The h Bar

John Rennie

2:11 PM

@0celo7 :-)

Noah P

@JohnRennie I love how this is your top answer

214

A: Why are four-legged chairs so common?

Suppose the leg spacing for a square and triangular chair is the same then the positions of the legs look like: If we call the leg spacing $2d$ then for the square chair the distance from the centre to the edge is $d$ while for the triangular chair it's $d\tan 30^\circ$ or about $0.58d$. That ...

And especially that it's on-topic

Jim

4500 answers, the top one's bound to be funky

Noah P

I'm surprised that john duffield hasnt posted an alternative answer

Attempting to debunk the theory of chairs

How come no one seems to ever agree with him?

John Rennie

@NoahP If a question gets on the hot network questions list then the top answer attracts shedloads of upvotes simply because everyone follows the herd.

Noah P

Ahh okay

John Rennie

2:25 PM

It would be nice if an answer I was really proud of was top, but well votes are votes :-)

Noah P

Votes are votes!

I'd love a few more myself

My biggest rep gain was the site association bonus

+100

Wooh!

John Rennie

The key to getting rep is to answer lots of questions. If you look at my average upvotes per question it isn't especially high, but I've answered lots and lots of questions ;-)

Noah P

Fancy answering mine? :P

John Rennie

@NoahP Link?

Oh, you mean the coffee pot?

Noah P

Indeed I do

Its a matter of utmost importance

John Rennie

2:31 PM

The trouble is that your question amounts to why does my coffee taste bad and I don't know what makes coffee taste bad.

Noah P

It doesn't really

John Rennie

The only two variables I can think of are the amount of water passing through the coffee and the temperature of that water.

Noah P

Its a why does my coffee machine not work properly

The coffee tastes bad because it is burnt, from a high temperature

coffee boiled is coffee spoiled

John Rennie

Suppose you did the experiment where you removed the coffee and (a) measured the build up of water in the top compartment and (b) the temperature in the internal tube.

You could do (b) with a thermocouple and (a) with some sort of dip tube.

Noah P

It wouldnt be a true representation without the coffee in there

John Rennie

2:34 PM

Then you would have hard data to show what the effect of filling past the valve was. This would be something I could get my teeth into.

@NoahP OK, so what effet on the water flow does the coffee have?

Noah P

The water is 'filtered' through the coffee

Slows the flow, and provides resistance to the flow, which in turn increases the pressure in the lower chamber

John Rennie

@NoahP that's a meaningless statement. What effect does the coffee have on measurable parameters like the flow rate and back pressure.

Noah P

You jumped the gun - see last message :P

John Rennie

@NoahP How do you know? Have you ever measured to see what happens without the coffee.

@NoahP Asynchrous communications rule :-)

Noah P

Well no, I havent explicitly, but I have observed the ground coffee after the machine is finished

Its tightly packed and compressed, and comes out almost as a solid block. When I put it in, its loose, with room left in the chamber

John Rennie

2:39 PM

Well if it were me, I'd be jumping up and down with enthusiasm to try some experiments. Then I'd have some hard data to look at. This is what 20 years as an experimental scientist does to you :-)

If you can't borrow some thermocouples from school I'm sure you could find them on ebay for not much.

Noah P

Thermocouples..?

John Rennie

You'd need a multimeter, but you'd be surprised how cheap they are these days.

@NoahP to measure the temperatures inside the pot.

Noah P

Oh, a fancy thermometer

John Rennie

See, not only can I teach you GR, but experimental physics as well. Now when they ask you in the Oxford (spit) interview what a thermocouple is you'll know :-)

Noah P

Lovely :)

John Rennie

2:42 PM

I am actually serious about this. Were I to be faced with something like this I would immediately be planning experiments to study it.

Noah P

So say I had a therocouple

I do have a multimeter

How on earth do I measure temperature?#

John Rennie

You could Google thermocouple ...

Noah P

I'd basically need to record the voltages and then use excel to make it into usable temperatures

Ah. There is a problem

There is no hope in hell of me getting a thermocouple in the lower chamber

There has to be a tight seal for the thing to work

John Rennie

@NoahP feed the thermocouple from the upper chamber down through the internal tube. Thermocouples are basically just a long thin piece of wire so you should be able to push it through the bed of coffee.

