The h Bar

Basically I give you a transformation that an n-dimensional matrix into a real number. This transformation is such that $f(AB) = F(BA)$, with B, A being n-dimensional matrices. Also $f(I)=n$ where $n$ is the order of the Identity. Prove that this transform is the Trace operator, and then prove the mentioned properties.

Kenshin

I've never seen a programming language do that

heather

what i'm really looking for is a logic error, because it isn't changing the first letter in the message i want to encode, which is nuts, because under no circumstances should it not be changed

Bernard Meurer

@heather Gimme teh codez

heather

gist.github.com/physicsnerd/c1cd7e057e88ce1ea0fdb3e0b589c44c @BernardMeurer

Slereah

3:14 PM

@ACuriousMind What is the difference between a diffeomorphism and a weyl transform in 2D

Kenshin

does it work for other letters @heather?

Bernard Meurer

@heather Let me change your life

ACuriousMind

@Kenshin It's called the off-side rule, Python and Haskell are probably the well-known examples of it.

Slereah

According to Some Paper, in 2D a weyl transformation is equivalent to the coordinate transform $z \to f(z)$ with $f$ some holomorphic function

But then isn't that the same thing as a diffeomorphism?

heather

@BernardMeurer, how so?

Slereah

3:16 PM

but it feels a bit odd because I don't think all conformally related metric as minkowski space are flat

ACuriousMind

@Slereah A diffeomorphism is a map $\phi : M \to M$ that changes coordinates, the metric and everything else by the standard rules, a Weyl transform is a direct transformation of the metric $g\mapsto \Omega(x)g$.

Kenshin

@ACuriousMind ty very interesting

Slereah

as they would be if it was just a coordinate change

I know, but the fellow says that for Minkowski space

Since the metric is $ds^2 = dz d\bar z$

Any weyl transform can be written as what I said

And then the Weyl transform is just $f'^2 dz d\bar z$ or something

which sounds reasonable

But does that mean that all metrics conformal to flat space are flat?

I would expect so if it's just a coordinate transform

ACuriousMind

@Slereah Indeed it isn't, not all conformally flat manifolds are flat.

Bernard Meurer

[chr(x) for x in range(97,123)]

This generates the alphabet

Slereah

3:19 PM

but it seems weird since from what I've gathered, all Lorentz surfaces are conformal to flat space

heather

wow

Bernard Meurer

@heather Go read on list comprehensions

it's some cool magic

Slereah

So is it more general than a coordinate transform?

ACuriousMind

@Slereah Yes, that is also true, and part of the reason gravity is trivial in 2D

Bernard Meurer

Now I gotta go back to chemistry

heather

3:20 PM

@BernardMeurer, I've used them before but I didn't realize they could be used for the alphabet!

@BernardMeurer, okay, thank you.

Slereah

Well, not trivial :p

Bernard Meurer

@ACuriousMind Go prove what I mentioned above

It's a fun exercise

Slereah

but quite simple

But anyway, what is the distinction here between a coordinate change and a weyl transform?

heather

@Kenshin, what do you mean "does it work for other letters"

ACuriousMind

@Slereah Have I not answered that?

The Weyl transform does nothing to the coordinates, it just directly changes the metric

I'm not sure what exactly you want to know

Slereah

3:24 PM

Well yes, but if mathematically it is identical, wouldn't that give them both the same curvature

ACuriousMind

@Slereah How could something that changes the coordinates and everything else be "identical" to something that just changes the metric?

Kenshin

@heather another quesiton about the code

your test is "if location > 26"

heather

@Kenshin, what?

right

Kenshin

and you use alphabet[location]

heather

indeed

Kenshin

3:28 PM

but doesnt the index of alphabet range from 0 -25

so if location = 26 it will error?

heather

I add one to location somewhere, i think...one moment let me look

yeah, i did

look at the definition of location

line 16

Kenshin

yes I saw it already

my point is you are passing that into alphabet[location]

and this parameter must be between 0 and 25, not 1 and 26

heather

oh, duh

(::face palm::)

Kenshin

so I think given this, you can just use location = location%26

and no need for the + 1

heather

or, one moment

hmm.

okay, updated

Bernard Meurer

3:34 PM

@ACuriousMind Why are you not loving me?

