The h Bar

ACuriousMind

14:02

@skullpatrol no

skullpatrol

ok

have you thought about writing something for their blog?

Balarka Sen

14:35

@ACuriousMind I just remembered you are a bot

Will you participate in my attempt at making four SE bots crash at once by looping them up?

Constantine Black

Hi to all. Is there ant reference on the procedure of triangulation of surfaces? A text or something showing how one is to produce such a homeomorphism between the surface-manifold and the polyhedron? I mean a description of a basic logic of how the triangulation is constructed- or is such an effort vain? Thanks.

Balarka Sen

Are your surfaces smooth 2-manifolds, or are you not assuming any regularity?

Sir Cumference

Morning

Constantine Black

Yes, I'm discussing for a beginning for 2-manifolds.

Balarka Sen

That doesn't answer my question though. Smooth 2-manifolds or topological 2-manifolds?

'Cuz the construction of a triangulation is easier for the first type

Constantine Black

14:45

Sorry; the first type.

Balarka Sen

There are a few ways to do it

One is Morse theory, another is Riemannian geometry

Which one do you want me to do tell you?

GPhys

hi

Constantine Black

Well, let me be a little more specific: I believe I understand the trivial example of the circle, but I would like how one is to produce a triangulation for a disc, a 2-sphere, a torus, the Klein bottle, the Möbius band. Is there a preferable way from those you mention? Although, I think I could grasp better the Riemannian case 'cause I have not studied at all Morse theory.
Am I making any mistakes on my statements?

Balarka Sen

I understand what you are saying; you want a systematic way to construct triangulation of a given surface.

Constantine Black

Yes, thanks.

Balarka Sen

15:00

So, let's see if I can coherently parse the Riemannian argument

So $S$ be a smooth surface

Maybe assume it's compact for now

Impose a Riemannian metric $g$ on $S$.

0celo7

impose?

Balarka Sen

Give it one

0celo7

yes, but it's hardly an imposition

Balarka Sen

Like the overlords imposing capitalism on us marxists

It's oppression

Anyhow, let's say $\varepsilon$ is (less than the) injectivity radius of $(S, g)$: this means the exponential map $\exp_p : T_p S \to S$ is defined on a $\varepsilon$-ball around $p$ in $T_p S$ regardless of $p$.

0celo7

@ACuriousMind Is it possible to decouple my MO account from the SE network?

Balarka Sen

15:10

Consider the cover by $\varepsilon/2$-geodesic balls $B_{\varepsilon/2}(p)$ around $S$

This has a finite subcover by compactness of $S$

Let's call the centers of the balls in that finite subcover $p_1, p_2, \cdots, p_n$

I think for any $p_i$, if $p_j$ is the point that's closest to it, $d(p_i, p_j) < 2 \cdot \varepsilon/2 = \varepsilon$

Which means I can join $p_i$ and $p_j$ by a unique geodesic

Secret

Strange concepts:

Non-Archimedean time

A non-Archimedean time theory of time is any theory that holds that there exist instants infinitely in the future or infinitely in the past. It is so called because, if the instants of such time are assigned numbers, the set of such numbers must be non-Archimedean. Non-Archimedean future time would entail the existence of a future moment T, such that for any finite duration y there exists a moment Now + y but less than T. Note that if such a future moment T existed, there would exist an infinity of moments such that for all finite moments y' , T − y' would be after every moment Now + y where y...

Will make sense of it later

Avantgarde

Anyone here who has drawn (neat!) diagrams for LaTeX?

If yes, what do you use/recommend?

Constantine Black

15:36

@BalarkaSen Are you going to define a triangulation as a union of "triangles" that equals S?

@BalarkaSen Also, I just found a proof for the Riemannian case in: "Compact Riemann surfaces" by Jost. Was that the discussion you were going to do?

Balarka Sen

15:54

@Constantine Yes.

I didn't finish what I was writing but you can go along the lines of what I was talking about to construct a sequence of points on $M$ such that any epsilon-geodesic ball around one point contains two more points

Then you join those three points inside the ball by geodesic segments to construct a geodesic triangle

Do this for every triad inside an epsilon-geodesic ball

You have constructed a triangulation of S by geodesic triangles

If you write a question on MSE I can explain the various different proofs in more detail

Constantine Black

I'll try and study the text and think about what you are saying; but let me ask this: shouldn't the triangulation capture in some sense the fact that some manifolds identify some points? For example, the 2-sphere?
Let me put it this way: why should the disc be a 2-simplex united with the triangulation of a circle and a two sphere be the boundary of a tetrahedron? I'll make a naive suggestion: should I understand that for a 2-sphere I need two "triangles" to fully cover it, and so I end up with a tetrahedron?

