Mathematics - 2020-02-27

Ted Shifrin

12:02 AM

I believe Fuzzy knows only standard Riemann integration. There's a problem with the domination at $1$, though, right?

Thorgott

12:18 AM

I haven't thought it through, but I think $\lvert t^{x-1}\sum_{n=0}^k(-1)^nt^n\rvert\le\frac{t^{x-1}}{1+t}+t^x$ works

Jack Ohara

12:53 AM

@TedShifrin I want to write a code on mathematica for the Miller Rabin test

I have everything but I dont know how to code it in mathematica

Ted Shifrin

1:17 AM

Mathematica has all sorts of tutorials and samples. I wrote a brief primer on Mathematica for my differential geometry students. It includes a number of the basics and some nontrivial little programs. If you send me an email (address in my profile), I can send you the Mathematica notebook. @JackOhara

Jack Ohara

@TedShifrin I shall thanks

But for my particular case

Do you know such code? I searched and found just nonsense codes

they wrote that they work on mathematica but failed all of them

Ted Shifrin

2:05 AM

@JackOhara: Nah, I don't even know the test or what you'd want to code.

Jack Ohara

@TedShifrin Okay,I sent you email !

I can check if I can use your pdf, either way It must be useful in general

user76284

2:35 AM

Does anyone know a good example of a simultaneous-move perfect information extensive form game? i.e. like chess or go but with simultaneous rather than sequential moves.

topologicalmagician

Every closed subset of an n-manifold is an n-manifold?

It is easy to show for open subsets

but what about closed?

Jack Ohara

@TedShifrin Thanks a million Prof ! :D

Ted Shifrin

@topologicalmagician: Absolutely not.

Open is way different from closed.

Sure, @JackOhara.

topologicalmagician

@TedShifrin why?

Ted Shifrin

What is an example of something that is not a manifold?

topologicalmagician

2:41 AM

@TedShifrin union of x axis and y axis

Ted Shifrin

But that's sitting inside $\Bbb R^2$ as a closed subset.

topologicalmagician

aha, I see.

anyways, I need to go now. I'm exhausted. Have an amazing day

Simple

2:58 AM

Suppose $(X,\mathcal{S},\mu)$ is a measure space and $f_1,f_2,\dots,$ are $\mathcal{S}$-measurable functions from $X$ to $\mathbb{R}$ such that $\sum_{k=1}^{\infty}\int|f_k|d\mu<\infty$. Prove that there exists $E\in\mathcal{S}$ such that $\mu(X\setminus\,E)=0$ and $\lim_{k\to\infty}f_k(x)=0$ for every $x\in\,E$.

can someone give me hint to start this problem

Thorgott

3:13 AM

Can you say something about $\lim_{k\rightarrow\infty}\int|f_k|\mathrm{d}\mu$?

Simple

since $\sum\int|f_k|<\infty$, $\lim\int|f_k|$ converges

Thorgott

converges to?

BananaCats Author

It converges to nothing

math is bs

@Ultradark

hey

:D

I should say math like everything else other than good food is bs :>

Simple

it converges to $f$

Thorgott

?

There is no $f$

Simple

3:22 AM

sorry, should be a finite number

Thorgott

yes, but which one

Simple

zero, I guess. cause we are looking for a set $E$ such that $\lim f_k=0$

Thorgott

Why does it converge to zero?

This has nothing to do with the measure theory side of things, for the record

Simple

I don't know

Simple

4:33 AM

@Thorgott I still didn't come up anything that gives zero

Ted Shifrin

5:28 AM

@Simple: Surely you know this. If $\sum a_n$ converges, what is $\lim\limits_{n\to\infty} a_n$?

Simple

@TedShifrin oh my god, I totally didn't recall this

Lucas Henrique

Hey chat

How do I prove that $\mathbb Q$, as a $\mathbb Z$-module with usual product, has no basis?

Without deep module theory arguments

Oops, linear dependence.

Secret

7:46 AM

A binary relation that is antisymmetric and reflexive only has no name

Stan Shunpike

7:57 AM

Can anyone recommend an introductory text on graph theory with emphasis on having plenty of exercises to do?

northerner

9:24 AM

Anyone here, @LeakyNun?

Rithaniel

10:55 AM

I hate when I solve a problem that says "find $Y$," I proceed to find what I believe to be $Y$, then next question says "Now perform these calculations with $Y$," which I also do without issue. But then, the next question says "Now perform those same calculations with $X$," but $X$ is exactly the same thing as I found for $Y$.

