The h Bar

Semiclassical

01:12

@EmilioPisanty welp

have fun with that

Semiclassical

01:36

@EmilioPisanty yeah, I mean, who wouldn't notice the heat equation in imaginary time? :P

Kane

02:44

hey can someone help me out with a simple power question?

$\eta = \frac{P_{out}}{P_{in}} = \frac{iV_{out}}{iV_{in}} = \frac{i^{2}R}{iV} = \frac{iR}{V}$
$i = \frac{\eta V}{R}$

That was the way I found the current in a wire starting with the efficiency

I was given the efficiency to be: 0.998 and resistance to be 1659 ohms

so 0.998 * 750000V / 1659 ohms = 451A

but the answer is incorrect, I don't see what's wrong with it. The question was to find the current in this wire with 1659 ohms resistance and 0.998 efficiency and 750000V

note: That eta, my mathJax seems to display 'eta' as 'or'

Secret

04:52

Black star (semiclassical gravity)

A black star is a gravitational object composed of matter. It is a theoretical alternative to the black hole concept from general relativity. The theoretical construct was created through the use of semiclassical gravity theory. A similar structure should also exist for the Einstein–Maxwell–Dirac equations system, which is the (super)classical limit of quantum electrodynamics, and for the Einstein–Yang–Mills–Dirac system, which is the (super)classical limit of the standard model. A black star need not have an event horizon, and may or may not be a transitional phase between a collapsing star and...

I want to mine that stuff in the core of these

John Rennie

06:13

@Kane does the 0.998 efficiency mean 0.2% of the power is dissipated in the wire and 99.8% in the load at the end of the wire?

user228700

Hello, everyone :-)

John Rennie

Morning. I see your new avatar now.

user228700

Great! :-)

user228700

How's your Saturday going?

John Rennie

@KaumudiH bad start - I had a server down. Now fixed though :-)

user228700

06:17

Ah, damn :-/

John Rennie

S'OK. All part of the job.

user228700

:-) Right.

John Rennie

That's why I start so early. Fix this stuff before anyone even knows it's gone wrong.

user228700

My Saturday is not looking up. I have several assignments to finish and it is my last day at home :-/

John Rennie

Last day :-(

user228700

06:19

Yep :-/

John Rennie

Are you going to your Gran's tomorrow, or straight back to college?

user228700

Straight back to college. Guess how I'm going back this time!

John Rennie

Bus?

user228700

No! Flight!

John Rennie

Wow!

user228700

06:21

Yep! :-) There was a cheap offer and we grabbed it!

John Rennie

Where is the airport in Kochi? Actually I guess I could Google that ...

user228700

@JohnRennie :-) Yes. Two bus-rides away from my hostel, I believe.

user228700

Did you know that Cochin International Airport is the world's first airport to be fully powered using solar energy alone?

John Rennie

CIAL Solar Power Project

The CIAL Solar Power Project is a 13.1 megawatt (MW) photovoltaic power station built at COK airport, India, by the company Cochin International Airport Limited (CIAL). Cochin International Airport became the first fully solar powered airport in the world with the commissioning the plant. == Overview == The plant comprises 46,150 solar panels laid across 45 acres near the international cargo complex. The plant has been installed by the German-based M/s Bosch Ltd. It is capable of generating 50,000 units of electricity daily, and is equipped with a supervisory control and data acquisition system...

user228700

Yep!

John Rennie

06:32

How come you're flying? It has to be much nicer than the train, but I'd guess it's a lot more expensive.

user228700

11 mins ago, by Kaumudi H

Yep! :-) There was a cheap offer and we grabbed it!

user228700

:-)

user228700

Not only is it much nicer, it is also much faster! Only one hour from here!

John Rennie

I've flown on an internal Indian airline once, from Mumbai to Pune.

(and back)

user228700

And that was a long time ago, too, wasn't it?

John Rennie

06:35

I was just thinking that :-)

1995 or 1996 or something like that

user228700

A few years before I was born :-)

Secret

3 years after rereading this, finally had enough background to comprehend it:

user228700

I have flown once before, when I was merely 8.

