Mathematics - 2015-12-20 (page 1 of 2)

Mary Star

01:22

Hello!!

What does it mean that a linear map is invertible?

@anon do you have an idea?

idonutunderstand

@MaryStar does that just mean it's a bijection? I'm not really sure either.

Mary Star

A matrix is invertible when its determinant is non-zero. Does something similar stands in the case of a linear map? @idonutunderstand

idonutunderstand

01:41

My guess would be that if the linear map is between finite-dimensional vector spaces then it would have a corresponding matrix the determinant of which would be nonzero. But I don't really know what I'm talking about.

@MaryStar what's the context of your question?

Mary Star

I am looking at the following exercise:

and there is the following proposition:

@idonutunderstand

idonutunderstand

02:05

Is a more general result, where the curve is not necessarily regular and the diffeomorphism is not necessarily local, false?

@MaryStar

PVAL

02:31

@MaryStar Just use the chain rule on $f \circ \gamma$ to show it is regular. The "proposition" is usually referred to as the inverse function theorem.

Mary Star

03:22

We have that $\frac{df(\gamma (t) )}{dt}=f'(\gamma (t))\gamma '(t)$, right? Since $\gamma$ is regular we have that $\gamma '(t)\neq 0$. How can we use the fact that $f$ is a local diffeomorphism? @PVAL

anon

05:05

@MaryStar A function is invertible if it has an inverse. Linear maps are functions. Given any choice of coordinates on a vector space, linear maps correspond to matrices, and the linear map is invertible if and only if its matrix has nonzero determinant.

Chat guidelines | $\LaTeX$ in chat

14

skill patrol

When do you expect to get the results from the Putnam? @anon

anon

spring break

L33ter

06:40

how did you do on your putnam exam @anon ?

anon

I submitted four answers, so I'm hoping for 30 points

user174558

In the Putnam exam, just Put your Name on the paper.

Huy

no

user174558

Each time @OFFSHARING announces a discovery in chat, she is not bragging, just being excited. There is nothing wrong with being excited, so we should not criticise her for that. She has probably produced more theorems than any of us will ever produce this lifetime.

L33ter

how is the exam graded the putnam exam ?

Huy

06:59

@MithleshUpadhyay: Why are you inviting me to a room?

anon

to answer his question

which I've now addressed in a comment

Mithlesh Upadhyay

Yes, I need special care for that problem. May be, I need to read given solution multiple time. I need a reliable reference :(.

user17629

Hello everyone

Guys, I was looking for some information about "Non-smooth manifolds with singularities"

Anyone knows some references about it?

In this page map.mpim-bonn.mpg.de/Manifolds_with_singularities considered the manifolds as smooth, and I'm trying to read about non-smooth manifolds

Someone can suggest me something?

Mithlesh Upadhyay

07:17

It's probability problem. Would you want to check this, please?

http://math.stackexchange.com/questions/1544460/group-of-r-people-at-least-three-people-have-the-same-birthday

Vrouvrou

08:17

please, what it mean: $I: W^{1,p}_0(\mathbb{R}^N})\rightarrow \mathbb{R}$ is rotationally invariant

How to see that a function is rotationally invariant

?

L33ter

Hey @anon awake ?

anon

yeah

L33ter

I wanted to ask a question so I am showing that if $|G| = pn$ where p > n > 1. I am proving that sylow p subgroup of G is normal in G without using Sylow theorems.

my idea for this is to prove that $|N_G(M)| = pn$

after using Lagrange theorem and stuff I get that $|N_G(M)| = p$ or $|N_G(M)| = pn$

anon

M is a p-sylow?

L33ter

so I want to show that $|N_G(M)| = p$ isn't possible but I am don't have a idea about why its not possible

yeah

I can't use Sylow theorems

anon

08:34

Well, if |normalizer|=p, then M has n conjugates, and G->Perm(conjugates of M) has image at least n (since the action is transitive) with kernel divisible by p (since the image is not), which forces the kernel to be a p-sylow P. Then PM would be a subgroup of order p^2 unless P=M.