ACuriousMind

@BernardMeurer What?

Bernard Meurer

@ACuriousMind I pinged you like thrice already, and you didn't give me love

Kenshin

@heather the program is working now correct?

Bernard Meurer

24 mins ago, by Bernard Meurer

Basically I give you a transformation that an n-dimensional matrix into a real number. This transformation is such that $f(AB) = F(BA)$, with B, A being n-dimensional matrices. Also $f(I)=n$ where $n$ is the order of the Identity. Prove that this transform is the Trace operator, and then prove the mentioned properties.

@ACuriousMind How would you go about showing that?

heather

@Kenshin, yes, I think I figured out all the problems

Kenshin

3:39 PM

good job

gtg

laterz all

heather

have a good day/night

ACuriousMind

@BernardMeurer Is it given that it's linear?

Bernard Meurer

@ACuriousMind Ya

When I say "n-dimensional matrix" I mean $A, B \in \mathbb M_{n\times n}$

ACuriousMind

Well, you decompose a matrix as $A = A_{ij} E_{ij}$ for the basis matrices $E_{ij}$ with entry 1 at ij. We have that $f(E_{11}) = f(E_{22}) =... = 1$ because a base change turns all of these into each other, and $f(B^{-1} A B) = f(A)$ follows from $f(AB) = f(BA)$.

And we get $f(E_{ij}) = 0$ because $f(E_{ij} E_{ji}) = f(E_{ij})$ but $f(E_{ji} E_{ij}) = f(0) = 0$.

So $f(A) = \sum_{i,j} A_{ij} f(E_{ij}) = \sum_{i,j} A_{ij} \delta_{ij} = \sum_i A_{ii} = \mathrm{tr}(A)$. Enough love for you, @BernardMeurer?

Bernard Meurer

4:00 PM

AHA

That's what I did too!

Well, minus the basis thing

I defined matrix multiplication as a sum

ACuriousMind

@BernardMeurer Not quite sure what you mean by that

Bernard Meurer

Imagine $A=[a_{ij}];B=[b_{ij}]$, the first diagonal entry of $AB$ is $\sum_{z=1}^{n}{a_{z,1}\times b_{1,z}}$

ACuriousMind

Yes, that's the definition of matrix multiplication, how does it help me here?

Bernard Meurer

Because you define the trace one the basis of this (with another sum)

and it becomes clear how $Tr{AB} = Tr{BA}$

ACuriousMind

Yeah, how does it help you to show that $f$ is the trace?

I.e. which step of mine do you want to replace here?

Bernard Meurer

4:07 PM

Eh, well, I just defined this function and showed that it: a) Summed across the diagonal b) Obeyed the desired properties, and thus was the Trace

I'm not replacing, I'm just saying how I tried to solve it :P

ACuriousMind

I'm not following you

Bernard Meurer

On Twitter?

ACuriousMind

Showing that some specific $f$ you defined that obeys the properties in the question is the trace doesn't show that all such functions are the trace

Bernard Meurer

Huh? I didn't define $f$, the question already said $f$ was the trace, you just had to prove it

i.e. prove that the trace of AB = trace BA, that the trace works for any given square matrix, that Tr(I)=n

ACuriousMind

No, that's not what the question is asking.

Bernard Meurer

4:11 PM

The one I translated no, the original one would be more precisely written like this

ACuriousMind

As you wrote it, it's asking to show that, if a function $f : M_{n\times n} \to\mathbb{C}$ fulfills $f(AB) = f(BA), f(A+B) = f(A)+f(B), f(\mathbf{1}_n) = n$, then $f = \mathrm{tr}$.