Secret

user228700

I don't remember the experience in the slightest.

John Rennie

06:37

I used to fly lots as a child. We lived in Khartoum in Sudan and we used to fly between London and Khartoum several times a year.

user228700

Oh, wow!

John Rennie

But like you I've long ago forgotten anything about it.

user228700

:-) Hmm.

Abcd

Is this video correct?

John Rennie

Scandium is [Ar] 3d1 4s2 according to Wikipedia

Secret

06:39

Basically, the easiest way to think about 2 time dimensions with causality, is that you are considering a vector field that is the gradient of the entropy described by two time variables $s$ and $t$, and for every initial condition, the evolution took the form of trajectories on the $s,t$ plane following the gradient field

John Rennie

@KaumudiH flying is good fun as long as you do it so rarely that it remains an adventure.

I hardly ever fly these days mainly because there's nowhere I particularly want to go.

Secret

which means, things are literally existing in multiple futures at the same time and these influence each other

Abcd

@JohnRennie okay to represent orbitals that way?

user228700

@JohnRennie I s'pose so, yes :-)

John Rennie

@Abcd it's a common way to represent orbitals, but of course orbitals don't have an edge. They are fuzzy objects that spread out to infinity. The usual way of drawing s orbitals as spheres, p orbitals as kind of figure eights etc is just a visual guide.

Secret

06:45

so, things looks really arbitrary with objects suddenly vanishing and appearing at times when only one time dimension is considered, but remains deterministic when both directions are considered

John Rennie

@KaumudiH what's waiting for you at college? What courses?

user228700

Oh, haven't I told you about this semester's courses yet?

user228700

Wait, no, I did...

Secret

In conclusion: For a 2D time that operates in such a model such that the arrow of time is given by $\nabla S(s,t,\vec{x})$ where $S$ is the entropy, the laws of physics itself and its evolution will be frame dependent in a consistent manner, given the same outcomes and initial conditions

So perhaps, a possible experimental evidence that our universe only has one time dimension is that we never see the laws of physics changing with velocity or acceleration

user228700

@JohnR: Ping me back when you're free.

bolbteppa

06:54

@Semiclassical I think I know where Cayley-Hamilton comes from... (!!!) If you consider the set of polynomials, we can set up the division algorithm, the remainder theorem, and the factor theorem for them. If we then generalize and consider matrices $A(\lambda) = [a_{ij}(\lambda)]_{nn}$ for square matrices whose entries are polynomials,

John Rennie

@KaumudiH I'm free now. My servers are all working fine :-)

bolbteppa

we can re-express these as polynomials with matrix coefficients, and so treat them as polynomials with non-commutative coefficients. We can now set up analogues of the division algorithm, remainder theorem and factor theorem, splitting things into two cases for left vs. right division. Once we do this, the factor theorem applied to the 'matrix times adjoint = determinant times I' relation shows, for $A - \lambda I$ that $A - \lambda I$ divides $\det(A - \lambda I) I$

which means the remainder is zero, and you can show the remainder is $f(A)$. This is done in Ayres Schaums matrices book, and we can see why Cayley-Hamilton generalizes to modules instead of vector spaces pretty directly, why it links to Nakayama is the next stage...

user228700

@JohnRennie Cool :-) If you remember, I have Engineering Mechanics this semester.

John Rennie

OK ...

bolbteppa

e.g. the example taken from Eves book

\begin{align}
f(\lambda) &= \begin{bmatrix} \lambda^3 - 4 & 2 \lambda - 1 \\ \lambda^2 + \lambda + 1 & \lambda^4 + \lambda^2 + 5 \lambda \end{bmatrix} \\
&= \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \lambda^4 + \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \lambda^3 + \begin{bmatrix} 0 & 0 \\ 1 & 1 \end{bmatrix} \lambda^2 + \begin{bmatrix} 0 & 2 \\ 1 & 5 \end{bmatrix} \lambda + \begin{bmatrix} -4 & -1 \\ 1 & 0 \end{bmatrix}
\end{align}

