Mathematics

Martin Sleziak

5:00 PM

ok

J. M.

(Hi @Matt!)

Chandrasekhar

@Mariano: Are you over here. Have a very small doubt

Mariano Suárez-Alvarez

Sure

t.b.

@MartinSleziak I think you have this in mind, as well as this meta thread

Chandrasekhar

@Mariano: Thanks. How did Mathematicians come up with this exact sequeces, short exact sequences and things like that...

Matt N.

5:02 PM

@JM Hey : ) Someone ate my donut!

Chandrasekhar

@Mariano: The definition of exact sequences and things like that..

Mariano Suárez-Alvarez

it is a very natural thing :)

if you have a group G, say

and a subgroup H

Matt N.

@Ilya No, he wouldn't.

Chandrasekhar

@Mariano ok

Mariano Suárez-Alvarez

the short exact sequence $H\to G\to G/H$

t.b.

5:02 PM

@Ilya hehe, told ya :)

Hi Matt!

Mariano Suárez-Alvarez

encodes the fact that to know something about $G$ you may work "up to details in H"

Matt N.

@tb Didn't know you swung that way...

Martin Sleziak

@tb Good find. I like that attitude: *Users are adults (well, okay, at least 13 years old). They should be able to better handle themselves without constant "adult intervention". *

t.b.

@MattN hm... Schoen's gyroid, lousy pun.

Mariano Suárez-Alvarez

this can be traced back to working modulo n in elementary number theory, for example

Matt N.

5:03 PM

@tb Hello Teddybear.

Mariano Suárez-Alvarez

or "almost everywhere" in measure theory

Chandrasekhar

@Mariano: What is that number theory example

Mariano Suárez-Alvarez

long exact sequences, on the other hand, arise because in nature we find lots of things that we would want something to be a short exact sequence but it isn't

but we find that we can continue it to get a long exact sequence instead

Chandrasekhar

@Mariano: Oh , i haven't heard that terminology Long Exact Sequence

Mariano Suárez-Alvarez

if you want to show that a number is not a square, it is enough to show that it is not a square modulo some other number

a long exact sequence is just like a short one but long :)

Matt N.

5:05 PM

Brian must be asleep.

Chandrasekhar

@MAriano: Ha, ha :)

Mariano Suárez-Alvarez

if youhave a group G, its quotients G/H are, in some sense, approximations to G

skullpatrol

like the digits of $\pi$? @MarianoSuárezAlvarez

Dylan Moreland

Conversely, you can always break up an LES into a bunch of SES's.

Mariano Suárez-Alvarez

likewise, but in a different sense its subgroups $H\subseteq G$ are also approximationsto G

exact sequences (and more complicated gadgets, like spectral sequences or even derived categories) encode that type of information

Chandrasekhar

5:07 PM

@Mariano: I didn't understand this: if youhave a group G, its quotients G/H are, in some sense, approximations to G

approximations to G

unNaturhal

However, I haven't understood if I have to search vertical asymptotes or not xD

Chandrasekhar

means

are u talking about their structure

Mariano Suárez-Alvarez

@Chandrasekhar, sometimes you want to know something about G and it is hard

but you can answer your question for some of its quotients

Chandrasekhar

@Mariano: Ok

yeah

understood

Mariano Suárez-Alvarez

if you are lucky, there is a way to relate the answer oft he question regarding G to the answer of the question regarding G/H

J. M.

5:08 PM

@MattN Sorry, it was time to move on with my avatar-change schedule... :)

Mariano Suárez-Alvarez

in that sense, you approximate G by G/H

Matt N.

@JM Np : ) This one looks quite fancy.

Dylan Moreland

A lot of the motivation must have come from algebraic topology, right? That might be more convincing, depending on your taste.

Mariano Suárez-Alvarez

sure

Chandrasekhar

@MAriano. Hmm... Thanks for the help. It doesn't appear natural to me because, I need much more maturity In the subject I guess.

J. M.

5:10 PM

@MattN Yup. Here is a bigger picture.

Mariano Suárez-Alvarez

I would suggest you stop worrying about it :)

Chandrasekhar

@DylanMoreland: Never learnt Algebraic Topolo.

John Smith

so my equation goes in a column, constant on the rhs?

Mariano Suárez-Alvarez

it is the sort of thing that one immediately gets once one sees it in its correct context

alg. topology, or representation theory, or...

