Mathematics - 2018-03-06 (page 2 of 3)

Alessandro Codenotti

10:00

Well... Quantifiers elimination in model theory

But I also want to see a couple of algebraic/geometric applications, in particular the Nullstellensatz and Tarski-Seidenberg

Balarka Sen

So yes :D

Aha

Slereah

What qualifies as a quantifier in model theory?

Like what unites $\forall$, $\exists$, $\Box$ and $\diamond$

Alessandro Codenotti

I think quantifier is just an informal term? Anyway I'm interested in FOL so I only worry about the first two

Slereah

How very amoral :p

(MODAL LOGIC JOKE)

Secret

there are amoral models?

Slereah

10:12

Modal logic has been used for axiomatization of morality

Balarka Sen

Jesus god everything I have been thinking for the last couple minutes is totally flawed

frikfraking hell

Slereah

that's math for you

Secret

There's some breakthrough today in office. After plotting a colorful scatter plot, I figure out what the trends should be and how to slice that high dimensional data

Balarka Sen

:applause:

Secret

I can make two plots of the same set of dots, but i can change the color gredient to run through the 3rd axis or 4th axis. The 2D graph plus the colors then give me the trends I am looking for and switching the color schemes effectively slice the dots along the 3rd and 4th direction respectively. Now to experiment further cause I have procratinated enough already

actually. post a better one:

so the color pattern determines whether the data is being sliced in the L2 direction or the L3 direction

Slereah

10:38

Why do physics people and math people use different symbols for the second fundamental form

Secret

10:49

I thought it is always $\mathrm{I\!I}$?

Balarka Sen

No, it's

Mar 2 at 16:43, by Balarka Sen

$\underline{I}\!\!\overline{I}\!\!\underline{I}\!\!\overline{I}$ is the second fundamental form

Slereah

Physicists usually use $K$

And call it the extrinsic curvature

Balarka Sen

what

Secret

I thought K is the gaussian curvature (or is that kappa?)

Slereah

Well $K_{ab}$

Hence not a very scalar quantity

Secret

10:52

make sense

Balarka Sen

@AkivaWeinberger You want to write a topology apologist answer on this old thread? Should be a good summary of all of what we discussed

It popped on my active feeds a few minutes ago

The two answers are basically elaborate calculations which does not really do the fact a justice

WhatsThePoint

11:11

hi I have a question, is the relational inverse just as simple as swapping all the ordered pairs?

Tobias Kildetoft

@WhatsThePoint Yes

Secret

Exponential factorial

An exponential factorial is a positive integer n raised to the power of n − 1, which in turn is raised to the power of n − 2, and so on and so forth, that is, n ( n − 1 ) ( n − 2 ) ⋯ . {\displaystyle n^{(n-1)^{(n-2)\cdots }}.} The exponential factorial can also be defined with the recurren...

tetration is weird

WhatsThePoint

11:33

@TobiasKildetoft thought so

Secret

11:51

[Random]

Akiva Weinberger

4

Secret

42224242422224244444222422242424444444224422224222242424222

Specker sequence

In computability theory, a Specker sequence is a computable, monotonically increasing, bounded sequence of rational numbers whose supremum is not a computable real number. The first example of such a sequence was constructed by Ernst Specker (1949). The existence of Specker sequences has consequences for computable analysis. The fact that such sequences exist means that the collection of all computable real numbers does not satisfy the least upper bound principle of real analysis, even when considering only computable sequences. A common way to resolve this difficulty is to consider only sequences...

Rotating surface in 4 dimensions

►

Fourth dimensional rotation of Z4

►

Balarka Sen

@AlessandroCodenotti Listening to Russian Circles rn

Alessandro Codenotti

They're great, which album?

Balarka Sen

Station

Akiva Weinberger

11:57

@Secret Very interesting

Alessandro Codenotti

@BalarkaSen the best one in my.opinion

Balarka Sen

Hah yeah it showed up in my youtube feed and I remembered you recommended it to me once

Akiva Weinberger

@Secret Death due to YouTube compression

Alessandro Codenotti

Youngblood and station are my favourite songs from the album

Balarka Sen

I'm on Harper Lewis. Amazing instrumentation

Great drums, good shreds

This borders on metal

Vrouvrou

12:02

@AkivaWeinberger there is no relation between $Y$ and $B$

Alessandro Codenotti

I think they're classified as post metal but they're pretty heavy compared to other bands in this genre

Balarka Sen

Mmm I see that

Secret

It's so sad that everytime the chat become active, I had to go to sleep

Someone please F888 timezones!