Secret

06:55

(Of course, the above argument does not invalidate thermal time dimensions proposed by bars, compactified time dimensions in F-theory and other multiple time dimension theory where the 2nd time dimension is proposed in a way such that it is microscopic and does not mess up causality)

user228700

And I have a quick-ish question. Can you tell me about the moment of a force? The way that it has been described by my prof./in the "text" isn't very clear to me.

bolbteppa

Nakayama's lemma

In mathematics, more specifically modern algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem — governs the interaction between the Jacobson radical of a ring (typically a commutative ring) and its finitely generated modules. Informally, the lemma immediately gives a precise sense in which finitely generated modules over a commutative ring behave like vector spaces over a field. It is an important tool in algebraic geometry, because it allows local data on algebraic varieties, in the form of modules over local rings, to be studied pointwise as vector spaces...

John Rennie

@KaumudiH I don't think there's very much to tell.

user228700

OK, let me rephrase my question so that it is more specific.

user228700

How about Varignon's Principle?

Balarka Sen

06:58

It's not very inspiring to prove the C-H theorem from Nakayama's lemma

bolbteppa

Can't make sense of what Nakayama is even saying heh

John Rennie

@KaumudiH never heard of it. Give me a Googlemoment ...

Balarka Sen

Yeah its rather algebraic. I do know of one interpretation of the lemma...

user228700

Ah, OK...

bolbteppa

But this whole $\lambda$-matrix perspective seems really easy, and apparently if you prove the Smith normal form which seems easy it leads really quickly to the rational canonical form, the jacobson canonical form (wtf) and then the JNF with no fluff

John Rennie

07:01

@KaumudiH that seems like a complicated way of saying that torques add together, which since they are vectors shouldn't be any surprise ...

user228700

Hmm, OK, but what is this about moving the point of action?

Balarka Sen

Yes but I mean it's really a hassle to set up the correct theorems in noncommutative rings

bolbteppa

I'm really happy with a hassle compared to having no idea what's going on :p

Balarka Sen

No but I mean there are simpler ideas which proves the same thing

Eg the diagonalization, and then the fact that diagonalizable matrices are dense

bolbteppa

haha

John Rennie

07:05

@KaumudiH I think that article isn't terribly clear.

Balarka Sen

That idea generalizes to arbitrary (algebraically closed) fields, actually. Instead of the real topology in $M_n(\Bbb R)$ you use the Zariski topology on $M_n(k)$

bolbteppa

It's interesting that they stopped at gcd's for these things and are going this division algorithm to Smith to RCF to JNF, while you can prove the JNF directly from gcd thinking also

Serre swan lurking around this stuff, god

John Rennie

@KaumudiH The moment of a force is always defined relative to some reference point. In mechanics that point is generally a pivot because we're interested in rotation about the pivot. But you can choose any point you want to define moments, even if the result isn't especially usefukl.

user228700

@JohnRennie No, it's not :-/ That's the trouble.

user228700

Right, right...

John Rennie

07:07

A moment is just a torque i.e. a twisting force

And obviously that's most usefully defined relative to a pivot because we'd be twisting around the pivot.

user228700

Right, of course.

user228700

Hmm, we use this principle to find the point of action of the resultant of a system of forces.

John Rennie

... I'm not sure if this has helped because I'm not sure what the confusion was

user228700

What are we doing there, exactly?

John Rennie

I'm not sure what is meant by the point of action

user228700

07:12

It's defined, I believe, to be the point at which the resultant of a system of forces can be assumed to act so that the reaction force on that system remains the same.

John Rennie

Ah OK. So if you have some object with several forces acting on it then you can replace the multiple separate forces with a single force acting at a single point?

user228700

Yep, precisely.

John Rennie

That single point being the point of action?

user228700

I think so, yes.

Dawood ibn Kareem

That just means you can add torques together.