Chandrasekhar

@Mariano: How can I stop to worry, when I see champs like you playing with and me struggling with it :(

Mariano Suárez-Alvarez

5:11 PM

explainign what exact sequences are for and where they come in the abstract, is doomed to fail :)

never worry

J. M.

@JohnSmith Coefficients on each row of the matrix; variables on the vector on the left; constant terms on the right-hand side.

John Smith

if my equation is 6x1 = 6 + x0 + 4x2 + x3 i rearrange to -x0 + 6x1 - 4x2 - x3 = 6

Mariano Suárez-Alvarez

worry does not help you learn anything

Chandrasekhar

@Mariano: Yeah true.

One day or the other by constant practise

John Smith

which implies a row/column? of -1, 6, -4, -1 and a rhs of 6

Chandrasekhar

5:12 PM

one is sure to master

whatever one wants

Mariano Suárez-Alvarez

later guys, I have to go to class :)

be nice to each other!!

Chandrasekhar

@Mariano: Thanks for spending ur time, I am blessed.

Mariano Suárez-Alvarez

I have to talk about mayer-vietoris, in fact :)

Matt N.

@JM Nice. On the small one I didn't realise it was all rainbow coloured.

skullpatrol

@MarianoSuárezAlvarez worry does not help you learn anything, but it helps to keep you working at it :D

J. M.

5:13 PM

@JohnSmith It's a row. One row for each equation.

Chandrasekhar

Hmm... By the way I don't understand why Mathoverflow doesn't allow comments to be edited. Does anyone know anything about it

Mariano Suárez-Alvarez

pleasure and interest keeps you working

J. M.

@Chandrasekhar The old SE engine didn't have that feature.

Chandrasekhar

@all guys MAriano has class lets not make him wait

John Smith

OK I understand now, thank you JM

J. M.

5:14 PM

The ability to edit comments only showed up in SE 2.0; MO is still in the earlier version.

Dylan Moreland

@JM Is there any timeline for the switch?

Chandrasekhar

@J.M Then has anyone taken an initiative to rectify that. I have asked a MEta question but don't know what will be the outcome

J. M.

@DylanMoreland There's a changelog somewhere in meta.SO, if memory serves...

Dylan Moreland

@JM Oh, I meant MO moving to 2.0.

J. M.

@Chandrasekhar I don't think they're keen to upgrade their SE installation any time soon, for a number of (good) reasons.

skullpatrol

5:15 PM

Hi @PeterTamaroff

Dylan Moreland

@Chandrasekhar The MO software is frozen. Won't happen, I'm afraid.

Peter Tamaroff

@skullpatrol Hey.

J. M.

@DylanMoreland Dunno. Wasn't there a long discussion at meta.MO that just went "pfft"?

Matt N.

If you buy travel insurance, do you buy it at a travel agency? Or insurance company?

t.b.

The first text here's a short note by Eckmann on arrows and exact sequences and how they were introduced in the thirties and forties.

Chandrasekhar

5:17 PM

@t.b thanks t.b

Dylan Moreland

The MO folks don't get access to the source, and I'm not sure that anyone there would have the wherewithal anyway. SE won't touch it; they're working on this now.

t.b.

@Dylan: are you suggesting ... :)

J. M.

@tb Can of worms. ;)

Dylan Moreland

@tb Hm?

t.b.

@DylanMoreland I was having this in mind. Never mind.

Peter Tamaroff

5:19 PM

@MarianoSuárezAlvarez Are you around?

t.b.

@PeterTamaroff he just left for class

Dylan Moreland

@tb Aha :)

Peter Tamaroff

@tb OK.

I just got my Analysis exam back

t.b.

@PeterTamaroff ...and? suspense

Dylan Moreland

I imagine that went well.

J. M.

5:20 PM

If memory serves, short of maintaining the Fog Creek servers holding the software, the SE guys aren't really obliged to support MO's software anymore... but they're being nice anyway.

Peter Tamaroff

@tb The professor has to change the grade

Chandrasekhar

@PeterTamaroff Ask t.b. t.b is also an expert

Peter Tamaroff

From 9 to 10 =)

Matt N.

@PeterTamaroff Congratulations! (Is 10 the max?)

Peter Tamaroff

@MattN Yeah, I guess.

J. M.

5:21 PM

@PeterTamaroff That is a good thing, I take it. Well done. :)

t.b.

Congrats from here, too.

leo

@robjohn are you arround?

Matt N.

