those numbers are the arities of the operations

+ is a binary operation, so its arity is 2

Okay.

1, on the other hand, is a constant, aka a 0-ary operation

Then what is 1-nary operation?

By the way, there is a mistake in my copy-paste. I would paste the correct formulation and then it will make sense and my question about $-1$ will be rendered moo(t).

Oh I see.

A field is therefore an algebraic structure $\langle F, + , \cdot, -, ^{-1} , 0 ,1 \rangle$ of the type $\langle 2,2,1,1,0,0 \rangle$.

@MarianoSuárez-Alvarez Is it necessary to provide inverse operation? Are there cases where the inverse operation is not characterized by the forward operation and the identity?

the question does not make sense, really :-)

there are algebraic structures where some operatons are not determined by others

don;t do that

use the normal definitions

bringing up universal algebra and what not in tthat context is simply silly

Okay. I won't.

All that only makes sense in some contexts, becase there it is useful

in all likelyhood, it is not useful to use than language in your context

Rats!

@JonasTeuwen No Juice. No juice.

"If you stimulate rats’ brains once with a small electric shock, then they’ll be like “what the hell was that” and then go back to ratty things like munching on pellets."

Too many electrical shocks and they will die.

It was in the test of kiddling of epilepsy.

power ~ current * voltage

my kid brother has epilepsy too bad we don't know why

i just got into the elevator and pushed 16 and dude enters and pushes 10

he says sorry to stop you halfway up

Okay.

i said it's five eights he said wut?

i said it's five eights

goodnight

then he said goodnight too

I am surprised he did not punch you.

duh

No problem!

everybody wins

No!

If everybody wins the game sucks.

People should be more correct with their fractions.

yeah but saying shit twice won't help

for that i need a speech therapist

i guess there is no game, everything is okay and the hotel is kewl

having engaged with someone in comments to (laboriously!) get to a point only to have the OP delete the question afterwards is rather annoying

Can you abuse the mod powers and just undelete it if there's worthwhile content there?

since this was not the first instance of this by that user, I did

@DanBrumleve Let's see of conditioning with some shocks help!

@MarianoSuárez-Alvarez Frustrating

new users won't understand the implications of actions, like value of content to search users

If it doesn't, then at least we had some fun, right X_X.

inthe past, that behavour has frequently been associated with people cheating in a test

They will get quite far in life, the little backstabbers :-/.

doh i hope it doesn't come to that

this is san fran*CISC*o

(shock sound)

Why are there always pigeons shitting on my balcony?

They do not even taste good.

I'm in San Francisco, but I don't quite follow...

it's your personality

it attracts them

What about my personality attracts them?

@JonasTeuwen I wouldn't know; I've never tasted pigeon poop....

dunno

@BenW punning on the sound and the idea that i need electroshock therapy

@robjohn You have to include the pigeon as well!

also that is where i actually am

No, not as a therapy.

Just to reprogram you.

So the first time you will be like 'what the hell was that!'.

Cool, SF is a nice city.

shrug it's okay i used to live here now i just go here to work a week or two every month

@MarianoSuárez-Alvarez I know this is kind of out of the blue, but do you have any suggestions on how to decide between grad schools?

difficult question :-)

@JonasTeuwen you can use that on your pigeons, they will run away from you shouting, here comes the what's hell guy?

Something I am unfortunately becoming aware of haah

You pick the best one you can get, period.

yeah it's never very therapeutic

Best in what way though??

What is best for you, hence your question is apples 8-(.

best can mean many things

It is like asking 'what cd should I put on to complete strangers'.

I know people who've been to the very best places in the world and hated it muchly

You'd get locked up for that if you do it too often, so why not with the grad school question?

Hmm, for me I in the end 'picked' the current location: I got employee status (much more €€) and I knew to advisor well already.

I assume that should have been parsed " 'what cd should I put on' to complete strangers"

Any fear about job placement afterwards?

find someone who want to work with, a place where you want lo tive (if it involved moving), and where you mightt develop a life (if you have one apart from math)

how to decide where you want to work?

people locations it's hard to choose

hm?

Is it really that hard?

Seriously?

You make like your list of priorities. And see how well it matches.