Hello John and Kaumudi.

user228700

07:14

Hello :-)

Dawood ibn Kareem

But it doesn't make sense to add a torque around one point to a torque around a different point. So before you do any vector arithmetic with torques, you need to define what point they're going to act around.

John Rennie

Morning :-)

Dawood ibn Kareem

I was going to point out that it's not morning where I am; but since I've never told you where I live, there's no way you could have known that.

user228700

@DawoodibnKareem Right, right...

Balarka Sen

@DawoodibnKareem I was ranting a few days ago how a physicist's vector space is honestly truly an affine space

Dawood ibn Kareem

07:17

@BalarkaSen Yes, I frequently rant about that exact same thing.

Balarka Sen

Aha, a comrade

Dawood ibn Kareem

Sorry @KaumudiH I don't think I answered your question. But I'm not entirely sure what your question was.

user228700

Me neither :-P But I think I've understood it.

Dawood ibn Kareem

OK, that's good. How are your other courses going?

user228700

Thanks, guys :-)

user228700

07:21

@DawoodibnKareem Boring, but OK, thanks :-)

Dawood ibn Kareem

Which one's the worst?

user228700

Chemistry, to be sure.

bolbteppa

@Semiclassical also here solitaryroad.com/c152.html

user228700

How're your pigeons?

Dawood ibn Kareem

When I studied first-year chemistry, another student set my lab notebook on fire.

user228700

07:23

x'D By accident, I assume?

John Rennie

@DawoodibnKareem I had the idea you're in the middle East somewhere. Offhand I don't know what the time difference is between the UK and that region.

Dawood ibn Kareem

They're very well; thanks for asking. I hand-raised a baby, and I'm just in the process of graduating him to the point of being able to live in the aviary with other pigeons.

Balarka Sen

I'd set the author on fire, not the notebook

Consider yourself lucky

Dawood ibn Kareem

Well, I hope it was by accident. He did follow it up by putting it in the sink and pouring water on it; so it seems that arson wasn't his actual intention.

Dawood makes mental note not to do chemistry with Balarka

user228700

@DawoodibnKareem :'-) Aw, nice!

user228700

07:25

@BalarkaSen Same.

user228700

@Balarka: How are ya, BTW?

Balarka Sen

Im good

Dawood ibn Kareem

Hey, this is really hostile! Two of the people in this room want to set me on fire? What did I do?

I'm not Ocelo7!

user228700

Two?

Dawood ibn Kareem

You and Balarka.

user228700

07:27

Hey! I expressed no such desire!

user228700

Oh :-P Scratch that.

Anonymous

@KaumudiH By "author" of the lab notebook, Balarka implied Dawood ;)

Anonymous

Hi btw

Dawood ibn Kareem

Hello Blue.

Secret

skullpatrol

07:28

\o

user228700

@Blue :-P Lab notebook, of course! No, Dawood, I wouldn't set you on fire!

Anonymous

@DawoodibnKareem Hey, so finally you did drop some hints about which part of the world you live in :D

user228700

@Blue: Hello! How are you doing?

Dawood ibn Kareem

@Blue Yes, I'm somewhere where it's not currently morning. That eliminates pretty much exactly half the globe.

Anonymous

It's night-time only in the North and South American region at the moment....

Dawood ibn Kareem

07:30

Is that a fact?

Anonymous

And I can guess that you won't be awake after 12-1 at night...(Given that you are a family man who likes to rest on weekend nights after working the whole week :P)

John Rennie

@DawoodibnKareem Q1: locate Dawood using bisection :-)

Secret

Anonymous

So that eliminates almost 3/4 th of the globe

Anonymous

Okay, I'll keep looking for more hints XD

Secret

07:34

Dawood ibn Kareem

This is my baby.

But he's much bigger now.

skullpatrol

Aaww...

user228700

Aww!

John Rennie

Goodness, baby pigeons are even uglier than baby humans

Secret

lol

skullpatrol

07:36

LOL

user228700

@JohnRennie John, have you ever seen a baby cat?