L33ter

well if $|N_G(M)| = p$ this means that N_G(M) = M, where M is the sylow p subgroup of G

oh I see

hm

also we can do it as follows

if |normalizer| = p, ofcourse we have $M \leq N_G(M)$ so we have $MN_G(M)$ is a subgroup of G of order $p^2$

what do you think ?

@anon

anon

huh?

|normalizer|=p is the same as M=normalizer, so M*normalizer is just M

that doesn't work

L33ter

oh yeah

kevin Tah N.

09:04

hey guys

can someone tell me what a clifford algebra really is, in very basic terms

anon

the free associative algebra generated by n anticommuting square roots of -1

kevin Tah N.

so generated by{1, e_i}

anon

yes

kevin Tah N.

I see

anon

if you want indefinite signature, you can make some sqrts of -1 and the others nontrivial sqrts of +1

kevin Tah N.

09:10

nice

I had encountered such in gamma matrices in QM

for Dirac eqns

I figured it might be best to look at them by themselves first

anon

Clifford Algebras and Spinors by Lounesto, is a great book on the topic. also see: "geometric algebra" (recommended search term)

L33ter

@anon I heard there is a topological proof of the fact that there exists infinitely many primes

Huy

@L33ter check Proofs from THE BOOK

anon

sure, you use residue classes to form a topology on Z

I've read it before

L33ter

oh cool

anon

09:13

don't remember it though

L33ter

THE BOOK ?

Huy

@L33ter springer.com/in/book/9783662442043

anon

> Paul Erdős, who often referred to "The Book" in which God keeps the most elegant proof of each mathematical theorem. During a lecture in 1985, Erdős said, "You don't have to believe in God, but you should believe in The Book."

L33ter

haha cool

kevin Tah N.

omg lmao "the book" .. . . I thought it did not even exist, but this is real! lol

Robert Cardona

09:43

Has anyone here worked through Loring Tu's introductory book on manifolds?

Or have it on hand? I have a quick question about an example.

It says the differential of left multiplication on the general linear group is also left multiplication. It's done using curves. So we choose a tangent vector $X_I$ and find a curve such that $c(0) = I$ and $c'(0) = X$ which we've established we can find.

Then there is this line: $$(l_g)_*(X_I) = \frac{d}{dt} \bigg \vert_{t = 0} l_g \circ c = \frac{d}{dt} \bigg \vert_{t = 0} g c = g c'(0) = gX$$

It's the reasoning for this second to last equality I'm not understanding.

It says $d/dt \vert_{t = 0} g c = g'(0)$ by $\mathbb R$-linearity and Proposition 8.15, which states that $$c'(t) = \sum_{i = 1}^{n^2} \dot c^i(t) \frac{\partial}{\partial x^i} \bigg \vert_{c(t)}$$

Balarka Sen

@anon you here?

Robert Cardona

I can't see how that proposition comes into play here.

It seems to me that all that is being used is the definition of $c'(0) = \frac{d}{dt} \bigg \vert_{t = 0} c$ because we pulled out the $g$?

Balarka Sen

@Huy What're you studying?

Pseudo-Anosov maps? :P

Enthusiastic Student

hi

I have a system of partial differential equations

with many dependent and independent variables

is there any standard way that I may organize these variables?

For instance, one may write: Yx,x +Yy,z +Yx,z +Yy,y

and another wite: Yx,x + Yy,y + Yx,z + Yy,z

is there any standard guide on this? any reference or paper about this...

Huy

10:09

@BalarkaSen: Just trying to finish preparing my stats class for next year, after that I can study without distractions for a month or two.

Otherwise I have to alternate all the time which is really annoying.

