3:05 AM
Does anyone know whether the classifying space functor from the category of groups to the category of topological space is a full functor?

How does one find the norm for a particular algebraic number field? Like $\mathbb{Z}[\sqrt[3]{-2},\sqrt{-2}]$ as an example.
(It always seems to happen. Two or three people with questions show up at the same time)

Let $f(x) = (x^ 2 − 2)(x^ 2 − 3)(x^ 2 − 6)$. For every prime number p, show that $f(x) ≡ 0 \pmod p$ has a solution in $\Bbb Z$.
I have no idea what to do

That sounds like a job for quadratic reciprocity, Silent.
At least I'm pretty sure. Just show that 2, 3, or 6 is a quadratic residue mod p for each prime p.

@Rithaniel I have encountered term 'quadratic residue' first time, so will you please explain a little bit more?

Hi everybody, I've posted a question here (https://math.stackexchange.com/q/3203462/662117),
could you please check it for any help

3:19 AM
Gotcha. So $x$ is a quadratic residue mod $p$ where $p$ is prime if there exists an $a$ such that $a^2\equiv x$ mod $p$.
You generally show a number is a quadratic residue via use of the Legendre symbol (if I've spelled that correctly). Which has it's own fairly simple (but detailed) arithmetic associated with it.
The Legendre symbol $\binom{x}{p}$ is 1 if x is a quadratic residue mod p, or -1 if it's not. If $x=a_1a_2\cdots a_n$ then $\binom{x}{p}=\binom{a_1}{p}\binom{a_2}{p}\cdots\binom{a_n}{p}$
(Typing this out on mobile is difficult)
There are other details, but that should be enough to solve this particular problem, I hope.
@Silent (In case you'd like a ping)

Thank you so much, Rithaniel, for going through all this pain.

3:37 AM
Ah, it's okay. Had some time to kill. :P

guys, take a few seconds to check my question, I lack a good maths base to be sure of my approach there

3:55 AM
hello guys
can a vector, not a scalar, follow a probability distribution?

4:20 AM
@Rithaniel, so even $x^2-2$, $x^-3$ and $x^2-6$ each is individually zero mod p for any prime mod p right?

4:57 AM
0

Please assume that this graph is a highly magnified section of the derivative of some function, say $F(x)$. Let's denote the derivative by $f(x)$. Let's denote the width of a sample by $h$ where $$h\rightarrow0$$ Now, for finding the area under the curve between the bounds $a ~\& ~b$ we can a...

1 hour later…
6:07 AM
@Ultradark You can try doing a finite difference to get rid of the sum and then compare term by term. Otherwise I am terrible at anything to do with primes that I don't know the identities of $\pi (n)$ well

6:22 AM
in The h Bar, 43 secs ago, by Secret
That is, given a language L where each word has a semantic meaning, is it possible to find some string s such that s cannot be ascribed semantic meaning for any extensions L' of L

1 hour later…
7:40 AM
@TedShifrin @Semiclassical mathcounterexamples.net/…

What an ugly function

8:07 AM
if $F' = f$, is true that $F' \circ g = f \circ g$?

@Odestheory12 yes

2 hours later…
9:57 AM
@Silent No, take for example the prime 3. 2 is not a residue mod 3, so there is no $x\in\mathbb{Z}$ such that $x^2-2\equiv 0$ mod $3$.
However, you have two cases to consider. The first where $\binom{2}{p}=-1$ and $\binom{3}{p}=-1$ (In which case what does $\binom{6}{p}$ equal?) and the case where one or the other of $\binom{2}{p}$ and $\binom{3}{p}$ equals 1.
Also, probably something useful for congruence, if you didn't already know: If $a_1\equiv b_1\text{mod}(p)$ and $a_2\equiv b_2\text{mod}(p)$, then $a_1a_2\equiv b_1b_2\text{mod}(p)$

10:21 AM
just a number theory question
remainder when $5^103$ divided by 14
I am trying to apply the congruence but getting no trcks

$5^{103}$?

oh
yes
sorry
$5^{103}$ is divided by 14

Divide $103$ by $14$ with remainder, use Euler's theorem

oh cool!
got the remainder as 5
cheers!

10:52 AM
:)

11:14 AM
is Q(π) simple extension of Q ?

yes
A simple transcendental extension

11:32 AM
@ÍgjøgnumMeg is it necessary in definition of simple extension of F that the element which we are going to adjoin to F is root of some polynomial of F ?

