The h Bar

Slereah

00:01

"Topological spaces E and F which are not homeomorphic even though there exists a continuous bijection from E to F and a continuous bijection from F to E"

What?

0celo7

$4 tax on this math book

government, are you happy???

@Slereah it's true

a homeo has to be continuously invertible

Slereah

Apparently!

0celo7

@Slereah if you have a bij $f:E\to F$ and a bij $g:F\to E$ it's not necessary that $fg=gf=1$

loltospoon

Friends, I am in distress

I still don't understand how to do this part of my project

Slereah

call 911

loltospoon

00:08

I wrote earlier about finding the energy of a wavefunction numerically

But

My professor said that for the first point we guess an energy value. Great, I did that. But now I want the total energy using $E=\int \psi ^* (-2mE/\hbar ^2)\psi dx$

But this doesn't make sense at all

How is it that by integrating something with $E$, I get $E$ back??

0celo7

Why not?

loltospoon

$E$ is on both sides of the equation!

0celo7

because the other stuff is one

@Slereah Springer sale does not include the big book of spacetime :(

loltospoon

@0celo7 you mean $\int \psi ^* \psi dx=1$?

Slereah

I know one of the guy who wrote in it

And yet I can't buy it

What a shame

0celo7

00:12

@Slereah Volume 3 of Hörmander is about PDE on manifolds...there's conormal distributions

sounds interesting

Slereah

it does not

on the other hand...

It's 1 AM and I can't sleep

0celo7

@loltospoon are you saying $\int\psi^*(-2mE/\hbar^2)\psi=E$?

Slereah

Perhaps learning about conormal distributions could lull me to sleep

loltospoon

@0celo7 when $V(x)=0$, yes.

Slereah

@0celo7 I can see one big flaw

It rhymes with shminearity

loltospoon

00:14

@0celo7 this is for the infinite square well

Slereah

$$\int \psi^* (-2 m E / \hbar^2) \psi dx = (-2 m E / \hbar^2) \int \psi^* \psi dx = -2 m E / \hbar^2$$

0celo7

yes, we can all do PhD measure theory

loltospoon

Ooooh crap, my mistake. Wow what an awful mistake.

I meant

0celo7

...

loltospoon

$$\int \psi ^* \frac{-\hbar^2}{2m}\frac{\partial ^2}{\partial x^2}\psi dx$$

0celo7

00:16

...

Slereah

yeah that sounds better

0celo7

you probably have some error like this in your code or whatever

loltospoon

So now we are at my point of confusion. Given that the equation above is correct, all I have to do now is multiply $(\psi (x))(-\frac{\hbar^2}{2m})(\psi '' (x))$ for each $x$ in my interval, correct?

And sum them up to get the total $E$, right?

Slereah

Should be fine, yes

Just the usual numerical integration

0celo7

don't you have to multiply by the interval width

@loltospoon you should figure out what kind of numerical scheme you want

Slereah

00:20

Although

0celo7

what lingo are you using

Slereah

A helpful tip

loltospoon

@0celo7 I'm using Swift 3.x

0celo7

tf is that?

loltospoon

I thought you were asking about the programming language?

Slereah

00:21

For a numerical integration, $$\frac{d^2\psi}{dx^2} dx = \frac{d^2\psi}{dx} $$

0celo7

yeah

loltospoon

It's based on obj-C

0celo7

lol, I just witnessed mod censorship on the apple forums

loltospoon

@Slereah ohh I think that's what @0celo7 meant about multiplying by interval width

0celo7

looks like mods are the same everywhere

Slereah

00:25

I wonder if that counterexample book has the $L^2$ function that doesn't go to 0

"Open set of $\Bbb R$ containing $\Bbb Q$ of arbitrarily small measure"

"Subset of $\Bbb R$ with empty interior of positive measure"

Aaaah

It is apparently $[0,1] - ([0,1] \cap \Bbb Q)$

0celo7

@Slereah Sure, irrational numbers.

@Slereah Btw, here's a neat proof. Claim: $[0,1]$ is uncountable. If it were countable, then $\mu([0,1])=0$. But $\mu([0,1])=1-0=1\ne 0$.