Bernard Meurer

It was me being a bad translator

ACuriousMind

Which is a nice exercise; what you said sounds as if you just showed $\mathrm{tr}(AB) = \mathrm{tr}(BA), \mathrm{tr}(\mathbf{1}_n)$, which is pretty boring.

Bernard Meurer

Wait

I gotta study Chem

We will continue this

Bernard Meurer

4:29 PM

@ACuriousMind Can you explain how is it that showing that the trace matches all the requirements specified does not prove that the function is the trace?

ACuriousMind

@BernardMeurer Say I dropped the requirement of linearity and value at the identity, i.e. only claimed that any function $f(AB) = f(BA)$ was the trace. You could still show that the trace fulfills this, but it's not the only function that does so (e.g. the determinant is another)

Bernard Meurer

@ACuriousMind Sure, but we don't drop any requirements

what I did was come up with a formal definition of the trace and then show that it had all the properties asked

ACuriousMind

@BernardMeurer Well, but how do you know there isn't another function that fulfills the same requirement?

Just because you can't think of one that doesn't prove the trace is the only possiblity

Bernard Meurer

Because those requirements define the trace!

ACuriousMind

How do you know?

Bernard Meurer

4:33 PM

heck that's how we defined the darn thing

ACuriousMind

That's the entire point of the exercise - prove that these requirements uniquely define a function, and that that function is the trace.

Bernard Meurer

Wtf

Screw this shit

How much is Burger King's wage?

heather

-2

A: Joule-Thomson effect of Van der Waals gas

Solve Van der Waal's equation and approximate the $PV$ term by $RT$, and you will have your required equation. Although we are introducing an error, it will hardly affect the final result since there will be both $a$ and $b$. For Van der Waal gas: $$\left(p+\frac{n^2q}{v^2}\right)\left(v-nb\ri...

@ACuriousMind, did I do the above edit alright?

Bernard Meurer

Come on man

I can never get any of this crap right

Sigh

ACuriousMind

@heather What do you mean? (I trust that you can read the image as good or bad as I can)

@BernardMeurer If it's any consolation, your error is one that many people make when first confronted with this type of exercise

heather

4:44 PM

@ACuriousMind, I suppose so, yeah. nevermind.

Secret

[Universal algebra] After consulted with the maths guys, hopefully the formalism makes sense now...

All elements in a cayley table:

Def 1: Given a magma $(M,\circ)$ of $n$ elements. Let $S$ be a $1 \times n$ matrix or $n \times 1$ matrix where its entries are elements $a \in M$. The cayley table is then given by the outer product $T=S \otimes S$, which is a $n \times n$ matrix. For all matrices, the tranpose is defined in the usual way, thus $(T^\textrm{T})^\textrm{T}=T$

Rows and columns

Def 2: Rows and columns of the cayley table are given in the usual way: For all $i\in M$, $T_{ai}$ is the row correspond to the element $a$ multiplied to $S$. i.e. $aS=T_{ai}$. Similarly, columns are g…

In english: multiplication $\circ$ maps rows to rows, columns to columns. If a row is guarentee to map to a row that is from the cayley table, then the magma is associative

Typo: $Sb=T_{ib}$

Bernard Meurer

@ACuriousMind I'm growing more and more dislike for formal maths :/

Proofs and things like that

I loved LA

but yet I can't prove the simplest thing

It seems like I'm building bicycles for fishes

Emilio Pisanty

omg, look at this: dafix.uark.edu/~danielk

someone should tell Kennefick that the nineties called and they want their web design back

Bernard Meurer

Oh god, why red?

Secret

eye is bleeding from staring into that endless red

Emilio Pisanty

4:55 PM

well, at least it's intentional dafix.uark.edu/~danielk/Politics/redpage.html

Bernard Meurer

This guy must be insane or something

Emilio Pisanty

I guess it's not quite so outrageous if you consider it's from 2008, but still

Secret

btw Emilo, a weird question, what is the context of that complex plot that is your avatar?