There is a strong sensation of "uh oh"

(At the same time, I'm nearly 100% certain that I did my math correctly. I feel that this is a question designed to try and trip you up)

Leaky Nun

11:38 AM

@LukasHeger this is the best I can do: if $\Bbb Q$ has a $\Bbb Z$-basis then $\Bbb Q = \Bbb Z^{\oplus I}$, but $\Bbb Q \otimes (\Bbb Q/\Bbb Z) = 0$ instead of $(\Bbb Q/\Bbb Z)^{\oplus I}$.

Mike Miller

Every two elements x = m/n and y= p/q of Q are linearly dependent. Here we have (np)x + (mq)y = 0. So if Q has a Z-basis, it must be one element. But Q is not cyclic: for any element x you think might be a generator, x/2 is also in Q, and is not of the form nx for some integer n.

And you meant the other Lucas

Slereah

12:09 PM

Is there an introduction to category theory which specifies along the way what the morphisms, objects, functors etc. are in ordinary mathematics

I'm not even sure what all of those are supposed to be for natural number objects so it's a bit tough

Leaky Nun

12:45 PM

@LukasHeger bist du hier?

@Slereah I think every introduction does that

Secret

So... I was doing some worldbuilding stuff with some community, and we end up building a weird "pre-topological" structure

Basically, there is a set S, which has some nets $\phi$ and some open sets $U$ but in general, all 3 axioms of topology fails, thus not all union and intersection of open sets always guarantee to be an open set

Circles are open sets, arrows are nets

Slereah

1:03 PM

@LeakyNun Kind of, but not usually for long

Secret

1:15 PM

Coherent topology

In topology, a coherent topology is a topology that is uniquely determined by a family of subspaces. Loosely speaking, a topological space is coherent with a family of subspaces if it is a topological union of those subspaces. It is also sometimes called the weak topology generated by the family of subspaces, a notion that is quite different from the notion of a weak topology generated by a set of maps. == Definition == Let X be a topological space and let C = {Cα : α ∈ A} be a family of topological spaces, as subsets of X (typically C will be a cover of X). Then X is said to be coherent with C...

this sounds like it

geocalc33

2:44 PM

@BananaCatsAuthor hey

pi-π

Hello

What is the difference between series and progression?

Rithaniel

4:48 PM

I could use some help with a stats problem

I'm trying to show that if $S^2$ and $\overline{X}$ are the sample variance and sample mean for a sample of Poisson random variables, that $E[S^2\mid\overline{X}]=\overline{X}$. But I can't wrap my head around a good approach

Ted Shifrin

5:03 PM

@Rithaniel: Dredging out of memory, if you have a Poisson random variable, it depends on a parameter $\lambda$, and the mean turns out to be $\lambda$. So does the variance. This must have something to do with your question.

nbro

Suppose I have a coordinate system $T_{x, y, \theta}$ represented by $x$, $y$ and an angle $\theta$. Then I convert some points $p_1, p_2, \dots, p_N$ to this coordinate system, i.e. $p_i^T$. Then I compute the distances between each $p_i^T$ and other coordinates represented as tuples $c_i=(x_i, y_i)$. I do this for different sets of points $p_i$, but $c_i$ are always fixed. $T_{x, y, \theta}$ can change.

My question is: what distance are we exactly computing when computing the distance between $p_i^T$ and all fixed $c_i$?

We're computing a relative distance, which depends on $T_{x, y, \theta}$

Rithaniel

5:19 PM

Yeah, I think I managed to get it, Ted. You actually did point me in the right direction. It's stuff with unbiased estimators and one-to-one functions of complete and sufficient statistics giving you an UMVUE

nbro

What I don't understand is whether $c_i$ is represented in the coordinate system of $T_{x, y, \theta}$ or not. $c_i$s are like $(0, 50)$, $(50, 50)$, etc. They are always fixed

Archer

5:53 PM

The eigenspace of A and A^k would be same right?

Because the same set of x's satisfying Ax = lambda x will satisfy A^x = lambda^k x

Could someone please verify?

Thorgott

Well, what's true is that if $x$ is an Eigenvector of $A$ with Eigenvalue $\lambda$, then $x$ is also an Eigenvector of $A^k$ with Eigenvalue $\lambda^k$.