Balarka Sen

Right.

skill patrol

What's the youngest class age you've taught at a school @Huy

Huy

10:26

@skillpatrol 14-15 year old kids

I only teach high school, and at the high school I teach at, students are at least 13-14 years old when they enter

L33ter

10:54

do you guys see any easy criterion for testing the irreducibility of $x^4 - 2x^2 + 5x + 3$?

@BalarkaSen ?

Balarka Sen

rational root theorem

+ no quadratic factors

that's what I usually do.

otherwise, go Eisenstein

L33ter

Eisenstein doesn't work

Balarka Sen

then do what I said above

L33ter

yeah

I will do the above

Mithlesh Upadhyay

11:08

Doubt on double integral; do you want check it, please?

http://math.stackexchange.com/questions/1579690/the-value-of-double-integral

Vrouvrou

11:47

Someone have an idea please: math.stackexchange.com/questions/1583104/rotationally-invariant

Karl Kronenfeld

12:48

@BalarkaSen I haven't a clue, sorry.

Agawa001

13:48

hmmm ramanujan formula applied one of my favorites

user174558

Hi @SohamChowdhury Sayan has deleted his account.

Huy

14:24

@DanielFischer: sorry to bother you with this but will I need complex numbers to teach differential equations for anything apart from computing roots of the characteristic polynom (HS level)? trying to figure out what to teach in which order.

Daniel Fischer

@Huy Not really. Of course it's convenient if you can use functions like $x \mapsto e^{\lambda x}$ for arbitrary $\lambda\in \mathbb{C}$, but not essential.

Huy

@DanielFischer: if you had the choice, which would you teach first? it seems more elegant but I don't know if it's that much of an advantage

(those are the last two topics I teach)

on the other hand, teaching it the other way around doesn't have any obvious advantage, but for some reason most teachers do diff eqs before complex numbers

Daniel Fischer

14:46

I'd do complex numbers first (we're not talking about complex analysis here, I suppose).

Huy

no, just basic complex numbers :P

Soham Chowdhury

@JohnNash why all of a sudden?

Brennan.Tobias

15:01

hello

Huy

16:00

@DanielFischer: do you know of any super-elemental way to show Euler's formula? my students don't know Taylor and the trigonometric functions were obviously not defined as power series.

Balarka Sen

@Huy One way to do it would be to recall polar coordinates.

user174558

@Huy Do you mean super elementary?

user153330

@Huy i think this is elementary without taylor or power series definitions of trig functions math.stackexchange.com/questions/1079121/… is that what you looking for?

Huy

@BalarkaSen recall?

Balarka Sen

Not sure what the question is.

ok, I have to go.

user153330

16:09

@BalarkaSen balarka, could you please send the link i posted to huy, maybe he blocked me and can't see it

@JohnNash could you please send the link i posted to huy, maybe he blocked me and can't see it

@Anubhav.K hi

Daniel Fischer

@Huy Introduce power series and define the functions by their power series. That makes it really simple. Of course it's not quite simple to show that the thusly defined sine and cosine are the same as the ones they saw in trigonometry. ;)

Agawa001

16:47

hey @Ramanewbie comment ca deroule

Ramanewbie

Bjr @Agawa001!

How come you speek French.

Agawa001

im known french speaker in this room

Anubhav.K

17:05

@user153330 hii

Alec Teal

17:19

I'm going to bump this here:

1

Q: When is a vector in the image of a bilinear map?

In this case I have a bilinear map defined as: $\varphi:V\times V\rightarrow U$ with: $$\varphi(x,y)=x_1y_1c_1+x_1y_2c_2+x_2y_1c_3+x_2y_2c_4$$ Here $\{c_1,\ldots,c_4\}$ is a basis of $U$ and $V$ has dimension 2 with basis $\{b_1,b_2\}$ (so you can think of $x:=x_1b_1+x_2b_2$ and "" for $y...

OFFSHARING

19:01

I wanted to change the username to Monica, but I still need to wait (a few days) ... @JohnNash

Frank Science

19:17

@Agawa001 Francophone?