Noah P

You cant

Theres then a filter between the coffee and the chamber

google.co.uk/…

John Rennie

2:48 PM

Drill a hole in the side of the lower chamber and epoxy the thermocople in place. But don't do it to your favourite pot :-)

Noah P

My mum would murder me

The pot is about 30 years old

John Rennie

A thermocouple is basically two pieces of wire soldered together. You could feed the two wires upwards through the small holes in the filter.

Noah P

They are very small holes

Secret

Johnrennie, chairs in (macroscopic) spatial 4D gets more interesting hi.gher.space/forum/viewtopic.php?f=27&t=1232

John Rennie

Commercial thermocouples may have insulation or some sort of cladding round the wires, but you could try and remove this or make your own thermocouple from enamelled wire.

Secret

2:50 PM

a typical 4D equivalent of a 5 legged office chair will need 12 legs

Noah P

Hmm

John Rennie

The thing is that you're giving strong vibes that you don't want to do the expt so you're looking for reasons why it won't work. That approach doesn't win Nobel prizes.

Noah P

I fear that it's not something I'm going to get round to

John Rennie

@NoahP then you don't really care about the answer.

Noah P

It's more a case of I'm back at school in a few days and I've got no hope of doing it then

'tis a boarding school, and although I don't board, im there from 6.45am to 9pm

John Rennie

2:53 PM

School is the ideal place to do expts like this because they'll have all the kit you need. Is there a science club? If not, can you sweet talk a science teacher into believing it would make a good project?

Noah P

ITs unlikely with the amount of work going into the last year of a-levels, and I'd also have to buy a new moka pot

They're not expensive, but im not in a position to really do that

I'd need to find a way to adapt it

Ill think about it

I really dont think temperature is the important factor though

Its linked to the valve, which would suggest pressure

John Rennie

My point (and I do feel like I've been ranting a bit) is that any answer to your question would be something like I think it's the pressure and this is why the valve affects it blah blah.

But if I don't know whether the pressure is actually different in the two cases the answer is, well, a bit pointless.

Noah P

But we do

John Rennie

@NoahP Proof?

Noah P

The ejection of water through a tight seal

The lack of coffee flowing through the pipe

And a bitter taste in my mouth

John Rennie

3:01 PM

Why wouldn't the water simply flow out of the valve until the level fell below the valve, then it would flow up through the coffee as before? Does the valve jam open when water flows through it?

Noah P

Its a tiny valve, no way of telling when its open

John Rennie

If it jams open you would expect that once the water level had fallen below it steam would continue to flow through the valve. You should be able to see the condensed vapour and hear the hissing as the steam flows through it.

0celo7

@ACuriousMind what do I mean by "due to symmetry" here

Jan 30 at 2:00, by 0celo7

@ACuriousMind Define the set $K:=\{d(x_1,y)-d(x_2,y)\mid y\in A\}$. Then $D(x_1)-D(x_2)=\inf K$ (due to $\inf(A+B)=\inf A+\inf B$). It can be easily seen that for all $k\in K$, $k\le d(x_1,x_2)$. Then, by the previous lemma, $\inf K\le d(x_1,x_2)$. Due to symmetry, $|\inf K|\le d(x_1,x_2)$. Let $\epsilon >0$ be arbitrary and let $\delta =\epsilon$. Then $d(x_1,x_2)<\delta$ implies $|D(x_1)-D(x_2)|\le d(x_1,x_2) <\delta =\epsilon$, showing continuity of $D(x)$.

Noah P

Steam goes through the valve anywya

John Rennie

OK here's an idea. Try the experiment without the coffee and see how much water flows into the upper chamber and how fast. If there's no difference for the different water levels then it must be related to the way water flows through the coffee.

I'm assuming that without the coffee present the water flow is fairly free i.e. unrestricted.

Noah P

3:11 PM

Ill have to try it out

John Rennie

Well you don't have to - no-one is paying you to - it's a matter of badly you want to know what is happening in your pot.

Noah P

No, I am interested

John Rennie

Though there are risks in doing experiments with a prized family possession. I might be inclined to see if I can find a second hand one to experiment with :-)

Noah P

Yes, thats probably a good idea :P

ACuriousMind

3:27 PM

@0celo7 Uh...I think you mean that you can do the same argument for the roles of $x_1$ and $x_2$ reversed. Then, if $\lvert \inf(K)\rvert$ were greater than $d(x_1,x_2)$, you could find an element in the $K$ with the roles of $x_1$ and $x_2$ reversed such that it's greater than $d(x_1,x_2)$, but all elements of $K$ are bounded by $d(x_1,x_2)$.

