Mathematics - 2019-05-18

2:45 AM

Let $x_n$ be a sequence of positive real numbers. If $\sum_{n=1}^{\infty} x_n$ convergent, then $\sum_{n=1}^{\infty} (x_n)^2$ convergent, too.

Please provide a counter-example to above statement

1 hour later…

4:06 AM

@Silent that statement should be true for positive numbers. If $\sum x_n$ converges, then $x_n$ converges to $0$. Choose some $N$ such that $x_n < 1$ for $n \ge N$, then $x_n^2 < x_n$

If these aren't positive then I think $\sum \frac{(-1)^n}{\sqrt{n}}$ converges, but then when you square terms you get the harmonic series

vzn

4:26 AM

in The h Bar, yesterday, by Ryan Unger

https://www.dropbox.com/s/8ufwclcy7aqdhw6/ThesisFinal%20Jan.pdf?dl=0

in The h Bar, yesterday, by Ryan Unger

Yes. Going to Princeton in the fall to work with Fernando Marques

2 hours later…

6:48 AM

What does 'roots occur alternatively' mean?

6:59 AM

So let's say $P$ has roots $\lambda_1 > \ldots > \lambda_5$. You want to show that $P'$ has 4 distinct real roots $\mu_1 > \ldots > \mu_4$ and that $\lambda_1 > \mu_1 > \lambda_2 > \mu_2 > \ldots > \mu_4 > \lambda_5$

@Daminark Where can i read more about this? I have encountered this kind of problem first time.

I dunno any source, but these you can just prove

The only two problems about interlacing that I know are the following

First is a generalization of yours, namely if you have a polynomial $p$ with real roots $\lambda_1 \ge \ldots \ge \lambda_n$ (not necessarily distinct), show that $p'$ also has real roots $\mu_1 \ge \ldots \ge \mu_{n-1}$ and that they interlace: $\lambda_k \ge \mu_k \ge \lambda_{k+1}$

Second, if you know Courant-Fischer (if not you can prove it too, good linear algebra practice), then let $A$ be a symmetric matrix and let $B$ be $A$ except that we killed the $i$th row and column

Then $A$ and $B$ both have all real eigenvalues by the symmetric theorem, show that they interlace

7:20 AM

Thank you!

5 hours later…

12:04 PM

I spent more than an hour to figure this out : Consider a sequence $a_n$ of positive numbers satisfying the condition $a_na_{n+2}\le a_{n+1}^2$ for all $n\in \Bbb N$ then $a_n$ is convergent sequence if $a_1=2a_2$. Is this true or false?

2 hours later…

1:42 PM

Hey anakhro

How are you?

Doing well, how about you?

Not horrible

What have you been up to?

Re Silent: Hmm, so just thinking out loud here, $a_1a_3 \le a_2^2 = \frac{1}{4}a_1^2$, meaning $a_3 \le \frac{1}{4}a_1$. And $a_2a_4 \le a_3^2 \le \frac{1}{16}a_1^2$, meaning $a_4 \le \frac{1}{8}a_1$. Conjecture: $a_n \le \frac{1}{2^{n-1}}a_1$. Should be easily provable by sufficiently careful induction

Not too much, having a bit of a hard time holding on to motivation to be diligent for classes but I've just gotta push through a few more weeks and I'll be done. Then it'll be getting ready for grad school

what classes

1:49 PM

Graph theory, number theory, ancient empires (required class, probably the one I'm most checked out of), and probability (close second, kinda just doing it for a requirement)

What are you covering in the number theory course?

Also maybe you can help me with a question with graphs: in colloquial usage, does "acyclic" already guarantee "simple"

My rough understanding that "simple" was no edges between the same vertex, and no double edges.

So those would both be covered under "acyclic"

Even though acyclic is much stronger.

I just want to be succinct and clear in my thesis....

It's been going around between things. Lectures were hard to understand + in the morning so I hadn't been good about going to class but the topic flow was basically commutative algebra (Dedekind domains, DVRs) -> p-adics -> Riemann surfaces = function fields in one variable (didn't really prove, just did the CP^1 example and said it holds) -> Riemann Roch and the number theoretic analogy. Now it's units, I think soon it'll be class groups

At the end I think the intent is to do brief intros to CFT and maybe Langlands

Nice! Also I will be back in at most 15 minutes, just gotta do a chore, sorry!

