sigh. i guess i should get back to work

Wash off your knuckles @Mike

@TedShifrin I didn't give you a knuckle sandwich, I only offered

Ah ...

@TedShifrin I wonder how little effort I need to do to get a D in this class

Aren't those just free nowadays?

What class? I'm giving a pile of F's in my diff geo, for the first time ever.

Sociology.

@TedShifrin CHEESUS OUR GRATE LORD.

If the prof thinks you're blowing it/him off, don't be so sure.

Say what @Pedro?

"Remember, essays have introductions, body paragraphs, conclusions, and flow well, please keep this as a structured essay. For a short essay, stick to the traditional 5 paragraph essay."

@TedShifrin You giving Fs?

Hmm... No, I'm fairly sure they don't, and haven't since I was in high school.

Comma splice @ that quote

Yes @Pedro

@TedShifrin Shouldn't they just stop taking the class for that?

@TedShifrin Maybe I should email the professor letting her know.

Tell her I said so, @Mike

Don't ask me, @Pedro. I'm appalled by the attitudes of a handful.

@TedShifrin Oh? What happened?

No effort, minimal work. F's on tests, homework. They seem to think that because they need the class to graduate I'll be Santa Claus.

@TedShifrin Well, that's what I'm hoping for sociology.

Other weak students making big efforts. Not sure they all will make it to C.

Well, @Mike, it doesn't work with me. Good I'm retiring imminently.

@TedShifrin I'll still hand in all my essays and do all the work. I just don't intend to write my magnum opus.

Fair enough for a core course @Mike

Now time to write this 3-page, double-spaced "essay" (lol) and insert the required references on Monday.

Ok, do it.

See ya.

See ya.

everyone took a look on this picture: i.stack.imgur.com/yzOKo.png

@seaturtles

what about it

@PedroTamaroff yes?

@seaturtles Do you know what an $i$-rowed minor of an $m\times n$ matrix is?

dunno what i-rowed means

Would it be a submatrix obtained by selecting $i$ rows?

Let me show you.

that would make sense

THM: Let $A$ be an $m\times n$ matrix with entried in a PID $D$, suppose $A$ as (row) rank $r$. For each $i\leqslant r$; let $\Delta_i$ be a gcd of the $i$-rowed minors of $A$ (I think these are determinants!). Then any set of invariant factors for $A$ differ by units from the elements $d_1=\Delta_1$, $d_2=\Delta_1^{-1}\Delta_2, \ldots, d_r=\Delta_{r-1}^{-1}\Delta_r$

@seaturtles

Yes, he means determinant of submatrices, I think.

"We recall that a matrix $A$ is said to be of determinantal rank $r$ if there exists a non-zero $r$-rowed minor of $A$ but every $(r+1)$-rowed minor of $A$ is zero."

@seaturtles computationally inefficient, it seems, though.

Well, unless one is witty and obtains coprime stuff, say.

@seaturtles Ugh, I have a question.

=)

k

@seaturtles After explaining the idea with the Euclidean case, Jacobson moves on to the PID general case.

Now, instead of using the Euclidean norm, he uses the "length" of the elements: the number of prime factors (with multiplicity, I think).

He says he will also use matrices of the form $$\begin{pmatrix}x&t&0 &\cdots\\y&s&0&\cdots\\0&0&1&\cdots\\0&0&0&\ddots\end{pmatrix}$$

Now, he uses that something is invertible.

But I don't see what's the reason this is so.

hi @karl

hey @mike

Last paragraph starts with PIDs

@karl if i defeat you in a duel do I gain your algebra ability?

Here he uses $\gcd(a,b)=d$ is invertible. Why is that so?

pedro pls

@Mike if you focused solely on algebra, a duel would be unnecessary, I am sure

@Karl I'm sure, but I feel it might be faster that way

@seaturtles Ping me when you want.

@Mike ok where do we meet

can you swing by my place

sounds fantastic; I'll be there sometime next week

@KarlKronenfeld If you can be of any help, I'd be glad.

wait so what do you get out of this

great

@PedroTamaroff how far up do I have to scroll?

@Mike ;)

@karl what do you win if you win the duel

is the goal just to destroy me

@KarlKronenfeld The two links.

Around 22:25.

@Mike your claim to Ted's library

@seaturtles For example, nothing forbids from having $a_{11}=2\cdot 3$ and $a_{12}=2\cdot 5$. Then $(a,b)=2$ is not invertible over $\Bbb Z$.

the duel is off

Oh.

@seaturtles I'm being silly.

He means divide by $d$.

Not invert it.

Sorry.

=)

Anyways, I'm watching Ender's Game now.

y

@KarlKronenfeld @Mike @seaturtles

I have a question for ya'll.