Kind of.

Gosh, now I'm getting nervous.

Make your list of priorities and then rank your choices between each of them.

for me it is i'm in sf right now to work but my home is in champaign il

Then you take a weighted average.

@BenW. Why?

But, I still think that will not work that well: can't plan it all.

exactly

you can t plan it all

Just make sure it is a location you feel in, and could be productive.

The rest is just details.

but if you are into mountainhiking do not go to utah

Well, I mean... if the first one fails 8-).

@AlexYoucis I too have to start deciding which grad school to choose. I guess I have a few months.

Yes, I think the choice is more on very personal things than objective facts like their ranking.

@BenW. A fellow sufferer! Cheers!

or whatever state it is that has no mountains, that is

@AlexYoucis Cheers!

The issue is how do you decide where you want to live? Did you guys visit the schools you ultimately ended up attending?

@BenW. Any frontrunners on your list?

use your educated guesses

So, for me: 1) knew advisor (really relaxed) 2) Pay 3x as high... plus all employee benefits 3) I like this place.

And he said 'I want to keep you here'. So I was like, alright, nobody else will say that.

well, the pigeons seem to like you

that makes several

Yeah.

@AlexYoucis Right now I kind of like UCLA, but I'm tempted by UW - Madison because I think I like group theory. You?

But I don't like them.

@JonasTeuwen That is a pretty unique situation right now!

@BenW. I'm not sure. UCLA and Berkeley are up there

one imprtant thing is, your choice will not be determinant for your future

Yes, so I figured that even going to a place with a higher ranking would be much worse - even since I am pretty much allowed to look at all of it.

@MarianoSuárez-Alvarez Kind of feels like that!

Have you actually already like... reduced it to three options?

If not: you must be female.

@AlexYoucis Do you think you'll visit UCLA in March?

@JonasTeuwen Who is that directed towards?

To you.

@BenW. I think so--Berkeley is like two days after it, so why not?

@DanBrumleve Are you okay?

You seem profoundly wasted.

@JonasTeuwen I have not reduced it to three options haha

you have a continent, at least?

First do that.

@BenW. You're into group theory? Where are you at right now?

First think about what is the most important to you, and then rank those and go on with the top 3.

Just decide fast and go for it.

For group theory, can also come to here. Lenstra and so on etc etc.

@AlexYoucis Berkeley actually. It's a nice campus.

i don't think so but it is bedtime

@BenW. Haha, do you know Landry?

@DanBrumleve Good night.

@AlexYoucis Landry? I don't think so. Is he/she a student?

@BenW. Yeah, a math student--senior. It was a long shot at Berkeley.

@AlexYoucis Ha, sorry, can't say that I do.

@BenW. I don't know how you faired, but this year was brutal. I think we decided to apply on the wrong year

@AlexYoucis Really? Were things tough for your colleagues? It sounds like you fared pretty well.

@BenW. Yeah, they kind of were. My acceptances were sparse--I applied to like 16 places.

Hi guys

night

@AlexYoucis And you've heard back from all 16? I'd say if you got into Berkeley/UCLA/others then you did more than all right.

@BenW. Were you at JMM?

@AlexYoucis No I wasn't, what's JMM?

@BenW. True, true. It was just wacky where I got accepted and where I got rejected.

@BenW. The Joint Mathematics Meetings--they were in San Diego. I was going to say that I saw Ellenberg's talk there--super, super cool. You into number theory at all?

There is a fun Banach spaces conference in Israel in May (memorial for Lindenstrauss).

@MarianoSuárez-Alvarez Any suggestions here?

@AlexYoucis I know a bit of number theory, I worked out of Ireland and Rosen's book. I audited the 254AB classes at Berkeley, but I'm no expert. Is that your prospective field?

@BenW. It depends where I end up going. I'm keen on Olsson at Berkeley.

@BenjaLim, gotta run now! I'll later

@AlexYoucis Ha, I took number theory with Olsson two years ago! He's a really nice guy. Humble too.

@MarianoSuárez-Alvarez Man mariano you abandon me.....

@BenW. Very cool dude :) So are you like straight up group theory?

@AlexYoucis Hi, did you see my comment to your answer here?