I don't think what I said is correct, but, I'm not sure how the above discussion is to be implemented in any case; I will try and think about it more.

@BalarkaSen Do you think an understanding of the proof of Jost could provide a good grasp of the subject such that I will be able to reproduce the above problems? Or the proof does not provide a way for solving specific problems?

Balarka Sen

16:11

@ConstantineBlack I am not really sure what you are confused about. What do you think a triangulation is?

Constantine Black

A homeomorphism to a polyhedron(a collection of simplexes)

Balarka Sen

So there is indeed a homeomorphism from S^2 to the tetrahedron.

I am not sure what your question is.

Constantine Black

@BalarkaSen Nash is discussing it as a crucial step of an algorithm for finding the fundamental group.
The confusion is on how I should produce the tetrahedron, in this specific example.
I'm sorry; is what I'm saying meaningless in some way I don't get?

Balarka Sen

Well I gave you an algorithm to do it for Riemannian manifolds

In most cases you can work it out using a trick

Eg if you are given a torus or a Klein bottle or whatever

Start from the fundamental square, then barycentrically subdivide things iteratively

Constantine Black

16:40

OK, I'll think more about it and see where I get; thank you very much for your help.

ACuriousMind

@0celo7 What do you mean by "decouple"? MO is currently part of the SE network and therefore all MO users are, too

0celo7

17:31

@ACuriousMind I want my MO profile to be professional

And my current account has a history of being not that

Anonymous

@0celo7 Why not make another account with your real name? A MO account with a video game character-type name isn't professional anyway

Cows

the plan

buy gas

interview with bank(3rd interview)

buy new shoes

compute qft things at starbucks

catch up on some tv

0celo7

@Blue I because I like having 130 rep there

And I was going to change my name but I don’t want to do that here

Anonymous

Getting 130 rep shouldn't be too difficult for you. Just ask a few good questions from a new account

Semiclassical

17:55

electrostatics question:

Suppose you put an uncharged conducting sphere in an external electric field. A textbook problem is then to compute the induced charge distribution; the net charge will be zero, but it will carry a certain dipole moment. If one computes the induced polarization (dipole moment per unit volume of sphere) this comes out as just $\vec{P}_{in}=3\epsilon_0 \vec{E}_{ext}$.

Why 3??

By comparison, a long conducting cylinder placed in an external electric field (perpendicular to its axis) will develop an induced polarization $\vec{P}_{in}=2\epsilon_0 \vec{E}_{ext}$.

(I think it should come down to the fact that the volume of the former varies as R^3 whereas the latter varies as R^2.)

Maybe I should do this as an actual on-site question.

0celo7

18:13

@Blue that’s harder than you might think :P

The interesting questions I have are not ones I want other people to know about

Semiclassical

Not so long as you don't have answers to them, at any rate :P

What I find myself curious about in a more general way: Suppose I've got a finite volume of uncharged conductor. I place it in an external electric field; this induces a charge distribution on the conductor so as to zero out the field inside. Then there's an induced dipole moment and therefore an induced polarization.

Is this polarization going to be $3\epsilon_0 \vec{E}_{ext}$ as well?

nbro

18:35

Wikipedia defines spin as an intrinsic angular momentum. I am reading the first chapter of the book "Quantum computation and quantum information" by Nielsen. In section 1.5.1., the authors say: "by constructing the Stern–Gerlach device appropriately, we can cause the atom to be deflected by an amount that depends upon the $\hat{z}$ component of the atom's magnetic dipole moment, where $\hat{z}$ is some fixed external axis.".

My questions are the following.

Why is $\hat{z}$ an external axis? External to what? To the atom of hydrogen? Why is it called $\hat{z}$ and not just $z$? Why exactly the $z$ axis and not the $x$ or $y$ axis? Does it depend on how we construct the magnetic device? What's the relation between this axis and the spin as a property of the electrons?