@PeterTamaroff Well done. : )

Dylan Moreland

@JM There was this. Anton saying, "I think we should do it" is a strong thing.

Mark Dominus

If it's an analysis exam, the prof will probably change the grade from 9 to 9.9, and then you will have to lobby to have it changed to 9.99, and be content that it approaches 10 in the limit.

Peter Tamaroff

5:22 PM

@MattN Yeah, I guess.

There is no more than that

Chandrasekhar

All right guys, thanks for spending time. Its 23:00 HRS IST and I would like to go to bed. See you later.

J. M.

@MarkDominus If the prof's name was Zeno, well...

Peter Tamaroff

@MarkDominus Oh, no no. We don't have much decimals. Its 9.00 9.50 10.00

t.b.

@Chandrasekhar see you, good night!

J. M.

@PeterTamaroff He was kidding. ;)

leo

5:23 PM

@Chandrasekhar sleep well

J. M.

See you later, Chandru.

Matt N.

Night Chandru.

Peter Tamaroff

@JM OH! Well there are places that grade with 97% for example

Cambridge gives grades in terms of X/100

Mark Dominus

I make joke. Laugh, Comrade, laugh!

Peter Tamaroff

@MarkDominus The charm is now gone. You'll have to go with another one.

J. M.

5:25 PM

@PeterTamaroff Yes. Mark was joking about grades for an analysis course being asymptotic... ;)

Matt N.

@PeterTamaroff There are also places who have 6 failing grades and 5 passing grades. It's always good if you can tell students just how much they suck...

J. M.

@MattN Six ways to fail? How... colorful.

@MattN Cooooool

: )

Germany has 1-15

5:26 PM

Perfect score is not in the base grade space, but only in its completion.

J. M.

Back in the day, we had to make do with "excellent", "satisfactory", "remedial", and "failure"...

Peter Tamaroff

I'll be leaving just for a while to get something to eat! BRB

Matt N.

@JM Sounds great. The good old times...

Martin Sleziak

Do you think it would be good to open a separate room for tag-related discussions? Something like Jury Duty used to be.

Dylan Moreland

@MarkDominus I Zeno joke.

2

Matt N.

5:27 PM

: D

leo

@tb I'm stuck with $(6)$ in the robjohn's answer. Otherwise it's pretty good. I have leaved some comments. But he ignores me. Is that a silly question? Or is it trivial?

anon

He was talking about that the other day. I don't remember what he said though.

J. M.

@MartinSleziak You're afraid of that long thread in meta too, I gather. :)

leo

uhmmm... :-\

Dylan Moreland

God I need to get out of here. If I come back within the next few hours, tell me to leave. The modular forms need me. Cheers all.

Martin Sleziak

5:29 PM

@JM Some thing are perhaps not worth meta thread. And it could also serve as a kind of noticeboard. (E.g. "I have retagged 5 questions today with blablah tag - so if you plan some big retagging, wait a little bit, so that we don't overfill the front page.)

Mark Dominus

What are the six different failing grades?

Matt N.

Dylan: Ok, will do.

t.b.

@leo basically the same trick as here

Mark Dominus

Are they qualitatively or quantitatively different?

J. M.

@DylanMoreland See ya, and good luck!

Matt N.

5:30 PM

@MarkDominus 1, 1.5, 2, 2.5, 3, 3.5. (where 1 is worst)

J. M.

@MartinSleziak Oh, sorry; I was thinking about that long meta thread for merges/synonyms...

Mark Dominus

Oh, that's disappointing. I was hoping they would have names like "bad," "awful," "ridiculous," etc., or else be for various humiliating reasons.

Matt N.

I think they need them so that one single prof can fail you if he wants to. Maybe. Not sure. If you take a bunch of exams the average of all grades has to be 4.0

@MarkDominus : )

J. M.

I once had a professor that wrote only two sorts of comments when checking our exams.

If an answer was nicely done: "Very good!"

If it was completely botched: "My goodness!"

David Wheeler

one fears that a student not versed in the subtleties of english might interpret "my goodness!" as a ringing endorsement of prowess.

Mark Dominus

5:33 PM

Is there anything interesting about matrices whose eigenvalues have all the same sign?

J. M.

@MarkDominus ...on that scale, I guess the worst grade is "Yuck!"

leo

@tb By the way the past year I got stuck in that question too. Finally I gave. I leave it to later. I never solver it. Thanks :-)

the linked question

Martin Sleziak

At meta.SO they have separate tags notifications and comment-replies. For some of things we have under notifications the tag comment-replies would be more suitable, I guess. But I am not sure if it's worth creating a new tag.

