Mathematics - 2016-04-11 (page 3 of 3)

Mary Star

20:01

Let $R$ be a finite ring. If $R$ is a division ring, does it hold that $R$ is non-trivial?

Andrew Thompson

Define non-trivial.

IdiotfromPrinceton

hello

Andrew Thompson

its me

IdiotfromPrinceton

is the group of odd permutations a subgroup?

Andrew Thompson

no

IdiotfromPrinceton

20:04

why?

Mambo

what is the identity?

IdiotfromPrinceton

$e$

Semiclassical

is $e$ an odd permutation?

IdiotfromPrinceton

i dont know

Mary Star

@AndrewThompson It is not the zero-ring.

Mambo

20:07

By def?

IdiotfromPrinceton

i cannot understand it

Andrew Thompson

@MaryStar In general dr's are defined to be nonzero.

Mambo

Do you know what a transposition i?

IdiotfromPrinceton

yes I do,

Mambo

How many number of transpositions does $e$ have

IdiotfromPrinceton

20:09

do you mean $$e=(aa^{-1})$$

hhh

Is someone able to explain binomial ideals that are not primary -- non-toric binomial ideals versus toric ideals. Moved this here, I am trying to visualise them in M2.

Mambo

Also for any $a$, $aa^{-1}$ will be what permutation ?

Semiclassical

more direct: the product of two odd permutations is an even permutation, as is the product of two even permutations. but the product of an even and an odd permutation is odd.

Mary Star

@AndrewThompson Ah ok... Thanks!! :-)

Mambo

$e$ has zero number of transpositions so even permutation

IdiotfromPrinceton

20:14

my head is going to blow up now :( so if I can write $e=aa^{-1}$, then there is 1 2-cycle, 1 is odd, so it is an odd permutation

Mambo

?

If $a$ is even , $a^{-1}$ is also even

Similarly for odd

Compute it

IdiotfromPrinceton

ok. i see. thanks a lot for your time

Mambo

By the way why are you idiotfromprinceton

IdiotfromPrinceton

because i cannot do mathematics well

anyway

whatever man

i feel that way most of the time

so at the wake of a minute, i needed to choose a name, and that name came to my mind

Mambo

You are just starting

Long way to go

Dont be prejudiced

IdiotfromPrinceton

20:20

i look at the ppl around me, they do it as it taking it out of their pocket, it is so frustrating

thanks a lot buddy

Mambo

Ignore them and study

Bye man

Akiva Weinberger

20:48

@IdiotfromPrinceton $e=(1~2)(1~2)$. That's an even number of transpositions, so it's even.

You can also view it as the composition of zero transpositions, and zero is an even number.

Another reason why the set of odd permutations is not a subgroup: $(1~2~3)(2~3~4)=(1~2)(3~4)$, so it's not closed under composition.

Krijn

@MikeMiller Thanks! I'll check it out!

Mike Miller

Sure. I've forgotten all my potential theory.

Krijn

You know a lot of different areas of math tho

Quite amazing sometimes

Mike Miller

Thanks. I wish I knew half of what some of my colleagues do.

What course is this for?

Akiva Weinberger

So, there are closed Jordan curves such that every ray from the origin intersects them infinitely many times. (I'm pretty sure.) Do these have a name?

(Assuming they exist)

user19405892

20:57

hi

Mike Miller

They're all topologically equivalent, so my name for them is "Jordan curve" :)

Krijn

@MikeMiller Riemann Surfaces. The course isn't bad but I thought it would follow an algebraic approach :/

Akiva Weinberger

@user19405892 Hello

user19405892

If $C$ is a compact subset of a CW complex does it follow that $C$ is contained in a finite union of open cells?

Mike Miller

There isn't really an algebraic approach, you need to get your hands dirty with some analysis eventually. You'll use all this harmonic analysis to prove the uniformization theorem.

There are definitely proofs I prefer to the potential theory proof, since I don't really understand it.

Krijn

20:59

Maybe approach is the wrong word, but you can definitely do more algebra on Riemann Surfaces than we are doing

Mike Miller

Sure, but supposing you want to do algebraic geometry, you'd need to know that every compact Riemann surface is projective, and that's not a triviality by any means.

Akiva Weinberger

@user19405892 My guess is yes

user19405892

we know that $X$ has the weak topology with respect to its skeleta

We can prove by induction that $A \cap X^n$ is closed in $X^n$ where $A \subseteq S$ and $S = \{x_1,x_2,\ldots\}$

such that each element of $S$ is in a different open cell

Krijn

@MikeMiller Is it me or is Conway's definition of a Green's function different from the usual definition?