Secret

3:37 PM

Weird metal compound saw in my dream last night

and yes that planar moiety has delocalised electrons

I think I have some idea on where my brain get this idea from:
1. When searching for a metalloporyphyrin supervisor as one of my possible pHD project, I stumbled upon this https://www.google.com/patents/WO2011028935A2?cl=en
2. The ethane side group is likely taken from this article, which is another possible supervisor group that interested me http://pubs.acs.org/doi/abs/10.1021/jacs.5b10583?journalCode=jacsat
3. The aluminium atom at the centre (and how I read about this compound within the dream as a possible project) is inspired from a professor who worked on main group metal complex che…

Interestingly, this is not the first time I saw compounds in dreams with this square shaped porphyrinoid moiety

Perhaps I am somehow subconciously drawn in interest to porphyrin structures?

DanielSank

3:53 PM

@EmilioPisanty cool!

Secret

and some 4 years ago, I had a dream where there's a lecture that mentioned about something called the I boson which has a very weird property to transmutate into a tachyon and back when passing near an atomic nuclei

John Rennie

It appears that it's theoretically possibly to have a massless charged particle though obviously since none have ever been seen they (probably) don't exist in practice.

Secret

Well, we never observed electrons decay (and in fact nobody ever have suspect electrons will decay, unlike the hunt for proton decay)

John Rennie

I say probably because the particle might carry some other quantum number that means it's hard to create. Actually it would have to carry some other conserved quantity otherwise electrons or protons would decay into it.

ACuriousMind

@JohnRennie Of course massless charged particles are possible; in the unbroken phase of the Higgs mechanism most charged particles are massless.

Bernard Meurer

4:01 PM

Hey guys

John Rennie

@ACuriousMind I've never been clear what happens to electric charge above the EW transition. Does it merge with isospin and whatever the other weak charge is to create some other form of charge?

Or does it remain as a conserved quantity we would immediately recognise as electric charge?

@BernardMeurer Afternoon/morning.

ACuriousMind

@JohnRennie Yes, you have the unbroken electroweak $\mathrm{SU}(2)\times\mathrm{U}(1)$ with its own notions of charge. The charge now associated to the U(1) is no longer electric charge but weak hypercharge.

John Rennie

So technically that isn't a massless electrically charged particle then?

ACuriousMind

@JohnRennie Well, it is "electrically" charged but it's unnatural to look at that charge in the unbroken phase

Since it's a combination of hyperchange and isospin.

However, since isospin and hypercharge are conserved, the electric charge is still conserved

It's just no longer the natural conserved charge to look at

Bernard Meurer

@ACuriousMind Do you like Ghost?

How are you guys

ACuriousMind

4:10 PM

@BernardMeurer What ghost?

Bernard Meurer

@ACuriousMind It's a band

Secret

@BernardMeurer sharing dreams that happened to bore out some of the h barers. Meanwhile preparing phD meetings tomorrow

ACuriousMind

I consider Faddeev-Popov ghosts a necessary evil

Bernard Meurer

some people say they're metal, not sure I would though

John Rennie

@BernardMeurer Got a cold. Probably fatal. If I'm not posting tomorrow it's because I turned into a 66kg ball of mucus and died.

Bernard Meurer

4:11 PM

@Secret Sounds cool, who was in your dream this time?

@JohnRennie LOL

ACuriousMind

@BernardMeurer never heard of them

Bernard Meurer

That guy's curse got to you :p

Secret

@BernardMeurer See above pile of post (Much shorter this time, basically chemsitry)

Bernard Meurer

@ACuriousMind youtube.com/watch?v=-0Ao4t_fe0I

John Duffield

@NoahP : I only debunk popscience woo, pseudoscience, and lies-to-children. And I am interested in physics, not chairs.

John Rennie

4:12 PM

This Ghost?

Bernard Meurer

@JohnRennie I never get colds, I'm uncoldable

John Rennie

Ghost (Swedish band)

Ghost is a Swedish heavy metal band that was formed in Linköping in 2008. In 2010, they released a 3-track demo followed by a 7" vinyl titled "Elizabeth", and later their debut full-length album Opus Eponymous. The Grammis-nominated album was widely praised and increased their popularity significantly. Their second album and major label debut Infestissumam was released in 2013, debuted at number one in Sweden, and won the Grammis Award for Best Hard Rock/Metal Album. The band released their third studio album, Meliora, in 2015, to much critical acclaim and high record sales, reaching number one...