@AlexYoucis I might like to get into representation theory. Out of all the professors at Cal, I probably know Vera Serganova the best, so I got a taste of what she did. But we'll see. What other potential interests do you have?

@BenW. When I first saw your name I thought you were Ben Wieland

@BenjaLim What you wrote is true though. An integral domain R is a Dedekind Domain if and only if every submodule of a projective is projective

R is projective so every ideal is projective

@AlexYoucis Yes but actually what I wrote is only true if the ideal is non-principal

because if it is principal it is already free of rank 1

@BenjaLim Sorry, I am but a mere humble student.

@BenW. I really like number theory, algebraic geometry, and complex geometry.

@AlexYoucis Do you have any ideas here?

@BenjaLim What am I supposed to be reading?

@BenjaLim Yes, I do. That is a formidable question though, and not one that I feel like I am the best (currently!) to answer.

I would take a look at the Vakil thread though, fo' real.

@AlexYoucis That we can always find an ideal $J$ os that $I + J = R$, and $IJ$ is principal. I think that is true only if $J$ is non-principal...

@AlexYoucis But a first course in schemes?

@AlexYoucis I can imagine studying number theory and algebraic/arithmetic geometry with Olsson would be quite something.

@BenW. Yeah man, I'm fairly psyched about Berkeley. Who at UCLA were you diggin' on?

@BenjaLim Where is this coming from? Did I use that implicitly/explicitly somewhere?

@AlexYoucis I don't think you used it in your answer, but for mine if such a $J$ always exists, then the method used in my answer shows that every ideal in a Dedekind domain is projective

Are you wanting to know whether it's true because you think that result is false

which it isn't

every ideal in a Dedekind domain IS projective

or because you think its a neat proof of that fact?

@AlexYoucis Of course it's true. In fact more generally the fractional ideals are projective. I just wanted to know if it was a neat proof :)

@AlexYoucis I was kind of interested in what Balmer and Rouquier were doing in algebra.

@BenW. High five for Balmer! Yeah, he seems pretty sweet. I like Khare though--the dude solved Serre's conjcecture.

@BenjaLim That is very close by.

All right friends, I'm heading out--it's like 5 am here.

see ya

'Night.

@AlexYoucis Bye. Good night.

@BenW. In case I don't talk to you before then, maybe I'll see you at the UCLA thingy!

@BenW. Can you help me with my question here?

@AlexYoucis Yeah, maybe I'll see you there!

Man wish I was in the states.....

@BenjaLim I'm not too keen on Algebraic Geometry, but here is the first half course Olsson taught at Berkeley: math.berkeley.edu/~molsson/256.html

@BenW. I have been advised that for a first course, it is not good to start with Hartshorne

The prereqs for that course are Math 250AB, which basically cover groups, rings, fields, commutative algebra, multilinear algebra at the graduate level.

@BenW. 1) I am not a graduate student 2) my category theory is not strong enough

@BenW. Oh wait I know the answer to the first problem in problem set 3. That's a problem in algebraic number theory!

@BenjaLim Sorry, there was no undergraduate level algebraic geometry course offered at Berkeley. I can't recommend anything else. Maybe supplementing yourself with Mumford's Red Book could help, and you could use the course page as a guide.

ok.

My bad, maybe someone who's taken an actual intro course can give you a better answer. (I haven't taken 256AB).

@BenW. It's ok. I understand you're more into group theory and stuff like that.

I really should buckle down and try to better learn some basic AG though.

Anyway, good night fellows.

@BenW. nite.

@MarianoSuárez-Alvarez When you have returned, please try to answer my question :)

@MarianoSuárez-Alvarez I would really appreciate it.

@anon Ok thanks, sorry for the late reply. So I could use this as a counterexample to the statement: there is a sequence of functions from $\mathbf N\to\mathbf N$ so that for every function $g:\mathbf N\to\mathbf N$ there is a function $f$ in the sequence with $\lim\limits_{n\to\infty}\frac{f(n)}{g(n)}=\infty$?, right?

@awllower It is typical post-it note stuff, I assume

Hmm, all right, I guess it must be some modern version of Fermat.

hello

Hi

Maybe it is too long to show that the number is indeed perfect.

