Mathematics - 2015-09-10 (page 2 of 2)

Ramanewbie

4:24 PM

Hi @evinda @Agawa001

Agawa001

hey @Ramanewbie

evinda

4:36 PM

Hi @Ramanewbie

@Ramanewbie What's up?

Ramanewbie

@evinda Not much, and you ?

evinda

The same @Ramanewbie

Do you have still vacations? @Ramanewbie

Ramanewbie

@evinda No, I'm back to school since beginning of September

Agawa001

5:04 PM

@Ramanewbie which class are you

Ramanewbie

@Agawa001 1°S

Agawa001

did you think of any thing to specialize in university @Ramanewbie

Ramanewbie

@Agawa001 Not yet, I don't really know how it works

Balarka Sen

@MikeMiller I'm kind of unsure about how to do this, because I'm not very well-versed with Lie groups. I think, because $g_1$ and $g_2$ are close enough, multiplication by $g_1$ and $g_2$ would induce same maps $H_n(G, G - \{1\}) \to H_n(G, G - \{g_i\})$.

Agawa001

@Ramanewbie you must think of it earlier

Ramanewbie

5:16 PM

@Agawa001 Why would I ?

Agawa001

@Ramanewbie if you think you dont have to, wait still choices are open for you

i think people tend to choose their future at the level of secondary school

Ramanewbie

@Agawa001 Yes, but It's totally useless to think about UNIVERSITY now for me since I don't have so many choices and I can't tell if I will be accepted or not.
It will think of it next year.

Mary Star

What does the symbol $\models$ mean?

Rigor

models

Mary Star

What is it used for? @Rigor

Rigor

5:22 PM

modeling :P

21

Q: What is the meaning of the double turnstile symbol ($\models$)?

What's the meaning of the double turnstile symbol in logic or mathematical notation? : $\models$

Double turnstile

In logic, the symbol ⊨, or is called the double turnstile. It is closely related to the turnstile symbol , which has a single bar across the middle. It is often read as "entails", "models", "is a semantic consequence of" or "is stronger than". In TeX, the turnstile symbols and are obtained from the commands \vDash and \models respectively. In Unicode it is encoded at U+22A8 ⊨ true (HTML ⊨) In LaTeX there is the turnstile package, which issues this sign in many ways, including the double turnstile, and is capable of putting labels below or above it, in the correct places. The article A...

Balarka Sen

@DanielFischer In this question, it is stated that any map $\Bbb RP^2 \to S^1 \times S^1$ is nullhomotopic. Can't we just do it as follows : it induces the zero map on $\pi_1$, as $\Bbb Z^2$ is free. hence, lift to universal cover, nullhomotope, and pushdown.

Asking for a sanity-check

Daniel Fischer

@BalarkaSen $\mathbb{Z}^2$ is not free. It's free-abelian. Torsion-free would suffice. If you have the theorem that (for sufficiently connected Hausdorff spaces at least) you can lift to a cover if $f_\ast(\pi_1(X,x_0)) \subset p_\ast(C,c_0)$, you can argue thus.

Balarka Sen

it's free in AbGrp. that's what I meant

Daniel Fischer

@BalarkaSen AbGrp is the wrong category when you argue with $\pi_1$.

Balarka Sen

fair enough. but thanks.

i just didn't bother about writing abelian, a both my groups were abelian

Daniel Fischer

5:35 PM

Coincidence, plain coincidence.

Balarka Sen

nod.

I wonder whether I should post an answer with "yes, you are correct" and my approach above.

@DanielFischer do you think that will be helpful to the OP?

Daniel Fischer

@BalarkaSen Not really, they know that every map from the projective plane to the circle is null-homotopic. That's the non-obvious part. After that, arguing with the product is much easier and more direct than repeating the argument.

But that's just my guess.

Balarka Sen

Hmm, I guess you are right. I wonder if he used $\pi_1$ and lifting lemma to prove that every map $\Bbb RP^2 \to S^1$ is zero. If so, I'd have found the same proof done with this map easier.

I'm posting an answer in any case. If nothing, it'll get off the unanswered queue.