No, just an extension by a single element
So it is either isomorphic to an extension obtained adding the root of a polynomial or to F(X)

2 hours later…
2:04 PM
Is there any book or article that explains the motivations of the definitions of group, ring , field, ideal etc. of abstract algebra and/or gives a geometric or visual representation to Galois theory ?

@Ted plug

what I am not sure though is how large the galois theory section is

2:24 PM
@Secret Thanks :)

2:39 PM
Hi, guys

3:26 PM
Hi guys

Hi guys

any patterns here?

3:47 PM
Yes
It looks like some of the numbers appear only twice
2, 5, 10, 13, 20, 28

all numbers except $1$ occur an even number of times
I don't know why though

4:40 PM
Hi, please, can anybody provide hint or suggestion regarding this question? math.stackexchange.com/questions/3204567/…

5:30 PM
@TedShifrin hey

5:45 PM
Hi @Leaky

@TedShifrin comment tu dis Sturm?

Huh?

comment prononces-tu le nom Sturm
'ya pour moi trop d' consonnantes

Shtoorm (auf deutsch) ... Sterm in English

il est francais non

5:47 PM
pas autant que je sache

Jacques Charles François Sturm ForMemRS (29 September 1803 – 15 December 1855) was a French mathematician. == Life and work == Sturm was born in Geneva (then part of France) in 1803. The family of his father, Jean-Henri Sturm, had emigrated from Strasbourg around 1760 - about 50 years before Charles-François's birth. His mother's name was Jeanne-Louise-Henriette Gremay. In 1818, he started to follow the lectures of the academy of Geneva. In 1819, the death of his father forced Sturm to give lessons to children of the rich in order to support his own family. In 1823, he became tutor to the son...

hmm, German/Swiss name ... not French

ah, people with inconsistent names
> The family of his father, Jean-Henri Sturm, had emigrated from Strasbourg around 1760 - about 50 years before Charles-François's birth

it's almost like people can move from one place to another

that explains things...
ok so it's German
and like "Paul du Bois-Reymond"

5:50 PM
Is that allowed, @Eric? Here, it's clear from Leaky's link that Geneva was then part of France. Switzerland is always murky, with all sorts of French/Italian/German mixtures.

@TedShifrin well give it time and it wont be allowed, at least not here

growls

if you looked at the name Paul du Bois-Reymond it would be undoubtedly french
what do you know, his whole family is German
and even from his full name, Paul David Gustav(e) du Bois-Reymond, you can't tell
Paul and David and Gustav are all French/German names

@TedShifrin it's actually becoming easier to move to america from brasil tho, bc the us likes horribly regressive political regimes

Oh, that's a lovely thought.
I can pick among Russia, Brazil, and Libya.

5:55 PM
hoorah

@TedShifrin oh did you see the link I sent you?

Yeah, I saw it.
I highly recommend Körner's book, regardless. It has fabulous stuff in it.

I'm requesting it of the library

OK.
It's one of the "few" books I kept when I gave away hundreds and hundreds of books.

that's nice
but wouldn't it be, you know, too analytical for your liking

6:00 PM
I like analysis, dopey.

you must like... Tate's thesis then

Know nothing.
And don't be stupid about over-generalizing statements ....
I no more like everything in analysis than I like everything in differential geometry or certainly everything in algebraic geometry. I know only so much.
And what I know is diminishing by the day.

oh :c

fourier analysis is also super cool and good and if anyone doesnt disagree they drool

I have no idea what my Erdös number is. Probably low, since he was a coauthor with lots of UGA folks and visited frequently during his lifetime.

6:08 PM
@TedShifrin it's four!
Theodore Shifrin -> Rémi Langevin -> Gilbert Levitt -> Jean-Louis Nicolas -> Paul Erdős

That's a crazy route. There might be a shorter one through Pomerance at UGA.

Pomerance who?

Carl Pomerance — he wrote zillions of things with Erdös.
He left UGA, went to industry, then many years at Dartmouth. Now retired retired.

that's crazy considering how completely different the field u work in is

Theodore Shifrin -> Robert Varley -> Robert S. Rumely -> Carl Pomerance
that's still 4

6:12 PM
I figured the link was Varley, but I don't remember the specific route. OK, 4 it is. Not that I give a damn.
The French route is through Langevin, who's a geometer, as is Levitt.
I don't know Nicolas.