Akiva Weinberger

@Vrouvrou Then $\partial B$ and $\partial(Y\setminus{\rm int}(B))$ need not be related

Take two disjoint closed disks

Oh wait sorry

It was $\partial Y$ and $\partial(B\setminus{\rm int}(Y))$

Same thing though

Vrouvrou

no relation ?

Akiva Weinberger

12:08

If $B$ and $Y$ are disjoint closed disks then $\partial Y$ and $\partial(B\setminus{\rm int}(Y))$ are disjoint circles

Vrouvrou

ok thank you @AkivaWeinberger

@AkivaWeinberger we must prove that $\partial Y\subset \partial (B\setminus{\rm in}(Y)$

Akiva Weinberger

@Secret "F888"?

Vrouvrou

not $\partial B\subset..$

Akiva Weinberger

Then there must be some additional conditions on $B$ and $Y$

Maybe $Y\subset B$, with $Y$ closed and $B$ open

Something like that

Vrouvrou

i have that $Y$ and $B$ are the two closed

but i don't see any inclusion

Balarka Sen

12:28

@Alessandro Youngblood is flipping awesome

Tobias Kildetoft

@BalarkaSen You mean the Brass Band?

Balarka Sen

Ah, no, a song of Russian Circles I'm listening to

nitsua60

12:41

So a kid in my HS class came up to me after the exam:
"Mr. Nitsua, I hope it's alright, but one of the proofs I found easier to explain if I used Kronecker deltas."
"Of course that's fine--Kronecker deltas are maybe my favorite- no, one of my favorite bits of simplification. Einstein summation-notation, though, is probably my favorite."
"Oh. I used that, too."

Some days it's really nice being a teacher =)

0celo7

@BalarkaSen I watched Serre again last night

I didn't see the part you mentioned

was it the $f=O(g)$ thing?

user193319

0

Q: Space of Sequences with Finitely Many Nonzero Terms is Paracompact

I just proved the following theorem: If $X$ is a regular space that can be written as a countable union of compact subspaces of $X$, then $X$ is paracompact. I am now working on the following: Show that $\Bbb{R}^\infty$ is paracompact as a subspace of $\Bbb{R}^\omega$ with the box top...

Balarka Sen

I'd have to find it. But yes, something like that; you have things like, let $\epsilon > 0$ blah blah blah $K_1 |f| < |g_\epsilon| < K_2 |h|$ say, but a major pitfall is not to mention that $K_1$ and $K_2$ depend on $\epsilon$ or if it's a uniform bound or whatever the hell

He explicitly mentioned it

"How to write mathematics badly" by Jean Pierre Serre (noise removed)

►

Found it

0celo7

@BalarkaSen I'm working on something where there's three parameters and I feel like it's cheating because I get a bunch of shitty error terms that they don't get in the paper, but I can send something to zero and works out in the end

Balarka Sen

lmao

Well Serre gives you a good notation to say it

0celo7

12:50

I've never seen that double less than for what he says

in pde you can write $f\lesssim g$

Balarka Sen

that's the same as little o or some shit?

0celo7

no, it means $f\le Cg$

Balarka Sen

oh just big O

0celo7

no!

he explains that people abuse big O all the time

Balarka Sen

modulo abuse i meant

i'm from the clan of gromov, don't show abuse to me

0celo7

12:54

if you want to be correct, you do something like

Balarka Sen

i'll $\mathcal{Op}(E)$ you up

@0celo7 yeah but that just looks horrible

But probably helpful

0celo7

Serre might object to the dependence on $\Omega$, like what properties of $\Omega$ does it actually depend on

@BalarkaSen I've found that to be a good notation for personal notes

I've tried to use it when explaining stuff to people but they think you're insane if you do

Balarka Sen

it's the origin of the abuse of statement of h-principle

really, a function on Op(A) is a germ at A

0celo7

right

@BalarkaSen look at this horrible shit i.gyazo.com/52dec34f36a6f5d5b35515b5af10369a.png

Balarka Sen

is the trickery in "assume delta is small such that ..." statement?

everything else seems carefully chosen

there's a little o hanging around but that's fine

0celo7

13:02

@BalarkaSen yeah you need to choose $\delta\ll \rho$ and then $\varepsilon$ will be $\ll \delta$ or something

the idea is that you make $\rho$ small s.t. the term in [] is positive

$A$ is positive

Balarka Sen

mhm

0celo7

and then for $A$ fixed, you can choose $\delta$ as small as you want

so the first term goes to zero, but the second one goes faster

so eventually the overall contribution is negative

Balarka Sen

That sounds fair

'cuz you have \delta^2 hanging around at []

0celo7

it's not so bad now, but it took me a long time to figure this out

the paper and book have typos, so...

but the thing you need to make absolurely sure about is that $C$ doesn't depend on anything

@BalarkaSen I'm debating writing out what the $o(\delta^2)$ term should be

Balarka Sen

I'll be back in 15 minutes after I do this diffraction computation

I am become physicist

6

0celo7

13:08

it's really like $\delta^{2+2/(n-2)}$ with some more crap

Balarka Sen

Destroyer of rigor

0celo7

good luck

Slereah

@BalarkaSen Destroyer of notation

I love physics notation so much

Akiva Weinberger

@Slereah ???