Balarka Sen

6:02 PM

hi @iwriteonbananas

iwriteonbananas

hi there, balarka

Balarka Sen

what're you upto?

iwriteonbananas

studying L2 invariants is frustrating me at the moment

Balarka Sen

well, why?

iwriteonbananas

well, i think it's mainly lücks book that frustrates me

it's very unintuitive

Balarka Sen

6:03 PM

possibly. what, in particular, is frustrating you? they seem like nice and powerful things to me

oh. well.

iwriteonbananas

i'm sure they are but i'm having trouble understanding the relevant defintions and their motivation

there's some functional analysis involved. i don't know any functional analysis

Balarka Sen

right. you should really discuss these with someone knowledgeable and pick a good book, i think. if it's atiyah who found them, then it's unlikely to be so easy

iwriteonbananas

i'm gonna try to find a couple more ressources on that subject

right

Mike Miller

@Balarka: I mean, it doesn't make sense to say that the maps to $H_n(G,G-g_i)$ are the same, because on the face they're different groups. You need to use the definition of local coherence and the fact that muldiplicafion is continuous.

Balarka Sen

I mean, the image of the $1$'s are the same after identifying the groups with $\Bbb Z$. Let me try this again.

Mike Miller

6:08 PM

@iwriteonbananas: I've heard that book called "Lueck's tomb" because you could die in its pages. (Alternatively, it was a misspelling of tome. I prefer my interpretation.)

2

Balarka Sen

lol!

Chris's sis the artist

@M.S.E your last question led me to some very interesting results.

Mike Miller

@Balarka: That still doesn't make sense and isn't the definition of local coherence.

Balarka Sen

ok, ok, let me write down everything before I make a fool of myself.

iwriteonbananas

@MikeMiller hahaha

:((

Mike Miller

6:12 PM

When you understand those, teach them to me.

iwriteonbananas

will do. hopefully i will have understood them by the end of my course. :)

it starts in a month and lasts 3-4 months.

Mike Miller

@iwriteonbananas: Who's teaching?

iwriteonbananas

@MikeMiller an academic grandson of Lueck

the topology department at my uni is headed by an academic son of Lueck

Mike Miller

Fair enough.

Agawa001

6:58 PM

@Chris'ssistheartist why you keep posting deleted lines :D

i cant even review the history

Chris's sis the artist

@Agawa001 I don't say (write - in this case) very interesting things. :-)

Agawa001

@Chris'ssistheartist , btw whats the detailed score of yesterday s match

Moses

@DanielFischer Hi. If you have a chance, please have a look at my post.

Chris's sis the artist

@Agawa001 Simona Halep won with 2-1 (sets). :-) Well, it was a hard game, and in the third set it was interrupted due to the rain. That break was pretty helpful for Simona since she had time to recover a bit.

Daniel Fischer

@Moses I will. The tab is already open, but I'm looking at some other posts first.

Moses

7:03 PM

@DanielFischer Okay thanks.

Agawa001

@Chris'ssistheartist lol, september began, i asked for total score (including points and sets)

Chris's sis the artist

@Agawa001 :D 6-3, 4- 6, 6-4. The thing is that before interrupting the game, in the third set Victoria Azarenka had 2- 0. :-)

Agawa001

nice

romanian revenge :D

Chris's sis the artist

hehe, indeed! I'm looking forward to the next game (that starts in 6-7 hours or so).

Moses

7:51 PM

@DanielFischer The $\mathcal{A}$ valued integral is the Holomorphic Calculus contour integral. I'm familiar fubini's theorem for real valued function it's sufficient to consider a bounded real valued function. But I'm not sure how Fubini's Theorem applies to one integral which is Banach space valued and the other which is complex valued. Is the the line between the third and fourth line $ \frac{1}{2 \pi i}\int_{\Gamma_{1}}\frac{1}{2 \pi i} \int_{\Gamma_{2}}f(w)(w-g(z))^{-1}(z-a)^{-1}dwdz$?

Daniel Fischer

@Moses Bochner integral, Pettis integral? There are more ways of defining Banach-space-valued integrals. But you always - well, at least for all variants I have some acquaintance with - have $$\biggl\lVert \int_X f\,d\mu\biggr\rVert \leqslant \int_X \lVert f\rVert\,d\mu,$$ so in our situation, you can uniformly approximate the integrand with piecewise constant functions and take limits to see that the interchange is legitimate.

Moses

@DanielFischer Okay not 100% sure of what you mean, but I'm quite sure that I am referring to a Bochner integral, since I think this is the type of integral used in the description of Holomorphic Functional Calculus. Is the step I mentioned in the last line of the previous message correct?