Does this make sense $D_f = w(u) *f \in \mathbb{R}^{J \times n \times F \times b}$, where $w(u) \in \mathbb{R}^{J \times n \times n}$ and $f \in \mathbb{R}^{n \times F \times b}$?
I essentially applying a matrix multiplication, but the dimensions are little bit more intricate

I have no idea what you're writing.
What does $\Bbb R^{a\times b\times c}$ mean?

@TedShifrin have you read Proofs From The Book?

No.

have you heard of it?

6:20 PM
Probably.

@TedShifrin It's a tensor

6:50 PM
I spent my career working with tensors. You have to be careful about defining multilinearity, domain, range, etc. Typically, tensors of type $(k,\ell)$ involve a fixed vector space, not so many letters varying.

Isn't that rare?
to just have a masters
and not a phd

depends where/what kind of institution

Probably not a professor. Lots of instructors/lecturers may have just a masters.

oh yeah you are right this person was indeed a lecturer
she studied banach spaces i recall

@TedShifrin isnt it weird just to have a masters in math
in the us i mean

7:00 PM
unless it's in math education

Nah. Lots of people end up there. People who want to teach at a small school, people who can't make it to finishing the Ph.D.
Even at Berkeley, a lot of my fellow grad students dropped out after their masters. A LOT.

well yeah i guess if u master out, but i dont even know if i can name a program that offers a terminal masters off the top of my head

West chester university in pennsylvania does

actually that's a lie, i know nyu does, but i hear it's a cashcow

UGA definitely grants a number of masters to people wanting only that (and sometimes admitted only for that). You people at fancy places think that every university is like Chicago, MIT, and Princeton.
NYU is probably in financial math or something.

7:02 PM
nah it's pure
@TedShifrin idt every school is like the kinds of schools i have experience with *i just dont know

OK
I'm saying that most likely when you get out of the top 10-20 schools, everyone does it.
Some faculty don't like it because they don't think it's what "we" are there for. I totally disagree.

@TedShifrin In my case, I am just using a library like TensorFlow to perform this tensor-like operations

I have no idea what any of that means.

Bonsoir @TedShifrin ca va?

Oui, ça va, mais je ne reste pas longtemps.

7:14 PM
ok

Pas autant que je sache.

comment est appliqué le théorème de sortie de compact?

Je ne connais pas ce dont vous parlez.

j'essaye de comprendre la preuve de ce théorème:perso.eleves.ens-rennes.fr/~afontain/…

Comme j'ai dit ailleurs, il me faut m'en aller. Pardon.

7:24 PM
pas de pb

8:15 PM
hi there,
I need to linearize nonlinear system about a fixed point. I've computed the jacobain matrix but one of the elements of this matrix is undefined at the fixed point. What is a better approach to solve this issue?
The element is (24*x_2 + 5cos(x_1)*x_2)/abs(x_2).
The fixed point is x_1=0, x_2=0

8:50 PM
@CroCo what is the system?

9:31 PM
@LeakyNun please see the following pic

10:02 PM
Consider the following integral: $\int 1/4*(1/(1+(u/2)^2)))dx$ Why does it matter if we put the constant 1/4 behind the integral versus keeping it inside? The solution is $1/2*\arctan{(u/2)}$. Or am I overseeing something?
*it should be du instead of dx in the integral
**and the solution is missing a constant C of course
Actually, don't bother...it solved itself!

10:23 PM
@TedShifrin can you tell me what does it take to do postdoc at Harvard, Princeton, MIT, etc.

10:55 PM
Is there a standard way to divide radicals by polynomials? Stuff like $\frac{\sqrt a}{1 + b^2}$?
My expression happens to be in a form I can normalize to that, just the radicand happens to be a lot more complicated. In my case, I'm trying to figure out how to best simplify $\frac{x}{\sqrt{1 + x^2}}$, and so far, I've gotten to $\frac{x \sqrt{1+x^2}}{1+x^2}$, and it's pretty obvious you can move the $x$ inside the radical.
My hope is that I can somehow remove the polynomial from the bottom entirely, so I can then multiply the whole thing by a square root of another algebraic fraction.
Complicated, I know, but this is me trying to see if I can skip calculating Euclidean distance twice going from atan2 to something in terms of asin for a thing I'm working on.
"... and it's pretty obvious you can move the $x$ inside the radical" To clarify this in advance, I didn't mean literally move it verbatim, but via $x \sqrt{y} = \text{sgn}(x) \sqrt{x^2 y}$. (Hopefully, this was obvious, but I don't want to confuse people on what I meant.)
Ignore my question. I'm coming of the realization it's just not working how I would've hoped, so I'll just go with what I had before.