The h Bar

$\textrm{diag}(-1,1,1,1)$

Not if you read Straumann, HE or others.

@Slereah oh.. sounds like GR!

obe

@TanMath You mean SR?

well both, technically

10:00 PM

@obe yeah.. SR! oops!

@Slereah true, but mainly SR...

@0celo7 Yes. They mostly don't do that stupid thing of defining them by their components and transformation laws, for one. :P

@ACuriousMind Neither does HE, Wald, Straumann, Carroll, others.

You're a physicistist.

What about that GR with matrices book

But really, I'm not very stongly convinced either way whether such questions belong here.

@Slereah Probably the pinnacle of linear algebra.

4x4x4x4 matrices are very difficult to deal with.

10:03 PM

@0celo7 OK.. I turned off ignore because I think you have stopped calling me stuff and other things... I don't like you because of your attitude and your language, that is why..ok?

al-Khwarizmi would be proud

Good old Muḥammad ibn Mūsā al-Khwārizmī

He's ready to do some algebra

@TanMath When did I call you stuff?

obe

@TanMath Deep down 0celo7 is a really nice and friendly person.

@Slereah where are you from?

France.

That's just the wikipedia picture for him

10:05 PM

@0celo7 chat.stackexchange.com/transcript/message/25258173#25258173

For some reason the soviet union had a stamp of him and that is what wikipedia went with

@obe I am very nice and friendly on the outside, too.

@Slereah al-Khwārizmī would be proud if modern math students knew who he was.

@TanMath Dead link? Doesn't lead to anything of mine.

@HDE226868 My math teacher in school used to drop his name quite often (probably because he was proud he'd memorized it :D )

10:07 PM

@HDE226868 Never heard of him.

He do anything notable?

@HDE226868 al-Khwārizmī is mostly known for being "that guy the word algorithm comes from"

@0celo7 wiping out proof? huh?

@0celo7 He wrote a big book on algebra during the middle ages

@0celo7 Go read a history of math book. I list a good one in my profile, I think. Let me find it.

@TanMath Sure.

@HDE226868 ~~Math book?~~ History?

10:07 PM

@obe no he isn't!

Although really Diophantus should be the father of algebra, really

@0celo7 putting you back on ignore!!!!!!!

I took the AP histories in high school. I've had enough of that.

@TanMath lol

You deserve it!

Science history is neat!

10:08 PM

The Norton History of the Mathematical Sciences

Sure.

Let's talk about science history!

Let's all go to HSM!

You keep telling yourself that.

VETO

Talking here's fine, though.

10:08 PM

Have you ever read a mathematical demonstration in ancient egyptian

It is excruciating

It's like "Take the whole, then the half, then the square, it gives 10"

@Slereah More than wading through the stuff Newton wrote? :P

@TanMath If you're reading this, I'm genuinely confused by what your problem is.

For $x + \frac{x}{2} + x^2 = 10$

@ACuriousMind ~~Or your SE posts~~ or HE

Newton's stuff is a breeze compared to ancient math

See the thing is

Mathematical symbols are a recent invention

Like fairly recent

Very few symbols before the middle ages

10:10 PM

what did they do?

pictures?

They wrote it in plain text

proof by picture is the BEST proof

oh god, that's amazing

Yeah they did also use a lot of diagrams

But also a lot of writing

Let's prove the EFE in plain text

I mean like

The + symbol is recent

10:12 PM

So I finally never got my tensor question answered, and I don't know what you guys are talking about, so I am leaving....

Before that they just wrote "and"

The variation of the integral of the scalar curvature and the root of the negative metric determinant with respect to the metric gives the Ricci tensor minus one half of the scalar curvature times the metric tensor.

Proof:

I really don't want to do this...

Proof: Left to the reader.

Diophantus is actually the first guy to have used extensive mathematical notation

But it didn't really catch on

Also it's a bit of an awkward notation

@Slereah That stuff was horrible.

DanielSank

@TanMath Hello.

10:14 PM

pic?

$x^2 + x + 3 = 2$ would be like

Descarte gave us the "x" :P

@DanielSank have you heard of FRET as well?

$\Delta^\Upsilon \varsigma \bar \gamma M \iota^\sigma \beta M$

That is how you write ancient algebra

are you kidding

I can't believe that

the ancient greeks had indices

looks like physicists are using 2500 year old notation

I FIXED IT

10:18 PM

$\iota^\sigma$ is short for iso

"equal"

$\Delta$ is for duo, 2

To indicate a variable squared

::grumbles something about chat being mean::

Not sure what the upsilon is for

wait

you were serious :o

Yes

I totally read Diophantes' book with the original notation

It is quite awkard to use

So they have a gamma matrix

what is M?

neutrino mass matrix?

10:20 PM

Ancient greek numbers were written with letters

$\Delta$ is the feynman propagator

$\alpha = 1, \beta = 2, \gamma = 3$

etc

I think the ancient Greeks knew QFT.