Daniel Fischer

8:12 PM

Not sure what you mean with "between the third and fourth line". You use Cauchy's integral formula to write $f(g(z))$ as a contour integral. Then you change the order of integration. Since everything is continuous, and the domains of the integrals are compact, a very weak version of Fubini's theorem suffices.

And actually, your integrand is $\mathcal{A}$-valued, $f(w)(w-g(z))^{-1}(z-a)^{-1}$. If you look at it as an iterated integral, in the one order of integration, the inner integral is $\mathbb{C}$-valued, but when you look at it as a double integral, you have to take the whole integrand into account. And that is $\mathcal{A}$-valued.

Moses

@DanielFischer I understand the process, it's just the fine details I'm having difficulty with since I'm not sure how to work with Bochner integrals and complex valued integrals, hence I was asking if $\frac{1}{2 \pi i}\int_{\Gamma_{1}}\frac{1}{2 \pi i} \int_{\Gamma_{2}}f(w)(w-g(z))^{-1}dw(z-a)^{-1}dz = \frac{1}{2 \pi i}\int_{\Gamma_{1}}\frac{1}{2 \pi i} \int_{\Gamma_{2}}f(w)(w-g(z))^{-1}(z-a)^{-1}dwdz$

Daniel Fischer

Yes, that is correct.

Moses

@DanielFischer So then your $\mathbb{C}$ integral $\int_{\Gamma_{2}}....dw$ becomes a $\mathcal{A}$ integral $\int_{\Gamma_{2}}....dw$?

Daniel Fischer

@Moses When we move the element of $\mathcal{A}$ into the integral to convert the nested integral into a double integral [and then that into a nested integral with the other nesting].

Moses

@DanielFischer Is that a yes or a no?

zounds

8:27 PM

are almost all functions f: R->R surjective? or is this just meaningless?

Daniel Fischer

@Moses A yes.

@zounds You need to define what "almost all" should mean. Then it may be true or false.

Moses

@DanielFischer Okay, but could I ask this more simply. If you didn't have the double integral and just had $\int_{\Gamma_{2}}f(w)(w-g(a))^{-1}dw(z-a)^{-1}$ would that be equal to $\int_{\Gamma_{2}}f(w)(w-g(a))^{-1}(z-a)^{-1}dw$?

zounds

given a measure u such that if S has less than beth two elements, then u(S) = 0, does the set of all surjections have positive measure?

or rather,

does the set of all functions R->R that are not surjections have measure 0?

Daniel Fischer

@Moses Yes. For any reasonable integral, you have $$\int_X h(x)\,d\mu \cdot b = \int_X h(x)\cdot b\,d\mu$$ whenever $h$ is a complex valued integrable function on $X$ and $b$ and element of a complex vector space.

@zounds Umm, what was $\beth_2$ again?

zounds

cardinality of all functions R->R

Daniel Fischer

8:37 PM

$2^{2^{\aleph_0}}$ then?

zounds

yeah

Daniel Fischer

Well, since $\mathbb{R}^{\mathbb{R}} \cong (0,1)^{\mathbb{R}}$, there are $\beth_2$ functions whose image is contained in $(0,1)$.

zounds

ah, right

oh wait, that doesn't mean that set has positive measure though

or maybe it does

Moses

@DanielFischer Okay thanks. The left hand side of what you wrote seems fine, since that is just a scalar times a vector but the right hand side seems strange (at least for me since I don't have experience with Banach space valued integrals). But as you say this holds for all reasonable integrals. So this holds for Bochner integrals?

Daniel Fischer

@Moses Yes.

Moses

8:41 PM

@DanielFischer Okay thanks again. Tricky business...

Daniel Fischer

@zounds Depends on the measure, possibly. So far, you only said that all sets of smaller cardinality shall be null sets.

Balarka Sen

9:30 PM

@AkivaWeinbergercolumbus Georges is just joking with that answer of his, I can assure you he hasn't forgotten his logic :P

(to add, he has an immense amount of knowledge in algebraic geometry)

ArtOfCode

Jesus.

Vogel612

what a flag-fest.

Danu

Someone is spam-flaggin'! Stop it! :P

Balarka Sen

what happened?

Almo

yeah what the hell

Vogel612

9:33 PM

somebody just flagged 6 or 7 messages..