John Rennie

@KaumudiH Yes. Do you mean newborn, or when their fur has had the chance to dry out and fluff up?

user228700

No, I mean a newborn.

John Rennie

Eyes still closed?

user228700

Yes. They look incredibly terrifying and ugly.

Dawood ibn Kareem

07:38

@KaumudiH Especially if you're a baby pigeon.

user228700

x'D Hahaha, I s'pose.

John Rennie

@DawoodibnKareem :-)

@KaumudiH they do look a bit alien :-)

Balarka Sen

Horror of infants is very Freudian

#thisisdeep

skullpatrol

hmmm...

Secret

books.google.com.au/…

Dawood ibn Kareem

07:42

I used to flat with some people who got themselves a black kitten. It ran away after a few months, never to be seen again. About a year later, after I had moved out, a black cat showed up on their doorstep. They were convinced it was the same one, until one of their friends pointed out that the original kitten was male, and the new arrival was female.

Secret

That's easy

Anonymous

I've heard far fewer stories of cats returning to their original owner, compared to dogs. There must be something...

John Rennie

@Blue cats aren't pack animals

They don't form attachments to people in the way dogs do

Balarka Sen

more importantly cats dont have owners

they have employees

Dawood ibn Kareem

@JohnRennie I missed the @ at the front of that on first reading, and wondered where on earth you've seen blue cats.

Loong

07:45

@JohnRennie lions are

Anonymous

@JohnRennie Makes sense

John Rennie

@DawoodibnKareem :-)

@Loong lionesses are. Male lions aren't.

Anonymous

@BalarkaSen lol

Loong

@JohnRennie true

Secret

skullpatrol

07:46

lol

Dawood ibn Kareem

Hmm, not exactly smurf-coloured though is it?

Secret

Well, more like grayish blue

the russian blues are a famous breed

Loong

@Secret Chartreux?

Dawood ibn Kareem

Fair enough. We talk about blue pigeons too, and they're a similar kind of colour I suppose.

skullpatrol

Russian Blue vs Chartreux - Difference Explained

►

:-)

Loong

07:53

@skullpatrol Now I want one.

skullpatrol

Me too.

Dawood ibn Kareem

The one on the right is blue.

user228700

I do not like cats. A cat which visits my gran's place everyday likes to follow me around and make a dash for my leg to rub its body against it. Very uncomfortable.

Dawood ibn Kareem

@KaumudiH They make me sneeze.

user228700

Oh, yeah, the stupid fur.

John Rennie

07:58

I admire cats for their elegance, but I've never felt the urge to own one.

("own" = house, feed, and pander to their every whim)

3

Dawood ibn Kareem

Do you think they admire you for your elegance?

John Rennie

I suspect cats have a very different dopamine response to humans or dogs. You have to approach their behaviour with that in mind.

Anonymous

Lazy as I am, I'd like to have any pet which can care for itself. Maybe the only things which fall in that category are mosquitoes, cockroaches, ants etc.

John Rennie

Rats?

Dawood ibn Kareem

@Blue If you give me your address, I'll arrange a mosquito to be sent to you.

Anonymous

08:02

Ah, rats sound nice. I hope they can feed themselves

Anonymous

:P

Anonymous

Of course they can...

Anonymous

@DawoodibnKareem Okay, but I will only accept mosquitoes which won't feed on me. ;)

user228700

@Blue So male ones, easy.

Dawood ibn Kareem

@Blue Tell you what - I'll send you one of each, just to make sure you get an acceptable one.

Anonymous

08:05

Lol

user228700

Dammit, I don't know how to edit images well enough :-( I have spent the past few minutes attempting to edit John's photograph onto a photo of a cat lady, but in vain.

skullpatrol

What happened to your avatar? @KaumudiH

user228700

I fell in love with Dwight Schrute, that's what.

skullpatrol

You look like a guy :P

Dawood ibn Kareem

@KaumudiH I assumed you had tried to edit your own photograph onto a photo of a cat lady, but failed.

user228700

08:08

-..-

user228700

Oh, that looks like a pig squinting!