Emilio Pisanty

in any case, he's the crack science historian who got to the bottom of the Einstein run-in with Phys Rev, cf. dafix.uark.edu/~danielk/Talks/PhysRev.pdf and physicstoday.scitation.org/doi/abs/10.1063/…, so maybe we oughta cut him a tiny bit of slack

JamalS

@ACuriousMind Have you ever come across the 'Arakelov' metric of a Riemann surface?

ACuriousMind

5:02 PM

@JamalS nope

JamalS

@ACuriousMind How about the Faltings invariant?

Emilio Pisanty

@Secret It's figure 11c in arxiv.org/abs/1507.00011

ACuriousMind

@JamalS No, haven't seen that either. I don't know much except for the basics about Riemann surfaces.

Emilio Pisanty

it shows the evolution of the roots of a certain complex-valued equation as a parameter is moved around in a loop

it shows that after one loop, the roots move around and exchange places, so they are all in some sense the "same" root

JamalS

Yeah, I guess there's nobody I could really ask here.

ACuriousMind

5:05 PM

@JamalS Danu, perhaps

Bernard Meurer

@JamalS Sadly @0celo7 is banned

AccidentalFourierTransform

::clears throat::

Emilio Pisanty

@Secret this episanty.github.io/Slalom-in-complex-time is also related

heather

2

A: How can we combat arrogance and ignorance on physics stack exchange?

Oh, the agony of questions like these. Arrogance, meh. Physicists should be a bit arrogant, they've certainly worked through enough degrees and years of maddening "when will I ever get my PhD?" craziness that, while it does sound incredibly fun, I think gives permission for a tad bit of arroganc...

this answer is probably way to sarcastic.

i just couldn't resist.

ACuriousMind

@AccidentalFourierTransform ::hands over lozenge::

3

AccidentalFourierTransform

5:10 PM

t̥̥͇ḫa͓̘̣͜ṋ̖͔͝ͅͅk̳̻̺͉̝̯͙s̝̦͕̯ͅ

JamalS

@AccidentalFourierTransform Has encontrado el variante de Faltings o de Kawazumi-Zhang?

AccidentalFourierTransform

no, why do you ask?

:-P

JamalS

In the evaluation of the two-loop graviton amplitude in type II string theory, the integration of the invariant of a Riemann surface times some particular modular form over all metrics on genus 2 surfaces arises.

AccidentalFourierTransform

Yeah, I know what a string is

I can help you with that

JamalS

The invariant was originally studied by Kawazumi-Zhang but is essentially equivalent to the Faltings invariant, and I was interested in how it can be computed explicitly, in terms of the period matrix of the Riemann surface.

Integrating it seems easy thanks to a method by Green et al, once you know its form of course.

@AccidentalFourierTransform Are you Spanish by the way, or just studying in Madrid?

AccidentalFourierTransform

5:14 PM

@JamalS I am from Madrid, living in Barcelona rn

JamalS

@AccidentalFourierTransform Oh, I was in Barcelona in August - I'll be back next year.

Secret

hmm, the two saddle points are very close together in Fig. S4, as if they are going to touch soon...

AccidentalFourierTransform

@JamalS Madrid is better ;-)

JamalS

@AccidentalFourierTransform I might be flying there just for 1 night for New Year's Eve.

AccidentalFourierTransform

actually, Barcelona is lovely. I really like it in here

JamalS

5:16 PM

I'll be going 5th to 20th August

Then probably down to Marbella, Andalucia (where my parents are), until the end of the summer.

AccidentalFourierTransform

for something in particular? or just tourism?

JamalS

@AccidentalFourierTransform An event.

AccidentalFourierTransform

Well, Spain is great in Summer. The nicest weather in Europe ;-)

And you are from the UK, right?