Balarka Sen

uh-oh

ArtOfCode

Whoever that is, stop flagging messages without reason. Misuse of flags is grounds for lengthy suspensions.

12

Vogel612

~grabs popcorn

Hosch250

WOW.

Alex A.

Flags should be used for serious issues only.

Hosch250

9:34 PM

Just flagging every message in the room?

Balarka Sen

i think you guys get the highest amount of flags from this room, mostly for messages which are not really flag-worthy.

Vogel612

@BalarkaSen hmm.. on this domain or including So chat?

Balarka Sen

i am not really sure. i have been year for about one and a half year, and have seen nonstop flagging in here most of the time. i think some mod also mentioned that we're highest(second highest? not sure) in flagging rate

Hosch250

Why doesn't the flagger get IP banned?

ArtOfCode

@BalarkaSen Probably second. You don't quite live up to Mos Eisley, here.

Vogel612

9:38 PM

@Hosch250 IPbans are a crappy tool

Hosch250

In one room, someone used a script to flag every message - twice.

Vogel612

also there is valid cases of a multitude of flags at once

Hosch250

Imagine the flag-fest then.

Balarka Sen

@ArtOfCode fair enough

i agree, this must be so annoying for 10k+ers and mods

Vogel612

also I want to (dis)honorably mention the Lounge in this context..

eh there's worse IMO. We get to eat popcorn and see burning fires

I like seeing fires burning until the firefighters arrive

sidenote: related MSE post

21

Q: Chat flagging is out of control

In room 10 of Stack Overflow we have had a troll (or multiple trolls) for the past few weeks casting random flags and generally being disruptive. Today for example I woke up to this. For us or room owners there's no way to handle this kind of disruptive behaviour, and moderators cannot help e...

support flags chat

aaand sayonara :D

M.S.E

9:45 PM

@Chris'ssistheartist please show me how you did it.

@Chris'ssistheartist and the interesting results

Chris's sis the artist

@M.S.E I attended a slightly different version. I still work on the results I got.

Maximilian

$f(g(x)) = \frac{2(\frac{4x-3}{2-x})+3}{(\frac{4x-3}{2-x})+4}$

What would be the initial step in this?

I tried doing it and leaving the 2 on the top for last, but it didn't really pan out to anything

This will end up just being equal to X

Balarka Sen

@Maximilian I don't understand the question. What do you want to do?

Maximilian

Verifying that f(x) and g(x) are inverses

Balarka Sen

ah, I see

Maximilian

9:49 PM

and I'm just wondering what I should do first in this. I tried it but didn't get it so maybe I started off wrong?

Balarka Sen

So you want to simplify that thing

Maximilian

Yup

What I did before was get +3 and +4 to have common bases of 2-x and then add those up and left the 2 on the top for last

and that didn't work out into anything I could simplify to just X with

Balarka Sen

First start by simplifying the numerator and the denominator

Almo

who keeps flagging stop it

Balarka Sen

C'mon. What the hell.

Hosch250

9:51 PM

And starring.

JohnP

who keeps flaggint normal comments?

Doorknob

/me slowly trudges through the "clear stars" buttons...

Hosch250

I have no idea. Can't mod's see this stuff?

ArtOfCode

-_- flag and star flood

JohnP

@Hosch250 not AFAIK, it just says "one user"

Maximilian

9:52 PM

$f(g(x))=\frac{(\frac{8x-6}{2-x})+3}{(\frac{4x-3}{2-x})+4}$?

Hosch250

OK.

ArtOfCode

Stars cleared.

Almo

room owners get notified of flags, which are anoymous to prevent bullying people with legitimate flags

Rainbolt

I don't understand the issue with the star flood (it's a privilege!) but the flags are annoying because I just can't ignore the little blue circle.

JohnP

@ArtOfCode - You'll have to show me how to do that sometime.

Doorknob

9:53 PM

@JohnP Click the little arrow next to a message on the starboard.

ArtOfCode

@JohnP On the star board on the right. Drop down menu on each post, hit clear stars.

Hosch250

@Rainbolt Some rooms aren't as star-happy as others.

Almo

i come from a pretty star-happy room (gamedev)

JohnP

gr0k, thanks.

Hosch250

I'm from The 2nd Monitor, which is notoriously star-happy.