2

skullpatrol

lol

Balarka Sen

good video

Fractals are typically not self-similar

►

skullpatrol

thnx for sharing

Anonymous

@BalarkaSen Is that about scale invariance and correlation length?

Balarka Sen

08:10

um no its about hausdorff dimension

Anonymous

Interesting, I'll watch

skullpatrol

Speluncean explorers

►

This was interesting.

Secret

Poincaré–Hopf theorem

In mathematics, the Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important theorem that is used in differential topology. It is named after Henri Poincaré and Heinz Hopf. The Poincaré–Hopf theorem is often illustrated by the special case of the Hairy ball theorem, which simply states that there is no smooth vector field on a sphere having no sources or sinks. == Formal statement == Let M be a differentiable manifold, of dimension n, and v a vector field on M. Suppose that x is an isolated zero of v, and fix some...

Just when I thought I am reading chemistry, I saw this utterly incomprehensible theorem

Balarka Sen

This is far from an incomprehensible theorem

Dawood ibn Kareem

It's quite hard to understand. How did it come up in chemistry?

Secret

08:21

My prerequisites on differential topology is close to none. So far I only knew the most basic notion of topology and is completely ignorant of things like mapping degrees, homotopy etc.

Dawood:

wiley.com/legacy/wileychi/ecc/samples/sample02.pdf

It's part of the theory of Atoms in molecules, which locates critical points in the electron density of molecules to establish the location of bonds

Balarka Sen

i can explain if you want

Secret

sure

Dawood ibn Kareem

@Secret It's really just talking about a generalisation of the Euler polyhedral formula, which is much easier to understand.

Balarka Sen

Take a surface $\Sigma \subset \Bbb R^3$ of genus $g$ sitting in the 3-space. A natural question to ask is if you can construct a nonzero vector field in the 3-space everywhere tangent to the surface

Dawood ibn Kareem

Very natural.

Balarka Sen

08:26

The reason why it's natural is the famous hairy ball theorem, which you all know, right?

That says you cannot have an everywhere tangent nonzero vector field on the sphere in the 3-space

Secret

yup (thought I have not read the proof in detail yet, but I knew the heuristics in terms of cowlick)

Dawood ibn Kareem

In a recent episode of The Chase, Mark Labbett didn't know the Hairy Ball Theorem. This surprised me greatly.

Balarka Sen

Hah

@Secret Right, that would suffice

What about the torus? Can you construct an everywhere nonzero tangent vector field on the torus?

Turns out you can

Dawood ibn Kareem

@BalarkaSen That's kind of trivial.

Balarka Sen

Once you see it of course

Secret

08:28

I am thinking about combing all circles of minor radii

Balarka Sen

That does the trick

There is an interesting point to be made. Suppose you take the torus in 3-space, and let $R_\theta$ be rotation of the torus along the axis passing through the hole by an angle of $\theta$

What it does is it "rotates the torus in the long direction"

Is this okay?

Secret

This?

Balarka Sen

Yah

Now think about "$d/d\theta (R_\theta)$"

You get an everywhere tangent nonzero vector field

Namely, the field which describes the infinitisimal motion of this rotation

Secret

looks fine

Balarka Sen

Well the field should be always perpendicular to the circles of the shorter radii

Because every such circle moves to the next

bolbteppa

08:36

There is a nice example of Poincare-Hopf in terms of maxima, minima and saddle points for like baby functions on the plane right?

Balarka Sen

@bolbteppa True

Anyhow, I got a little side-tracked there.

bolbteppa

Where did this come up in chemistry

Balarka Sen

So it turns out the torus is the only surface on which there exists an everywhere tangent nonzero vector field

Secret

@bolbteppa Theory of Atoms in molecules by Bader, a model that uses gradient of molecular electronic density to assign bonds in molecules

wiley.com/legacy/wileychi/ecc/samples/sample02.pdf

Balarka Sen

This is a surface of genus 4 and an attempt at drawing a nonzero tangent field on it

But alas, it has 10 zeroes

Secret

08:42

The zeros are located at the 8 saddles on the surface?