$M$ is a monad

Which means that it's just a number

$x$ was written as $\varsigma$, for $\alpha\rho\iota\theta\mu`{o}\varsigma$

Aritmos, the number

yeah, right

Occam's razor

The Greeks knew QFT.

your theory is just too elaborate, sorry.

10:22 PM

Fractions were like

$a/b = a^b$

Or for $1/a$, $a^{o\nu}$

that's ridiculous

the inverse function is $^\chi$

$a^{\omicron\nu}$ is a tensor

Yeah it's a pretty awful notation

Quite inconsistent

$\chi$ is a spinor index

10:25 PM

But then what would that symbol be in QFT

(It's a minus)

hmm

upside down psi

it's the wavefunctional

They even rejected negative numbers :P

They do mention them

ἀδύνατος

IMPOSSIBLE

negative numbers don't even make sense

Although

10:27 PM

Show me a negative apple!

When Arithmetica was written

^^

Negative numbers already existed

show me a negative book

It's been in China for a while

10:27 PM

or a negative horse

show me

^^

also

4-Dimensional Rotation

►

what does $2^\pi$ even mean

is that even well defined

you can't multiply $2$ $\pi$ times

@0celo7 Joking or serious?

10:29 PM

@ACuriousMind the fact that you have to ask that is sad

that's why you don't troll by pretending to be stupid

^

Trolls are know it alls

hello @skullpetrol !

Hi pal @TanMath

@skullpetrol on your phone? when do you ever use SE chat on your computer?

10:31 PM

@ACuriousMind $2^\pi=\exp\pi\log 2$

how is $\exp$ and $\log$ defined

and what about $2$ and $\pi$

@TanMath when I'm home.

@skullpetrol oh... i never catch you at night

Yup

@0celo7 grid.ece.ntua.gr/sites/metamathsite/mpeuni/df-ef.html

hmm

10:34 PM

grid.ece.ntua.gr/sites/metamathsite/mpeuni/df-log.html

grid.ece.ntua.gr/sites/metamathsite/mpeuni/df-ppi.html

what the hell

grid.ece.ntua.gr/sites/metamathsite/mpeuni/df-2.html

there you go

you're a scary man, @Slereah

Oh wait

Wrong pi

that's the pi function

grid.ece.ntua.gr/sites/metamathsite/mpeuni/df-pi.html

that's the pi

⊢ π = sup((ℝ+ ∩ (◡sin “ {0})), ℝ, ◡ < )

how is sin defined

10:37 PM

⊢ sin = (x ∈ ℂ ↦ (((exp ‘(i · x)) − (exp ‘(-i · x))) / (2 · i)))

grid.ece.ntua.gr/sites/metamathsite/mpeuni/pilem3.html

Who writes this crap

Have you read any Bourbakian math?

No

Only 109 steps!

Nicolas Bourbaki is the collective pseudonym under which a group of (mainly French) 20th-century mathematicians, with the aim of reformulating mathematics on an extremely abstract and formal but self-contained basis, wrote a series of books beginning in 1935. With the goal of grounding all of mathematics on set theory, the group strove for rigour and generality. Their work led to the discovery of several concepts and terminologies still used, and influenced modern branches of mathematics. While there is no Nicolas Bourbaki, the Bourbaki group, officially known as the Association des collaborateurs...

10:39 PM

@TanMath : a tensor is in essence a matrix. See stuff like this and this and this and this.

3

OH DEAR GOD

I...I...I...

@JohnDuffield I thought a tensor was a multilinear map on a combination of vectors spaces and dual vector spaces. (Into the reals or complexes.)

I think that's a record: 2 inline links to Wikipedia in one sentence :-)

A matrix has the same relationship to a tensor as a tuple has to a vector. In both cases the latter can be represented by the former, but the latter has structure that is not uniquely captured by the former.

That kind of advice is perhaps defensible as a first introduction if you know that the students have been show the appropriate mathematical opperation on tuples and matricies, but it will teach them lessons that have to be unlearned later.

::voice of bitter experience::

@0celo7 : no. It's just a generalization of a matrix. Woldemar Voigt introduced the word in its current meaning in 1898. He was working on stress and strain in crystals.

@dmckee Strictly speaking, only true for tensors of rank 2. For higher-rank tensors you'd need multidimensional arrays which you can't really visualize for rank higher than three.

10:49 PM

Oh the pain

@ACuriousMind Yeah. What you said.

Also, one can have tensors on Hilbert spaces. "Infinite dimensional matrices" is a bit silly.

Especially if they have continuous dimensions

time for some pasta

@0celo7 This is one of the lessons that must be unlearned.

@dmckee Poor Dave

DanielSank

10:54 PM

@TanMath Nope.

@JohnDuffield What's wrong with the usual definition as a multilinear map?

@dmckee who's fault is it for having to unlearn it?

@skullpetrol Don't know that there is blame to assign. I was ready for matricies, and not ready for abstract tensors when I first needed 2nd rank tensors.

@dmckee When did you need them

I think the concept is quite simple

Later I had to decouple the properties from the representation, and I'm dull enough that this was a struggle.