Agawa001

@FrankScience sometimes

Frank Science

@Huy What's the definition of $e^z$ for $z\in\mathbb C$?

Alec Teal

@FrankScience why are you asking one person that?

Frank Science

@AlecTeal That's context-specific.

Alec Teal

@OFFSHARING you're a girl? in that case "Damn gurl, you love 'dem integrals" and "integral slut" makes sense! :P (this is an inclusive teasing)

@FrankScience but other people know $x^{a+b}=x^ax^b$ too :P

Frank Science

19:26

@AlecTeal You can click the arrow before my question to see what happened before.

Alec Teal

$e^z$ doesn't require me to do that

dorothy

if you take $n$ steps on a symmetric random walk on the number line, what is the probability distribution called for the position you end up at?

Frank Science

You mean, the probability distribution of $S_n=X_1+\dotsc+X_n$ for i.i.d. $X_j$'s?

dorothy

@FrankScience exactly

Alec Teal

@FrankScience you should see an arrow by what Dorothy said.

dorothy

19:31

@FrankScience also.. am I right that as $n$ tends to infinity this distribution tends to the normal distribution with std \sqrt{n}?

Frank Science

@dorothy Central limit theorem.

dorothy

@FrankScience yes :) So about the name?

Alec Teal

@dorothy just compute the variance: maths.kisogo.com/index.php?title=Variance it's easy for the sum of uniform variables.

dorothy

@FrankScience is it a shifted binomial distribution?

Alec Teal

@dorothy it's just called "symmetric random walk" you don't have to name every distribution.

No

dorothy

19:34

@AlecTeal what is the relationship to the binomial distribution?

Frank Science

Consider $2^{-n}(X+1/X)^n$.

Agawa001

@FrankScience trigonometric amount ? like sin(sthing) or cos(sthing)

Frank Science

@Agawa001 He needed a proof for Euler's identity, so I need to ask for his definition to clarify. / Il avait besoin de une demonstration pour l'identité d'Euler. (My French is very bad)

Agawa001

i can understand english

Frank Science

*d'une

Agawa001

19:42

tbh, i think euler id or demoivre or whatever based on that is just a shot at random

dorothy

@FrankScience was that for me?

Frank Science

@Agawa001 But what if he defines $e^z$ as $\lim_{n\to\infty}(1+z/n)^n$?

Agawa001

that is with absolute real numbers

Frank Science

What?

Agawa001

z,n defined as real

Frank Science

19:46

But it works for complex numbers $z$.

Agawa001

limit for an imaginary number ? never saw that

Frank Science

Limit could be taken in any Hausdorff topological space.

Dans ce cas, $z_n\to z_0$ ssi $\Re z_n\to\Re z_0$ et $\Im z_n\to\Im z_0$. @Agawa001

Agawa001

for example , what is $lim_{x->\infty} xi$

Frank Science

Quel est $\lim_{n\to\infty}n$? Est-ce que c'est bien défini?

Agawa001

n*i

ya -t- il une chose telle $\infty*i$

?

OFFSHARING

19:54

@AlecTeal I'm a girl, of course!

Frank Science

En fait, il n'est pas bien défini dans $\mathbb R$ pour $\lim_{n\to\infty}n$

Agawa001

reste un point de vue personel

Frank Science

$S^1\cong\mathbb R\cup\{\infty\}$ is a compactification of $\mathbb R$, but not the only one.

OFFSHARING

@AlecTeal You can call me Monica.

Agawa001

@OFFSHARING wish u never read that,such wierd tense accented language

OFFSHARING

20:01

@Agawa001 :-)))

Agawa001

@FrankScience ur point is $\infty$ doesnt belong to R ?

Frank Science

@Agawa001 The point is that, convergence to $\infty$ is not convergence in $\mathbb R$.

Agawa001

i know its divergence

Frank Science

So when you want to determine $\lim_nin$ in $\mathbb C$, you need to check whether it's convergent.