Bernard Meurer

@JohnRennie Yep, that one

@Secret You see specific metal compounds in your dreams dawg?

Secret

yes

Bernard Meurer

Cool :D

Secret

4:13 PM

and a database search don't found exact matches, but a couple of simialr ones

ACuriousMind

@BernardMeurer meh, much too soft for my taste

John Rennie

There is no bad prog metal (actually that's not true :-)

Bernard Meurer

@ACuriousMind So I thought :p

@JohnRennie Are they prog metal though?

@Secret What's your favorite element from the table?

John Rennie

From the video they seem fairly prog metalish ...

I can't speak in general since that's the only thing from I've heard

Painting your face to look like a skull and wearing a silly hat is a bit of a giveaway though.

Bernard Meurer

lol, yeah

I really like that videoclip though, the kids look so funny

Hari Prasad

4:16 PM

what is prog metal?

Bernard Meurer

@HariPrasad youtube.com/watch?v=-0Ao4t_fe0I

I think Kiss falls under prog too

Secret

@BernardMeurer Nothing in particular, each chemical element is interesting in its own way. But for compounds, my interest is strongly on metal complexes because they are so colorful and with a much diverse variety of bonding modes

John Rennie

@HariPrasad This is prog metal:

Tool - Vicarious

►

Bernard Meurer

@JohnRennie What the fuck is this video I'm watching

John Rennie

What! WHAT!

You've never heard of Tool?

Bernard Meurer

4:18 PM

I'm a summer child :p

ACuriousMind

Tool are great

John Rennie

Bow down. You are unworthy!

Hari Prasad

@JohnRennie music is good but the video is awful what the heck was that :(

Bernard Meurer

This video is so disturbing lol

It's better than Guillotine's video still

Death Grips - Guillotine (It goes Yah)

►

yuggib

Tool are great, but 10000 days is a shitty b-side of Ænema

Bernard Meurer

4:20 PM

Worst music video

GUILLOTINE

YAH

Hari Prasad

@BernardMeurer haha

yuggib

@BernardMeurer look for "schism" or "stinkfist" ;-)

John Rennie

@yuggib I was about to say the opposite. I really, really like 10,000 days but the other albums haven't grabbed me the way 10,000 days did.

Bernard Meurer

@yuggib I don't trust you :p

Hari Prasad

@BernardMeurer what was that?

omg thats it

ACuriousMind

4:21 PM

@BernardMeurer What, you're not going to search for "stinkfist" because someone on the internet told you to? :D

Hari Prasad

i am done

Bernard Meurer

@HariPrasad IT GOES IT GOES IT GOES IT GOES GUILLOTIIIIIIIIIIIINE

prokaryoticeukaryote

@JohnRennie Yeah, in particular, Wings for Marie is probably one of their most powerful songs

Bernard Meurer

@ACuriousMind Yes. That's exactly it :P

prokaryoticeukaryote

well, couple of songs really

Hari Prasad

4:22 PM

@BernardMeurer that is not cool.

Secret

@BernardMeurer I am slightly not a fan of dubstep because the reverb is weird sometimes

yuggib

@JohnRennie 10000 days is too commercial

Bernard Meurer

@HariPrasad YAH

ACuriousMind

@BernardMeurer I'd say you need to live more dangerously, but given what I know about you that's not true

prokaryoticeukaryote

also, they have definitely mellowed out somewhat

John Rennie

4:22 PM

For the record:

Stinkfist

"Stinkfist" is a song by the American rock band Tool. It is their first industry single and first music video release from their third major label album Ænima. == Interpretation == The song title, the lyrics and the perceived subject matter caused changes made, to the originally released version, by TV and radio programmers, who also shortened the track. The track has been remixed by Skinny Puppy. Keenan said the use of the words "stink" and "fist" and the resulting perception of "fist fucking" is actually symbolic in dealing with a friend of drummer Danny Carey who "isn't afraid of getting his...

Bernard Meurer

@Secret Believe it or not that's categorized as Hip-Hop

experimental Hip-Hop

@ACuriousMind I live dangerously enough already

Holy shit

I'm moving in less than a week

John Rennie

@BernardMeurer Cool :-) The visa and everything is sorted then?