Semiclassical

external to the system.

it's something that the experimenter chooses.

the reason it's z-hat is because it's a direction vector i.e. positive z direction not negative z direction

the reason it's the z-axis is because the experimenter chose it like this.

you could have chosen the x- or y-axis just as well

nbro

19:05

It doesn't make much sense that sentence (to me): "$\hat{z}$ component of the atom's magnetic dipole moment, where $\hat{z}$ is some fixed external axis.". They first say that is a component of the atom's magnetic d.m. and then they say it's some fixed external axis...

Semiclassical

You pick an axis, and then ask about the component along that axis.

nbro

Ok

Semiclassical

$\hat{z}$ is not a component of the dipole moment. $\mu_z=\vec{\mu}\cdot \hat{z}$ is a component of the dipole moment vector, namely the $\hat{z}$-component.

nbro

Why didn't they explain it as you're explaining it to me?

Semiclassical

Because they assume you understand vector algebra well enough to follow it.

nbro

19:09

I don't think that's my problem. I didn't even know that a vector is associated with the dipole moment.

Semiclassical

Then you need to review your physics. The dipole moment is a vector quantity by definition.

Anonymous

I have to admit that Nielsen and Chuang's textbook isn't very easy to follow for beginners

Semiclassical

(I mean, sometimes you'll see people refer to the "dipole moment" when they mean the magnitude of a dipole moment. That's not great in an intro text, but in context it usually makes sense.)

Similarly, the quadrupole moment of a charge distribution is a (rank-2) tensor quantity.

It's like talking about linear momentum. That's properly a vector quantity, not a scalar quantity.

Same for angular momentum.

nbro

Yes, I really need to study "a little bit" more about physics

Physics Meta

0

Q: How do I delete my account on se?

I know this question is irrelevant, but please tell me how to delete my account.

0

Q: How do I delete my account on se?

Please tell me how to delete my account on se? (leaving because using it too much.)

support user-accounts

CooperCape

19:21

ohai

Emilio Pisanty

@CooperCape why are you speaking to a bot?

CooperCape

Mmm not

just saying ohai

alerting those who dislike me of my presence

Emilio Pisanty

@CooperCape =P (context)

CooperCape

I see

I was about to say I haven't been around here long enough then I realised it was ~2 weeks ago

maybe less than 2 weeks

Tanuj

How is it going @Semiclassical

Anonymous

19:30

@BalarkaSen Is norm defined only for vector spaces over a field of real/complex numbers? Or is there a more general definition?

Anonymous

(over any arbitrary field i.e.)

2

loocsieulb

why is that starred

dmckee

111

Q: Allow users to hide connections between accounts

This feature request is inspired by the question on Programmers.SE on whether it is appropriate to ask an interview candidate for their Stack Overflow user. Stack Overflow is a site aimed at professional programmers, and it is used by some employers and some employees to examine or showcase the...

feature-request user-accounts linked-accounts

Anonymous

29

Q: Inner Product Spaces over Finite Fields

Inner product spaces are defined over a field $\mathbb{F}$ which is either $\mathbb{R}$ or $\mathbb{C}$. I want to know what happens if we try to define them over some finite field. Here's an example: Let $\mathbb{F} = \{0,1,a,b\}$ be a finite field with + and * defined by the following Cayley ...

abstract-algebra vector-spaces finite-fields inner-product-space

Anonymous

This looks interesting

dmckee

19:41

And links therein, of course.

Not sure how to solve your issue, but you're not he only one thinking about it.

Ah. This might be the most helpful link:

2

Q: How do I hide a community from my network profile?

I want to hide an account from my network profile. I would like to reference my Stack Exchange profile in a professional setting. I do not want one of my accounts publically displayed for employers to see. I've heard that it's possible to add accounts to a list of hidden communities, but I hav...

support profile-page network-profile hidden-communities

John Duffield

20:09

Hi guys. How's tricks?

7

0celo7

@JohnDuffield how are you?

John Duffield

Good thanks. Busy though. And you?

@Slereah : this is for you.

0celo7

Doing ok, very busy

John Duffield

Good stuff. I've now written 42 of those essays. It's been informative. I've learned a lot in the past year.

0celo7

Not being in this place is probably good for productivity.

John Duffield

20:15

It is. To be honest I was finding that I was spending an awful lot of time on the internet talking to people.

0celo7

Who is starring?

John Duffield

Check this out: Pascual Jordan’s resolution of the conundrum of the wave-particle duality of light by Anthony Duncan and Michel Janssen dating from 2007.

