5:00 AM
I would like to know more about the geometric side/representation theoretic side, but it's a bit beyond me at this point.
@Ethan, yes.
You can produce n-th power reciprocity law from it.

@Ethan You can find it on Library Genesis. Do you know the site?

@BenW. No what is that?

haha

@Ethan Google it. I'll probably remove my comment soon.

The answer should be clear from Ben W.'s comment.

user19161
5:03 AM
The thing is, any sharing site contains many "legal" materials too, so there is nothing wrong with providing a link to such a site.

It's there, I'm just telling you about it. It has some nice resources.

user19161
Has any of you seen Ahlfor's Riemann Surfaces? It has gone out of print.

@JasonBourne Why do you know so much about books?

user19161
@Eric Well, I don't know. I like books.

user19161
@Eric Are you planning to apply to graduate school?

5:10 AM
@JasonBourne Well I don;t know if I should.
My schools math program seems like a joke to be honest.

user19161
@Eric Hmm, OK. You can always study on your own before entering.

I know that sounds very disrespectful to my institution, but I really feel like that.

Where do you go to school @Eric, if I may ask?

user19161
@Eric It's OK, I feel like that about mine too, but I won't say where...

I feel like if I went to and grad school and told them I never had topology, didnt read rudin, and every other course was taught without proofs, U would be laughed out of the school
^I would be
@BenW. I go to the University of North Florida

5:13 AM
I think it really doesn't matter. If you want to go for grad school, just do these on your own/under supervision of professors in your dept.

user19161
@Eric Just ace the GRE.

Is GRE of any use at all?

To be fair, some of the professors are too smart and shouldn't be there.

user19161
And also, try to show them any undergrad work you have done.

user19161
Have you written any undegrad papers?

5:14 AM
@Sanchez They really hate that

@Eric, that's weird.

Hate what?

@JasonBourne Papers? You mean something published in a journal? No, no way.

user19161
@Eric No, I mean trivial writings.

I don't think the general GRE matters much. The Math GRE only matters slightly, so I hear.

5:16 AM
@BenW. Independent study, you have to get permission from the head of the math department. Because officially independent study is only allowed for graduate student

user19161
Ultimately, how students are chosen is very subjective. It's a combination of many factors.

Have you taken the GREs @Sanchez?

@JasonBourne Can you define "Trivial writings"?

@Ben W., yes.
@Ben W. and I have been under the impression that it's basically useless.

user19161
@Eric For example, some places have a final year project for the students.

5:17 AM

Oh yeah. My department has that too.

user19161
Also, try to get good letters of recommendation.

@Sanchez I see, that's more or less what I've heard too.

@Eric, you don't need to make it official, if you just want to learn.

user19161
@Sanchez Well, not really advice. I am not qualified to speak. But I think all other factors being equal, everything plays a part.

5:18 AM
@Sanchez Thats exactly why I am thinking of not even going to grad school

When you apply to grad school, recommendation letters are the most important. It doesn't really matter whether those courses are official, as long as you learn what you need and leave a good impression to your professors.
@Jason, fair enough.

My professors tend to like me alot
Christ, having even heard of baby rudin will you on their good side

@Eric, depends on what you go to grad school for. If you want to do research, you do need an advisor - doing individual research is almost impossible.

I've heard that letters might be the most important factor, math grades coming in second.

user19161
If you have the backing of a famous prof, that is it.

5:20 AM
@BenW. Well honestly, I am married, and will not be going out of state unless it is a "free-ride" type of situation.
@Sanchez ^that was for you
BUt in the future, I would like to go to grad school

I see. Good luck then.

user19161
I would like to apply in future too.

user19161
But I am not sure if I would.

@Eric Most Ph. D. programs pay your tuition, and offer a teaching stipend on top of that. So you don't have to pay a lot like one does for undergrad.

the only way i will go after i graduate is if I am made some offer which would be absurd to turndown
@BenW. yeah i have heard

5:22 AM
Oh ok, free-ride situations are probably more common than not in that sense.

user19161
@sanchez So how many more years till you get your PhD?

user19161
@Sanchez HAHAHAHAHA

I am earning two degrees, one in CS one in math, i may go to grad school in CS if i ever go

quite a bit of time to go anyway.

user19161
5:23 AM
@Sanchez A worse question is: how many more chapters to write? =)

:S

user19161
Then one starts to hang out on TeX SE.