Bernard Meurer

@JohnRennie Yep, less than a week left in Rio

I'm lowkey freaking out

@ACuriousMind Is this heavy enough for you? youtube.com/watch?v=uvxkEvM81tU

^ Best music video

ACuriousMind

what the...

Bernard Meurer

@ACuriousMind youtube.com/watch?v=2MHhLDCJ57E

John Rennie

4:28 PM

Not really metal though ...

Bernard Meurer

@JohnRennie No, they're just bizarre

0celo7

@JohnRennie I'M ALIVE

@ACuriousMind uhhh

John Rennie

@0celo7 Since there haven't been any news reports of catastrophic explosions in the US I assumed you hadn't done the experiment yet.

Bernard Meurer

@JohnRennie lol

0celo7

@JohnRennie I tried to cut cerium with the diamond band saw

I lived

It was unclear what would happen

John Rennie

4:31 PM

@0celo7 sparks?

0celo7

@JohnRennie Yeah, a lot

John Rennie

Excellent :-)

0celo7

I stopped cutting

I'll get another blade first, I'm not sure this will be good for the saw

Don't want to ruin the only blade

Bernard Meurer

US Gov. Expenses in 2016 so far:
1. @0celo7
2. Military

Secret

I sometimes have a hard time understanding heavy metal. It takes an insane amount of concentration for me to hear the melody buried under the static

ACuriousMind

4:34 PM

If your metal has static, stop listening to crap recordings :P

John Rennie

@Secret Music? There's music there?

Secret

there is, it is usually something powerful impression buried within the melody

My opinion might be partly influenced on how video games tend to use heavy metal as background music for scenes full of tension however

after all, how most people felt about a music is often shaped by the culture itself

if a certain interpretation is commonplace, you tend to associate to it automatically unless you are some die hard fan that has a view very different from the social norm

0celo7

@BernardMeurer my budget is pretty nice

Secret

A similar principle underlies all artwork in general. Music is one of the arts

My music preferences is highly determined by my emotional state, but independent of that, the classical music stood out.

My least favourite is some type of pop music with cliche melodies

In general, I hate repetative things, and extremely addicted to novelty. This is why science interested me because there's always something eye opening in the discoveries, even if it takes a lot of effort to get to them

ACuriousMind

@Secret What? I would associate metal in video games much more with sequences of destruction or power fantasies (like the entirety of Doom). Name one game that uses metal to communicate "tension".

Secret

4:43 PM

I think I am trying to said something similar, but I felt the term boss music is a bit too broad

yuggib

@Secret google.it/url?sa=t&source=web&rct=j&url=http://…

John Rennie

@ACuriousMind Is Nine Inch Nails metal?

Secret

@yuggib It's a pretty common type of melody or motif. While I don't have any comments for or against it right now, I am pretty sure based on past experience I will get bored of it quickly

ACuriousMind

@JohnRennie I'd say that's "just" rock, mostly.

John Rennie

Now I think about it NIN did Quake not Doom.

0celo7

4:50 PM

Quake? Doom?

John Rennie

Prehistoric video games

0celo7

Oh, I don't play those.

Secret

https://en.wikipedia.org/wiki/Doom_(2016_video_game)
There's a mordern version YMMV

Secret

This music describes my default emotional state, which explains why most of the time I have no particular feelings for or against many things
https://www.youtube.com/watch?v=WZKW2Hq2fks

Hint: There are 3 things that can disrupt that however, and 0celo7 knew half of the 1st one (because he experienced that before)

ACuriousMind

@0celo7 Aha.

Secret

4:54 PM

These music are also very effective in keeping one focused on their work and some atmospheric games like to use them to give a scene of some alien space

0celo7

@ACuriousMind yes?

On mobile

ACuriousMind

I just remarked on your declaration that you don't play video games

0celo7

I don't.

John Rennie

The evidence would suggest otherwise

0celo7

Not me.

That was the other 0celo7

I've quit.

