The h Bar

@Qmechanic We use diff, presumably because the diffeomorphism group is Diff.

Qmechanic

@0celo7 : That's confusing. I use the abbreviations diff=differentiable and diffeo=diffeomorphism.

@0celo7 I trust you'll figure it out eventually, I'm not in the mood to decipher a proof step for you

@Qmechanic Differentiable is $C^1$, we already have symbol for that.

@Qmechanic Same thing to me, I didn't realize "diff" is also used for differentiable.

5:14 PM

$\mathcal C^1$

Just call it "sufficiently smooth", that also prevents possibly demanding one derivative too few :P

@ACuriousMind Do you know of a simpler proof that $\Bbb R^n$ is not homeomorphic to $\Bbb R^k$ unless $n=k$ than using de Rham cohomology?

none I could recall

So your proof would use de Rham as well?

Well, it would use (co)homology. Which way you choose to compute it is pretty irrelevant

5:20 PM

I think a homological proof would be considered "simpler"

Not that I know anything about homology...

Wait a moment

@ACuriousMind I can prove, using dR cohomology, that the dimension of a smooth manifold is homeomorphism invariant.

Ah yes, the "simple" homological proof is to one-point compactify and then use the knowledge of the homology of spheres.

The problem with that proof is that you rely on the smooth structure.

And I think there are exotic $\Bbb R^n$s.

@ACuriousMind Hmm

I don't understand what that does but ok

@ACuriousMind Hmm, maybe you're supposed to assume it for $k=k$ and then prove that case proves it for $k-1$.

Not really induction but it seems more plausible.

Then you descend to $k=1$ and prove it from there.

I'm not convinced a traditional induction is even possible

5:42 PM

@0celo7 As a matter of course: $C^k$ merely means "all derivatives up to the $k$-th order are continuous", and nothing more. A $C^2$ function, like a cubic spline would be differentiable as well, so just using the symbol $C^1$ is rather restrictive.

@J.M. To me "differentiable" means "the first derivative is continuous"

And that's precisely $C^1$

and that's alright. But don't use $C^1$ to denote differentiability, unless you're in the business of conflating concepts.

What do you mean?

To me "let $f$ be a differentiable function" and "let $f$ be $C^1$" are the same

I'm not sure what your point is.

One does not say $\sin x$ is $C^1$, because it is in fact a $C^\infty$ function. But we do say that it is differentiable.

but it is $C^1$

5:49 PM

We will have to agree to disagree, then.

@J.M. ??? Not every differentiable function is $C^1$ (because some functions are differentiable but the derivative is not continuous), but every $C^k, k> 1$ function is also $C^1$. When people say a function is $C^k$, they mean at least, not exactly $k$-times continuously differentiable

@ACuriousMind Is the countable cartesian product of countable sets countable?

Qmechanic

@0celo7 : $C^1$ is different from diff=differentiable. $C^1$ means differentiable with continuous derivative.

user54412

@0celo7 what's your definition of $\mathbb{R}$ again?

@ChrisWhite I don't have one

6:00 PM

@0celo7 What do you think?

@ACuriousMind Yes

What is the "counting function"?

That's what I'm asking you

Why do you think it's countable, then?

gut feeling

6:02 PM

Ah, but @ChrisWhite's question already gave it away - you may see $\mathbb{R}$ (or $[0,1]$, if we don't to deal with digits in front of the dot) as the countable product of the digits $\{0\dots 9\}$ for instance.

really?

I don't see it

What is a real number (in decimal presentation) other than a countable ordered collection of digits?

oh

@ACuriousMind Ok I'm trying to show that the balls with rational radius and rational centers form a countable basis

(sanity check for some measure theory stuff)

I'm just miffed on how to do this

I can see that it's a basis

Um, the rational centers in $\mathbb{R}^n$ are $\mathbb{Q}^n$ and the radii are $\mathbb{Q}$ so what you need is that finite products of countable sets are countable.

o.o

@ACuriousMind Ok, I can see that there is a bijection between the balls and $\Bbb Q^{n+1}$

So why is $\Bbb Q^{n+1}$ countable

proofwiki.org/wiki/…

I was able to come up with their "informal" proof on my own.

Q: How can I make my younger brother's interest in physics at home?