@Sanchez I got it at lunch I think.

@jason lol
@BenjaLim, nice. I typed a solution above if you are interested.

I used to think that that there was something that happened in a classroom that could never be replicated by reading a book. However, I find that I was wrong. I really do enjoy talking to the professors and other people who like math, but at the end of the day I just want to learn.
A book and the internet is all you need to learn really

user19161
5:26 AM
@Eric Yes, I learn most things by myself. The lectures were a waste of time.

@Sanchez What about $(\sqrt{x}) \subseteq (x^{1/4}) \subseteq (x^{1/8}) \ldots..$?
@Ethan I can give you a proof of quadratic reciprocity using the fact that the Galois group of the $p$ - th cyclotomic field is cyclic.

@JasonBourne Yeah, there are really cool people at universities, but I think that the school gets in the way of education

user19161
@Eric I know of many who dropped out because they could not produce research and they graduated with a Masters instead.

Oo that's a lot simpler than what I was thinking. Nice.

@Sanchez The problem is right now I don't know how to show that it doesn't terminate.

5:27 AM
@JasonBourne You mean PhD students?

user19161
@Eric Yeah, something like that.

@BenjaLim, each inclusion is strict, so you are done.

something like that?

user19161
@Eric Well, different places have different systems.

@Sanchez well if could be that say $x^{1/4} = \sqrt{x} f(x)$

5:29 AM
What country are you in?

@BenjaLim lol I don't think I would understand it

@Ethan But I know a proof that avoids the Legendre symbol and only uses some Gauss sums.

user19161
@Eric Haha, I prefer to keep that a secret. But in general, yeah different universities have different systems.

user19161
@BenjaLim Me.

@JasonBourne Ok well can you at least answer this: What is the official language of the country you live in? If the place you live in is not recognized as it's own country, the what is the language used in daily life of the are you live in?

user19161
5:31 AM
@Eric English.

Ok.
@BenjaLim It was for jason, but you are welcome to reply

@Sanchez I think I have an idea as to why such an $f(x)$ can't exist.

user19161
@Eric By the way, you should know that Jason is a movie character. =)

It is?
^he is?

user19161
HAHAHA, yes, Jason Bourne is a movie character.

5:33 AM
Oh

user19161
Google it and you will see.

Are you living in a movie?

user19161
I have also used Clark Kent and Will Hunting as usernames.

user19161
Again, all movie characters. =)

user19161
@benja Are you still on vacation?

5:34 AM
yes.

No Andy Dufresne? Come on man
I thought you were cool :(

user19161
By now I have used over 9000 names, so nobody knows who I am anymore, not even myself. =)

Hah

user19161
Haha @benja that's what you think! =)))

OK guys the inverse of the following permutation is: (a_1,a_2,...,a_n)^-1 = (a_n,a_2,...,a_1)?
correct?

user19161
5:38 AM
@Eric You typed wrongly.

@Sanchez Hmmm can I ask you if this reasoning is correct?
If $x^{1/4} \in (x^{1/2})$

They should really have latex support for this chat room, that be great

then there is $f(x)$ so that $f(x)x^{1/2} = x^{1/4}$

user19161
@Eric There is, see chatjax on the right or on meta.

@Sanchez I want to say from here that $f(x)$ is not defined at $0$.

user19161
5:40 AM
@benja I am going to sleep now, good night, I will try not to quarrel with you next time. =)

user19161
@BenjaLim Yes, I sleep in the day. =) And you like to reveal all my details don't you?