Does your handwritting change every now and then?

not really

For example, I have changed the way I write down my $j$s.

I change it occasionally for clarity when writing for others

every once in awhile I learn how to write a new strange letter, like $\wp$ or $\frak P$

Hehe, I love writing in fraktur.

I cross my "Z"s, curve my "t"s, and such

I pronounce them both out loud as "strange P"

@Mike Ah, but I mean when you write your notes say.

@seaturtles strrrrrange P. russian accent

once i change my writing it changes everywhere

The first one is weiertrass P

indeed

holy cow that second letter isn't a B

holy smokes

:)

I have read it as B all my life.

inorite

never wondered why $\frak P\mid p$? :)

no.

i just accepted it

@PedroTamaroff I've been writing my notes in a small (1/3-1/2 of a line), precise cursive that has changed only as I have improved at it.

@KarlKronenfeld Ah! Well, my handwriting has been decreasing size too. I cannot afford to waste too much paper, mainly because of room issues.

There was a time I didn't write in both faces, now I do.

my notes don't matter since if they matter i type them.

why hasn't someone made a counterexamples in alggeo book yet

@PedroTamaroff You should type up your notes and save them.

This means that the paper is a temporary storage.

I typed my notes for my first two courses at uni

That was a terribly bad idea

I am very close to typing everything up these days. It is much, much more convenient.

@FernandoMartin why?

@FernandoMartin your first two courses probably weren't worth keeping notes for, long-term :p

I spent a lot of time typing them up and I gained nothing in return

@Mike exactly

every course I've taken so far was based on books I can get on the internet

so it's a waste of time to type up my notes

I still take handwritten notes in class though

just because you have a reference deosn't mean notes aren't worth it

the point of notes, to me (when speaking of classes) isn't to be able to look up theorems

it's to internalize them

But I already have handwritten notes for that :)

I handwrite 10x faster than I type

meh

only time i would hesitate to type it too is if there are diagrams

besides, I get really perfectionist when dealing with .tex documents

I tend to rewrite every sentence like 10 times until it sounds good

nerd

I like being able to prove and insert lemmas that come logically earlier only after completing the proof of the main theorem: keeps me focused

you hurt my feelings @Mike

that's how it goes.

@FernandoMartin That's something I have to actively resist doing.

another thing I like to do is when reading a book, take notes to pare out anything I don't find interesting, or cull statements in proofs that I find trivial

and then when I type those notes up I end up thinking "how the hell do I do that step?" and in figuring out how it went again, i realize it's nontrivial, but now have an appreciation for the argument and will remember it

Do you guys have really shitty long-term memory?

I can remember most results from earlier courses and some vague idea of their proofs

But I can recall nearly no proof by heart

no? but that doesn't mean it's not helpful

if i don't spend some time thinking about a proof i'm not going to remember it.

unless it's analysis

then i'll remember it because the first step is to fix epsilon and the second step is to find an appropriate delta

hahaha

@FernandoMartin but seriously you should check some of the proofs in the c* algebras class i'm in. if i didn't spend time thinking about them i wouldn't even remember the IDEA in a month

either way i'm not remembering the details

Yeah I know but

even for "small" results

let me give you an example

I took a standard complex analysis course last year

exactly a year ago

I know what the identity theorem says

I don't even know what that theorem IS

if $f$ and $g$ are holomorphic and $f-g$ vanishes in some set that accumulates then $f=g$

see, i don't remember how the proof goes in that generality

cool

if the set is open it's something to the effect of trivial

I know it works because non-zero holomorphic maps have isolated zeroes

and that works because of the Taylor expansion (I think)

:p

but I can't remember the proof of holomorphic <-> analytic at all

I remember it was technical

define holomorphic

complex-differentiable in an open set

continuously or no

let's say continuously

@FernandoMartin the trick is to use the other way to get taylor coefficients

Well, apparently one uses Cauchy's integral formula

by integrating instead of taking the derivativ e

if you don't require continuously it's a goddamn hassle

I never grokked why that works

if you don't require continuity you need to start by proving the theorem that if a function's integral vanishes on every triangle, it's holomorphic

good thing I don't plan to be an analyst

and vice versa

Nice, I proved that

that theorem sucks

that theorem is really cool brah

it's really cool but i don't remember how it goes

I don't either

hence it sucks

but like i said, a lot of it's a matter of maturity

What do you mean?

maybe the proof that euclidean domains are PIDs was hard the first time around but now i can reconstruct that myself

Use the division algorithm :D

@FernandoMartin Anyway if I was to distill this conversation to a single point I would tell you that Gattaca is on netflix now and you should go watch it

@Mike I save my notes.

=)

Don't worry.

