Mathematics - 2016-11-14 (page 1 of 7)

Alessandro

00:00

As long as it doesn't distract you from the road!

DHMO

because I ran out of symbols

Ted Shifrin

Yeah, Danu ... You should get signs as a result, but nothing unexpected.

DHMO

a hyperfield lol

arctic tern

okay. unless you have the third operator interact with + and x in some way, the answer is completely trivial and uninteresting: the third operation is just any random old operation

DHMO

has anyone studied such things?

@arctictern i know

has anyone studied things with 3 operators?

Balarka Sen

00:01

why should I care about such a thing?

Danu

@TedShifrin Other question @Tedshifrin: I realized that in the definition for the Euler-Poincare characteristic for arbitrary bundles, we sum only over the $h^k(X,E)$ for k less or equal to dim_C X, instead of dim_R X. Why is that?

arctic tern

if you mean to ask "are there interesting algebraic structures with addition, multiplication, and more operations" then the answer is yes

DHMO

because why should we limit ourselves to 2 operators?

Danu

I mean, don't (p,q)-forms for p+q>dim_C X with values in E still make sense, too?

DHMO

@arc such as?

arctic tern

00:02

bbl dinner

Danu

Mike said something about holomorphic forms, but I'm not sure I completely get what's going on

arctic tern

plethysm in the ring of symmetric functions comes to mind

universal enveloping algebras have regular old associative multiplication on top of a lie bracket operation

Ted Shifrin

@Danu: You're doing the EC of the holo complex $\Omega^\dot(E)$.

Danu

@TedShifrin So perhaps my question is: Why am I actually always computing cohomology of holomorphic forms?

saturatedexpo

would defining exactly 0 devided by exactly 0 equals 1 be a definition in a field that could make sense?

(a field which mimics R)

DHMO

00:05

@saturatedexpo field doesn't have division

Alessandro

Vector spaces with an inner product are a maybe more familiar example of an algebraic structure with $3$ operations. (Yes, they require $2$ sets, but why should we limit the number of underlying sets if we don't want to limit the number of operations?)

Ted Shifrin

Because you're working with a resolution of $\mathscr O(E)$.

DHMO

you can't say 0•0^-1 = 1 because 0 is absorber

Ted Shifrin

Alessandro, you're asleep!

Alessandro

I'm supposed to be

DHMO

00:06

if 0=1 then every element is the same

Danu

@TedShifrin Right, I'm working with holomorphic vector bundles. But could I in principle also do something with non-holomorphic things?

DHMO

bye

saturatedexpo

@DHMO so that would only make sense for $\{0\}$ which is.. boring?

Alessandro

I'm going to for real this time or I'll end up sleeping in class tomorrow though!

saturatedexpo

ah well

even for only the 0 set that doesnt make sense

Ted Shifrin

00:08

Then you're back to the usual topological EC, @Danu.

saturatedexpo

because $1_0$ is 0, so that would be a tautology

Ted Shifrin

I guess smooth ...

Danu

Hmmkay

Balarka Sen

@TedShifrin Been flipping through Milnor's Morse theory lately, as a way to procrastinate on studying for the exams.

user2154420

Question about the quantum torus: I want to show the quantum torus, $A_q=\mathbb{C}[x^{\pm 1},y^{\pm 1}]/(xy-qyx)$ has no finite dimensional representations without using the fact that it's simple. Does this work?

If it had a finite representation then there would exist $\varphi: A_q\to\mathbb{M}_n(\mathbb{C})$ for some $n$, where $\varphi$ is a $\mathbb{C}$-algebra homomorphism. But then $XY-qYZ=0$ for some $X,Y\in\mathbb{M}_n(\mathbb{C})$. Then $\text{tr}(XY)(1-q)=0$. Since $q\neq 1$ this implies that $\text{tr}(XY)=0$ ...wait, is this necessarily contradiction?

Typo: $XY-qYX$

Oh, also there is a condition on $q$: nonzero and not a root of unity

Ted Shifrin

00:21

@Balarka, you always get an A in procrastination, but Milnor is good.

Balarka Sen

:P

Yeah, I like what he writes a lot

meow-mix

Must the subring of a ring with identity contain $1$?

if so, why?

saturatedexpo

@meow-mix just try to find a subring in a big ring

@meow-mix proofwiki.org/wiki/Definition:Subring

meow-mix

im confused

how does that answer my question

user2154420

@arctictern Any ideas?

saturatedexpo

00:26

well, can you give me a subring of a "sufficent" big ring, that does contain $1_R$, then the assumtion is near that it has to be. The proof WILL 100% contain the definitions

Ted Shifrin

Depends on the definition. Is $2\Bbb Z$ a ring?