@Sanchez ah but I think here's the problem
for $x\neq 0$ our $f(x)$ is $x^{-1/4}$

At least, you can define your $f(x)$ for $x \neq 0$, and you know $f$ is supposed to be cts at 0

there is no way to extend this to a continuous function on $[0,1]$.
@Sanchez there is no value that we can define for $f$ at $0$ to make it continuous :D

5:43 AM
:)

I think that's it :D

@JasonBourne Well that is what i meant to type is this, $(a_1,a_2,\ldots,a_n)^{-1} = (a_n,a_2,\ldots,a_1)?$

user19161
@benja You have not said good night to me. =)

@JasonBourne good afternoon.

@JasonBourne And i got the chatjax working now its great

user19161
5:44 AM
@Eric No problem. Now back to your books.

@Sanchez my example was a lot simpler!!!!

@JasonBourne wait, do you mean my answer was right or, no problem as I dont have to thank you?

@Eric What is the inverse of $(234)$?

user19161
@Eric The latter, I am too tired now, let them answer you...

@Sanchez How did you come up with all the highfalutin examples?

5:45 AM
@BenjaLim $(342)$?

@Eric No.
$(342) = (234)$.

ok wait

@Sanchez If you replace $x$ with $2$ in my example above this shows that the ring of all algebraic integers in $\Bbb{C}$ is not Noetherian :D
@Sanchez So in some sense the example I gave is a "general example" to show that a class of rings is non-noetherian.

@BenjaLim, it was the same idea, you want something that goes faster and faster to 0.

@BenjaLim $(342)^{-1} = (243)$?

5:50 AM
@Sanchez you mean faster and faster to $1$?
@Eric yes.

ok so

Now try for a permutation in $S_4$.
@Sanchez and then generalise.
@Sanchez Sorry for $C[0,1]$ those fucntions approach $1$.
But for the algebraic integers
yes it is going to zero.

$(a_1,a_2,…,a_n)^{−1}=(a_n,a_{n-1},…,a_1)$?
@BenjaLim That correct?

@Eric looks correct to me.

ok, so its just the elements in reverse
got it

5:52 AM
$(a_1a_2a_3\ldots a_n) ( a_n a_{n-1} \ldots a_1)$ = ?
$a_n \to a_1 \to a_n$
$a_{n-1} \to a_n \to a_{n-1}$
continuing this proccess
we see they multiply to the identity @Eric
so indeed it is the inverse.

Yea

:D

THats pretty simple

yea.
@Eric My personal advice for you is:

@BenjaLim THank you

5:54 AM
You need to learn to self - study many topics.
For example, I learnt commutative algebra under the supervision of a supervisor. There was no such course at my university.

Well this summer I am going to self study topology, regardless if it is approved for self study
I am taking 19 credits this semester, so i am too busy to self study now

@Eric Ok. fair enough. are you a freshmen? junior? senior?

But i completely understand what you are saying,
Well I 3 years left, but i am earning 2 BA's, but i am taking senior level classes
so i guess I am like 2-nd and 1/2 year

ok.

@BenjaLim At the end of this semester I will finish everything required to graduate for a math major, except the number of credits
@BenjaLim And the other courses for undergrads are not rigorous, Complexe analysis, number theory, etc, are all taught without proofs

5:59 AM
right. But what I want to say is the major is not as important as your experience in maths. At my uni lots of people are qualified to graduate with a math major even without knowing what a ring is.

So my plan is to take graduate level courses and/or do independent study

@Eric yes.
@Sanchez @Eric I should go now.

Also the undergraduate math adviser wants me to take graduate level classes, and another professor still wants me to do self study this book with him, regardless if it is approved
so i think i am ok

@Sanchez I started reading up on the Buchsberger algorithm. That should be handy in computing zero sets of polynomials.

@BenjaLim goodnight, it was nice to meet you.

6:03 AM
@Eric Are you sure you want to start with lee's smooth manifolds? How is your general topology?

Crap sorry i meant this book
that book is basically undergrad topology with a lean towards manifolds

well the first half at least is.

yes

Later on when you get to things like algebraic topology it can be quite hard going.
@Eric But I believe it is better than Hatcher. I used that book for last semester's algebraic topology and didn't like it that much.

well that is where my professor is supposed to come in
Also my professor says "All math books should be written like this one."
So i guess that is a good sign

6:06 AM
@Eric Well the thing I should say about lee's style is that he gives you a lot of details. Which is a good and bad thing.
Bad because it can overwhelm a beginner
good because it doesn't leave one hanging like Hatcher.

and i do know a little but of very elementary topology, I read about half of bert mendelsons intro to topology

ok.
@Eric I really should go now. Bye!