Mike Miller

What's the usual definition?

Krijn

21:13

It is me.

Akiva Weinberger

@user19405892 See Proposition A.1 from here

Mike Miller

That was a quick mystery.

Krijn

In relation to the Dirichlet problem, a Green's function for a domain is the function such that $u(z) = \int f(s) G(x,s) ds$ is the solution to the dirichlet problem

That was a quick reference.

Mike Miller

Sure, but that follows because convolution with the Dirac delta is the identity.

and the laplacian of that log is the delta function at that point

Krijn

I think it works out

Yay!

Obliv

21:26

$a$ > $b$ the g.c.d of $a$ and $b$ is $d$. the g.c.d of $a+b$ and $b$ is still $d$. $a - b$ and $b$ still $d$. $10a$ and $b$ is still $d$. however, when you multiply $b$ the g.c.d changes. How can someone describe the relationship between the g.c.d and the smaller number $b$?

for ex: the g.c.d of $a$ and $2b$ is $2d$.

Krijn

@MikeMiller How is the Laplacian of $\log |z - z_0| $ the delta function at that point?

I'm a retard.

Mike Miller

@Krijn If that's $g$, integrate $\int_{|z| \geq \varepsilon} (\Delta f) g$ by parts to see that, as $\varepsilon$ goes to zero, this goes to $f(0)$.

Because $g$ is integrable on the whole region, the limit as this integral goes to zero is the same as the integral over the whole region.

Krijn

21:50

That finishes my math for today :)

arctic tern

22:10

@MaryStar pick an element of $xI$, pick an element of $R$, multiply them together, explain why the result is also in $xI$. will need to use the associative property. hint: what do elements of $xI$ look like?

IdiotfromPrinceton

hi

i have this proposition i cannot understand, it says 3 cycles in S_4 are conjugate. can someone pls clarify this?

Mary Star

The elements of $xI$ are of the form $xi, i\in I$.
Let $r\in R$.
So, $xir\in xI$, because $I$ is a right ideal and so $ir\in I$, right? @arctictern

arctic tern

yes

Benjamin R

Hi Everyone

Ted Shifrin

hi anon, @BenjaminR

Benjamin R

22:22

Given that $\mathbb{Q} \subset \mathbb{R}$

Ted Shifrin

@IdiotfromPrinceton: All $k$-cycles in $S_n$ are conjugate. Do you know the rule for what happens when you conjugate a $k$-cycle by a permutation?

Benjamin R

surely the difference $\mathbb{Q} - \mathbb{R} = \emptyset$?

Q - R I mean

Ted Shifrin

Are you asking, @Benjamin?

Yes, surely.

Benjamin R

Yeah... I kinda am

Thanks

Ted Shifrin

Since $A-B$ means the set of things in $A$ that are not in $B$, yes, surely.

Benjamin R

22:24

Sometimes my Linear Algebra prof asks such seemingly basic questions that are so "obvious" that I doubt myself

Ted Shifrin

Some things in linear algebra are very basic, but others aren't, and sometimes students get thrown off, yes.

Benjamin R

Thanks Ted

IdiotfromPrinceton

@TedShifrin yes I know the rule for doing it

Ted Shifrin

OK, @IdiotfromPrinceton. Then you know that when you do it, a $k$-cycle transforms into another $k$-cycle.

IdiotfromPrinceton

do we assume that k is smaller than n in that case?

arctic tern

22:26

@TedShifrin hi

Ted Shifrin

And you can cook up a permutation that will take any given one, say $(1\,2\,3)$ to any other one, say $(4\,2\,5)$.

Yes, you can't have a cycle of length larger than $n$. So $k\le n$.

IdiotfromPrinceton

ok alright. Thanks a lot for your attention. really appreciate it

Adeek

@TedShifrin hi

Ted Shifrin

I wish you'd change your name, @IdiotfromPrinceton :P

Adeek

I have a quick question

Ted Shifrin

22:27

Don't have to be like Karim, who changes his name every month.

Adeek

haha

I have a question @TedShifrin

Ted Shifrin

I'll probably ignore it, Karim, but go ahead.

Adeek

Suppose you consider for example the two sheeted covering $P : \tilde{X} \rightarrow X$ and consider $\sigma : \Delta^n \rightarrow X$

I want to show that each $\sigma$ has at least two lifts

here is how

Ted Shifrin

I don't agree with that.