ACuriousMind

4:56 PM

Can we get the other 0celo7 back? :P

0celo7

He asks more basic topology questions

Wouldn't you rather have me

Secret

When my tranquil state got disrupted, suddenly music like these became addictive until I return to normal again
https://www.youtube.com/watch?v=tlf09gtll7U

But the above is not the worst case scenario, and for "safety reasons" we are not allowed to disclose the music that represent my most disrupted emotional state joke lol

Secret

@0celo7 Give me an example of a topology where there is only one neightbourhood

0celo7

Trivial topology

On any set

Nonempty set

Secret

Is it the coarsest possible of any topology as it has the least number of open sets?

0celo7

5:07 PM

You're going to have to tell me what coarse means, we don't use that in our course because it's confusing

Secret

https://en.wikipedia.org/wiki/Coarse_topology
It basically is a comparison of how many subsets are containined in a topology. Thus coarse topology is a topology which is a subset of another topology

And nvm, I think I accidentally found one of your answers here

0celo7

I know it's a comparison, but I never remember which one coarse is and which one fine is

Secret

I mostly learn the terminology on wikipedia when ACM answered my question about the "pile of lines" $\cup_{i\in S}\mathbb{R}_i$ and I am trying to work out the difference in topology of the product and the discrete union

In one of the wikiepdai artciels (to be quoted shortly) there's an intuition that a coarse topology has fewer open sets and neighbourhoods than a fine topology

0celo7

@ACuriousMind do you really want the old me back :(

Secret

Comparison of topologies

In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. == Definition == A topology on a set may be defined as the collection of subsets which are considered to be "open". An alternative definition is that it is the collection of subsets which are considered "closed". These two ways of defining the topology are essentially equivalent because the complement of an open set is closed and vice versa. In the following, it doesn't matter which definition is used. Let...

> One can also compare topologies using neighborhood bases. Let τ1 and τ2 be two topologies on a set X and let Bi(x) be a local base for the topology τi at x ∈ X for i = 1,2. Then τ1 ⊆ τ2 if and only if for all x ∈ X, each open set U1 in B1(x) contains some open set U2 in B2(x). Intuitively, this makes sense: a finer topology should have smaller neighborhoods.

This picture is quite visual that I can illustrate it as follows

0celo7

5:20 PM

Good god

Can you take a topology course lol

Secret

So in a sense, think of the set as a cake, the coarseness or fineness of a topology is basically how small you cuts on the cake are

Or in optics like terms, how well your "resolution" is

I am currently reading this math.colostate.edu/~renzo/teaching/Topology10/Notes.pdf, but my progress was sluggish because I have an unfinished book to read on art forms and also my pHD application chores

0celo7

A far better terminology is larger and smaller. A topology is larger if it contains another.

Secret

In that case, the right diagram will be the larger topology as it contains those on the left

NeuroFuzzy

Started to try to answer physics.stackexchange.com/questions/165156/… and I got down the rabbit hole...

I can't tell if a bunch of code I wrote a few years ago is correct or not...

Relativistic optical effects are weird.

0celo7

5:39 PM

Bah, physics

NeuroFuzzy

but look, spheres! en.wikipedia.org/wiki/Relativistic_Doppler_effect#/media/…

Emilio Pisanty

@NeuroFuzzy cute

but

A Slower Speed of Light Official Trailer — MIT Game Lab

►

A Slower Speed Of Light

►

NeuroFuzzy

@EmilioPisanty I'm supposed to be studying for my physics GREs, don't inspire me to code games!!!

Emilio Pisanty

@NeuroFuzzy just sayin'.

NeuroFuzzy

I really want to make a relativistically invariant RTS

where you and the AI only have access to information in their past light cones

ACuriousMind

5:54 PM

@0celo7 Well, I enjoyed the video game discussions...

0celo7

@ACuriousMind video games are boring compared to geodesic balls

I will replay the ME series over Christmas, don't worry

Gotta get a rogue character

@ACuriousMind Consider a metric space and two nonequal points. Suppose I have balls around these points and that the difference in radii of these balls is greater than the distance between the points. Is one ball inside of the other?

What the hell is a nonesuch, autocorrect?

I take it back, I'll probably be reading Ricci flow over Christmas.

I lied. I'll be playing the new Pokemon games

Emilio Pisanty

@0celo7 Yes. Suppose $R-r>d(p_1,p_2)=D$ and $p\in B_r(p_1)$. Then $d(p,p_2) \leq d(p,p_1) + d(p_1,p_2) < r + D <R$ so $p\in B_R(p_2)$.

DanielSank

@ACuriousMind What game are we discussing? Smash Bros.?