J. M.

@DavidWheeler I suppose, but the paper would be soaking in the red ink too, so the notion of endorsement won't be likely in your thoughts.

anon

@MarkDominus The matrix is either positive or negative definite?

Martin Sleziak

5:34 PM

But I guess that more questions concerning both of them will come in the future.

J. M.

@MarkDominus Like SPD matrices?

David Wheeler

well, in china, red= good

t.b.

@MartinSleziak aren't notifications the new greyish things that pop up instead of the banners?

Mark Dominus

So the answer is yes?

J. M.

@MarkDominus Definitely. ;)

anon

5:36 PM

ba dum tss

Mark Dominus

Thanks!

Martin Sleziak

@tb The way I understand it, notification is everything which goes to your inbox.

t.b.

maybe.

J. M.

@tb Yes, they got rid of those dropdowns. Thank goodness.

David Wheeler

@MarkDominus yes I said yes I will Yes

Peter Tamaroff

5:37 PM

@DylanMoreland LOL!

t.b.

@JM the only minor gripe I have is that I almost can't see the new notifications.

David Wheeler

you know what my biggest gripe with mathstackexchange is?

J. M.

@tb Oh, the color? I have to agree...

Mark Dominus

@David: Let me guess: You hate mathematics?

David Wheeler

it's that it's more about mathstackexchange than it is about math.

skullpatrol

5:39 PM

Good point^

J. M.

@DavidWheeler I think I know what you mean, but could you elaborate a tad more?

skullpatrol

Politics =/= Math

David Wheeler

the finer points of site maintenance, ettiquette, reputation, editing, closing, and duplication often make the communication of ideas take a back seat

Peter Tamaroff

Seen this?

I'm curious about the nature of $1/x$

J. M.

@DavidWheeler Ah, yes. On the other hand, we still have things a bit better than, say, on SO...

Peter Tamaroff

5:44 PM

How all primitives are "the same" for $x^a$ $a \neq -1$ but when one goes to try integrate $1/x$, we get a new function.

It is obvious the problem is from $x^{a+1}/(a+1)$

J. M.

@PeterTamaroff There was a question on main about that...

Peter Tamaroff

but what interests me is the structure behind that

David Wheeler

0 is different than other numbers?

Peter Tamaroff

@DavidWheeler What?

David Wheeler

hmm...when you multiply by 0, there's no problem. when you divide by 0, things go south....

skullpatrol

5:48 PM

@DavidWheeler Actually, there is a hidden problem when you multiply by 0; you lose the original number.

David Wheeler

there's a certain parallel between the relationship of differentiation and integration as between multiplying and dividing. i'm going to express this badly , but it's something like this: when you multiply, you just calculate what you're going to get

Peter Tamaroff

@DavidWheeler I don't follow.

David Wheeler

when you divide, you essentially guess what number would give you 1 if you multiplied by it

Peter Tamaroff

@DavidWheeler What is 2348974897/2348975?

David Wheeler

for example, although the euclidean algorithm somewhat "mechanizes" the process of division, the calculation of the quotients still requires "guess-work"

Zhen Lin

5:52 PM

No, no. The problem is that you've expressed the integral wrong! Clearly, we should integrate from $+1$ rather than $0$. So the antiderivative is $\frac{x^{a+1} - 1}{a+1}$. Now take the limit $a \longrightarrow -1$...

David Wheeler

differentiation is a more-or-less mechanical process, integration is more or less guess-work

skullpatrol

hi @Eugene

@DavidWheeler Isn't that why they are called "inverse" operations? :P

Eugene

@skullpatrol hi

David Wheeler

@ZhenLin ok, i get the limit is ln(x)

skullpatrol

Operations that "undo" each other :D

J. M.

5:57 PM

@Peter: found it

David Wheeler

it's interesting though, integration respects polynomials, but not rational functions

J. M.

@DavidWheeler Yes, one has to introduce the logarithm (and the arctangent if one insists on keeping to the reals) to integrate them... more or less the same idea with a number of the usual special functions.

David Wheeler

bah. there's too many functions. at one point i had a great big book full of 'em. it's worse than organic chemistry.

Zhen Lin

Agreed. Which is why I only study abstract arrows! :D

Transcript for

$Mathematics$ Mathematics