Mathematics

1:45 AM

@robjohn Thanks! I have found over the years that many of my attributions are incorrect, so appropriate, pithy quotes are now surrounded by legal disclaimers :-).

Q: Show that $G:[c,d]\to \mathbb R$ defined by $G(x)=\int_0^{f(x)}F(t)dt, x\in [c,d] $ is differentiable and $G'(x)=F\circ f(x) \cdot f'(x).$

2:58 AM

Henlo, anyone here?

Unknown x

3:11 AM

1

Let $F$ be continuous on $[a,b].$ let $f:[c,d]\to \mathbb R$ be differential function satisfy $f([c,d])\subseteq [a,b].$ Show that $G:[c,d]\to \mathbb R$ defined by $G(x)=\int_0^{f(x)}F(t)dt, x\in [c,d] $ is differentiable and $G'(x)=F\circ f(x) \cdot f'(x).$ My attempt $\lim_{h\to 0} \frac{ G(x...

Q: Show that two orthogonal probability measures have the following representations using Vitali covering

3:33 AM

why are you posting the question here?

@robjohn is the mean green for Christmas?

Mike

3:49 AM

Hello, can anyone take a look at my post regarding orthogonal probability measures?

0

Problem: $\alpha$ and $\beta$ are two mutually orthogonal probability measures in the sense that for some Borel subset $E\subset[0,1]$, $\alpha(E)=0$ and $\beta(E)=1$. If $F(x)=\alpha\{[0,x]\}$ and $G(x)=\beta\{[0,x]\}$ for $0<x\leq 1$, show that for any $\varepsilon>0$, there is finite collectio...

real-analysis probability-theory measure-theory solution-verification

4:26 AM

Hi All.

In group $G = \{ 1 , -1 , -\iota , -\iota \}$

$O(1) = 1$

$O(-1) = 2$

$O(\iota) = 4$

$O(-\iota) = 4$

Question is that $n$ is the order of $a \in G$ where $n$ is the least positive integer.

How do we apply $a^k = e$ Where $k$ is some integer.

$k = nq + r , 0 \le r < n$ Where $q$ & $r$ integers.

(1) What is $q$ and $r$ as in my example???

(2) Also what is the meaning of $0 \le r < n$. Why we take $r$ value less than $n$ and greater than equal to $0$???

Pls explain these questions with my example.

love_sodam

4:44 AM

I know that if $R$ is a PID, then any submodule of a finitely generated free R-module is free

Can we drop the condition 'finitely generated?'

4:59 AM

beautiful

does someone know anything about the number of ways of representing a number as sums of cubes?

Hi @TedShifrin Good Morning.

hello @TedShifrin

hello @123

you new to this room?

Hello @LeonhardEuler . Yes i am new in this room.

okay

which field of math you study?

maybe algebra

University Math. But now i am focusing on math more deeply rather than computation.

5:09 AM

I study NT

Analytic NT

Want to know about the number of ways of representing a number as sums of squares

@LeonhardEuler Great

yes sure definitely.

But I can't find much information

they are not studied very much

@LeonhardEuler Pls see my question. If you can answer.

which question?

Pls see few comments above.

5:13 AM

@123 this one?

Yes this one.

okay let me read it

Sure Thanks.

$k=nq+r,\,0\le r<n$ reminds me of division algorithm

it's like we are dividing $k$ by $n$

you can think $q$ to be the quotient

and $r$ to be the remainder

this is what division algorithm says

a very trivial answer is $$q=\lfloor\frac{k}{n}\rfloor$$

@LeonhardEuler Yes it is Euclid division algorithm.

5:22 AM

I dunno much about group theory

Yes but in group theory, to find an order of element what $q$ and $r$ in my example?

@LeonhardEuler Okay.. no prob.

please ping me if someone gives a proper answer

@LeonhardEuler Sure. Yesterday i have discussed with @TedShifrin may today someone gives me answer.

5:37 AM

@123 Your question is very unclear.

5:47 AM

Hello @copper.hat

I want to understand theorem for finding the order of element of group with example.

In this theorem we used division algorithm. If $n$ is order of element $^n = e$. Why we use $a^k = e$ Where $k = nq + r$ and $0 \le r < n$.