Balarka Sen

Yes, 8 saddles plus the source and sink

(Forgot to include them)

So it's an interesting question to ask what is the total number of zeroes of a certain tangent field on a surface

But again, that turns out to be the wrong question

The usual picture of a tangent field on the sphere has 2 zeroes, right? The "cowlicks"

Secret

yeah, at the poles

Balarka Sen

Here's one with 1 zero

It's cheating, kinda, because you're smashing the two zeroes together to get this picture

Secret

looks like some kind of saddle. I can see vectors flowing in and out, and the rest circles around

John Rennie

Isn't that a bit of a cheat? That's a dipole i.e. you've combined the source and sink.

Balarka Sen

08:46

@JohnRennie Great minds think alike

;)

John Rennie

Sorry. I hate people who jump ahead when I'm telling a story :-)

Balarka Sen

@Secret It's a saddle, but of the bad kind. It's analogous to the Monkey saddle with four hills and four passes

The right saddle should have 2 hills, 2 passes

These are what is known as the "non-generic singularities" or in fancier language "non-Morse singularities".

Burn them. We don't want to deal with this.

Dawood ibn Kareem

It's not bad if you have a two-headed monkey.

Balarka Sen

lol

But yeah, you can fix this. Instead of doing a total over zeroes for arbitrary vector fields, we could do the following

skullpatrol

@JohnRennie you lost me, what story were you telling?

Balarka Sen

08:51

Say $X$ is your vector field on the surface $\Sigma$. You can wiggle $X$ a little so that all the non-Morse singularities become (1) a source (2) a saddle or (3) a sink. This is what @JohnRennie said; you can uncombine the bad zeroes to get the simpler zeroes

Now consider the ALTERNATING SUM: number of source - number of saddle + number of sink

John Rennie

@skullpatrol I was saying I shouldn't have interrupted Balarka's story - like we're doing now :-)

skullpatrol

Right, right.

Balarka Sen

This number turns out to be independent of the chosen vector field. This is the Euler characteristic $\chi(M)$ of $M$.

Secret

Is there a reason why we need a minus sign for the number of saddles?

Balarka Sen

Indeed :) The point is a saddle singularity has index -1. If you go clockwise on a loop around a saddle singularity, the vectors rotate in the counterclockwise direction (I encourage you to draw a picture to understand this)

skullpatrol

08:54

Have you heard of the speluncean explores case before? @JohnRennie

John Rennie

@skullpatrol let's keep it until Balarka has finished ...

skullpatrol

ok

Balarka Sen

INDEED, this notion of index is crucial to the ultimate formulation of the Poincare-Hopf theorem

If $X$ is a vector field on $\Sigma$, the sum of index of the zeroes of $X$ is $\chi(M)$. No other assumption required.

The "index" somehow keeps track of the multiplicity of the zeroes

@BalarkaSen This dude, eg, has index 2

On the other hand in the standard cowlick picture, the total index = (index of a source) + (index of a sink) = 1 + 1 = 2

Indeed, euler characteristic of a sphere is 2

Check with wikipedia

$\chi$ of a surface of genus $g$ is $2 - 2g$, in fact. This gives a proof of the earlier fact: If it had a everywhere tangent nonzero vector field with no zeroes, the total index would be 0 (it has no zeroes!!!). That would force $\chi = 0$

But $2 - 2g = 0$ implies $g = 1$

Thus the only possible example is the torus

Any tangent vector field on any other surface has to have a zero

Ok, I'm done

Secret

Ah I see

Thanks. I think with that knowledge, I will be able to understand this formula in the chemistry context, since I have a maxima, a minima, and two kinds of saddle points in 3D space

John Rennie

Klein bottle has EC of zero ...