2

On page 47 they quote Jordan saying this: "Einstein drew the conclusion that the wave theory would necessarily have to be replaced or at least supplemented by the corpuscular picture. With our findings, however, the problem has taken a completely different turn. We see that it is not necessary after all to abandon or restrict the wave theory in favour of other models; instead it just comes down to reformulating the wave theory in quantum mechanics.

The fluctuation effects, which prove the presence of corpuscular light quanta in the radiation field, then arise automatically as consequences of the wave theory. The old and famous problem [of] how one can understand waves and particles in radiation in a unified manner can thus in principle be considered as solved".

They're talking about Pascual Jordan's final section of the Dreimännerarbeit in 1925. Interesting stuff.

0celo7

20:37

@dmckee Hmm

@dmckee should I hide my MO profile?

John Duffield

OK I gotta go. Bye.

bolbteppa

21:03

"They find some support from ancient publications by famous physicists; in the first decades of the 20th century, indeed, Karl Schwarzschild, Hermann Weyl, and even Albert Einstein, had misconceptions about the theory, which at that time was brand new, and these pioneers indeed had not yet grasped the full implications. They can be excused for that, but today’s professional scientists know better."

staff.science.uu.nl/~hooft101/gravitating_misconceptions.html

3

nbro

21:20

I have just found a decent explanation of the Gern-Sterlach experiment in the book Modern Quantum Mechanics (section 1.1) by Sakurai: fisica.net/quantica/…

Sir Cumference

21:47

Oh, John Rennie's back

Anonymous

@CooperCape Completed reading Axler?

CooperCape

22:09

@Blue aha sadly not. Got distracted by exams and coursework - only just started reading it again... just got onto linear maps etc. (Yeah, not very far through)

But it’s getting more interesting now, and I feel I understand it a lot better than last time

bolbteppa

@Blue one can set up normed spaces over valued division rings

Valuation ring

In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x −1 belongs to D. Given a field F, if D is a subring of F such that either x or x −1 belongs to D for every nonzero x in F, then D is said to be a valuation ring for the field F or a place of F. Since F in this case is indeed the field of fractions of D, a valuation ring for a field is a valuation ring. Another way to characterize the valuation rings of a field F is that valuation rings D of F have F as their field of fractions, and their ideals are totally...

dmckee

@0celo7 Not a clue. I just saw that you were wondering about multiple accounts and such like.

skullpatrol

Aren't "tricks" for kids...

...you silly rabbit.

dmckee

I know a long rambling joke with the punchline "Silly, Rabbi. Kicks are for Trids."

skullpatrol

Yeah, that's the one :-)

thnx

dmckee

22:20

I was telling it to a group of fellow grad students once, and half-way through I realized that nearly half the audience hadn't grown up in the US.

What the heck do you do then?

Anonymous

@CooperCape Tell me when you get to chapter 3. There are some very interesting facts about dual spaces, quotient spaces, operators etc in that. Took me quite some time to understand their significance (geometric too)

dmckee

The whole thing turns on a childhood memory that they didn't have.

skullpatrol

Yeah, tough crowd.

kinda like here, sometimes...

Anonymous

@bolbteppa How do you define the norm there?

bolbteppa

In basically the same way, the point is that a division ring allows you to abstract properties of the norm beyond fields like R, as one does in p-adic analysis it seems

The whole thing about using a division ring is ensuring you can measure $0$ to be zero and not worry about $2 \cdot 3 = 0 \mod(6)$ etc

You mentioned finite fields above, 'All finite division rings are commutative and therefore finite fields'

Anonymous

22:31

@bolbteppa Umm, that makes some sense

Anonymous

Field norm

In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield. == Formal definition == Let K be a field and L a finite extension (and hence an algebraic extension) of K. The field L is then a finite dimensional vector space over K. Multiplication by α, an element of L, m α : L → L given by m α ( x )...

Anonymous

Found something

bolbteppa

This is what it generalizes to, that's like a special case I guess

Absolute value

In mathematics, the absolute value or modulus |x| of a real number x is the non-negative value of x without regard to its sign. Namely, |x| = x for a positive x, |x| = −x for a negative x (in which case −x is positive), and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero. Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings,...

CooperCape

@Blue I will most likely start it tomorrow :) School is cancelled due to snow or some other weather excuse like that...