6:13 PM

heheheh

balls

user116211

6:34 PM

2

This is a soft question. I have a younger brother who is in 10th standard but he don't like subject of physics at all. When I asked reason for it he says he gets bored in mugging the formulas of physics theorem that are taught in his school. He don't like theortical explaination but when I explai...

user116211

I really don't understand why they ask such things ;(

user116211

Well, one thing for sure; he wanted more reps.

user116211

@Chris White I really don't understand why people here behave like short minded . If you see other stackexchange sites like mathstackexchange ,there are many similar questions with huge reputation like question1, question 2 ,question 3 and many more — pandu 3 hours ago

user116211

q1 and q2 are not at all the same query OP is making and alleging those questions were making the same query he is making; the third one is likely to be the same as that of OP and it is closed there.... though after it gained +200 (Maths SE is incredible ;\ ).

user116211

However, the most efficient technique to encourage his brother:

user116211

6:38 PM

He's 15/16? Find some cute girls who think physics is cool. — mmesser314 5 hours ago

user116211

No need to respond to OP's rant as he was all the way in making quick reps....

Q: What is the difference between a hard and a soft fork?

o.O

7

There are currently different scenarios examined which include soft or hardforking the ethereum blockchain to mitigate the DAO attack. What plans are there exactly and what are the difference between the hard and the soft fork?

the-dao forks

What on earth is this about

user116211

@0celo7 What this SE is about.

Finally: arxiv.org/abs/1606.05721

2

user116211

@DanielSank you wrote it!

6:46 PM

@MAFIA36790 Yeah. Took long enough.

It's sick though.

We solved a major outstanding problem that one of our field's leading figures called "the skeleton in the closet of our field".

So @DanielSank's real name was Zijun Chen all along

@Slereah Wat?

I did not see it coming

It's a joke

The ol' switcharoo, you see

Switcheroo

A switcheroo is a sudden unexpected variation or reversal, often for a humorous purpose. It is colloquially used in reference to an act of intentionally or unintentionally swapping two objects. As a comedic device, this was a favorite of Woody Allen; for a time, he used so many switcheroos that friends referred to him as "Allen Woody." Some of Allen's switcheroo gags were: Carrying a sword on the street; in case of an attack it turned into a cane, so people would feel sorry for him Carrying a bullet in his breast pocket; he claimed someone once threw a Bible at him and the bullet saved his life...

^this

ahahahahahaha

Thanks ;)

In his book Gödel, Escher, Bach, Douglas Hofstadter names one of the rules in his version of Propositional calculus the Switcheroo Rule, apparently in honour of an Albanian railroad engineer, name Q. Q. Switcheroo, who "worked in logic on the siding".[5] This is in reality the Material implication.

6:54 PM

...what

Just spent a billion years proving that the intersection of open sets is open.

@Slereah Hahahaha

That reminds me of my favorite science joke

"Q Q Switcheroo"

It was an article on Uncyclopedia

It was the article on the Erlenmeyer flask

It started with "The Erlenmeyer flask, invented by Julius Flask"

@0celo7 Oof. Have you tried doing physics instead? :)

6:57 PM

> whereby the openness of the latter has been shown.

damn Germans and their translations

@NeuroFuzzy I am

It was

the ol' switcharoo

Currently reviewing metric topology so I can prove some measure theory stuff so I can prove some differential topology stuff so I can do morse theory so I can do more GR

@0celo7 I see nothing wrong with that sentence fragment

u can use morse theory to prove the theorem on Lorentz cobordism

@0celo7 Neat! So what application?

7:00 PM

@NeuroFuzzy not sure...

I think morse theory can be applied to GR somehow

I'm not clear on the details :P

@ACuriousMind Of course you don't

Wow. A well laid plan.

Well what I said

Oh, right.

Anonymous

Hi.I'm new to the chat room.Can someone help me with this question? physics.stackexchange.com/questions/263918/…

@NeuroFuzzy no I'm just taking a class on differential topology

@NeuroFuzzy I had to prove that $\Bbb R^n$ is $\sigma$-compact for a measure theory proof.

and I figured I might as well prove that the metric topology is actually a topology

was a nice exercise

7:13 PM

@0celo7 Ah yeah, that's good to know how to do

7:27 PM

I wonder if there's a trick to make a Good CTC Spacetime

Like

Compactly generated

No closed causal geodesics

And with topology $R^4$

Well, more to the point, all CTCs are deformable to a point

I suspect that "no closed causal geodesics" is hard to do in a spacetime that's very symmetric

The only compactly generated example is the rotated wormhole

not v. symmetric

but who knows

Godel has none

What are even some non bullshit compactly generated CTCs

Wormholes

Alcubierre

That guy

Alcubierre could be interesting, maybe

I hate analysis.

