The h Bar

Bastian Treichler

22:00

thanks

Slereah

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

0celo7

Not if you read Straumann, HE or others.

TanMath

@Slereah oh.. sounds like GR!

obe

@TanMath You mean SR?

Slereah

well both, technically

TanMath

22:00

@obe yeah.. SR! oops!

@Slereah true, but mainly SR...

ACuriousMind

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

0celo7

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

You're a physicistist.

Slereah

What about that GR with matrices book

ACuriousMind

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

0celo7

@Slereah Probably the pinnacle of linear algebra.

4x4x4x4 matrices are very difficult to deal with.

TanMath

22:03

@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?

Slereah

al-Khwarizmi would be proud

Good old Muḥammad ibn Mūsā al-Khwārizmī

He's ready to do some algebra

0celo7

@TanMath When did I call you stuff?

obe

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

TanMath

@Slereah where are you from?

Slereah

France.

That's just the wikipedia picture for him

TanMath

22:05

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

Slereah

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

0celo7

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

HDE 226868

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

0celo7

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

ACuriousMind

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

0celo7

22:07

@HDE226868 Never heard of him.

He do anything notable?

Slereah

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

TanMath

@0celo7 wiping out proof? huh?

Slereah

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

HDE 226868

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

0celo7

@TanMath Sure.

@HDE226868 ~~Math book?~~ History?

TanMath

22:07

@obe no he isn't!

Slereah

Although really Diophantus should be the father of algebra, really

TanMath

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

0celo7

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

@TanMath lol

TanMath

You deserve it!

Slereah

Science history is neat!

HDE 226868

22:08

The Norton History of the Mathematical Sciences

0celo7

Sure.

Slereah

Let's talk about science history!

HDE 226868

Let's all go to HSM!

0celo7

You keep telling yourself that.

VETO

HDE 226868

Talking here's fine, though.

Slereah

22:08

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"

ACuriousMind

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

0celo7

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

Slereah

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

0celo7

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

Slereah

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

0celo7

22:10

what did they do?

pictures?

Slereah

They wrote it in plain text

0celo7

proof by picture is the BEST proof

oh god, that's amazing

Slereah

Yeah they did also use a lot of diagrams

But also a lot of writing

0celo7

Let's prove the EFE in plain text

Slereah

I mean like

The + symbol is recent

TanMath

22:12

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

Slereah

Before that they just wrote "and"

0celo7

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.

Slereah

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

HDE 226868

@Slereah That stuff was horrible.

DanielSank

@TanMath Hello.

0celo7

22:14

pic?

Slereah

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

skull petrol

Descarte gave us the "x" :P

TanMath

@DanielSank have you heard of FRET as well?

Slereah

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

That is how you write ancient algebra

0celo7

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

Slereah

22:18

$\iota^\sigma$ is short for iso

"equal"

$\Delta$ is for duo, 2

To indicate a variable squared

0celo7

::grumbles something about chat being mean::

Slereah

Not sure what the upsilon is for

0celo7

wait

you were serious :o

Slereah

Yes

I totally read Diophantes' book with the original notation

It is quite awkard to use

0celo7

So they have a gamma matrix

what is M?

neutrino mass matrix?

Slereah

22:20

Ancient greek numbers were written with letters

0celo7

$\Delta$ is the feynman propagator

Slereah

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

etc

0celo7

I think the ancient Greeks knew QFT.

Slereah

$M$ 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

0celo7

yeah, right

Occam's razor

The Greeks knew QFT.

your theory is just too elaborate, sorry.

Slereah

22:22

Fractions were like

$a/b = a^b$

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

0celo7

that's ridiculous

Slereah

the inverse function is $^\chi$

0celo7

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

Slereah

Yeah it's a pretty awful notation

Quite inconsistent

0celo7

$\chi$ is a spinor index

Slereah

22:25

But then what would that symbol be in QFT

(It's a minus)

0celo7

hmm

upside down psi

it's the wavefunctional

skull petrol

They even rejected negative numbers :P

Slereah

They do mention them

ἀδύνατος

IMPOSSIBLE

0celo7

negative numbers don't even make sense

Slereah

Although

skull petrol

22:27

Show me a negative apple!