…?

Slereah

arrow goes up

arrow goes down

is it not clear

Slereah

13:40

also great : birdtrack notation

Balarka Sen

Oh please stop

The agony

0celo7

please continue hurting Balarka

15

Slereah

speaking of agony

Akiva Weinberger

@Slereah Right OK whose book is this

Slereah

@AkivaWeinberger mathinsight.org/double_integral_introduction

Ancient egyptian math

Lozansky

13:52

Does anyone know what a temperate solution is?

In the context of diff eqs

Akiva Weinberger

Sounds cool

Mike Miller

@Balarka My officemate was showing me a paper in which they write $k^{-\epsilon}$ where $k$ is a constant that depends implicitly on $\epsilon$

0celo7

@Lozansky like a tempered distribution?

Lozansky

@0celo7 Is that when $\varphi_n \to 0 \Rightarrow U[\varphi_n] \to 0$ as $n \to \infty$?

Balarka Sen

@MikeMiller Oof

Mike Miller

13:54

The dependence on epsilon was stated on a different page

Lozansky

Sooo... a solution bounded/tend to $0$ at $\pm \infty$?

Alessandro Codenotti

Also Geneva is pretty good if you want to listen to another Russian Circles album @Balarka

Lozansky

The context is: "We look for tempered solutions to $-\Delta u=f$ when $\Omega = \mathbb{R}^n$. Linear system theory tells us solving $-\Delta u = \delta$ is enough and we call these tempered solutions fundamental solutions"

Balarka Sen

@MikeMiller That's like seriously mean

0celo7

@Lozansky what?

Balarka Sen

14:00

Thanks @Alessandro, I shall check it out

0celo7

@Lozansky They mean a solution in the sense of a tempered distribution, I think.

So in the dual of Schwarz space

Lozansky

@0celo7 I don't know what "dual of Schwarz space" is

0celo7

Um

What does $-\Delta u=\delta$ mean to you

Lozansky

They claim the Laplace operator has the fundamental solution $K(\textbf{x}) = -\dfrac{1}{2\pi} ln|\textbf{x}|$ in $\mathbb{R}^2$

@0celo7 Some kind of point-source problem

0celo7

use \log

Lozansky

14:04

Oh.. too late

Slereah

\operatorname{logarithm}

$$\operatorname{NeperFunction}(x)$$

Semiclassical

For comparison: In electrostatics, the potential of an infinite line charge $\lambda$ along the z-axis is $V(\mathbf{x})=\frac{\lambda}{2\pi \epsilon_0}\ln s$ where $s$ is the distance from the z-axis.

0celo7

I call the log the Napierian operator and write $\mathrm{Nap}(x)$

Semiclassical

And the potential of a line charge is a harmonic function away from $s=0$.

0celo7

@Slereah great minds...

Lozansky

14:06

@Semiclassical But what does tempered mean? I thought it meant bounded of some sorts

Semiclassical

It's a definition.

0celo7

It means that it's a solution in the sense of tempered distributions.

Semiclassical

details here: en.wikipedia.org/wiki/…

Lozansky

Oh that looks difficult

0celo7

I'm trying to write a paper for this environmentalism class I'm taking

The literature in this field is so bad

Semiclassical

14:07

Tempered distributions are nice because you can do calculus on them (i.e. derivatives) without running into problems.

Lozansky

But it is not a condition that is easily checked?

Semiclassical

well, check out the paragraph that starts as "The tempered distributions can also be characterized as slowly growing..."

0celo7

I'm actually not 100% certain that the Green's function is a tempered distribution. It probably is, but...

You'd have to do the estimates

Lozansky

@0celo7 This is a chapter under "Green's function" so I guess it is :P

0celo7

Well your book doesn't seem to be terribly precise

Lozansky

14:10

No it is meant for engineers

0celo7

then you should ignore that line and move on lol

Semiclassical

The main thing is whether or not it makes sense to do differentiation (and therefore integration by parts) on your functions. If you're doing tempered distributions, then that's always safe.