10:59 PM

You have a PhD, that's smarter than like most people

@dmckee Question time!

Have you ever met a student and thought "this kid is going places"?

GTG.

evading the question

Both our teams lost today :(

UT won yesterday

that's all that matters

how are the Raiders this season, anyway

Have gone to a second game yet?

11:03 PM

playoff material?

@skullpetrol Nope, I'm not smart enough to think about reserving a ticket in time

Nope

@0celo7 : it defines it in terms of things that aren't defined. IMHO it's important to note the continuum mechanics origins.

Note the shear stress.

::giggles::

I see shear stress, yup.

::makes popcorn::

Honestly, what the heck does shear stress have to do with math?

11:06 PM

That's what you get before the exam.

@0celo7 : Einstein treated space as an elastic continuum. Note that people have tried to model space as a crystal. For example see epola. I'm not keen on this because it models space as an electron-positron lattice.

What the heck does Einstein have to do with this?

What does space, crystals, epola(?), electrons or positions have to do with this?

ONLY! He has The Evidence to back him :P

What does evidence have to do with this?

It begins with the same letter as Einstein. :P

11:12 PM

Uh, sure.

@0celo7: for what Einstein's got to do with this, see Wikipedia: "In the 20th century, the subject came to be known as tensor analysis, and achieved broader acceptance with the introduction of Einstein's theory of general relativity, around 1915." He treated space as an elastic continuum. Voigt introduced the word tensor for crystal stress & strain. Look up Voigt and relativity.

Uh, sure.

OK I'm off to bed. Good night all.

Later pal.

Phew

Now we can talk physics

11:25 PM

You can talk all you want, just not to him ;)

I am wondering what the Ozsváth–Schücking metric would correspond to in a graviton sort of description

Is it some kind of bound state

Wait

Are connected sums defined for the empty set

The empty set is a manifold

Do you have $S^2 \# \varnothing = \mathbb{R}^2$

@Slereah No, the connected sum requires non-empty manifolds since its construction involves picking a point on each manifold where they are to be connected.

o

@Slereah formally, taking the connected sum of $M$ and $N$ requires two gluing maps $D^n\to M$ and $D^n \to N$, but there are no maps into $\emptyset$.

11:40 PM

Well, the empty set is a map from open sets of a manifold to the empty set :p

What? No, there are no maps into the empty set.

I have an embarrassingly rudimentary physic question.. I'm doing some mechanics (math course) and whilst I got the correct answer to a question in my book, their approach in the solution is quite different than mine. And it confused me about a certain aspect of static friction.

Well yes, but the empty set is the function with no domain to no range :p

It's an initial object, it has exactly one map into everything else - the empty map - and no other map.

@Slereah That's not true.

Isn't it?

11:42 PM

What does that even mean?

Well functions are usually defined as the set of ordered pairs from the domain to the range

Hence the empty set can be defined as the empty map under that definition

@Slereah Well, but how are you defining "no domain" or "no range" in the first place?

basically, i had the understanding that F_s <= u_s|F_n| is the maximum force that can be applied due to friction, and this would be applied in the opposite direction to the force being applied

The domain and range are both the empty set

You need the notion of the empty set already for that, you can't define it such.

11:44 PM

so if the force being applied is less than this maximum, it stays put

I didn't say the empty set wasn't already defined!

@Slereah Okay. And now you need to carefully step through the definiiton of a map and convince yourself that there's exactly one map $\emptyset \to M$ and none $M \to \emptyset$ for every $M$.

Probably yeah

Vacuous truths are tricky things

It's because the map is required to assign an element of the target for every elements of the source. This is trivially so for $\emptyset\to M$, but can never be fulfilled for $M\to\emptyset$.

I guess not :p

especially not for a homeomorphism

11:46 PM

So you can't glue the empty set to anything because you can never have a gluing map

Wait

Can you glue the empty set to itself

Why can't you map something into the emptyset

What are you gonna map it to

The empty set

@Slereah Well...perhaps, but that gives just the empty set :P

11:48 PM

You can't

It depends how exactly you defined the gluing maps

Functions map members of a set

And the empty set has no member

now, in their solution, they state that for the equilibrium condition to hold, f_s <= u_s|F_n|, and use that to derive the solution. I don't entirely get this? wouldn't this just mean that if the above holds, AND the force acting in the opposite direction is larger than F_s, THEN the equilibrium won't hold?

@JustDanyul What are F_s, u_s and F_n?

friction, static friction coefficient and normal force

11:53 PM

@Slereah I'm not convinced.

A function $M \rightarrow N$ is a relation

A relation is a subset of $M\times N$

If $N$ is the empty set, $M \times N$ is also the empty set

@JustDanyul Okay. You are correct that F_s <= u_s F_n just says the force of friction is at most u_s F_n.. It's not an equilibrium condition, the equilibrium condition should be that the net force except friction on the object is smaller than u_s F_n.

@Slereah Is this some ZFC paradox?

No

It's just basic set theory