Balarka Sen

9:54 PM

@Maximilian OK. Now note that $$\dfrac{\frac{8x-6}{2-x}+3}{\frac{4x-3}{2-x} + 4} = \dfrac{\frac{8x-6}{2-x}+3\frac{2-x}{2-x}}{\frac{4x-3}{2-x}+4\frac{2-x}{2-x}}$$

Now simplify this thing.

Maximilian

Yup, finished and got X

Balarka Sen

Excellent.

Maximilian

lots of fractions :P

M.S.E

@Chris'ssistheartist cool show me please please?

Chris's sis the artist

@M.S.E more work is needed. I was trying to say that I still work on those results. When there is something to show I'll let you know.

Maximilian

9:58 PM

and just did G(F(X)) and got x as well

Balarka Sen

That's a relief :P

M.S.E

@Chris'ssistheartist oh sorry I misunderstood. Yay!

Patrick Stevens

Can we construct the set complement in a categorical way? Working in the universe $U$, the set intersection is the categorical product in the obvious boolean algebra, and the set union corresponds to the coproduct - does the complement correspond to any particular categorical notion? I'm a beginner at category theory

M.S.E

@Chris'ssistheartist Thanks

Balarka Sen

@PatrickStevens More precisely, intersection is categorified by pullbacks and unions are categorified by pushouts, but yeah. I don't know the answer to your question - I pondered on this sometime ago but couldn't come up with much.

Joe Stavitsky

10:03 PM

where is chatjax reference please

ArtOfCode

@JoeStavitsky Links on the starboard -->.

Balarka Sen

Hmm, I wonder what was the reason I was pondering on this.

Patrick Stevens

@BalarkaSen For my part, I was wondering if de Morgan's laws were true for a categorical reason

Balarka Sen

Oh, yeah, primes and knots. I wanted to know what the correct analog for knot complements was for primes. OK, there's definitely a way to do this.

Something something localization

@PatrickStevens a piece of suggestion : don't just categorify things for the sake of it. while it's very fun to do so, things might not be as interesting as you think.

Patrick Stevens

@BalarkaSen I'll bear that in mind, thanks - I'm still learning basic category theory so I'm trying to practise by treating everything categorically wherever possible

Balarka Sen

10:08 PM

ah, I see.

Joe Stavitsky

so where do I look to see how to input math - syntax etc

Maximilian

$x=\frac{2y}{3y-1}$. Having trouble with problems with Y in the numerator and denominator.

I got to $3xy-x=2y$

if I divide by 2 I still have y on the other side?

and if I divide by y I can't do anything because there is -x with no y?

Balarka Sen

Again, what do you want to do?

Express $y$ in terms of $x$?

Maximilian

I'm finding the inverse of functions

Original is $f(x)=\frac{2x}{3x-1}$

I flipped x and y

Balarka Sen

OK, right. you've made it till $3xy - x = 2y$. Why don't you shift $x$ to the RHS?

Maximilian

10:12 PM

then multiplied both sides by the denominator

and then stuck

oh.. wait.. let me try that.

morphic

Hi @Bal

Balarka Sen

heya

Joe Stavitsky

@Maximilian not all equations of x and y are functions and not all functions are reversible

Maximilian

I know it has a answer because my teacher gave me them

I just don't know how to get there for some of these

Ya, stuck again. pulled out the x, got $3y-1=\frac{2y}{x}$

Balarka Sen

Urm, I didn't mean divide by $x$

I meant you add both sides by $x$

Maximilian

10:17 PM

Answer is \frac{1}{x-3}

Well.. that would be a better plan than what I did lol

Balarka Sen

@PatrickStevens Dunno, maybe something like : if $A \to B$ is a morphism (this is the analog for a subset), then complement is some other morphism $A' \to B$ such that pullback is trivial, which is "maximal" among all such morphism... not sure how to make that "maximal" property rigorous

maybe you'd want a universal property here

Mike Miller

As someone with 10k it seems like there are a trivial number of flags in here.

Balarka Sen

and stars, which has been removed

@PatrickStevens You definitely cannot do this in completely general categories. You want it to have pullbacks/pushouts, for one

i gotta go and have some sleep

Maximilian

I got a answer but doesn't match up with the answer the teacher gave me

He could have written the answer down wrong though

I got $y=\frac{x}{3x-2}$?

ya his answers are in a weird order