Lozansky

0celo7

wtf

Semiclassical

yeah, that fits with the 'slowly growing' line I quoted earlier

0celo7

14:14

10 seconds ago you'd never heard of tempered distributions

Semiclassical

lol

0celo7

I'm more concerned with the pole than the long-range growth, anyhow

Lozansky

No I had heard of tempered distributions but not functions :P

Balarka Sen

Fantano says "ION" by Portal is the best metal album this year so far

I hope the melonboy's right

'cuz I'm going to listen to it

Semiclassical

hmmm

how do I pick a random pair of orthogonal 3-vectors

hmmmm

Lozansky

14:18

So if $K$ is a tempered solution (in distributional sense) to $-\Delta u = \delta$ then the function $u = K * f$ is a solution to $-\Delta u = f$?

@Semiclassical Matlab?

0celo7

@Lozansky yes

Lozansky

Is this due to Schwartz kernel theorem?

Semiclassical

well, i could do either matlab or mathematica

it's the method I'm not sure about

Tanuj

Guys , I'm stuck on this for a while now , any hints how to proceed ?

Semiclassical

Also, meant to say that they should be unit vectors

Tanuj

14:21

Lozansky

Why don't you just iterate random 3-tuples until the inner product between them is $0$?

Semiclassical

Well, there's some probability distribution on their inner product presumably

in which case the probability of getting exactly zero is nil

Tanuj

I have no idea how to proceed , it is $\frac{0}{0}$ form , but I can't even use L' Hospital

Semiclassical

But one can instead demand it be close to zero and do a lot of random samples

Slereah

What is "greatest integer"

Tanuj

14:22

@Slereah the floor function

Slereah

Shouldn't that be $\lfloor \cdot \rfloor$

Semiclassical

So I think that method works.

Lozansky

@Semiclassical So random integers in some closed interval?

Tanuj

@Slereah It's called that and written like that here in Iindia (at least in high school mathematics)

Semiclassical

@Lozansky What?

Slereah

14:23

I mean if it's the floor function

Wouldn't $\lfloor x/2 \rfloor$ be a constant function around $\pi/2$?

Tanuj

@Slereah how ?

Semiclassical

it'd be constant for $4>x>2$, yeah

0celo7

@Lozansky why does an engineer know about the schwarz kernel theorem

Semiclassical

Engineers are allowed to know things :P

0celo7

Like, how have you seen SKT but not Green's function

Slereah

14:25

Hence it's just $0$ in that region

0celo7

That's worse than Balarka learning calculus after homotopy theory

Slereah

Or at least it has zero left and right limits

Tanuj

@Slereah yes , it is 0 in that region, completely agreed

Slereah

then what is the issue

0celo7

@Lozansky this Green's function calculation is done in every book. Gilbarg-Trudinger, Evans, Shankar, Straumann, pretty much every book in every field has this

Balarka Sen

14:26

@0celo7 Probably not.

I think we both secretly agree on that

Tanuj

@Slereah how do I solve it or simplify it from $\frac{0}{0}$ form ?

Lozansky

@Semiclassical Say you choose a closed interval and fix a random 3-tuple of integers and then iterate a new 3-tuple of integers until their inner product is $0$?

Slereah

@Tanuj You don't have to

Lozansky

@0celo7 They mention it in my book

Semiclassical

I'm not really seeing why you're suggesting random integer 3-tuples

Slereah

14:27

For any $x < \pi/2$ it is equal to zero

Hence the limit by inferior values is zero

and same for the limit by superior values

So the limit is just zero

Semiclassical

But, I mentioned something in passing that I should've emphasized

I want these to be unit vectors as well

Tanuj

@Slereah but isn't the denominator also getting 0 for those intervals ?

Lozansky

Oh

Slereah

@Tanuj Doesn't matter here

Semiclassical

The easiest way I know how to do that is to create triplets of standard normal random variates and then project them onto the unit sphere

Tanuj

14:29

@Slereah why is that so ?

Slereah

You can use the proper definition of limits to convince yourself that it is true

Lozansky

Guess you need to impose some error limit :O

Semiclassical

Yeah

Which is fine

Slereah

Check that for all values of $x$ arbitrarily close to the limit, $f = 0$

hence it is the limit

Balarka Sen

God this album is full of abrasive sounds

100% noise metal

Tanuj

14:31

@Slereah I'm just not getting it how .Isn't $\frac{0}{0}$ by an indet form ?

Slereah

I give up

RIP me

user280247

Hello..just want to make you a question..