0celo7

@EmilioPisanty Awesome.

One step closer to world domination...

Emilio Pisanty

6:11 PM

Is this a good place to do a happy citation dance, btw?

0celo7

No.

vzn

@NeuroFuzzy nice diagram & hints at some of the strong connection between relativistic theory & fluid dynamics which seems widely unacknowledged/ underappreciated

0celo7

@EmilioPisanty what citation

Emilio Pisanty

@0celo7 arxiv.org/abs/1608.03844

0celo7

what about it

you're [24]?

Emilio Pisanty

6:25 PM

@0celo7 Precisely

0celo7

dance

ok, yes

Emilio Pisanty

4

0celo7

6:49 PM

@BalarkaSen hey

Balarka Sen

hi's. can you tell me if Milnor is the right place to look for the existence of flows along vector fields?

0celo7

No, Lee.

Milnor has one very basic result but there's lots of detains he omits

Balarka Sen

G-P says one needs to solve a differential equation and leaves it at that (they use a weaker notion of flow being tangent to vector field at time t = 0).

0celo7

Page?

Balarka Sen

@0celo7 I just need existence.

0celo7

6:51 PM

@BalarkaSen Short-time existence follows from Picard-Lindelöf

You can always solve smooth ODEs in small neighborhoods

Balarka Sen

page 134.

0celo7

Milnor does not given any more details than that IIRC

pull the flow ODE back onto a chart, then apply PL

@BalarkaSen But given the fact that you're not going to prove PL for coupled equations on $\Bbb R^n$, just accept this and move on

Balarka Sen

I can guess that much but I'd rather work out the details.

0celo7

I just gave you the details up to the gorey analysis

Do you want a reference for the analysis?

Balarka Sen

No.

0celo7

6:56 PM

@BalarkaSen Lee does it very carefully.

Read Milnor's two pages on it, and if that's not to your liking, check Lee.

Balarka Sen

I don't like Lee's writing, I'll fall asleep if I try to read him :S

OK.

0celo7

I can give you about 10 books that cover this :P

Hirsch assumes you know it already

Balarka Sen

lol.

@0celo7 I'll ask you if I can't figure out the details Milnor leaves to the reader.

0celo7

Hirsch is a reference book with problems for people with too much time :P

Balarka Sen

It'd be a good exercise to try out.

0celo7

6:57 PM

Ok.

Balarka Sen

Thanks.

0celo7

7:26 PM

@BalarkaSen Reading Milnor?

It's well known that if the bounded sets of a metric space are precompact, the metric is complete.

But can we weaken that to the open unit balls being precompact implying completeness?

Balarka Sen

@0celo7 What does precompact mean again?

0celo7

Compact closure

Balarka Sen

Oh. I agree.

@0celo7 No, I doubt this.

0celo7

@BalarkaSen I feared.

I'm trying to draw inspiration for my proof by understanding the classical proof of Nomizu and Ozeki.

But they claim that all $1/3$ balls being precompact implies completeness.

(you can make 1/3 balls into unit balls by multiplying the distance function by 3, obviously)

ams.org/journals/proc/1961-012-06/S0002-9939-1961-0133785-8/…

Balarka Sen

7:43 PM

No, I eat my words. If every open unit ball has compact closure, so does every open set, yes? Then given any bounded set S, closure of it's interior (compact by previous statement) contains closure of S. Cl S is closed in the closure of it's interior: closed subsets of compact sets are compact.

So does every bounded open set, I meant to say.

0celo7

Thinking

Every open set is a union of open balls, sure

@BalarkaSen Does closure commute with infinite union?

Balarka Sen

That's too hard. If every open unit ball has compact closure, so does every open ball. Bounded means fits inside a large open ball. If U is a bounded open set which fits inside a large ball B, you can make B large and fit cl U inside that (closure of bounded set is bounded).

cl B contains cl U, cl B is compact. Again, closed subset of compact set is compact. Same argument.

0celo7

Whoa why does every open ball have compact closure?

@BalarkaSen It's clear that if every unit ball has compact closure, every ball with radius < 1 does too

Balarka Sen

If you have a ball of radius $r$, you can cover it by finitely many unit closed balls, no?

0celo7

I've never heard of that one, please elaborate

Balarka Sen

7:59 PM

I don't know what to elaborate. I am claiming every closed ball of radius $r$ is a subset of a finite union of closed unit balls.

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