What is $q$ and $r$ here as in my example for elements in group???

6:12 AM

@123 i don't know what theorem you are referring to or how the division theorem is used.

@copper.hat I sahred yesterday. let me tag.

ok, just fyi i'm leaving in a few mins.

@TedShifrin Pls the theorem above this comment.

robjohn

@copper.hat See here and here.

:-)

@123 sry, how do i find the theorem you are talking to. i don't have the energy to trawl through the comments looking

Q: Submodule of free module over a p.i.d. is free even when the module is not finitely generated?

6:23 AM

what would be the most important consequences of a formula for primes?

just wondering

Leaky Nun

36

I have heard that any submodule of a free module over a p.i.d. is free. I can prove this for finitely generated modules over a p.i.d. But the proof involves induction on the number of generators, so it does not apply to modules that are not finitely generated. Does the result still hold? What...

abstract-algebra ring-theory modules principal-ideal-domains

Q: What would be the immediate implications of a formula for prime numbers?

ah found it:

10

What would be the immediate implications for Math (or sciences as a general) if someone developed a formula capable of generating every prime number progressively and perfectly, also able to prove (or disprove) the primality of every N-th number. I know this is a very large and subjective answer,...

number-theory prime-numbers philosophy number-systems prime-twins

does someone want to see an exact fromula for the nth prime?

crazy

@Measureme Theorem is below this comment.

feynhat

7:13 AM

I have a nonsense question. When we say 'a contravariant functor F is left exact' we mean that given an ES A -> B -> C -> 0, then 0 -> F(C) -> F(B) -> F(A) is exact. Is 3 a special number here? Will this not imply A->B->C->D->0 is ES then 0 -> F(D) -> F(C) -> F(B) -> F(A) is ES.

ES = exact sequence

10:46 AM

Hey all, I was wondering whether anyone took Apostol's calculus

11:05 AM

@AlexD I have that book on my laptop but didn't read it

Could you check whether he covers infinite series, please?

If so, would you mind sharing on what chapter it is?

Okay it would take some time lemme check

Thank you

Tom M. Apostol calculus volume 1 One-Variable Calculus, with an
Introduction to Linear Algebra has a chapter on infinite series

Chapter 10 @AlexD

Thank you

11:16 AM

You're welcome

Hello

Henlo

Yeah I greet everyone

I like that :-)

I'm trying to solve a problem on implicit differentiation using partial derivatives but I keep getting the same wrong answer, obviously something escapes me

Can't I use the simple division rule we use in single variable calculus with partial derivatives?

What is the problem you are trying to solve?

I'll Latex it. It'll take me a few minutes

Suppose $f: \mathbb{R}^3 \to \mathbb{R}$ is $\mathcal{C}^2$ and $\frac{\partial f}{\partial z} (\Bbb{a}) \neq 0$, so that $f=0$ defines $z$ implicitly as a $\mathcal{C}^2$ function $\phi$ of $x$ and $y$ near $\Bbb{\bar{a}}$.

$\Bbb{\bar{a}}$ is $\Bbb{a}$ without the third component.

Show that:

Rithaniel

11:39 AM

How about that for a fractal?

$$ \frac{\partial^2 \phi }{\partial {x}^2 } (\Bbb{\bar{a}}) = - \frac{\frac{\partial^2 f}{\partial z^2} \left(\frac{\partial f}{\partial x}\right)^2 - 2 \frac{\partial^2 f}{\partial x\partial z} \frac{\partial f}{\partial x} \frac{\partial f}{\partial z} +\frac{\partial^2 f}{\partial x^2} \left(\frac{\partial f}{\partial z}\right)^2}{\left(\frac{\partial f}{\partial z}\right)^3} $$

I can't believe I did it (more or less XD)

What I did is use implicit differentiation with partial derivatives on $\phi$ to get:

Oh the partial derivatives on the right are all evaluated at $\Bbb{a}$

with implicit differentiation I get:

$$\frac{\partial \phi}{\partial x} = - \frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial z}}$$

But if I just differentiate a second time using the rule for fractions I get the wrong result. so that's clearly the wrong way to go about it. what is it that escapes me?