I hate it so much.

OH MY GOD

@ACuriousMind I think I figured out the measure theory thing?

But once again I don't get the induction

So no, I don't understand it

7:48 PM

Wow... chat.stackexchange.com/transcript/message/30498051#30498051

top notch physics.

@ACuriousMind Ahhh, the induction step follows from the fact that $k+1-k=1$.

@ACuriousMind danke badeente

8:03 PM

Apparently the CTC spacetime for Alcubierre is basically Minkowski + a bubble + a second bubble going the other way

Perhaps using odd paths for the bubble we can make it so that no CTC is a geodesic

Although... If they have to cross at some point, it might be impossible to avoid

But maybe it could be like

1) particle rides the bubble

2) particle has to make a turn when it gets off and go a little way

3) particle takes the warp bubble back home

Could work mb

The interstellar bus stop

Tho it's not quite a bubble going the other way

It also needs like

a slight boost

Which I guess makes sense

If it was just going the other way, it would never get back at the original point

With a boost though, since the bubble goes in a "spacelike" direction, it will be able to point slightly backward

BUT WAIT

Warp bubbles are causally isolated from the outside

How can I make a particle travel from one bubble to the other

Argh

Building time machines is hard

I guess it might work better with a Krasnikov tube

I guess I could just have the bubble going back and forth but then I'm pretty sure it would have closed causal geodesics

arxiv.org/ftp/gr-qc/papers/0511/0511086.pdf

Oh no

It's like if Secret wrote a paper

2

"However, a spaceship captain hoping to make use of a region of spacetime with a suitably warped metric to reach a star at a distance D in a time interval ∆t < D must, like the potential trolley car passenger, hope that others have previously taken action to provide a passing mode of transportation when desired."

The ol' space trolley

8:33 PM

good lord

Bass

@ACuriousMind When I first learned about this, I became really sad. I know it's kind of good news, because we're able to make very precise predictions with a theory we know is not complete, but it also means chances are we get old and die without knowing a complete theory. Experiment and theory got out of sync.

Oh no

@ACuriousMind Is "A closed subset of $\Bbb R^n$ is the union of a countable collection of compact sets. " true?

Krasnikov's metric has Heaviside functions

How gauche

Gentlepersons, how do you set up hyperlinked references in revtex so that they actually work?

8:35 PM

A gentleman smoothes out his discontinuities

Any example code will be welcomed by me and my collaborators.

@Slereah facepalm

You know

A little convolution with a bump function

Something

Leaving a metric discontinuous is a pretty big faux pas

Oh wait, he does smooth it actually

nvm

@DanielSank I'm guessing just using hyperref like normal people is not an option? Or is there something wrong with the links it generates?

Kudos Krasnikov

You are a gentleman and a scholar

@ACuriousMind Can you provide a scmwe?

@Slereah rekt

@ACuriousMind revtex is a particularly terrible class.

8:39 PM

@DanielSank Um... you do \usepackage{hyperref} in the preamble and then \cite{YourCitationLabelHere} should automatically hyperlink to the corresponding item in the bibliography.

Bass

@0celo7 Let $X$ be a closed subset of $\mathbb R^n$ and $D_i$ be the closed ball with radius $i\in\mathbb N$. The collection of sets $X\cap D_r$ is a countable collection of compact sets (because they're closed and bounded), and their union is $X$. So, yes.

@ACuriousMind I see.

uhhh, hang on

@Bass Ah, taking $r\in \Bbb N$ is the trick, thanks.

Here's what my collaborator used to get the hyperlinks:

\usepackage[pdfpagemode=UseNone,pdfstartview=FitH,colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
\usepackage[all]{hypcap}

The references do show up in blue.

However, when I click on them, nothing happens.

Interestingly, it does work on my collaborator's machine (mac), but not on mine (Ubuntu).

@DanielSank Hmmmmm...that sounds more like a problem with the reader than with the TeX/pdf. What are you using to view the pdf?

8:51 PM

@ACuriousMind Document Viewer.

:|

Here, @ACuriousMind, try this

arxiv.org/abs/1606.05721

Do the ref's work on your machine?

