah

I understand now

@Huy it is indeed the inverse, also the map defined above is a loop based at $x_1$

Huy

yeah, it has to be for the map to be well-defined

L33ter

yeah

hm that is weird, but why is it like this what if there is no path from $x_0$ to $x_1$

what will happen then ?

Huy

15:13

@L33ter: you'll have to consider the fundamental group of each path-connected component

L33ter

I see

Chris's sis the artist

Without touching pen and paper ...

$$\lim_{n\to\infty} \int_{\pi-\epsilon}^{\pi+\epsilon} \frac{\sin(nx)}{\sin(x)} \ dx$$

A bit of care to the generalization $$\lim_{n\to\infty} \int_{\pi-\epsilon}^{\pi+\epsilon} \frac{\sin(nx)}{\sin(m x)} \ dx$$

Cortizol

How to determine if series $$\sum_{n=1}^{\infty} \frac{(-1)^{n(n+1)/2}}{\sqrt{n}}$$ converges or diverges. I mean, I see that $n(n+1)/2 = 1+2+\ldots+n$, but what then...

Agawa001

15:35

cortizol the numerator is a series of -1,1,1,-1,-1,1,1...

L33ter

can you have path connected space which doesn't have fundmental group being trivial ?

Chris's sis the artist

@Cortizol Once you notice the pattern of the signs you group the terms pairwise, and then

$$a_1-a_2+a_3-a_4+a_5-a_6+a_7-a_8+\cdots$$

Now make the series higher

Paradox 101

@Huy I meant if it's a sequence in terms of sin or exponential multiplied by something like $nx$ and the integral can't be solved simply

Chris's sis the artist

Replace $a_3$ by $a_2$, then $a_5$ by $a_4$ and so on

@Cortizol Q.E.D.

Mike Miller

@L33ter: Yes, the circle. Or any number of spaces. The fundamental group is only defined if you specify a path-component in the first place so it would be a worthless invariant if it was zero for every component.

L33ter

15:47

yeah I see

robjohn

@Cortizol That is equal to $$\sum_{k=0}^\infty\left(-\frac1{\sqrt{4k+1}} -\frac1{\sqrt{4k+2}} +\frac1{\sqrt{4k+3}} +\frac1{\sqrt{4k+4}}\right)$$ So consider $$\sum_{k=0}^\infty\left(-\frac1{\sqrt{4k+1}} +\frac1{\sqrt{4k+3}}\right)$$ and $$\sum_{k=0}^\infty\left(-\frac1{\sqrt{4k+2}} +\frac1{\sqrt{4k+4}}\right)$$

Chris's sis the artist

@Cortizol To bound it from below you do the same.

@robjohn First I thought of it, but the general case works nicer.

robjohn

@Chris'ssistheartist general case?

Chris's sis the artist

@robjohn when the conditions of the alternating test are fulfilled. Due to the signs you only group the terms pairwise, and the rest flows naturally.

Agawa001

@Chris'ssistheartist does it go to -inf ?

Chris's sis the artist

15:58

@Agawa001 It converges.

Agawa001

to a negative value

robjohn

@Chris'ssistheartist I don't think that works any better, but differently. One uses the alternating series test, the other the convergence of $\sum\limits_{k=1}^\infty\frac1{k^{3/2}}$

the only difference is how you group the terms.

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