Mathematics - 2019-04-27

user131753

03:05

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

Rithaniel

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)

Silent

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

Rithaniel

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.

Silent

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

lio

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

Rithaniel

03:19

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)

Silent

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

Rithaniel

03:37

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

lio

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

Eiffelbear

03:55

hello guys

can a vector, not a scalar, follow a probability distribution?

Silent

04:20

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

user8718165

04:57

Hello can somebody please help me with this question?

0

Q: Question about definite integrals

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...

Secret

06:07

@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

Secret

06:22

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

Leaky Nun

07:40

@TedShifrin @Semiclassical mathcounterexamples.net/…

Zee

What an ugly function

Odestheory12

08:07

if $F' = f$, is true that $F' \circ g = f \circ g$?

Leaky Nun

@Odestheory12 yes

Rithaniel

09:57

@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 $a_1a_2\equiv b_1b_2\text{mod}(p)$

BAYMAX

10:21

just a number theory question

remainder when $5^103$ divided by 14

I am trying to apply the congruence but getting no trcks

ÍgjøgnumMeg

$5^{103}$?

BAYMAX

oh

yes

sorry

$5^{103}$ is divided by 14

ÍgjøgnumMeg

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

BAYMAX

oh cool!

got the remainder as 5

cheers!

ÍgjøgnumMeg

10:52

:)

Glator

11:14

is Q(π) simple extension of Q ?

ÍgjøgnumMeg

yes

A simple transcendental extension

Glator

11:32

@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 ?

Alessandro Codenotti

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)

tarit goswami

14:04

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 ?

Secret

Abstract Algebra: A Geometric Approach

ÍgjøgnumMeg

@Ted plug

Secret

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

tarit goswami

14:24

@Secret Thanks :)

nbro

14:39

Hi, guys

Ultradark

15:26

Hi guys

Dair

Hi guys

Ultradark

any patterns here?

nbro

15:47

Yes

It looks like some of the numbers appear only twice

2, 5, 10, 13, 20, 28

Ultradark

all numbers except $1$ occur an even number of times

I don't know why though

Angie

16:40

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

Leaky Nun

17:30

@TedShifrin hey

Ted Shifrin

17:45

Hi @Leaky

Leaky Nun

@TedShifrin comment tu dis Sturm?

Ted Shifrin

Huh?

Leaky Nun

comment prononces-tu le nom Sturm

'ya pour moi trop d' consonnantes

Ted Shifrin

Shtoorm (auf deutsch) ... Sterm in English

Leaky Nun

il est francais non

Ted Shifrin

17:47

pas autant que je sache

Leaky Nun

Jacques Charles François Sturm

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...

Ted Shifrin

hmm, German/Swiss name ... not French

Leaky Nun

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

Érico Melo Silva

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

Leaky Nun

that explains things...

ok so it's German

and like "Paul du Bois-Reymond"

Ted Shifrin

17:50

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.

Érico Melo Silva

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

Ted Shifrin

growls

Leaky Nun

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

Érico Melo Silva

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

Ted Shifrin

Oh, that's a lovely thought.

I can pick among Russia, Brazil, and Libya.

Érico Melo Silva

17:55

hoorah

Leaky Nun

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

A continuous function with divergent Fourier series

Ted Shifrin

Yeah, I saw it.

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

Leaky Nun

I'm requesting it of the library

Ted Shifrin

OK.

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

Leaky Nun

that's nice

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

Ted Shifrin

18:00

I like analysis, dopey.

Leaky Nun

you must like... Tate's thesis then

Ted Shifrin

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.

Leaky Nun

oh :c

what's your Erdős number?

Érico Melo Silva

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

Ted Shifrin

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.

Leaky Nun

18:08

@TedShifrin it's four!

Theodore Shifrin -> Rémi Langevin -> Gilbert Levitt -> Jean-Louis Nicolas -> Paul Erdős

Ted Shifrin

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

Leaky Nun

Pomerance who?

Ted Shifrin

Carl Pomerance — he wrote zillions of things with Erdös.

He left UGA, went to industry, then many years at Dartmouth. Now retired retired.

Érico Melo Silva

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

Leaky Nun

Theodore Shifrin -> Robert Varley -> Robert S. Rumely -> Carl Pomerance

that's still 4

Ted Shifrin

18:12

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.

nbro

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

Ted Shifrin

I have no idea what you're writing.

What does $\Bbb R^{a\times b\times c}$ mean?

Leaky Nun

@TedShifrin have you read Proofs From The Book?

Ted Shifrin

No.

Leaky Nun

have you heard of it?

Ted Shifrin

18:20

Probably.

nbro

@TedShifrin It's a tensor

Ted Shifrin

18:50

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.

Ultradark

I had a proffesor that had a master's in mathematics

Isn't that rare?

to just have a masters

and not a phd

Érico Melo Silva

depends where/what kind of institution

Ted Shifrin

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

Ultradark

oh yeah you are right this person was indeed a lecturer

she studied banach spaces i recall

Érico Melo Silva

@TedShifrin isnt it weird just to have a masters in math

in the us i mean

Ultradark

19:00

unless it's in math education

Ted Shifrin

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.

Érico Melo Silva

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

Ultradark

West chester university in pennsylvania does

Érico Melo Silva

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

Ted Shifrin

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.

I'm not talking about that.

Érico Melo Silva

19:02

nah it's pure

@TedShifrin idt every school is like the kinds of schools i have experience with *i just dont know

Ted Shifrin

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.

nbro

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

Ted Shifrin

I have no idea what any of that means.

Vrouvrou

Bonsoir @TedShifrin ca va?

Ted Shifrin

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

Vrouvrou

19:14

ok

connaissez vous le théorème d'Hadamard-Lévy?

Ted Shifrin

Pas autant que je sache.

Vrouvrou

comment est appliqué le théorème de sortie de compact?

Ted Shifrin

Je ne connais pas ce dont vous parlez.

Vrouvrou

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

Ted Shifrin

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

Vrouvrou

19:24

pas de pb

CroCo

20:15

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

Leaky Nun

20:50

@CroCo what is the system?

CroCo

21:31

@LeakyNun please see the following pic

schn

22:02

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!

Newbie

22:23

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

Isiah Meadows

22:55

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.

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$Mathematics$ Mathematics