Adeek

hmm ?

Ted Shifrin

22:31

Oh, I guess I do. Take two different lifts corresponding to the preimages of the basepoint.

Adeek

yeah

exactly

Consider the constant map $K : \Delta^n \rightarrow X$ there is precisely two ways to lift this map

Since $\Delta^n$ is contractible then $\sigma$ is homotopic to the K

Ted Shifrin

Yes, although the image may be far from contractible in $X$.

Adeek

by homotopy lifting lemma applied to both of the lifts then we get also that $\sigma$ must also have atleast two lifts.

hmm?

Ted Shifrin

Can you think of an example, however, where both lifts cover the two points mapping down to the basepoint?

Adeek

what do you mean ?

that is both lifts give the same base points ?

Ted Shifrin

22:40

No, but both lifts hit both the points covering the basepoint. That's why I hesitated at the beginning.

Adeek

oh I see

Mary Star

When $I$ is a right ideal and $y\in I$ with $y\neq 1$, does it holds that $1-y$ has no right inverse?

Ted Shifrin

Why do you think that's true, @MaryStar?

Mary Star

If this is true, I want to apply it at an exercise... I think I have seen that statement somewhere, but I don't remember if $1-y$ has or not a right inverse.

Suppose that $1-y$ has a right inverse, then it exists $a\in I$ such that $(1-y)a=1$.
What do we get from that? @TedShifrin

Ted Shifrin

Have you tried examples?

Mary Star

22:51

No, what ideal could we take? @TedShifrin

Ted Shifrin

I'm not going to do this for you. Try simple examples.

Benjamin R

how do you prove that $(-a)x = -(ax) = a(-x)$ for any vector $x$ and scalar $a$?

I mean, it's a statement that is true...

arctic tern

the same way you prove it for scalars :P

Benjamin R

okay, just go blatant, fair enough

Ted Shifrin

You need to not assume that $(-1)a = -a$, by the way. That takes a proof :P

arctic tern

22:54

additive inverses are unique (know why?), showing two things are both additive inverses of the same thing thus entails those two things are equal

Benjamin R

Hi Ted, I have done that one already, thankfully

ah, yep, so use the same approach, gotcha

Ted Shifrin

Aha. What sort of course are you taking? A very pedantic linear algebra course?

arctic tern

heh

Benjamin R

Yes.

Ted Shifrin

Probably with no geometry whatsoever in it :(

Benjamin R

22:56

I mean, it is right and important to be able to do these things, I completely concede that

Ted – BINGO.

Ted Shifrin

My courses used to be by default very geometric, which bugs some students.

Adeek

I am the following courses @TedShifrin in my first semester

Ted Shifrin

You told me, Karim, and I asked why no analysis.

Benjamin R

And taught by a lecturer who says "I never attended class, I didn't understand things when I was first exposed to it." as if because he had a crap teacher makes it okay to continue the tradition...

Adeek

No I will take analysis actuallyu

Ted Shifrin

22:58

What book are you using, @Benjamin?

Adeek

I am required to take it

Ted Shifrin

Good, Karim.

Adeek

Classical Banach spaces. Hahn-Banach, open mapping and closed graphs theorems. Hilbert spaces, orthonormal bases. Elements of spectral theory, spectra of compact operators, spectral theorem for compact self-adjoint operators. Prerequisite: MATH 417. Corequisite: MATH 447.

what book would you recommend for this ?

I am gonna study over the summer

Benjamin R

Linear Algebra by Freidburg, Insel, Spence

Ted Shifrin

Wait, you're doing that before you take basic Lebesgue integration and measure theory?

Benjamin R

22:59

Lecturer admits it is really only useful for him to plan his lectures.

Adeek

I did take measure theory and Lebesgue integration

Ted Shifrin

Ah, @Benjamin, that's meant to be an upper-division course for people who've already had computational linear algebra. Is that your situation?

Adeek

But I forgot it

Benjamin R

No Ted, it's just a lecturer $\neq$ teacher situation. That's okay, I have notes from a much

Ted Shifrin

Doesn't hurt to take basic, important stuff again once you're a grad student, especially since you must pass qualifying exams, Karim.

Adeek

23:00

Linear algebra by Freidburg that is for me ? @BenjaminR ?

Benjamin R

better lecturer on campus that takes care of me.