Slereah

"Infinitely differentiable function at $0$ that isn't infinitely differentiable on any neighbourhood of $0$"

what

0celo7

01:13

@Slereah what is it?

Slereah

It's complicated

It's like four function compositions

0celo7

Ugh

the worse are the series counterexamples

hard to capture the pathological behavior of those in a picture

Slereah

$f_n^{(k)}$ is the $k$-th derivative of $x^n$, $g_n(x) = f_n(x - 1/n) + f_n(x+1/n)$, $u_n = g_n(x) / n!$ and $h(x) = \sum u_n(x)$

0celo7

interesting measure theory proof that harmonic series diverges

@Slereah ugh

Slereah

yeah it's a bit hard to get a good sense of it

You know who I found, though?

Our old friend $\sin(1/x)$

0celo7

01:20

of course

Slereah

In "Function admitting a Taylor series of order 2 around 0 but isn't twice differentiable at 0"

$x^3 \sin(1/x)$ to be exact

0celo7

jesus

why does physics math even work?

everything is analytic, all operators are compact, self-adjoint

Slereah

Mostly because almost everything is smooth as a baby's bottom

0celo7

you can have baby butt smooth stuff that's pathological though

Slereah

yeah but physics is mostly very well behaved

at least the approximations people do

0celo7

01:22

$p^2+x^4$ is not self-adjoint!

it's a pretty reasonable Hamiltonian

Slereah

Well apparently not!

0celo7

the perturbation series diverges too, for all $\lambda$, if you do $p^2+\lambda x^4$

Doesn't $\phi^4$ diverge?

Slereah

It's complicated

$\phi^4$ is probably trivial

But $\phi^4$ plus some gauge fields might not

0celo7

Trivial?

Slereah

Trivial as in its renormalized coupling constant is $0$

0celo7

01:24

ah

Slereah

Hey, it's the Dirichlet function!

"Periodic function that isn't constant and doesn't admit a smallest period"

0celo7

it's a good function

dude, did you not see my $[0,1]$ is uncountable proof above?

it's pretty lit

Slereah

I did

What is the Fourier series of the Dirichlet function

0celo7

oh god

zero?

Slereah

I'm guessing it's probably not well defined

Although... isn't Dirichlet $\approx 1$

Almost everywhere

0celo7

01:28

No, it's $0$ a.e.

Slereah

Oh

then yeah probably 0

0celo7

It's $\chi_{\Bbb Q}$.

Yeah it should be zero. $d=0$ in $L^2$, so in terms of Fourier it should have the same expansion.

($d$ is the Dirichlet function)

It's still strange that $\Bbb Q$ is so small compared to $\Bbb I$

@Slereah I got an emoticon in an email from a prof...what is the world coming to?

Slereah

"Series $\sum f_n$ of sum $F$ on $[0, \infty]$ for which $\lim F(x) \neq \sum \lim f_n(x)$"

hey it's a partition

$f_n = F(x)$ on the interval $[n,n+1]$

Then the limit is always 0

Noice

0celo7

@Slereah huh?

oh, limits and infinite sums don't work well?

Slereah

It's a nice counterexample

Simple enough

On the other hand there's a proof of $e$ not being algebraic

A bit long in the tooth

Secret

01:39

[Philosophy(?)]

The illusion of choice:
It will seemed that we are free to believe whatever we want
But the reality is that reality as a complex system impose restriction and pressures on some beliefs

This is what it might means for a belief to be objectively wrong

Since our macroscopic world quantum can do so little (having nearly everything decohered and cannot stay coherent for more than microseconds to establish superposition), our world follows mostly classical mechanics, and thus objectivity and realism holds mostly as far as policies and decisions of groups goes

Slereah

Also that book uses $\mathfrak A$

0celo7

@Slereah I would like to understand that one day

Slereah

Some of the counter examples are boring

0celo7

such as?

"not every real number is rational?"

Slereah

Like there's some example of $a^4 + b^4 + c^4 = d^4$

And yeah, there is that too

Although not quite

it's "constructible real number that isn't rational"

(though it's still $\sqrt 2$)

0celo7

01:48

constructible?

Slereah

Constructible is a geometric sort of concept

A number you can make using ruler and compass

0celo7

take a square with side length $1$

the diagonal is root 2

Slereah

"In terms of algebra, a number is constructible if and only if it can be written using the four basic arithmetic operations and the extraction of square roots but of no higher-order roots."