Slereah

When Arithmetica was written

0celo7

^^

Slereah

Negative numbers already existed

0celo7

show me a negative book

Slereah

It's been in China for a while

0celo7

22:27

or a negative horse

show me

skull petrol

^^

0celo7

also

Slereah

4-Dimensional Rotation

►

0celo7

what does $2^\pi$ even mean

is that even well defined

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

ACuriousMind

@0celo7 Joking or serious?

0celo7

22:29

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

Slereah

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

skull petrol

^

Trolls are know it alls

TanMath

hello @skullpetrol !

skull petrol

Hi pal @TanMath

TanMath

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

0celo7

22:31

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

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

and what about $2$ and $\pi$

skull petrol

@TanMath when I'm home.

TanMath

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

skull petrol

Yup

Slereah

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

0celo7

hmm

Slereah

22:34

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

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

0celo7

what the hell

Slereah

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

there you go

0celo7

you're a scary man, @Slereah

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, ◡ < )

0celo7

how is sin defined

Slereah

22:37

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

0celo7

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

Who writes this crap

skull petrol

Have you read any Bourbakian math?

0celo7

No

Slereah

Only 109 steps!

skull petrol

Nicolas Bourbaki

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...

John Duffield

22:39

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

3

0celo7

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.)

skull petrol

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

dmckee

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::

John Duffield

@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.

ACuriousMind

@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.

0celo7

22:49

Oh the pain

dmckee

@ACuriousMind Yeah. What you said.

0celo7

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

Slereah

Especially if they have continuous dimensions

time for some pasta

dmckee

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

0celo7

@dmckee Poor Dave

DanielSank

22:54

@TanMath Nope.

0celo7

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

skull petrol

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

dmckee

@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.

0celo7

@dmckee When did you need them

I think the concept is quite simple

dmckee

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

0celo7

22:59

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"?

dmckee

GTG.

0celo7

evading the question

skull petrol

Both our teams lost today :(

0celo7

UT won yesterday

that's all that matters

how are the Raiders this season, anyway

skull petrol

Have gone to a second game yet?

0celo7

23:03

playoff material?

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

skull petrol

Nope

John Duffield

@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.

0celo7

::giggles::

I see shear stress, yup.

skull petrol

::makes popcorn::

0celo7

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

skull petrol

23:06

That's what you get before the exam.

John Duffield

@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.

0celo7

What the heck does Einstein have to do with this?

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

skull petrol

ONLY! He has The Evidence to back him :P

0celo7

What does evidence have to do with this?

skull petrol

It begins with the same letter as Einstein. :P

0celo7

23:12

Uh, sure.

John Duffield

@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.

skull petrol

Uh, sure.

John Duffield

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

skull petrol

Later pal.

Slereah

Phew

Now we can talk physics

skull petrol

23:25

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

Slereah

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

ACuriousMind

@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.

Slereah

o

ACuriousMind

@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 $\emptyset$.

Slereah

23:40

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

ACuriousMind

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

JustDanyul

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.

Slereah

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

ACuriousMind

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.

Slereah

Isn't it?

ACuriousMind

23:42

What does that even mean?

Slereah

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

ACuriousMind

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

JustDanyul

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

Slereah

The domain and range are both the empty set

ACuriousMind

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

JustDanyul

23:44

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

Slereah

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

ACuriousMind

@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 $M$.

Slereah

Probably yeah

Vacuous truths are tricky things

ACuriousMind

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 $M\to\emptyset$.

Slereah

I guess not :p

especially not for a homeomorphism

ACuriousMind

23:46

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

Slereah

Wait

Can you glue the empty set to itself

0celo7

Why can't you map something into the emptyset

Slereah

What are you gonna map it to

0celo7

The empty set

ACuriousMind

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

Slereah

23:48

You can't

ACuriousMind

It depends how exactly you defined the gluing maps

Slereah

Functions map members of a set

And the empty set has no member

JustDanyul

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?

ACuriousMind

@JustDanyul What are F_s, u_s and F_n?

JustDanyul

friction, static friction coefficient and normal force

0celo7

23:53

@Slereah I'm not convinced.

Slereah

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

ACuriousMind

@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.

0celo7

@Slereah Is this some ZFC paradox?

Slereah

No

It's just basic set theory

0celo7

I'm convinced set theory is PhD level.

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