Mathematics - 2022-10-12

02:17

@Thorgott Ok now I think it doesn't work... Should follow Baire category theorem

User1865345

04:09

When an author with a good sense of humor writes a book:

> Probability and Statistics used to be married; then they separated; then they got divorced; now they hardly ever see each other.

David Williams.

copper.hat

04:21

good and a bit sad

si-LV-er_and_b-LA-ck

04:53

but they're still good friends

copper.hat

but by some measure they are still singular

si-LV-er_and_b-LA-ck

and casinos prey upon that singularity

copper.hat

they thrive on large numbers

si-LV-er_and_b-LA-ck

the larger the better

PNDas

Is it true that if distribution has finite order then it has compact support?

I found citeseerx.ist.psu.edu/viewdoc/…, see theorem 3.16 but I don't understand why $||\phi||_m$ is zero if support of $\phi$ doesn't intersect $[-m,m]$.

Isn't $||\phi||$ max of the supremums of derivatives of $\phi$?

Gotta go. I have a lecture now.

An_Elephant

07:33

@robjohn Yes I know product rule.But I am asking that suppose I forgot that d(xy) = xdy + ydx. Then how would I do it? The matter is not of only d(xy) but others also. For example, if I got somewhere written an expression like : (x^2)dy + 2xydx , then am I supposed to remember beforehand that this expression is also equal to d(yx^2) ?

@PM2Ring Yes I know product rule.But I am asking that suppose I forgot that d(xy) = xdy + ydx. Then how would I do it? The matter is not of only d(xy) but others also. For example, if I got somewhere written an expression like : (x^2)dy + 2xydx , then am I supposed to remember beforehand that this expression is also equal to d(yx^2) ?

@TedShifrin Yes I know product rule.But I am asking that suppose I forgot that d(xy) = xdy + ydx. Then how would I do it? The matter is not of only d(xy) but others also. For example, if I got somewhere written an expression like : (x^2)dy + 2xydx , then am I supposed to remember beforehand that this expression is also equal to d(yx^2) ?

Not Tfue

08:17

math.stackexchange.com/questions/102874/….

In the comment Jonas said L is also subset of S.

Is Jonas right?

I was also confused about rudin theorem 1.11 about it.

Fargle

Take one of the terms (say, x^2 dy), and "partially integrate" it, with the other variable held constant, to get yx^2 + C(x) (since we're treating x as a constant, our "constant of integration" can in principle depend on x). Then take the x-derivative of this and set it equal to the dx part:

2xy + C'(x) = 2xy, so C'(x) = 0 or C(x) = C, just a constant. Thus d(yx^2 + C) = x^2 dy + 2xy dx, and we can just take C = 0 in particular.

(This technique only works for expressions f dx + g dy such that the y-partial of f is equal to the x-partial of g, but if that's not the case, you're already in trouble trying to write f dx + g dy as dh for some other function h.)

As an example, take (2xy + 6) dx + (x^2 - 3y^2) dy. The y-partial of the first guy is 2x, and that's the same as the x-partial of the second guy.

Integrating the first term gives yx^2 + 6x + C(y), and taking the y-partial and setting it equal to the second term gives x^2 + C'(y) = x^2 - 3y^2, so that C(y) = -y^3 + C, and we can in particular take C = 0, so that (2xy + 6) dx + (x^2 - 3y^2) dy = d(yx^2 + 6x - y^3).

An_Elephant

09:12

@Fargle I think I can understand it a little . I haven't yet learn this type of calculus yet ( perhaps it's called multivariable calculus). I got these things in electric field and potential in physics. Thanks very much

robjohn

09:36

@An_Elephant I never mentioned the product rule. I said integration by parts. That would handle both of your questions.

THC

11:41

Is it true in general that a function X -> R with (X, A, mu) a measure space can be approximated (with the p norm, 1<p<infty) by functions with support of finite measure ? I feel like this requires sigma-finiteness of X, can this still be done without this constraint ?

An_Elephant

@robjohn Yes thanks

Thorgott

@THC I agree that $X$ need be $\sigma$-finite

THC

Is there a simple counter example in non sigma finite measure ? I can't think of one

Thorgott

but I also think you want the function to have finite p-norm, otherwise this isn't even true for sigma-finite spaces

THC

Yes i'm dealing with functions in L^p(X), forgot to mention it

If f is approximated by (f_n)_n with mu(supp(f_n))<infty, then supp(f)=\cup_n supp(f_n). In particular, if supp(f)=X, then mu is sigma-finite

is there always f in L^p(X) such that supp(f)=X ?

Thorgott

12:04

sorry, the missing integrability condition threw me off

I think the result is true

since integrable functions in L^p can be approximated by integrable step functions and those necessarily have support of finite measure

THC

right! thank you

Thorgott

12:22

an interesting corollary of this I've never thought about is that integrable functions necessarily have $\sigma$-finite support

Unknown x

15:30

What can I say about the extreme of convex functional? I come

I know about the extreme of functions on convex set in real analysis.

How things change here?

Can you give some hints?

Akiva Weinberger

15:53

Suppose we have an n-dimensional cube with diagonal 1 (and therefore sidelength $1/\sqrt n$). Empirically, as $n\to\infty$, the average distance between two randomly-chosen points in the cube approaches $1/\sqrt 6$

I don't know why that is

Ted Shifrin

16:32

@An_Elephant It’s all product rule. Of course you don’t memorize all possibilities. The list is very infinite.

copper.hat

17:06

@AkivaWeinberger If $X_k$ is distributed uniformly on $(-1,1)$ the the weak $L^2$ law shows that (a little bit of work) ${1 \over n} (X_1^2+\cdots + X_n^2) \to {1 \over 3}$ in probability.