Anonymous

@CooperCape Oh wow, we could meet up here for one hour and I could brief the chapter to you, then. Or atleast a part of it. (Will help me in revising, too)

CooperCape

22:42

@Blue Well if that’s fine with you I’m sure it would help my understanding a lot, thanks :)

skullpatrol

How much snow?

CooperCape

@skullpatrol Maybe a few centimetres... it’s rrally annoying

Was supposed to have a recital for my A level today but it was snowed off...

skullpatrol

hmm, more time to practice :)

Anonymous

@CooperCape Yes, sure. Around 4:30-5:30 pm IST (11:00-12:00 in UK) would be best for me :)

Anonymous

It's a big chapter. Maybe to cover whole of it will take 3 days. But let's start at least

Anonymous

22:47

I was revising section 3F a while ago

CooperCape

@Blue nice one, I should be up by then ;) I guess just ping me here when you’re ready...

@skullpatrol I don’t want more time :p

I’m getting worse by the day I swear

Anonymous

@CooperCape Okaies

CooperCape

@Blue I don’t think I’ve seen an ‘okaies’ in a long while...

Anonymous

My school physics teacher would use it a lot :P

CooperCape

Oh wow

Fair enough... it feels like a sort of sarcastic ok

Dunno

Sir Cumference

23:04

No, the TA said we will be quizzed on chapter 4, even the stuff that wasn't coverred in class. And it'll be on exams

skullpatrol

Ask him which parts.

Sir Cumference

Sigh, I thought I messaged that to someone

I think I'm out to lunch

skullpatrol

np pal

(my question still stands :)

Sir Cumference

@CooperCape Ugh. I have 2 midterms tomorrow and the winds are going to be dangerously high

As the warning reads, "Damaging winds will blow down trees and power lines. Widespread power outages are expected."

Sounds like fun

CooperCape

@SirCumference Brings a more realistic sense to those who “would rather die” etc..

Enjoy your travel though

skullpatrol

23:13

come on guys, if you can go out in the midday sun, you can handle this :P

Sir Cumference

@skullpatrol Can't really relate, since it's rainy 99% of the time here

Hell it's raining as we speak

CooperCape

One up you it’s still snowing here I think

Sir Cumference

@CooperCape At least snow isn't depressing :P

Avantgarde

hey

skullpatrol

at least you don't have to shovel rain @SirCumference

Avantgarde

23:15

@SirCumference I'd say it is. Well, not depressing, but painful.

CooperCape

@SirCumference Aha yeah

Sir Cumference

@skullpatrol I'm from NY. I know snow :P

Though my uni isn't in NY

CooperCape

According to my parents I’ve never liked snow so I’m proud of my younger self

Sir Cumference

@CooperCape I think my astro lab has been postponed for like a month because there haven't been any good nights to use a telescope

CooperCape

@SirCumference ooooooh that’s really annoying

Then again I’ve applied for a degree that minimises labs so...

Sir Cumference

23:19

You're not physics?

CooperCape

Well technically a degree that minimises labs would be English or something

skullpatrol

lol

CooperCape

Nah I applied theoretical... still got labs but like cut down, a lot.

I think

If I get in

Sir Cumference

Oh wait, are you in high school?

skullpatrol

A level

CooperCape

23:21

Yup

Sir Cumference

I was confused for a second there. "Applied theoretical"

Sounded like an oxymoron to me :P

CooperCape

Ohhh right yeah

Hah apologies

Applied for ...

Sir Cumference

Tbh labs aren't that bad. They're usually 1 credit so they're not the biggest deal

CooperCape

I do break everything I touch hoever

I managed to make capillary action not work for an hour

Might just be high school experiments thogh

Sir Cumference

So wait, then what's your (prospective) major?

CooperCape

23:23

(Theoretical) physics

Sir Cumference

In my uni there's just "applied physics" and "physics". I'm in the latter but we still need to do at least 3 labs

CooperCape

Oh right... I’m in the UK so I’m not sure how different it is... (probably not that much)

Avantgarde

23:41

What are A-levels? I've been forever confused.

CooperCape

23:54

@Avantgarde I’m not sure if I can explain this well but they are the exams we have to take in the last year of high school that for one dictate our entrance to university and obviously if you don’t go to uni employers etc.

I don’t know the alternative in other countries hiwever

Avantgarde

Oh ok I understand now.

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