SteamyRoot

Just a bit of a thought, but

Take the ring

$R = \{0,1\}$. And then take the subring $\{0\}$?

Ted Shifrin

That's not a ring, by most definitions.

saturatedexpo

@TedShifrin you mean 2Z?

meow-mix

@saturatedexpo use chatjax

Ted Shifrin

00:30

I meant what Steamy wrote.

meow-mix

oh nvm

saturatedexpo

@meow-mix is use it ;) (why you think i dont?) im just too lazy to write 2\mathhbb{Z}

Ted Shifrin

@meow: Did I answer your question?

SteamyRoot

Zero ring

In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rng of square zero, i.e., a rng in which xy = 0 for all x and y. This article refers to the one-element ring.) In the category of rings, the zero ring is the terminal object, whereas the ring of integers Z is the initial object. == Definition == The zero ring, denoted {0} or simply 0, consists of the one-element set {0} with the operations + and · defined so that 0 + 0 = 0 and 0 · 0 = 0. == P...

meow-mix

that doesnt have an identity

actually

it does

SteamyRoot

00:32

The zero ring has an identity, but not the identity of the ring it's a subring of

Unless you really want a subring without identity itself

?

Ted Shifrin

Steamy: as I said originally, it's all a matter of definitions. Many good texts require $1$, so that is ruled out.

Anyhow, I'm out.

meow-mix

a ring with identity is supposed to be where$ 1 \neq 0$ right??

SteamyRoot

Fair enough

user2154420

Any thoughts on $A_q$ question? :D

meow-mix

so, is my statement true?

saturatedexpo

00:34

we don't know, prove it to us

meow-mix

>.>

if i knew how to prove it

i wouldnt come here asking if it was true...

saturatedexpo

if i say its true, would you believe me?

meow-mix

given sufficient evidence, yes

saturatedexpo

well i guess on that part i must pass ;-)

Fargle

hello @Ted

(And hello to the chat at large.)

Oh he left.

saturatedexpo

00:40

@Fargle hi :)

meow-mix

hi fargle

Fargle

How is everyone doing?

meow-mix

hey guys if i make this year's IMO team i'll give you all a shoutout /s

@fargle im good how about you

Fargle

@meow-mix Quite well.

Maks

01:01

Guys, you know anything about algorithms? Like programming ones, imperative programming algorithms

Fargle

@Maks What do you need?

Balarka Sen

@Fargle hey

Maks

"Derivate a program that calculates the lower possible $ x $ that complies with $ x^3 + x \geq N $

Fargle

@BalarkaSen Howdy.

Balarka Sen

What's new?

Ted Shifrin

01:09

@Fargle, no fair saying hello after I've left!

Fargle

@TedShifrin Haha, my bad!

Semiclassical

It's not efficient, but wouldn't the simplest just be: Start at some $x_0$, test $x_0^3+x_0\geq N$, and increment $x_0$ if it fails. Repeat until you get a value which satisfies the inequality

Ted Shifrin

@meow: Yes. $0\ne 1$.

Maks

@Semiclassical yes, but on coding (I think its more "in logic" ) , you have to prove that it fulfills a post and pre condition, and do some other steps on the middle, so you end up with a code

Ted Shifrin

@Fargle: You have analysis stuff typed up

Semiclassical

01:11

eh. if you've done the above, you'll have shown that some $x$ satisfies the inequality but not $x-1$.

Ted Shifrin

email

Fargle

@TedShifrin It's not quite done but I'll show you what I have so far.

Ted Shifrin

Make it interesting stuff, please ;)

Fargle

@TedShifrin It's rudimentary! I'm still an undergrad. >_>

Ted Shifrin

There's tons of interesting problems in Rudin (and, immodestly, in my books).

But there are also duller ones :)

Fargle

01:13

@TedShifrin I just went through all of them. I'm going to transcribe the problems themselves for ease of reading

Ted Shifrin

LaTeX just a few you want me to criticize.

saturatedexpo

in which context(s) does $10^9\approx 10^{10}$?

Ted Shifrin

Only when a factor of 10 is negligible.

Balarka Sen

@Fargle You thinking about analysis?

Fargle

@BalarkaSen A bit. I'm not very far.

Ted Shifrin

01:16

Not counting psycho- ...

Fargle

I've got a better grasp of topology than I do of analysis, funnily enough.

Balarka Sen

He studies that too? Good god.

Fargle

Which seems about as backwards as it is.