It is basically because $E X_k^2 = {1 \over 3}$.

Ted Shifrin

Interesting, copper. Now what?

copper.hat

:-) Just giving Akira a plausible answer...

sorry, i meant plausible approach.

Ted Shifrin

How do we get the expected value of the sum of squares of difference of two?

copper.hat

17:23

Same approach except replace $X_k^2$ by $(X_k-Y_k)^2$.

Ted Shifrin

What’s E[X_iY_i]$?

0 by symmetry?

Thorgott

17:38

whats Y_i?

Ted Shifrin

An independent RV with the same distribution as $X_i$

copper.hat

They are independent with zero mean. There are many details, wasn't really meant to be an answer, just some 'intuition'. almost thermodynamic in nature...

Ted Shifrin

You mean dominated by entropy?

copper.hat

17:54

i am speaking very loosely this morning... i was thinking of the high school derivation of boyles law?

Thorgott

then yeah

E[X_iY_i]=E[X_i]E[Y_i] by independence and both factors are 0 (by symmetry)

Ted Shifrin

Hmm … not sure what to do about the $\sqrt{}$.

Obliv

21:01

Are lines, planes, spaces all just sets of numbers

Xander Henderson

@Obliv Yes? No? In what context?

Jakobian

@Obliv we think of them as sets of tuples of numbers, yes

geocalc33

21:16

3

Obliv

21:33

sorry had to brb, follow-up question: what is the difference between geometry and algebra?

I can't tell which one is more fundamental, like abstract algebra objects seem to encompass geometric ideas but I haven't studied enough geometry to know the extent of the overlap

PM 2Ring

21:59

Bear in mind that people were doing geometry for a couple of thousand years before coordinates were invented. And sets were invented less than two centuries ago.

Xander Henderson

@Obliv What do you mean by "more fundamental"?

monoidaltransform

0

Q: Smooth function defined on smooth submanifolds extending

Problem: Assume that $S$ is an embedded submanifold of $M$. Further assume $f\in C^{\infty}(S)$ then show that for each $p\in S$ there exists a neighborhood $U$ of $p$ in $M$ and a smooth function $\tilde{f}:U\rightarrow \mathbb{R}$ such that $\tilde{f} \equiv f$ throughout $U \cap S$. My attempt...

Ted Shifrin

22:18

Not much of an attempt.

Unknown x

23:03

I don't have much knowledge about the area, extremum of convex functional. Can anybody help me to learn the topic? Please suggest reading for learning the topic. I searched in basic functional analysis textbooks. I couldn't find. In which area, could I find? What can I say about the extreme of convex functional?

Ted Shifrin

23:27

You need to address that question to our convexity guru, @copper.hat.

Pedro Urday

Hi!

Unknown x

@TedShifrin okay.

Pedro Urday

I need to ask a question but its not easy for me to write Latex code. Can I ask using an image to show formulas?

leslie townes

unknownx when you get to the guru it may also help to specify any conditions you care about, e.g. what kind of space is the functional on, what kind of subset do you want to study extrema on, maybe even max vs. min (this sometimes makes a difference)

my naive guess is that the most general results possible are not strong enough to be interesting, and that the ones people use have hypotheses that prevent them from applying everywhere

Ted Shifrin

@PedroUrday if you plan to use MSE regularly, you should learn the basics of MathJax. But for one question, go ahead.

Pedro Urday

23:39

Can i ask here?

Ted Shifrin

Go ahead.

leslie townes

we may even be able to assist in offering suggestions as to mathjaxing it, or where to learn more about mathjaxing things like it

Pedro Urday

ok

Ted Shifrin

@leslie Has the coyote retired?

leslie townes

hard to know. we last saw it on a trip to the pool, and since it gets dark so soon after munchkin gets home from day care, she doesn't want to go to the pool anymore. even though we still need to kill the same time before dinner.

lots of great hawks lately though. on the neighbor's house. in our tree right next to our balcony. on our roof. must be good squirrel season.

Ted Shifrin

23:45

May the hawks hunt happily!

robjohn

@PedroUrday You can, but it might not be well received. There will be people telling you to use MathJax, and there may be downvotes because of it.

and as leslie says, people here can help with MathJax

Ted Shifrin

I thought he wanted to post in here

robjohn

If he does, then there will be no downvotes, but he may get thwapped.

Ted Shifrin

We gave royal permission for one time!

Although I was expecting it would be before the winter solstice.

leslie townes

i was really hoping it was a buildup to some kind of prank image. that would have been pretty funny, given the setup. not to give anyone ideas.

Pedro Urday

23:57

Can you see the image?

Ted Shifrin

Even in images, parentheses are needed in mathematics.

Yes, it’s horrible, but we can see it.

leslie townes

yes, we can. the first line could be "x = x_3 2^3 + x_2 2^2 + x_1 2^1 + x_0 2^0" in mathjax.

Ted Shifrin

Dollar signs, of course.

leslie townes

i didn't want it to render, in case he'd somehow had it on. but good point, " needs to be dollar sign there.

Ted Shifrin

Yes.

Pedro Urday

23:59

Ok. I will try it

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$Mathematics$ Mathematics