I think with my patience professor, and the fact that the ideas introduced wont be 100% foreign to me, i should come out ok
yeap
me too bye!

6:19 AM
@BenjaLim, cool, didn't know those stuff before.

6:33 AM
@Sanchez what stuff?

That algorithm you mentioned @BenjaLim

@Sanchez ah it allows one to compute Groebner bases of an ideal. It's cool :D

:) I see.
Cool!

@Sanchez lemme get back to AG now :D

sure.

6:36 AM
@Sanchez Yay now I know another example of a non-noetherian ring

@Sanchez $C[0,1]$

Oh, haha.

bye @Sanchez !!

Bye! Good luck in AG!

6:39 AM
I think I understand the expansion/contraction part here, but the explanations of how Cl measures failure of UF seems way too crude (namely: if |Cl|=1, then UFD, if |Cl|>1, then not UFD; this test measures only the bare minimum there is to say about failure of UF), or unsatisfactorily indirect (diophantine equation solutions).
(I have not studied AlgNT though.)

Why is it too crude? How would you measure it then?

I would hope for something like an explicit relationship between instances of failures of unique factorization and relations/elements/properties of the group.

Hmm I see.
That seems to be too complicated I guess.

Saying that Cl measures something should be much more than the order |Cl| measuring something, as there is way more information in a group than its size.

Oh, it does carry more information than that, but that may not be related to UFD though.

6:43 AM
Right. Thinking Cl is all about factorization might be too narrow of a perspective on it.

I guess it's more like the historical motivation.
And it is natural in that regard, as the basic obstruction to the domain being a PID.

7:22 AM
@Charlie Do you like this?

8:11 AM
Oh blimey.

8:35 AM
Short for gorblimey "God blind me."

3 hours later…
11:22 AM
Blimey!

@JonasTeuwen Hey

Hi.

@JonasTeuwen Are you all set to coming?

Better just mail me.

ok.

1 hour later…
12:26 PM
(removed)

@JasonBourne Good timing.

user19161
(removed)

Heya'

user19161
@Eric Since you ask me so much about books, I will give you the list I think is the best. Read Cohn's Classic Algebra, Basic Algebra, and Further Algebra; Rudin's Mathematical Analysis, Real and Complex Analysis, and Functional Analysis; Lee's Topological Manifolds, Smooth Manifolds, and Riemannian Manifolds. That is a total of 9 books to last all the way from first year undergrad to second year grad.

12:39 PM
Hi, gang; wazzup?

@skullpatrol The roof

the roof
the roof is on fire!

user19161
-5

Most of the math courses I am currently taking in university are packed with students from China, Hong Kong and Korea. They do seem to be extremely hard working and disciplined, and seem indeed to be good with doing fast calculations. I also met a Chinese woman recently doing her master's degree ...

user19161
This question is so silly, minus infinity!

12:48 PM
@JasonBourne Hahahahahahahahahahahahahahahahahahahahahahaha.
Irrespective of reasons etc, one thing I can say is, if he is a statistics student (He mentions statistics), he does not understand the concept of randomized samples well!

user19161
@OrangeHarvester I also think people should stop referring to Terence Tao for every thing. They worship him as some god and quote him everywhere however irrelevant.

user19161
See the comment there about him?

Generally, the asians who have migrated to US/Europe are the ones who have been largely successful in their home country. So, when Americans see the Asian population, they mainly see the hand-picked lot so to say. Its ridiculous to compare a good hand-picked lot as opposed to random lot.
@JasonBourne I don't care what people do with Terence Tao as long as people do not ping him and bother him too much.

user19161
First, race is largely a cultural instead of biological classification.

@JasonBourne Yes.

user19161
12:51 PM
Second, mathematical ability of people in a country is largely a cultural and not biological effect.

And economical. And social. And historical. And geographical. And political.

user19161
QED.