Tanuj

@Slereah I'm sorry mate , it must be something simple , but I just don't get what you're trying to say

user280247

Is there any book in history of maths with nice explanations on the discoveries?

0celo7

@BalarkaSen Die Melone loves that crap

Balarka Sen

14:34

i prefer some r a t t l i n g h i g h h a t s

Akiva Weinberger

There's a lot of pop math books (like, math for a popular audience), you mean like that? @santimirandarp

Slereah

@0celo7 is that German for "the melon"

Akiva Weinberger

Like The Pythagorean Theorem: A 4,000-Year History by Eli Maor, and various others

Slereah

I hear that Pythagoras got into his bathtub

Balarka Sen

I have a headache from listening to "Prey for sickness" being whispered into my ears with abrasive guitar riffs and white noise on the background

jesus christ

Slereah

14:36

And then realized that the water formed a perfect right triangle

So he leapt out, screaming Εύρηκα!

Balarka Sen

accurate

Lozansky

And then they made the movie "The Shape of Water" based on that event

Slereah

most mathematican discoveries were done in the bath

I don't even dare tell you how they found the hairy ball theorem

0celo7

@Slereah yes

Tanuj

@Slereah wasn't that Archimedes who shouted Eureka when he calculated the mass of gold stone by displacing water in his bath tub ?

Slereah

14:38

that is true

Akiva Weinberger

@Tanuj He's just being silly

Tanuj

Okay @AkivaWeinberger , I'm too silly to realise someone is being silly

Balarka Sen

@Slereah old jok

Slereah

the old joke I've seen is that Archimedes Εύρηκα a nail in his bath

Hence why he got out

Semiclassical

@Lozansky think I got it working in the way I outlined. (I was using a similar approach for a different aspect of the problem, so it was easy to to adapt it.)

Tanuj

14:45

@AkivaWeinberger could you help me with this

Lozansky

@Semiclassical Cool, not too long running time?

Semiclassical

nah

though I"m using a pretty generous error tolerance

Lozansky

I think you could do it pretty easily in Python

Semiclassical

yeah

Lozansky

There's a lin alg package in numpy

Semiclassical

14:48

I did it in mathematica

The main problem is that, if I do a more restrictive error tolerance, then I have to do a lot more samples

But that's sorta inevitable with this kind of approach.

In any case, I can use it to verify (within the error tolerance) a claim I thought should be true

So that's good

Lozansky

Neat :)

Akiva Weinberger

@Tanuj Should be 0

Tanuj

@AkivaWeinberger I don't get why

Akiva Weinberger

$[x/2]$ is zero when $x$ is between $0$ and $2$, right?

Tanuj

@AkivaWeinberger the numerator should always be zero , for left as well as right handed limits

@AkivaWeinberger yes

Akiva Weinberger

14:55

Which means that, on that interval, $\frac{x/2}{\ln(\sin x))}$ just equals $\frac0{\ln(\sin x)}=0$

Tanuj

@AkivaWeinberger I don't get why denominator isn't always 0 in the vicinity of $\pi/2$

Akiva Weinberger

and $\pi/2$ is in that interval

Tanuj

@AkivaWeinberger doesn't $\sin(x)$ approach 1 as we move on its graph from left hand and right hand sides towards $x=\pi/2$ ?

Akiva Weinberger

@Tanuj $\sin(\pi/2)$ is $1$, which means that $\ln(\sin(\pi/2))=0$. But when $x$ is slightly different from $\pi/2$, $~\sin(x)$ is gonna be slightly different from $1$ and $\ln(\sin(x))$ is gonna be slightly different from $0$

Semiclassical

Suppose you were to do $x=\pi/2-0.1$. What would you get on the top and bottom?

Lozansky

14:57

Uhh.. my book says $\dfrac{\partial}{\partial \textbf{n}} = -\dfrac{\partial}{\partial r}$ for a disk

Semiclassical

On the edge of a disk or on the surface of it

Tanuj

Okay guys , thanks for help , I got it that 'slight' error in the logic ! @Semiclassical @Slereah @AkivaWeinberger :)

Lozansky

Edge

0celo7

@Lozansky Yes.

Semiclassical

Then that's not so strange. A disk is a cross section of a cylinder, and the surface normal in that case is just $\hat{r}$

0celo7

14:59

Um, wait

Lozansky

The minus sign

0celo7

Which orientation for $\mathbf n$

Semiclassical

yeah, I was about to say

But it could make sense in context.

Lozansky

That's the thing, they say like 100 pages before that the outwards direction is standard

Transcript for

$Mathematics$ Mathematics