Balarka Sen

09:05

@Secret Ah, sounds simple enough

@JohnRennie Oof, yes, but the Klein bottle is also badly nonorientable so "genus" doesn't really makes sense per se ;)

It does admit a nonzero tangent vector field, though

Dawood ibn Kareem

@BalarkaSen If you ever wanted to do chemistry in four dimensions, you could store all your reagents in klein bottles.

Balarka Sen

@DawoodibnKareem Actually, that'd work. Surprisingly there's a way to put stuff inside a Klein bottle so that it doesn't come out and make a mess

(Unlike what you'd think nonorientability would give you)

Secret

@DawoodibnKareem Actually, that's a bad idea, since klein bottles as containers are basically like a looped container, The klein bottle is basically a moebius strip fatten into a cylinder

Thus I will suspect it is not very efficient usage of space

Balarka Sen

@Secret See above ^^^

The mathematical way to say it is to say that there is a 3-manifold $M$ with boundary $\partial M = K$ the Klein bottle

Quite surprising

Secret

though synthesizing 3D chiral molecules will become very easy with a klein bottle apparatus and tubing

Balarka: But the 3 manifold is basically tubular in shape with a twist, so it is still not very efficient in terms of amount of surface to volume of space being used?

Balarka Sen

09:09

Oh uh maybe

I suppose so. I haven't thought about what could be the volume of that 3-manifold embedded in R^4

Secret

Dawood ibn Kareem

It would be interesting to try to develop theories of organic chemistry in four dimensions.

Secret

The volume is something like the shaded, analogous to the rectangular region of the moebius strip

Balarka Sen

True

@DawoodibnKareem Interesting idea

Dawood ibn Kareem

Although I suspect just the four-dimensional analogue of a hydrogen atom would be complicated enough.

Balarka Sen

09:12

I suspect the immediate complexity is that there are way way way many regular 4 dimensional polytopes than regular 3 dimensional polyhedra

Secret

@DawoodibnKareem Actually, one of the groups in my uni are developing molecular dynamics that make use of the 4th dimension to hasten protein folding

Dawood ibn Kareem

@BalarkaSen Are there?

Secret

but for true 4D chemistry, it will probably be very alien to us since maxwell equations falls apart in 4 spatial dimensions

Dawood ibn Kareem

It's difficult to imagine more than two.

Balarka Sen

I'm far off on way many

There are just 6

Dawood ibn Kareem

09:14

What are the non-obvious four?

Balarka Sen

Still more than 5, which is in 3 dimensions

Secret

the 24 cell is the most interesting one since its closest analogue, the rhombic dodecahedron, is not regular

Balarka Sen

Right, the other three are the 4-orthoplex (analog of octahedron), the 120-cell (analog of the dodecahedron) and 600-cell (analog of the icosahedron)

Secret

@DawoodibnKareem Tetrahedron->Tetrachoron, cube->tesseract, octahedron->16-cell, 24 cell, dodecahedron->120 cell, icosahedron->600 cell

in fact, 4D has the 2nd most number of regular polytopes

Dawood ibn Kareem

OK, I'm reading about the hexadecachoron now.

Secret

09:16

(the no. 1 of course is the polygons in 2D, which there are countably many)

Dawood ibn Kareem

I can kind of visualise the hexadecachoron now, but I want to digest it a little before I move onto the last three.

Balarka Sen

I wish I knew much about geometry of polytopes

Secret

Dawood: It is kinda worth it in trying to visualise and interpret 4D shapes, it helps in the intuition from multivariable calculus to geometry and also 3D scalar functions in physics

But again, when the shape gets complicated, fall back to the algebra is the safest way cause we can be fooled by optical illusions of some 3D shapes, thus it will be worse when the 4D shape gets more complicated than prisms and the regular polychorons

Dawood ibn Kareem

09:39

It just requires patience and imagination.

skullpatrol

13:14

The last message was posted 4 hours ago.

same message in the math room

this much simultaneous quietness is rare

0celo7

13:30

@BalarkaSen dude

Why didn’t you respond to my Christmas comment

I’m offended

Secret

14:40

Reslts chang=/=chane resuls

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