0celo7

Hmm.

Slereah

There's a pretty old proof that $\sqrt[3]{2}$ isn't constructible

It's the old doubling of the cube problem

oh wait, the proof was actually made in 1837

Over 2000 years to prove

Tough nut to crack

0celo7

01:54

is that the longest standing conjecture?

Slereah

Depends how you define it

It could be "Euclid's 5th axiom is independant"

Also squaring the circle has about the same timeline

I think those are the three big problems that were posed in antiquity and took 'til the 19th century to get solved

0celo7

@Slereah which one is that?

Slereah

If you have a circle, find a square with the same area

Basically finding a rational expression for $\pi$

(hint : it doesn't work very well)

Well not necessarily rational but at least algebraic

0celo7

so, area of circle is $\pi r^2$

area of square is $l^2$

so, $l=\sqrt{\pi}r$

what's wrong with that?

Slereah

Well this is old timey geometry

0celo7

02:02

Do you have to construct it with the tools?

Slereah

It is assumed that the sides are proportional

yes

0celo7

Ah, well

I'm sure it can be done

Slereah

euclidian geometry basically has no link to $\Bbb R^2$

There's no notion of distance

You can only construct something of length 2 by defining something of length 1 and doubling it

0celo7

jeez

Slereah

people had a huge geometry boner back in the days

They didn't care much about numbers 'til like the 15th century

numbers were only good for merchants

0celo7

02:05

how much geometry did they even know?

what were the big, exciting theorems?

did they mail them to each other on wax tablets?

Slereah

A lot of the big theorems were about finding volumes of shapes

Or constructing shapes of same volumes as other shapes

that kind of stuff

I think Archimedes did tons of that

0celo7

if I went back and called Newton out on his infinitesimal shit, what would happen?

Slereah

Dunno

Not sure how infinitesimals were received back then

I remember that Newton's gravity was kind of poorly received

Nick Alexeev

What's the relationship between the strength of reflected ray and refracted ray?

0celo7

@NickAlexeev Are you supposed to be a Hanukkah candle?

Slereah

02:10

Apparently it's the logo of his fancy company

prolifictec.com

0celo7

@Slereah Well sure, because it's wrong.

Nick Alexeev

@0celo7 ... folded for easier transportation.

0celo7

@NickAlexeev Very efficient.

Slereah

@0celo7 It was more because it was an action at a distance effect

People weren't big on that notion

0celo7

@Slereah But did 17th century mathematicians understand Rigor?

Slereah

02:11

All forces were contact forces

0celo7

@Slereah That's what I'm referring to.

Slereah

The big rigor thing was still euclid by that time, I think

Nick Alexeev

@0celo7 It's actually a tree. I drew a logo for my ex-girlfriend's company, but she didn't like it, and I've kept it for myself. (Her company didn't go anywhere either.)

Slereah

Even though euclid's axioms didn't actually work as an axiom system in the modern sense

0celo7

why not?

it's not like we define sets

Hmm, unless. Is every set constructed from the empty set?

Slereah

02:13

There's a couple of axioms that are implicit in proofs but never stated

0celo7

In Euclidean geometry?

Such as?

Slereah

@0celo7 In most set theories, yes

0celo7

I'm sure they let their intuition guide them a bit at points.

Slereah

Pasch's axiom

Not to be confused with Pasch's theorem regarding points on a line In geometry, Pasch's axiom is a statement in plane geometry, used implicitly by Euclid, which cannot be derived from the postulates as Euclid gave them. Its essential role was discovered by Moritz Pasch in 1882. == Statement == The axiom states that, Let A, B, C be three points that do not lie on a line and let a be a line in the plane ABC which does not meet any of the points A, B, C. If the line a passes through a point of the segment AB, it also passes through a point of the segment AC, or through a point of segment BC. The...

Here's one of them

"Euclid assumed implicitly that any line contains at least two points, but this assumption cannot be proved from the other axioms, and therefore must be an axiom itself."