And so, if $V_{\lambda,A}$ denotes the $\lambda$-Eigenspace of $A$, we have $V_{\lambda,A}\subseteq V_{\lambda^k,A^k}$, but the second one can be strictly larger.

Archer

@Thorgott when can it be larger?

Lucas Henrique

Hey there chat

I was looking up some info about groups (def. of subgroup generated by a set, generating set of a subgroup)

And this question appeared

18

Q: Subgroup generated by a set

A subgroup generated by a set is defined as (from Wikipedia): More generally, if S is a subset of a group G, then , the subgroup generated by S, is the smallest subgroup of G containing every element of S, meaning the intersection over all subgroups containing the elements of S; equival...

group-theory

Specifically, if $S = \emptyset$, those two definitions are not equivalent, right?

Thorgott

one thing that can happen is that you have two Eigenvalues $\lambda_1,\lambda_2$ with $\lambda_1\neq\lambda_2$, but $\lambda_1^k=\lambda_2^k$

another thing that can happen is nilpotency

you can put the matrix in JNF and then you just have to observe what taking powers does to the individual blocks

@Lucas they are equivalent

Lucas Henrique

6:09 PM

But the set of the union of finite products would be $\emptyset$

Thorgott

you're forgetting the empty product

Lucas Henrique

Ugh, that's unfair

:p

Thanks, @Thorgott.

Thorgott

lol, np

Peter

Who wants to help search a prime factor of ( 512! ) ^ 512 + 1 ? We only need to check the primes of the form 1024k+1 , but I nevertheless did not find a factor yet.

Ted Shifrin

6:41 PM

@Lucas @Thorgott: YUCK.

geocalc33

6:55 PM

I don't want to search for primes numbers Peter^^

Thorgott

Hey Ted

geocalc33

I will however, help you find someone who does :)

just kidding.

What is a decomposition operator in the spectral theorem sense

Ted Shifrin

Hi, Thorgott :)

geocalc33

does it basically decompose something into spectra in the form of eigenvalues or something?

hi @ted are you ready for super tuesday?

Ted Shifrin

Already mailed in my ballot.

geocalc33

7:01 PM

nice lol. I'm voting this year actually

I can't wait for the debate between the nominee and Trump

the older I get the more interesting politics gets

Hopper

If I have a set of matrices and their rref are identical (linearly dependent), does that imply that the original matrices must therefore be linearly dependent?

Ted Shifrin

No, @Hopper.

Hopper

Do you have a counter example? @TedShifrin

Ted Shifrin

Yes. Write down the simplest matrix with 2 rows that has rank 1 and write another row-equivalent version.

northerner

Hi, anyone here?

Ted Shifrin

7:14 PM

No, no one.

northerner

perfect

geocalc33

perf

northerner

Regarding equations like $k ≡ ( k \mod 2^n ) + ⌊ k / 2^n ⌋ ( \mod 2^n − 1 ) $ what is the difference between the two $mod$? Is the first a binary operator where the second is saying the equation is congruent?

Ted Shifrin

This doesn't even make any sense.

northerner

It's from here en.wikipedia.org/wiki/…

Ted Shifrin

7:18 PM

This is computer notation, not mathematics. The first mod means take the remainder when you divide by $2^n$. The second one means that we work mod $2^n-1$ and then $k$ is equivalent to its remainder when you divide by $2^n$ added to the floor of $k/2^n$. No math person would write stuff like this.

northerner

I had asked a question here but haven't yet got a response to my comment math.stackexchange.com/questions/3559467/…

Ted Shifrin

So, let's see. Take $n=3$ and $k=28$. Then $\mod 7$ they're noting that $28\equiv 4+\lfloor 28/8\rfloor = 4+3$.

Repeated division by 2 is used a lot for effective computation.

northerner

Would the $mod 2^n-1$ at the end ever make a difference to the calculation? That's more saying that it's a congruency, right?

Ted Shifrin

Yes, look at the example I wrote down.

Do your own examples.

BananaCats Author

@geocalc33 are you UltraDark?

northerner

7:30 PM

@TedShifrin ok thanks I think I get it

geocalc33

why yes!