As for simple graphs, I haven't ever worked with non-simple graphs but acyclic seems to imply simple so it's probably not necessary. Might not hurt to throw in a remark to clarify though?

Q: Rigorous Proof of Binomial Theorem for Fractional Powers

2:13 PM

@Daminark pfft, who needs clarifications!

I will leave it as a "tree" being an acyclic, connected graph.

baxx

3:12 PM

-1

We know that the binomial theorem and expansion extends to powers which are non-integers. For integer powers the expansion can be proven easily as the expansion is finite. However what is the proof that the expansion also holds for fractional powers? This question was previously asked here, and...

what's supposed to be done with a question like this?

it was suggested that instead of a new question (as created there) the original was extended, so my question was closed. Then when I tried to edit the original question to include what I was asking my edits were refused

Thorgott

3:30 PM

I'm afraid I can't comment on the policies regarding closed questions, but the proof you are looking for can be taken care of by an application of Taylor's theorem and Bernstein's theorem on real analytic functions

Mathphile

Q: Is there a maximum number of consecutive sexy prime pair sums that are all divisible by $10$?

I was finding the sums of pairs of sexy primes (prime numbers that differ by 6) and noticed that there are a lot of pairs who's sum is divisible by $10$. Ex. $(7, 13)$ as $7+13=20$ and $20$ is divisible by $10$. I then wondered if there are consecutive sexy prime pairs who's sum is divisible by...

elementary-number-theory prime-numbers conjectures

any help on this question?

3 hours later…

baxx

6:09 PM

@Thorgott what's the done thing though?

I mean - it seems a bit daft to have something suggested then the implementation of the suggestion refused

Thorgott

Idk, as I said, I can't comment on site guidelines and whatnot. In the worst case, you can always try asking a question on meta

2 hours later…

Leaky Nun

8:02 PM

@Semiclassical hey

Semiclassical

Leaky Nun

@Semiclassical have you read Rudin's Fourier Analysis on Groups?

Semiclassical

nope

Leaky Nun

@Semiclassical are you interested?

Semiclassical

not really

8:17 PM

anybody read any interesting mathematics articles lately? I really like Quanta Magazine

Sebastian Alexander B Nielsen

8:38 PM

Does this make sense? Say you have a coordinate plane and you apply a linear transformation that sends each negative number $K$ units to the right so that each negative number is identified with a corresponding positive number

Rithaniel

9:08 PM

Define negative on a plane. Also, linear transformations fix the 0 vector. Are you talking about a vector space and interpreting it as a plane?

Am I correct in saying that the following differentialequation is of type 2?

$$(dy/dx)=3x+2y$$

@Rithaniel the negative x axis

imagine taking the x-axis and cutting it at x=0

then placing the two sides in superposition

Rithaniel

9:25 PM

So, take (x,y) to (|x|,y)?

say you're working with a finite interval $(-10,10)$

$-10 \mapsto 0$

and generally you have $-n + 10$

for any integer $n$

$0\mapsto 10$

Rithaniel

Ah, gotcha.

So (x)mod(10)

You get a whole lot more overlap with that, but if you're just working with a finite interval (-n,n) it should be okay.

Nisman

Hey, is anyone familiar with probability and statistics ?? I have a couple doubts

yeah @Rithaniel

@Nisman what's your question

I'm going to make a plot of some points with -xmod(10) and -ymod(10)

Nisman

10:31 PM

@Ultradark Its actually a theory question, I have to use the rejection method to generate values of a continuous distribution X with function f(x) using an auxiliar distribution Y with function y(x)

The problem is my textbook says to find the c such that $\frac{f(x)}{g(x)} > c$ I need to find the critical point of $f'(x)$, but I have another textbook that says I need to use $\frac{f(x)}{g(x)} d/dx$

Which one is the correct one ?

They're probably both correct, just slightly different methods

I would use the first one