@BalarkaSen Oh lord no.

saturatedexpo

@Fargle i got an equal grasp of both. (hint: i dont know much of either xd)

Ted Shifrin

I favor learning analysis before topology, as most

of Point set generalizes results/problems in analysis.

Balarka Sen

01:17

I learnt topology before analysis

saturatedexpo

@BalarkaSen do you feel ok with it?

Balarka Sen

but I learnt everything backwards so forget I said that

@saturatedexpo Sure

Fargle

Certainly I should have. But I was allowed into a graduate-level topology course teaching Munkres when I went to UT (which, as you may remember @Ted, Dr. Knox taught me)

Ted Shifrin

Yup. I remember.

But he took two courses simultaneously from

me as his baptism by fire :)

He would tell you to learn multivariable thoroughly and rigorously :)

Balarka Sen

Still streetfighting with your phone I see, @Ted

Ted Shifrin

01:20

Ugh, yes.

Can't even use my iPad :(

Fargle

@TedShifrin A sentiment that echoes yours. Perhaps I should listen.

Ted Shifrin

Go to sleep, Balarka!

Fargle

I've still got that youngster's impulse to shut out advice.

Balarka Sen

Multivariable calculus is cool and mysterious

Ted Shifrin

You'll pay for it. Even Balarka finally succumbed.

Fargle

01:23

@TedShifrin I do intend to heed your advice--it's a question of "when".

No time like the present.

Balarka Sen

Speaking of multcalc, where's Brody? I gave him homework for the next time we talk.

saturatedexpo

my understanding of topology: loops. Now laugh

Ted Shifrin

Brody's not very far, Balarka.

Fargle

@saturatedexpo That's the base part of algebraic topology, but there's a lot of point-set knowledge before that.

Ted Shifrin

I have several generations of ex-students who stubbornly didn't heed, @Fargle :)

Fargle

01:26

i.e. what a topology actually is, what connectedness and compactness are, the separation axioms, the Urysohn lemma and Tychonoff theorem...

Balarka Sen

@TedShifrin Were he supposed to meet with you?

Ted Shifrin

I meant far in the material, Balarka. ... up to him if we meet in GA or not.

Balarka Sen

Oh. Yeah, he's like learning dot products and I am telling him what a Riemannian metric is... :P

Semiclassical

Whenever I hear people talk about point-set, I just can't stay interested.

Fargle

@Semiclassical It's definitely dry, but I like it. Rather like poorly prepared turkey.

Semiclassical

01:28

hah

Ted Shifrin

@Fargle: Second countable and paracompact are important for geom/top too.

Fargle

@BalarkaSen Be careful you don't drown him before he learns to tread water!

@TedShifrin Yeah, that's what the dots were. ;)

Balarka Sen

@Fargle Nah, he knows all of this albeit from a computations-focused point of view.

Ted Shifrin

Don't behave like Mike, Balarka.

Ali Caglayan

hello @TedShifrin

Fargle

01:29

@Ted is like the personified wagging finger of MSE chat.

saturatedexpo

are computations even that necassary if the right approach is choosen?

Ted Shifrin

Wow, Ali's up all night again.

Balarka Sen

@Fargle lol

Ted Shifrin

@Fargle Not sure that's a compliment!

Ali Caglayan

@TedShifrin i slept all day today so it cancels out

Balarka Sen

01:30

wagging finger with eight and a half eyes

Fargle

@TedShifrin I mean it with a great deal of respect, Professor.

Ted Shifrin

Not exactly, Ali. It compounds :)

Fargle

Every group needs a wagging finger, just as every group needs an identity.

Ted Shifrin

Ha ha @Balarka @Fargle

saturatedexpo

im the additive neutral element

Ted Shifrin

01:31

LOL

Change your name to null, then.

saturatedexpo

haha

Fargle

I'd be shocked if that weren't already taken, @Ted

Ted Shifrin

Some variant can be found, I'm sure.

Balarka Sen

I suggest more creative usernames

Ali Caglayan

@TedShifrin i'm studying linear algebra atm so everything I do must be linear

Ted Shifrin

01:33

Make sure you learn interesting stuff :)

Ali Caglayan

@TedShifrin null2

saturatedexpo

success

mmh

it displays satur.. , but on my profile im now null lol

Ted Shifrin

Nullify? Nullity?

Ali Caglayan

@TedShifrin i'm working through 'linear algebra done right'

saturatedexpo

math.stackexchange.com/users/346682/null?tab=profile

its a me, null :D

Ted Shifrin

01:35

Not my fav, but ok, Ali. I think my books have some better exercises, even though his book

is more sophisticated.