I want to finish going through my ntoes so I can get back to the number theory :)

It says an odd perfect number has to be bigger than 10^300

Hmm, then it is undoubtedly running out of place!!

hehe

I wonder about the fermats last theorem

You mean modularity theorem?

What are you wondering? :)

well i don't know about modular forms

they seem to be natural, so I want to know about it

Hmm, temporarily off

no one asks about modular forms on MSE

Heya

Could someone help me quick ?

with what

wiki.math.ntnu.no/_media/ma1103/2013v/ma1103-04.pdf Number 4 here

can you formulate it as a calculation?

Oh, I mean 2.6: 4

do you know what it is you're trying to calculate

No!

ok

I guess I need to find a parametrization of x and y, such that $f(x(t),y(t)) = 13$, for all choices of $t$.

sort of

No idea how to find that one though

you could just consider lines that pass through (2,1) in direction (u,v): differentiate f(2+ut,3+vt) with respect to t

Okay, now what ( it turns into a mess) =)

solve for u,v such that height doesn't change (meaning derivative is zero)

Hmm

the idea of f(x(t),y(t)) is right, any smooth path that goes through 2,1 at t=0 will do

oh lol you have computer do it

Ofcourse, it turns into garbish

why did you add pi?

It says so in the problem ?

There should be a $\pi$ in there

you take the derivative then set t=0

because we chose lines that pass through 2,1 at t=0

So

$f(2+t,1+t)$ ?

this is the idea, but we're on a different surface than the sphere

Yeah, I understand the point

so you see why we differentiate then set t=0? that's the point at which we are interested in. We just chose a line because that's the simplest smooth curve that goes through a point

Yeah, but what I am supposed to take the derivative of?

differentiate f(2+ut,3+vt)

|It should be a 1 and not a 3, but give me two seconds

And this gives me $ u + v$

So all points where $u = -v$ would work then.

it would be nice to find one with u^2 + v^2 = 1

To transform it into spherical coordinates?

no just because then it's a unit length vector

which is what I mean when I say "direction"

Using $f(1 + t, 2 - t)$ only works when $t$ is even

what

"even" doesnt make sense for real numbers

Okay

are you trying to check that an answer is correct?

Yes

If I use an even number then I get z=7, now if I choose an odd number i get 13

you just need to check the derivative of f(x(t),y(t)) is zero at t=0

If I choose anything between, I get a z value between 7 and 13.

@n3b how are you?

where x,y are any smooth curves with (x'(0),y'(0)) = (u,v) and (x(0),y(0)) = (1,2)

but you know this is true already because that's exactly how you found u,v

But it does not work =(

My derivatives are not zero, which means my parametrization is off.

your derivatives can't not be zero because you just solved for u,v such that the derivatives are zero

math.stackexchange.com/questions/301205/… why is it downvoted?

There is no downvote?

welcoem back

Thanks!!

@awllower the answer was deleted

I see.

it's interesting how much information the zeta functions on the real line gives

for example, it can prove infinitely primes in progressions, infinitely many zeros, class number formula....

if its real you get the fact the zeros are mirrored around the real line

Hm, very interesting.

Which book is good to learn about Zeta function?

but prime number theorem comes from the imaginary axis

:-(

@awllower, do you have some use of modular forms?

:D

More precisely, it is as described in the book by Weil, Number theory from Hammurapi to Legendre, a historical approach.

ah

?

I think this is a really good book, I remember reading early chapters

Indeed, it is always a pleasure to read books by Weil.

Time to sleep. Bye!!

bye

I'd be grateful for two more people to vote to reopen. There are 3 votes so far. I'm still interested in an answer. Preferably the other question could be marked as duplicate of my question since the other asker seems no longer active but I am.

Is there a problem with this solution? :-|

@N3buchadnezzar, did you get it?

y!

Goodmorning

good morning

Hi :)

Wow I am so happy I can hear someone's bass-line from down the street right now.

what is going on

lucky you @EdGorcenski

That person is so cool. They have the loudest speakers. I think they should get a prize.

How can we proof that $$ 2^0 = 1 $$ ?

@devWaleed, its just a computation

@user58512 What do you mean by that?