Ted Shifrin

No, that's answering my question, Karim.

Benjamin R

Yes, sorry Karim, just responding to Ted

Ted Shifrin

@Benjamin: You're welcome to look at my books :P Also my YouTube lectures on linear algebra and (rigorous) multivariable calculus. The linear algebra stuff is mostly free-standing.

But on average professors suck, @Benjamin. Sorry to break it to you.

Adeek

measure theory is in winter

I will take it in winter

Benjamin R

23:01

I will, Ted, thanks. Gotta go to said class now though.

Ted Shifrin

Several of my former students are discovering that now they're grad students TAing for professors that don't care. It's driving them nuts.

Have fun, @Benjamin. :)

Adeek

@TedShifrin what book would you recommend for that class ?

Benjamin R

@TedShifrin I only call a Prof by Professor when he really deserves it. Like Strang at MIT, for example.

Ted Shifrin

Strang is a dear friend of mine and influenced my linear algebra writing, but I actually don't like his teaching style :P

Karim: I'm no expert on that stuff. Rudin's Functional Analysis is certainly a standard.

A classic is Riesz-Nagy.

Why don't you see what your professor uses?

Benjamin R

:)

Adeek

23:03

it is not posted yet

and it will be posted like 1 month before not yet

I am finishing this semester on 28th of april

I would like to start preparing for grad school on may 1st

Ted Shifrin

Well, preparing should mean filling in things like multivariable analysis, which you've never learned. Rather than getting ahead to courses you're taking in the future.

Adeek

yeah

I have 4 month to study and I will only dedicate 1 day for break

Ted Shifrin

Well, I would advise you to take a real break, too. Grad school is a long haul.

Adeek

I took a break in december. I want to do really well first year of masters, then I will take a break for a month during the summer after next one.

Ted Shifrin

I'm not saying to take the whole summer off. I'm saying you need a week or a bit more for your brain to recharge, or you're going to just burn out.

Adeek

23:07

okay yes your right

I will take a week off after the exams are over

Ted Shifrin

Yeah, Ted won one.

Adeek

and then go back to math :D

I have been having back problems, because of carrying heavy books to library, which sucks.

Ted Shifrin

I'm having horrendous back problems. Don't start having them at your age ...

Adeek

I will just carry them with my hand.

Ted Shifrin

Do lots of exercises.

Adeek

23:08

Yeah I will just carry things with my hand not with stupid backpack

Ted Shifrin

I mean physical exercises, not math exercises.

Adeek

haha

yeah

Ted Shifrin

That still stresses your back, btw.

Adeek

Yeah after this semester is over I will go back to excerising.

Ted Shifrin

Anyhow, I want to think about an interesting question on main, so I'm leaving for now.

Adeek

23:09

It is just I have to do stupid electives which takes a lot of time.

ok ok

I am off to studying 2

btwww

Ted Shifrin

@Benjamin: Talk to you more later.

Adeek

I am sure you will like this 1 sec ted before you go

I am also taking this course

"
Finite dimensional manifolds/submanifolds; tangent bundle, differential, inverse, and implicit function theorems, partitions of unity; imbeddings, immersions, submersions; vector fields and associated flows; Lie derivative, Lie bracket; tensor analysis, differential forms, orientation, integration, Stokes' theorem; basics of smooth bundle theory, Riemannian metrics; notion of a Lie group with basic examples, smooth Lie group actions, principal bundles"

see something geometrical you will like

Ted Shifrin

Good, but that's why I'm telling you to learn multivariable analysis.

Adeek

yeah

Ted Shifrin

Most math students have no idea what the derivative is in higher dimensions.

Adeek

23:11

yeah

Jessy Cat

23:24

0

Q: Sequence of functions that fails certain conditions of Arzela-Ascoli theorem

For a closed, bounded interval $[a,b]$, let $\{ f_{n}\}$ be a sequence in $C[a,b]$. If $\{f_{n}\}$ is equicontinuous, does $\{f_{n}\}$ necessarily have a uniformly convergent subsequence? I would think not, because according to the Arzela-Ascoli Theorem, $\{f_{n} \}$ also needs to be uniformly ...

Jess's got questions again. Take a look if you can :)

Semiclassical

23:52

hi @chat

Akiva Weinberger

hello

So, $\Bbb R$ is a metric space, and a totally ordered space.

I think it's the only thing that's both? Well, it and its subsets

hhh

Is sum of toric ideals ideal?

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$Mathematics$ Mathematics