12:06 PM

If I have a linear operator $T : Z^r \to Z^r$ such that $T^3 = Id$ and 1 is not an eigenvalue of T. How do I show 3 divides the index of the image of I-T ? So far I've shown that 3 divides the norm of all the eigenvalues of I-T and that I-T is diagonalisable if we extend the map to $\Bbb Q[w]^r$ (where w is a primitive third root of unity).

Also it's clear that the eigenvalues come in pairs: the dimension of the eigenspace for w is the same as the one for w^2

Measure me

@123 @123 I don't really see a Theorem.

12:24 PM

@Astyx the index of the image of $I-T$ is precisely $|\det(I-T)|$ as long as that is nonzero (which is the case since $1$ is not an eigenvalue of $T$)

Oh yeah

Thanks!

robjohn

12:35 PM

@LeonhardEuler That looks to be based on $$\pi(n)=\sum_{k=2}^n\left\lfloor\cos^2\left(\pi\frac{(k-1)!+1}k\right)\right\rfloor$$

12:45 PM

@EdwardEvans Re this: I'm not using the fact that B is a DVR am I?

I'm just thinking about how it works with the valuation

err

Maybe it's a superfluous hypothesis

Yeah I don't think you need that B is a DVR, it's just any old ring between A and Frac A I think

It's just that $Frac(A) = A[p^{-1}]$ where $v(p) =1$

1:02 PM

what about smth stupid like Frac(A) = A[X]/(pX - 1) lmao

I gotta go listen to someone talk about L-functions now

what's stupid about that, that's how you localize at an element smh

No Frac(A) is formed by writing A x 1/A

even for 0

@EdwardEvans direct product, right

4:35 PM

@LeonhardEuler giphy.com/gifs/cauliflower-xasZeFOhWS5cQ this is smth for you

uh

How do I show that there are an inifnite number of primes of Z that totally split in $Q[\sqrt{d}]$ ?

I've shown that for every nonconstant polynomial P, there's an infinite number of primes p such that there exists an n such that p|P(n)

uhh

I can't remember

In other words I want to show that the decomposition group is trivial

@Simone I haven't read everything, but since you already did some work yourself, this might be a good question for MSE.

I did precious little I'm afraid

I'm only confused as to how I should go about to do a second order partial derivative of phi again

Why can't I use the calculus 1 division rule (converted to partial derivatives)?

Even if I use the more rigorous approach of the Jacobian matrix and the differentiation rules on those I get the same (wrong) result.

I think I haven't yet digested partial differentiation very well

4:48 PM

If you include your attempt at calculating the second derivative, then it might be a good question on the site, just saying.

I just tried the division rule, getting:
$$\frac{\partial^2 \phi}{\partial x^2} = (-1) \left( \frac{\frac{\partial f}{\partial z} \frac{\partial^2 f}{\partial x^2} - \frac{\partial f}{\partial x} \frac{\partial^2 f}{\partial x \partial z}}{ \left( \frac{\partial f}{\partial z} \right)^2} \right)$$

Which is wrong.

thanks for the consideration btw, you guys always help.

I also tried some reverse engineering on the solution, and I saw that what I got is a part of the solution, but that didn't give me an epiphany.

5:07 PM

@Simone I didn't do anything to help you :D

then at least give some moral support damn you!! XD

jk

Rithaniel

What about amoral support?

Fine I'll have a look at it

sure, or immoral support... if you meet me at the back of the alley...

@supinf thank you.

@Simone What you're missing is the points at which you actually take those derivatives. The complete formula reads $\frac{\partial\phi}{\partial x}(x,y)=-\frac{\frac{\partial f}{\partial x}(x,y,\phi(x,y))}{\frac{\partial f}{\partial z}(x,y,\phi(x,y))}$, so if you differentiate again, you don't only have to use the quotient rule, you also have to use the chain rule and the Jacobian of $(x,y)\mapsto(x,y,\phi(x,y))$ will come into play

5:15 PM

riiiiiiight.

I didn't think of that

that's why I didn't ask this on MSE, I knew I was being stupid

Thank you Thorgott.

ok someone was faster than me :)