Ali Caglayan

@TedShifrin it was a Christmas present so I can't really change it, but I have been getting through it quickly which is good (and doing exercises)

Ted Shifrin

@Ali: If you remind me, I'll send you some exercises at some point. Plus, Sheldon's book assumes you actually

know basic computational stuff.

Ali Caglayan

@ted what is basic computation stuff consisted of

@saturatedexpo your name won't change in chat until tomorrow

saturatedexpo

@AliCaglayan thanks^^

Ali Caglayan

axlers book is also the only colour print maths book I own that isn't high school algebra

Ted Shifrin

01:40

Echelon form, Finding bases for kernels and images, eigenspaces, diagonalization, matrix exponential, etc.

There's colors in my diff geo :)

Ali Caglayan

@TedShifrin I thought this was common knowledge among highschoolers?

meow-mix

:[ i only got a 22/25 on the amc 8

Ted Shifrin

@AliCaglayan hell not in this country.

Semiclassical

echelon form would be

Ted Shifrin

No.

Semiclassical

01:42

beyond that, nope

Ali Caglayan

@TedShifrin maybe in india, definitely not in countries beginning with u

Semiclassical

I'm pretty sure I did see RREF in high school.

saturatedexpo

what is the most hipster sign or definition for 0 lol. additive identity doesn't quite strike xd

Ted Shifrin

Maybe 2x2 Cramer's rule.

Semiclassical

hmm, maybe

Ted Shifrin

01:42

You're not everyone.

Semiclassical

true.

meow-mix

@TedShifrin so was i right? does a subring of a ring with identity always contain the identity?

Ali Caglayan

In high school I came across diagonalisation but that was it

Most people here don't see matricies until uni

Semiclassical

I remember 'augmented matrices' from high school more than matrices proper

saturatedexpo

we used matrices to calculate populations of rabbits

(or trees, or or)

Ted Shifrin

01:43

No, @meow. I gave you a counterexample .

meow-mix

i didn't see it

Semiclassical

Which I suppose means we were really doing more 'systems of equations' than matrices per se.

Ted Shifrin

Markov processes... Cool beans.

Semiclassical

I think you do see Cramer's rule in high school, or at least can.

Ted Shifrin

Yes, you did, because saturated repeated it without ChatJax.

Ali Caglayan

01:45

I have come across most things in axlers book before, but it is night to make sure there are no gaps

Semiclassical

I'm not sure what I learned in high school math per se, though, versus stuff I learned for math team (which I did do, not surprisingly)

meow-mix

so how am i supposed to prove that the intersection of subrings of a ring with identity is non-empty?

Ali Caglayan

I can't remember who or where said it but I heard that differential geometry consists of people trying to turn geometry problems into linear algebra

meow-mix

what

Ted Shifrin

Not true, Ali, but linear alg is essential.

Dunno, meow.

Ali Caglayan

01:48

@meow-mix do subrings contain the identity?

Ted Shifrin

No, Ali. Not if rings needn't.

meow-mix

oh wow

the ring containing all the subrings doesnt even have identity in this problem

Ali Caglayan

oh so these are rngs

Ted Shifrin

I can see it's true for examples ($\Bbb Z$, $k[x,y]$).

meow-mix

its just " the finite intersection of a subcollection of subrings of a ring R is a subring of R"

arctic tern

01:50

@user2154420 Take determinants of XY=qYX, conclude one of X or Y is singular, but then this contradicts all of x,x^-1,y,y^-1 having images in End(V).

Ted Shifrin

Oh, do they allow $0$ to be a ring?

meow-mix

yeah

Ted Shifrin

Just check the requisite properties — like proving intersection of vector subspaces is a subspace.

Ali Caglayan

@meow-mix is the empty set a ring?

Ted Shifrin

No. it won't be empty!

Ali Caglayan

01:52

then showing the intersection of two subrings is also a ring is sufficient

meow-mix

in my book, "A subring is an additive subgroup closed under multiplication"

Ted Shifrin

All subrings contain $0$.

So do what I said :)

meow-mix

oh my god i just realized

they all contain $0$ because subrings are additive subgroups

lol sorry for the hassle

Ali Caglayan

i count 13 cans of energy drinks on my bed side table

saturatedexpo

how was HelloWorld not already taken yet?

Ali Caglayan

01:56

a group of order 13 is cyclic

so I will keep drinking energy drinks with no end

my logic is flawless

saturatedexpo

@AliCaglayan reality beats logic tho...

Ali Caglayan

Thats logic abuse

Transcript for

$Mathematics$ Mathematics