I don't

lol

@FernandoMartin What are you talking about?

@Mike

what is the definition of $f_\infty$

multiplication by seven

Perhaps some author decided to use that to denote the limit of a given sequence of functions.

Btw, I prefer $f_{\tilde\infty}$

$\widetilde{\tilde f_{\tilde \infty}}$

@Mike A guy gave a talk titled 'multiplying by 2 and 3' at our uni last year

Geniuses, what structure is this? math.stackexchange.com/questions/759971/…

If I ever write something worth reading, I will be sure to include a note about notation reading "I use tildes everywhere because they look pretty."

it was about ergodic theory

here is one of my favorite paper titles

A minus sign that used to annoy me but now I know why it is there

instant classic

never heard about that one

there was one called 'You could have invented spectral sequences'

or something to that effect

there's a whole thread at MO about funny paper titles

i think that one's different

it's just meant to say "spectral sequences aren't so scary, let's have a nice chat about them"

my goal below is to make you, the reader, feel that you could have invented spectral sequences (on a very good day, to be sure!)

hahahaha

A Trivial But Notable Observation About a Relation Between the Twin Primes and the Number 14

vixra is full of funny titles

those dont count

from the first 500 numbers which are lesser in a pair of twin primes, 66 of them have the following remarkable property: the sum of their digits is equal to 14.

wowee

lol

Do there exist functions $A \to A$ that act as identity only on some of the set of all maps?

whats a

A finite set

sure

let $f$ be the function that maps $A$ to some element

then any function $g$ that fixes that point will have $g \circ f = f$

What about $g \circ f = f \circ g = f$, then only one such map $g$?

for all of the maps?

what i just described has $f \circ g = f$ for any $g$

if you want $g \circ f = f$ for any $f$, then the only such $g$ is the identity (let $f$ be the identity...)

I.c. so there exist local-only identities in semigroups

@Mike, what do you make of this, then? math.stackexchange.com/questions/759971/…

i don't

Woohoo

I just proved Stone duality

gross

any boolean algebra is isomorphic to the lattice of clopen sets in some compact Hausdorff space

how cool is that?

I think the other way is more interesting

Given compact Hausdorff space does there exist Boolean algebra st...

@FernandoMartin that's my amateur opinion. Good job proving!

Well, the lattice of clopen sets of a space is already a Boolean algebra

Oh, so there's a bijection between the two concepts?

I have no clue why this is relevant though

@FernandoMartin Boring as hell.

@Pedro please, dualities are sexy

no, twins are.

oh Bill

always generlizing

I don't like Bill's style

at all

He's too dry.

But, oh well.

One gets used to it.

@Mike @FernandoMartin Jacobson missed something in his proof.

"In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is named after the algebraist Richard Brauer. The group may also be defined in terms of Galois cohomology. More generally, the Brauer group of a scheme is defined in terms of Azumaya algebras."

PLOP.

A lo Condorito.

what

is a condorito

It's a comic strip.

There was a course in Brauer groups last semester

I used Tietze's extension theorem twice today, for totally different things

That's what I call a useful theorem

PLOP

it's too useful

Why is that child a chicken?

Condorito means little condor

condors are a species of vultures

Still, condors don't look like Condorito at all

I don't get it

But hey, I don't get 70s humor at all

he's a chicken

he's definitely a chicken

but he should be a condor

assuming the comic is consistent with its title

i need karl back

someone summon @KarlKronenfeld

How big is $\ln(p!)$?

about $p(\ln p - 1)$ big

@PedroTamaroff @FernandoMartin Let's get ironic

link please

Nah, found it, nevermind

does anyone know if commutative law applies to gcd(a,b) = gcd(b,a) ?

huh?

what do you mean by "applies to"? the equality is already itself a commutative law

join and meet are commutative in any lattice too

i don't think that last sentence is going to help :p

well we know commutative is A+B=B+A right

would that also apply to gcd (a,b) stuff or now?

not?

i don't know what you mean

are you asking if gcd(a,b)=gcd(b,a) is true, i.e. if gcd(-,-) is a commutative binary operation?

then obviously yes

yessssss

k ^^

gcd(a,b)=gcd(b,a) says that gcd is commutative (like seaturtles says)

that's what I forgot to write in my proof

as of yesterday you said this was a theorem in your book

someone was asking about commutative... and it just clicked

done ^^

Hi, when you say that a & b are coprime with respect to d...what does it mean?

never heard that terminology

context?