OFFSHARING

@robjohn Hey. I wanna send you something in db. Are you around? Let me know when you're available.

anon

20:08

@BalarkaSen am now

OFFSHARING

@robjohn I sent to you the very little file.

@anon how often do you laugh a day? Not decided yet how the major part of the mathematicians are in terms of the sense of humour.

I have a lot of fun and laugh a lot while playing with my stuff.

Maybe I should have also addressed that question to @BalarkaSen.

@Agawa001 how is your sense of humour?

Karl Kronenfeld

lol.

Enjoys Math

Anyone know natural maps? math.stackexchange.com/questions/1583625/…

(category theory)

I've shown that there exists a family of maps, but can't tell if it's a natural transformation

NVM

someone answered

Agawa001

20:26

@OFFSHARING i laugh as much as i haircut myself

OFFSHARING

@Agawa001 lol :-)

Agawa001

the sad part is i rarely take a haircut (maybe once upon two months)

anon

@OFFSHARING depends on what I do that day. If I see coworkers I laugh at a few jokes here and there, and make some in return, mess with some of the people I work with in the lab. I enjoy a number of comedy shows regularly that always get laughs out of me, and laugh with friends regularly.

OFFSHARING

@anon Nice. Well, often the internet can create a wrong image of the other person, you seemed to me more serious, and less inclined to laugh (for some reasons).

Brennan.Tobias

hello

Thijser

20:34

Anyone here know what the : in j:i∈T_j might mean?

anon

@OFFSHARING in my infinite boredom it takes less trivial things to make me laugh. when people type lol on the internet, it's usually with a straight face, since their laugh is just on the inside. such is desensitization.

OFFSHARING

@anon hehe, whenever typing lol or anything like that means that I laugh (that's true for me). It's hard to type lol in other conditions. :-)

anon

@Thijser : means such that, especially in set-builder notation. would help with more context.

Thijser

Ok so it would mean j such that i is in T_j?

Thanks @anon

anon

I assume you're missing {} around it as well, @Thijser

Thijser

20:37

@anon It's written under a summation

anon

why don't you tell me your full question

it annoys me when people leave out potentially important context in their questions...

Thijser

I have a condition that increase y k until there exists an i ∈ T k such that j:i∈T j y j = c i

Alec Teal

Guys, what's offensive about:
"Have the 10 upvoters done ANY linear algebra at all?"
and "Oh crap. I trusted dot product. thanks for finding this exception! Maths would be better if we proved stuff."
and "This isn't well researched, it's just not read. 0!=10!=1 and 1010 is 11. The numerator is 01=001=0 so this is 00 - WTDuck is the problem?"
and "The question itself is phrased like a manual from something that one buys off ebay and ends up arriving from China."
and "Have you tried using a search engine for the definition of "Cartesian product" and "set intersection" - seriously, lazy."

Brennan.Tobias

@AlecTeal where are you finding these

Thijser

expect that there should be a sum written above the j:i but I can't seem t ocopy that over

Alec Teal

20:39

@Brennan.Tobias I've been suspended for these comments.

Brennan.Tobias

ok do you want advice to not get suspended again?

Alec Teal

the first one was a question about "why do we need span in linear algebra"

Frank Science

How can we show that every commutative unitary ring has a prime ideal using ultrafilter lemma?

robjohn

20:51

@OFFSHARING I just got your file. Sounds good to me.

@OFFSHARING any idea when your book will come out?

Thijser

so does Σ{j:i∈T_j }y_j = c_i mean that for every j where there is an i inside T_j sum up y_j until the sum is equal to c_i? Or do I also have to sum up the c_i?

Karl Kronenfeld

@Thijser i is a constant relative to the expression.

The sum is over all j such that this particular i is in the set T_j.

The equation is a claim: that the sum actually turns out to be equal to c_i.