Hmmm, no they don't

ok

@DanielSank They don't work on my Mac.

@DanielSank: Try loading the hyperref package after all other packages, cf this tex.SE post

@ACuriousMind Tried that. No dice.

I'll try setting up a super duper minimal example and see if I can get that to work.

Thanks for your help.

9:01 PM

@0celo7 I'm fairly sure there is a statutory requirement that place the back of one hand against your forehead, reach the other toward the uncaring stars and declaim 'Oh! Woe, is me!' before preceding at time like that. Just sayin'.

@DanielSank Might be a slightly insulting question, but you are compiling multiple times right? And you've tried deleting the aux files?

@ACuriousMind I swear that latex causes people to develop a bad case of cargo-cult debugging procedures. And this is one of mine: delete everything but the source files and re-compile afresh.

@dmckee I have no clue what that means. If you're going for a zinger make sure it's at the level of intelligence of the other person.

@dmckee Well, but it works!

It's not on the level of xkcd workflows, at least

Hm

It is quite possible that a system of two Krasnikov tube would be doable to compute the vacuum for

It is moslty Minkowski space

9:06 PM

@dmckee Edit did not help.

@0celo7 ANd probably the re-edit doesn't either. Just goes to show how lucky you are or how devoid your life is of melodrama.

Maybe I could even pretend

ArtOfCode was just doing you a favor by adding some back in.

THAT THE METRIC IS DISCONTINUOUS

Oh

So wrong

@dmckee What did he do?

9:08 PM

and yet so right

looks on back

Oh no, did I get stickied!?

@0celo7 Added some melodrama to your life.

@dmckee Hmm?

I wasn't aware he did anything.

@ArtOfCode What did you do?

@Slereah What happens between you and your metric should stay between you and your metric.

@ACuriousMind Should I refrain from asking you all measure theory questions?

9:10 PM

Hmmm ... somehow I was under the impression that ArtOfCode issued your long suspension. So it was DavidZ who did you a favor.

@0celo7 Yes, please

@dmckee "favor"

You surely have a strange worldview.

That I do.

I don't see what favor he did, my behavior has not changed (don't change a good thing I say)

And I've hung out with theater geeks who are a whole 'nother species.

9:11 PM

All he did was inconvenience me honestly

ArtOfCode

@0celo7 hm? I don't know, you tell me

@dmckee my girlfriend is a theater geek

not "'nother species" crazy

@ArtOfCode That comment was caused by my misunderstanding.

@ACuriousMind ok just 1 minute of your time, should the $U$ in the first line on page 205 of GP be $\bar U$?

ArtOfCode

@dmckee oh, I haven't accidentally nuked someone? Shame.

9:17 PM

^my point exactly

@ACuriousMind To see why, consider $U$ the open unit disk in $\Bbb R^2$ and $A$ the interval $(-1,1)$ along the $x$-axis

Enter The Wheel of Random Suspensions! Wheel, turn! Turn! Turn! Tell us the user that we should burn!

...you people are sick

@0celo7 I'm not opening GP now, I've shut off my brain for the day. Figure it out yourself

@dmckee Siunds like a nice premise for a game show

ArtOfCode

@dmckee this might not be such a bad idea

I stole the kernel of he idea from The Animaniacs.

9:28 PM

@ACuriousMind Not insulting at all.

Yes I was doing those things.

I just made an incredibly simplified example, and it still doesn't work.

I guess there's something screwed up in texlive.

Howdy

hey @bernard how's it going ?

@Obliv It's alright

How have you been?

did you give your calculator an a.i. yet? I've been good just studying algebra on my downtime

@Obliv No, but it has wifi and I'm close to getting online games working

I mean, doom already works

9:42 PM

that's awesome... how do you play it o_o @bernard

With a mouse and keyboard :)

It took a lot of kernel fiddling, I'm writing a blog post on it

I don't believe you. Does it have usb ports? :O

it has one mini-usb port, to which I connect a USB-OTG connector that get's plugged on a USB-hub and then I have 4 USB ports

but the voltage isn't enough, so the hub need's a 5v ps

:v

what's a 5v font?