JamalS

@AccidentalFourierTransform Yes, but I grew up in Spain. Only went back to the UK for university.

AccidentalFourierTransform

Cool, you're the first one here in SE whos been to Spain that I know of :-)

JamalS

5:19 PM

@AccidentalFourierTransform My favourite country; more of a home to me than the UK.

AccidentalFourierTransform

I've never been to Marbella, but I bet its beautiful

Where in the UK do you study?

JamalS

@AccidentalFourierTransform In the west mids

@AccidentalFourierTransform Marbella is great, but just don't go during tourist season, it's awful.

It's very easy for you to go to Marbella, if you take the train from Barcelona station to Malaga station (renfe), it's about five hours. Then just go by car down to Marbella area.

The train goes through Madrid, so you could stop there too

AccidentalFourierTransform

@JamalS and I'll probably go back to Madrid for the summer, so it is actually easier to go down there some time

But yeah, in summer it is too crowded

DanielSank

5:37 PM

@AccidentalFourierTransform How so?

I was in Madrid for a few hours...

I was in Barcelona twice, both times for a few days. Barcelona was cool, but I didn't have enough time to take in Madrid.

Sevilla was neat.

JamalS

@DanielSank Lots of British, and Northern European tourists during the summer - you'll see everything packed, even supermarkets, at least in Marbella as they often rent or have places down here.

AccidentalFourierTransform

@DanielSank Madrid is a very nice place to live actually

I've lived in several places and I always go back to Madrid

@JamalS Everywhere near the coast in Spain is full of people from the UK and Germany

JamalS

@AccidentalFourierTransform Well yeah, when I say Marbella I mean Costa del Sol :)

Naveen Balaji

5:57 PM

Hey guys! Do anyone of you know where I can get data regarding time taken for orbital decay p(due to gravitational wave radiation) of binaries. Wiki doesn't have much info.

Bernard Meurer

Does anyone here currently live/is in London?

@JohnRennie? Hmmm @EmilioPisanty?

Physics Meta

6:28 PM

0

Q: Is it acceptable to quote a Nazi claim on Physics SE?

I have no intention being rude, and have been hesitating a long time before posting this question. The fact is that a user of the Physics SE, MAFIA36790, includes this quote in his public profile: "There will come a day, when all the lies will collapse under their own weight, & truth will again...

discussion

Emilio Pisanty

6:39 PM

@BernardMeurer not no longer, no

moved out in February

why?

Bernard Meurer

6:53 PM

@EmilioPisanty I have a 12 hours layover at Heathrow

Thought I'd use the chance to buy some fellow h-bar member a beer

ACuriousMind

@BernardMeurer flying home for Christmas?

Bernard Meurer

@ACuriousMind Yeah :)

Christmas and New Years

I'm back on the 4th because my analysis exam is on the 9th

I foresee getting rekd

ACuriousMind

exams around New Year? Ew.

They're usually end of January/start of Februrary here

I don't need to take any anymore, though :)

Bernard Meurer

You lucky dog

I have Analysis on the 9th and Digital Systems on the 12th!

Digital Systems is the subject where I broke the matrix

I have a 10/20 (E) in theory, and a 19.2/20 (A+) on lab, lol

So now I also have to do the oral exam

ACuriousMind

What sort of things do you do in "Digital Systems"?

Bernard Meurer

7:05 PM

We saw boolean algebra, hmm, most fundamental components of circuitry (Multiplexers, decoders, logic gates, encoders, demultiplexers, counters, shift registers, flip-flops, RAM, ROM, ...), finite-state machines with state diagram and fluxogram descriptions, hmmm

VHDL programming and intro to FPGAs

AccidentalFourierTransform

why?

Bernard Meurer

Why what?

AccidentalFourierTransform

why everything

ACuriousMind

@AccidentalFourierTransform 42

AccidentalFourierTransform

all you said, all of it

why?