Thijser

@KarlKronenfeld alright thanks

Karl Kronenfeld

21:16

@BalarkaSen (Re: Logic + Topology) This question may interest you: math.stackexchange.com/questions/262654/…

OFFSHARING

@robjohn Glad for that.

@robjohn If all is fine, next year. I cannot give you now more details, but later on.

Balarka Sen

@OFFSHARING I don't laugh.

robjohn

@OFFSHARING Just wondering if there was a plan, that's all.

OFFSHARING

@robjohn Sure. There is always a deadline one must to respect but from the point of sending the final form of the manuscript to the moment the book is published it may take some months.

robjohn

@OFFSHARING of course.

Balarka Sen

21:22

@KarlKronenfeld Cool, thanks.

@anon ok, let's penrose.

OFFSHARING

@robjohn The nice thing was that I contacted a genius, a person I talked to a long time ago, it was a very nice person, and I had talks to him in the past about other kind of math, and he was so suprised when I told him about my proposal to him. He immediately accepted. :-) That is we didn't keep in touch for long periods of time, he was interested in other kind of stuff.

Balarka Sen

if you're free that is.

OFFSHARING

@robjohn The thing is that I never forget the precious (or say, nice people) people. ;)

Perhaps he will think for a long while to what I said to him these days, he was so pleasantly surprised. :-)

robjohn

@OFFSHARING Sounds nice. It's nice to catch up with old friends.

OFFSHARING

@robjohn Yeap.

@robjohn When I met him first he was like a machine to me (at that time I never met anything like that before). I wished much to do things like him.

Nice stories anyway (all that happened a long time ago).

Balarka Sen

21:34

@anon Ok, so you're here?

humph. apparently not.

Agawa001

@OFFSHARING how long u v been here ?

OFFSHARING

@Agawa001 Oh, I met that person in a different environment. I'm here since the summer of 2012 if I'm not wrong.

Agawa001

ok ok i could see that thru one's profile :p

Mike Miller

@Balarka: Remind me, are you done with whatever exercises I posed you recently?

Balarka Sen

I don't remember any exercise you told me recently.

I am done with all the Diff(S^n) exercises you set me.

Mike Miller

21:47

Last month, say. But I'll take that as a yes.

Thijser

Anyone here know how to see if a primal dual solution has zero slackness?

Karl Kronenfeld

Even though I know you're talking about linear programming, you still have to be more specific @Thijser

Balarka Sen

Prof called today. Heard of things called "stratified spaces". Sounds fun.

Mike Miller

Here's more, in increasing order of difficulty. 1) Consider $\alpha \in H_2(\Bbb{CP}^2)$, corresponding to CP^1. (So a generator of this group.)

a) Find an embedded copy of $S^2$ representing $2\alpha$. b) Find an emvedded surface representing $3\alpha$. c) Find an emvedded surface representing $n\alpha$. This should recover your answers to a and b. These should be done in order.

d) Calculate the genus of your answer to c. Can you rhink of any ways to make it smaller?

Balarka Sen

I remember you told me (a), but didn't get to think about it in the flurry of Diff(S^n) problems.

I am doubting if there is such a thing as in (a).

Mike Miller

21:52

2) (This requires you know a very famous result in 4-manifold theory, which I will not name.) Let $P$ be the Poincare sphere.

I am quite confident that I have made no error above.

2a) Prove that $P$ does not embed into $S^4$. b) Prove that $P \# P$ doesn't embed into $S^4$ by essentially the same technique.

c) See why your technique does not work for $P \# \overline P$.

d) (Much harder than the rest) Prove that $P \# \overline P$ does not embed into $S^4$.

2), even given what theorem I'm referencing, is probably harder than 1).

Balarka Sen

Whoa. That'd be a month's worth of problems. Thanks!

Mike Miller

When you finish 1) and if you've worked a bit on 2a), feel free to ask what theorem I'm referencing. G'luck.

Balarka Sen

Definitely. Thanks again.

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$Mathematics$ Mathematics