And intelligence.

@skullpatrol bleh to that.

user19161
Therefore, Tao is not a god, only anon is a god to me. =)
2

12:54 PM
@OrangeHarvester Why do you say "bleh" to that?

@skullpatrol Intelligence in a conventional setting is a very complex symptom of genetics, upbringing, exposure, health during infancy and pre-infancy etc. Relegating it to only genes is an insult to all the other factors that have contributed to it.

Mathematical ability has nothing to do with intelligence?

Atheistzilla did not destroy tokyo, but godzilla did.
Therefore: Atheists:1 , God:0

user19161
Haha, I thought my above line would be starred by now. =)

@N3buchadnezzar Did you like the song?

12:58 PM
@skullpatrol If Intelligence was an infinite dimensional vector space, Mathematical Ability is one of the components of one of the many possible bases.

@skullpatrol Old song, heard it before.

@OrangeHarvester That is all I'm saying: it is one of the components.
@JasonBourne Are you happy now?

user19161
@skullpatrol Yes.

:-)
anything special?

@JasonBourne I'm happy that you're happy.

1:09 PM
hello

1:40 PM
hi

2:05 PM
Anyone around?

hehe
I found a counter-example to the theorem and now everyone is adding the caveat to their answers math.stackexchange.com/questions/296255/…

2:18 PM
but they areant upvoting my answer :(

2:30 PM
I have a quick question: Is $A\neq B$ considered to be the negation of $A=B$? As in, does it mean "A is not equal to B in at least one case"? Or does it mean "A is never ever ever not equal to B"? Here, A and B can be functions of some variable.

@Manishearth, what are A and B?

some expression
Over a set, say

like if A=5 and B=3, then we have $5 \neq 3$. But if A and B are functions of time t, say .. then there only needs to be one moment, e.g. t=3 when $A(3) \neq B(3)$ - they could be equal everywhere else - for $A \neq B$ to hold

@Manishearth a statement does not have a contradiction

@Tobias Not saying it needs to

2:33 PM
he means the negation

I'm just confused over the meaning of the $\neq$ symbol

@Manishearth I mean that the word contradiction does not fit where you put it

@Tobias oh oops. was thinking of something else at the time :P

@Manishearth negating $a = b$ does indeed give you $a\neq b$

So when not specified, it means "may be equal to"?

2:37 PM
when what is not specified?

Hmm.

$a\neq b$ means that a and b are not equal

Yeah, but I'm putting it in a larger context
if A and B are some functions or properties
The exact question is
If one assumes that $\neq$ means "may not be equal to" (negation of 'always equal to'), we get b as the correct answer

but that is not what it means
it means "is deinitely not equal to"

If one assumes that it means "will never be equal to", we get e

2:40 PM
definitely*

@Tobias yeah, that's what I wanted to clear up

@Manishearth, it means: are not equal to

in this context there is an implicit "for all A" in front of those statements

Any source for this? A lot of folks are saying that "being the negation of $=$, $\neq$ means 'may not be equal to'"
@Tobias exactly

@Manishearth who are those people

2:41 PM
no it doesn't mean that
did you see what I said about $A \neq B$ when they are functions

@Tobias one teacher, some classmates
@user58512 I know :)

the question is poorly worded

true
that's my argument

if A is given, then the statements make sense

(the one I posted)

2:43 PM
but clearly A is supposed to be arbitrary
and then it becomes a matter of what quantifiers one adds where

exactly
gtg, food
thanks guys!

I need to do my math work but I feel lazy
do people think ther is a mistake in my answer because of the confusing comment someone wrote? IO don't get why they are upvoting a copy of my counterexample

@Manishearth Then option e would be a joke...

yo can anybody here do complex analysis?

@user58512 I believe when $k$ is even we have a different expansion than what you got.

2:56 PM
i just got a basic question, can't do $\int_{|z|=1}\frac{1}{\sqrt{z}}dz$.

group of units of Z/(p) is C_(p-1) = C_{2k} = C_{2^something} x C_{something odd} so you're rigt

@user58512 why does $k$ have to be odd?