@BananaCatsAuthor

Ted Shifrin

You can change your name but your character shines through regardless. :D

geocalc33

geocalc was my original name on this site and I wanted to take a break and turn a new leaf at

@TedShifrin

you don't know me. lol

Ted Shifrin

Yeah, I remembered geocalc.

geocalc33

I'm pretty nice actually

Ted Shifrin

7:32 PM

I never said you weren't nice.

geocalc33

this is one of my faults. I extrapolate things poorly

sometimes I reactly rashly online

but not in person.

like when I miss translated leaky nun's french and yelled at him and got banned for an hour

Ted Shifrin

Luckily for me, I missed that — I can translate French just fine :P

FuzzyPixelz

When talking about an ODE of the form $$y'=a(t)(y)+b(t)$$, is it just abuse of notation not to write it $y'(t)=a(t)(y(t))+b(t)$?

It really messes with my brain to write the $t$ for $a$ and $b$ but not $y$

Ted Shifrin

8:15 PM

@Fuzzy: I agree with you (except for your extra parentheses around $y$ and $y(t)$). I complained to my diff geo students when they did inconsistent stuff like this in their homework.

Rithaniel

9:23 PM

You guys ever get a student who is clearly not trying, maybe just putting math-like symbols on a page? I don't know how to grade that, cause my instinct is to point out where a student went wrong.

Thorgott

if there's no other option, you can always put a question mark

Rithaniel

Yeah, that's my default. But I wish I could help them

geocalc33

See them after class

Rithaniel

I'm a grader. My only contact with them is what I write on the paper

Ted Shifrin

9:40 PM

Write a note and tell the student to go to the professor's office hours (or to other help rooms) for help. It's not your responsibility to offer free tutoring to the class.

topologicalmagician

Hello

Let $X=$ $\{$ $f\in C[0,1]$ $:$ $f(1)=0$ $\}$. Let $A=$ $\{$ $f\in X$ $:$ $sup_{x\in [0,1]}|f(x)|$ $\leq 1$ $\}$. Show that this set is closed with the $d_1$ metric. What I have done is take a sequence $(f_n)_n$ in $A$ that converges to some f with $f(1)=0$ . So now I must show $sup_x|f(x)| \leq 1$ . My idea consists of taking $x\in X$ and $\epsilon>0$ and show $|f(x)| \leq 1 + \epsilon$. Are there other ways>

Thorgott

10:00 PM

looks like the preimage of a closed set under a continuous map

topologicalmagician

@Thorgott I'm trying to do it with inequalities. So, let $x\in [0,1]$ let $\epsilon>0$ . Since $f_n\rightarrow f$ converges to $f\in X$, there exists $N$ such that $n\geq N$ implies $\int_0^1|f_n-f|<\epsilon$. So $|f(x)| \leq |f_N(x)-f(x)|+1$. But not sure what to do next

Taking the integral doesn't do much

unless there's an inequality between $|f_N(x)-f(x)|$ and the integral

FuzzyPixelz

10:54 PM

for $x \in [0,1]$, you can write $$|f(x)| \le |f(x)-f_n(x)|+|f_n(x)| \le d_1(f_n,f)+1 \to 1 \text{ when } n \to \infty$$ Hence $\sup_{[0,1]}f \le1$

I don't think integrating can help

@topologicalmagician

Ted Shifrin

What is the $d_1$ metric?

topologicalmagician

@FuzzyPixelz you're confusing $d_{\infty}$ with $d_1$

$d_1$ is $\int_0^1|f-g|$

Ted Shifrin

You should have put that in your question.

The $L^1$ metric on $C[0,1]$.

topologicalmagician

@TedShifrin I thought it was clear, my apologies

Ted Shifrin

Nope, not remotely clear.

Also, note that because the functions are continuous on a compact set, the sup in the definition is actually the max.

FuzzyPixelz

11:05 PM

That's weird, my text calls $d_{\infty}$ $d_1$ and $d_2$ the integral one

Ted Shifrin

No, that would be $L^2$.

He's not integrating $|f|^2$.

topologicalmagician

@TedShifrin I'm aware that the sup is the max in this case. But, I'm still not sure how to finish off the problem

Ted Shifrin

So if the max of $f$ is actually greater than $1$, this max occurs at some point $c\in [0,1)$. What happens to the integral of $|f_n-f|$ in a neighborhood of $c$?

Hey there, DogAteMy.

topologicalmagician

@TedShifrin not sure

attains its max there?

Ted Shifrin

Does it go to $0$ as $n\to\infty$?

topologicalmagician

11:18 PM

@TedShifrin yes

Ted Shifrin

Nope.

Are you drawing a picture?

Transcript for

$Mathematics$ Mathematics