If I had to guess, I'd say it means "every common factor of a and b is also a factor of d". Just a guess.

my money is on "(a,d),(b,d)=1"

eats turtle

ok. This is a different question. Why do all medians in a triangle meet at one point (the centroid)? I mean why is this so? And why is it that that point divides the median in a ratio 2:1?

who knows bro

who knows

I got the definition of $f_\infty$ it's the restriction of f, with domain and codomain restricted

More complexity theory: math.stackexchange.com/questions/760119/…

hello, can someone help me: math.stackexchange.com/questions/760139/theorem-with-an-example

?

thank you

@Vrouvrou It would be easier to answer if you said what paper it was, so one could figure out what they're going for

(I'm not going to answer it either way, but it could be helpful to someone whowants to)

Hi @will

@user127001 Hey, still haven't thought of a username?

@seaturtles My money is in the bank.

@will no

How to simplify $\sum_{k=1}^n k^2\binom nk$

you have your $k$s and $n$s switched

@Mike, O sorry!

:)

How did you know that without knowing the question?@Mike

because your binomials don't make sense otherwise ;)

Ok!

@Sush Consider what happens when you take derivatives of $(x+1)^n$; use the binomial theorem

It'll be a 3 or 4 step process

Ok, trying.

@Mike, is it $n\times(n-3)\times2^{n-2}?$

Not quite

Ok!

two of those factors are correct :)

@Sush How about a direct combinatorial attack $\sum\limits_{k=1}^n k^2 {n \choose k} = \sum\limits_{k=1}^n k^2 {n \choose {n-k}}$ .. the later can be interpreted as the number of ways of choosing a team of $(n-k)$ along with a captain(in $k$ ways) and a fund-raiser(in $k$ ways) from the remaining $k$ guys. Now the same could be done in $n2^{n-1} + {n \choose 2}2^{n-2}$ ways (why ??)

@Mike, sorry, I mistaked at k+1.

I took k-1.

I'm not a hue fan of doing combinatorial attacks on these things

doesn't mean you shouldn't do it tho

@Mike I love combinatorial arguments .. :)

@hawk You didn't give the answer!

@r9m What happened to your avatar?

Yay! I got a gold badge just now.

@Sawarnik changed it :P God knows :P

@r9m What is it now?

Your avatar always seems to be cryptic :P

@r9m No you knows.

@r9m Was Hawk able to complete his solution to your inequality prob?

@Sawarnik IDK .. didn't post anything new yet

@Sawarnik did you see the problem where $\{a_i\}_{i=1}^n$ are distinct integers, then $\prod_{1 \le i < j \le n} (j-i)$ divides $\prod_{1 \le i < j \le n} (a_j-a_i)$ ?

@r9m Ok, r9m :D Any health problems after your fall from the window? :P

Seems beyond my level.

@r9m, does r9m mean Ramanujam?

@Sush no way :P

But nice spot!

just asked.

@r9m Hawk gave us a not soo tough question yesterday, which both SK and PK can't solve. Could be a nice timepass for you, want it?

Sush, sush, sush!

@Sawarnik okay ..

@r9m You must have seen it. Prove without using any trigonometric functions, that the integral of $\frac1{1+x^2}$ from 0 to 1 is the same as the integral of $\frac1{1+x^2}$ from 1 to $\infty$.

@Sawarnik divide numerator and denominator by $x^2$

@Sawarnik Make a substitution $u = 1/x$

@r9m, how did you get ${n \choose 2}2^{n-2}$?I get $\sum_{k=1}^{n-1}nk\binom{n-1}{k}=n(n-1)2^{n-2}.$

@Sush Those two expressions are very close, since $\binom{n}{2}=\frac{n(n-1)}{2}$

a two got lost in there somewhere

So i did mistake!

@Sush if you use ${n \choose k} = \frac{n}{k} {n-1 \choose {k-1}}$ then you should get $\sum_{k=1}^{n}nk\binom{n-1}{k-1}$

@Sawarnik Wow, that's an interesting question!

Yo, wazzup pal? @parth

@skullpatrol Not much.

@skullpatrol Are you sure? Because I've been very active.

I guess we're there at different times :-)

Yes, that seems it.

Where am i maisking?

Please help!

@Sush I don't think you maisking :P

@ParthKohli ruhere?

@Sawarnik, i also think so. I tried some values and i seem true and r9m wrong.

@Sush Oh, it was a joke on your spelling, which question are you talking about?

$\sum_{k=1}^n k^2\binom nk$ this question.

Oh sorry, I don't know binomials.

Ya, i know mistaking is not a good word. What can i use instead of it?@Sawarnik

@Sawarnik no

@Sush Why don't you mark yourself here? mapsengine.google.com/map/edit?mid=zg6p6CxQUjpY.ktFFDHMUvfso

@ParthKohli say no.

@Sawarnik no

@ParthKohli Are you still blocking me on fb?

@Sawarnik yes

@ParthKohli so reverse it.