5 volt power supply

9:46 PM

oh did you connect that to the terminals with electrical tape or something @bernard

battery terminals, that is

Nah, my USB hub had a power jack on it :v

I just found a plug that worked

that's neat. how much ram does it have @bernard

@Obliv A whooping 64MB

:)

10:06 PM

Guys

To prove that no closed geodesic exists

Is it sufficient to show that a coordinate of the curve has no local maxima

That is, $\ddot x \leq 0$

For $\dot x = 0$

Or something along those lines

@Slereah How large is your chart?

to show that there's no "turning back"

Well the manifold is $\Bbb R^4$

Pretty large

Given that my manifold is almost Minkowski I think I can even show that the coordinates of geodesics are even monotone increasing (or decreasing, depending on the direction)

I think that should be enough to show that there's no closed geodesic if it's simply connected

But that's just a gut feeling

@Slereah I'm so fat I'm sweating while sitting

maybe eat a whole carton of ice cream

to cool down

that might work

10:15 PM

Wait no, being simply connected isn't relevant

Since that wouldn't work for the sphere

But for $R^4$ I think it does

10:43 PM

Hm

What would be the best way to write a mollified heaviside function

Maybe just piecewise linear

Ohhh @dmckee were you saying I should stop whining?

It would still be Lipschitz continuous and solving equations should be easy enough on it

Just that you're being melodramatic. But everyone needs a little melodrama now and again.

Facts are not melodramatic...

Everett recommends $$\frac{1}{2} (\tanh[\frac{4\xi}{\varepsilon} - 2]+1)$$

But that's a tad calculation heavy

10:47 PM

$\tanh$ is just terrible

If something has to be roughly limited to a finite region of spacetime, I wonder if a Gaussian would be a good idea

It is $\approx 0$ outside of 3 sigmas

what about a mega Gaussian

$\mathrm e^{-x^4}$

ur a mega gaussian

girls in the library talking about physics

Tanh isn't even what he said it was, it's not 0 and 1 outside of a compact interval

David Z

10:58 PM

@0celo7 the point of a suspension is to show a person that their behavior is inappropriate. If you don't change your behavior after the suspension, you are not welcome to return to the site.

Only asymptotically so

There are probably better activation functions

@DavidZ I begrudgingly come here because of the chat.

I do not enjoy it.

Crape I need an analysis text

What's a good smooth tophat function

Something that's good analytically

tophat?

Gaussian is fine

11:06 PM

$\mathrm e^{\cosh x}$.

A function that is ~1 inside a compact set and ~0 outside

Like compact support but less of a pain

I just gave you one

$\cosh x$ grows a lot faser than $x^2$

I said "good analytically"

So you get less error as you go out

I don't want to have to solve PDEs with that monstrosity

I guess a gaussian will do, if that tanh monstrosity doesn't pan out

knzhou

11:11 PM

@DavidZ If somebody posts an incorrect answer, and somebody else edits it to be correct, should I accept the edit or reject as 'conflicts with author's intent'?

I also sometimes see really extreme cases of this, where barely a single original sentence remains...

sweats

Time to prove this...

@Slereah is there another set minus symbol besides $\setminus$ in TeX

something along the lines of

I think you have to define one yourself

Q: Redefining \smallsetminus by smaller \setminus

There's a thread on the latex SE for it

16

I don't like the appearance of \setminus (it's big and doesn't look good), and also don't like the appearence of \smallsetminus (its size is perfect but, in my opinion, it's too horizontal and it's not vertically center). I found that in normal math this is what I can use: \[ A \mathbin{\v...

math-mode

good lord

> \documentclass{article}

\usepackage{amssymb,tikz}

\newcommand{\mysetminusD}{\hbox{\tikz{\draw[line width=0.6pt,line cap=round] (3pt,0) -- (0,6pt);}}}
\newcommand{\mysetminusT}{\mysetminusD}
\newcommand{\mysetminusS}{\hbox{\tikz{\draw[line width=0.45pt,line cap=round] (2pt,0) -- (0,4pt);}}}
\newcommand{\mysetminusSS}{\hbox{\tikz{\draw[line width=0.4pt,line cap=round] (1.5pt,0) -- (0,3pt);}}}

\newcommand{\mysetminus}{\mathbin{\mathchoice{\mysetminusD}{\mysetminusT}{\mysetminusS}{\mysetminusSS}}}

user54412

11:46 PM

@DanielSank Word of caution in your debugging enterprise: most pdf viewers these days automatically detect links anyway, even if they are simply plaintext as far as the pdf is concerned. I recently had a case where I thought hyperref was working, when it turns out it was my viewer picking up the slack.

2