That scoundrel

"The very first geometric proof in the Elements, shown in the figure above, is that any line segment is part of a triangle; Euclid constructs this in the usual way, by drawing circles around both endpoints and taking their intersection as the third vertex. His axioms, however, do not guarantee that the circles actually intersect, because they do not assert the geometrical property of continuity, which in Cartesian terms is equivalent to the completeness property of the real numbers."

Euclid was kind of a hack

0celo7

:)

Springer sale is the best

@Slereah I mean...c'mon

You have to be pretty nitpicky to come up with that

@Slereah Ah, yes. I do remember that one.

Euclid and Einstein were both crackpots.

Slereah

02:17

Well without some of those axioms you can come up with geometries that aren't the plane geometry but still obey all axioms

0celo7

I can imagine Ioannes Duffieldus running down the via screaming "Euclid and the Evidence."

@Slereah Such as?

loltospoon

Should I be able to tell the value of $n$ in $E=\frac{n^2\pi ^2 \hbar ^2}{2ma^2}$ simply by looking at the graph of the wavefunction for any $a$ or does it explicitly depend on how big the well is? This is for the infinite square well.

For example, this is what my wavefunction looks like for a well length of 10. Simply by inspection, can I say that $n=2$??

0celo7

By inspection?

This requires a rigorous Proof

loltospoon

Crap....

0celo7

idk, ask your prof

Slereah

02:23

Hm, which geometry was it

loltospoon

Take a look at this image: i.sstatic.net/iIQEL.gif We can tell just by looking at the wavefunction that the green wavy line is $n=2$ because there is 1 node.

Slereah

it was a weird ass geometry

0celo7

@Slereah

260

Q: How do I draw a pair of buttocks?

I'm trying to develop a function which 3D plot would have a buttocks like shape. Several days of searching the web and a dozen my of own attempts to solve the issue have brought nothing but two pitiful formulas below. They have some resemblance to the shape I want, though not quite. Could you...

^that one?

Slereah

close but not quite

It's hard to find because "non euclidian geometry" usually gives curved spaces

0celo7

are you talking about a non-manifold geometry?

Slereah

02:31

8

Q: Are there simple models of Euclid's postulates that violate Pasch's theorem or Pasch's axiom?

While reading a paper (pdf) about the history of modern logic, I learned that some opinions (about deductive/axiomatic mathematics) typically attributed to David Hilbert can be traced back to Moritz Pasch. After googling for Moritz Pasch, I was surprised to learn that he had found important impli...

geometry euclidean-geometry

there's an example

"Her research at this time included work on the axiom of choice, but it was interrupted by the 1939 Invasion of Poland."

Damn you Hitler!

0celo7

lol

Slereah

But yeah to make geometries that follow Euclid's axioms and fail to be plane geometry you need to go pretty weird

isa-afp.org/browser_info/devel/AFP/Tarskis_Geometry/…

What the fuck is this paper

0celo7

category theory

by god it's a long paper

I like how the 200+ page paper has three references

Slereah

"But Hilbert's axiomatization is really not suitable for a further discussion of the OP's question. This is because Hilbert's axiomatization of geometry very much shows its age. His Axiom of Continuity (completeness) is second-order. Hilbert developed the axiomatization many years before he started to take a serious interest in Logic."

what a scrub

0celo7

Hilbert was all over the place

functional analysis, geometry, logic

algebra

Slereah

02:45

Apparently the good geometry is Tarski

"The only point between the segment xx is x"

Well I agree

Tarski geometry is all about congruence and betweenness

0celo7

@DanielSank Hey

Do you want to know how to solve that integral equation?

Slereah

$$ \exists a\,\forall x\,\forall y\,[(\phi (x)\land \psi (y))\rightarrow Baxy]\rightarrow \exists b\,\forall x\,\forall y\,[(\phi (x)\land \psi (y))\rightarrow Bxby]$$

Pretty bad for geometry

0celo7

what does that even mean?

what is $\phi,\psi,B,a,x,y$?

what is all of that?

Slereah

Apparently this means that if you have a point $a$,a set of points $X$ and a set of point $Y$, such that for all $x \in X$ and $y \in Y$, $x$ is between $a$ and $y$, then there's a point $b$ between every $x$ and $y$

0celo7

@Slereah what?

Slereah

02:56

It is to express continuity

Let's make a GR entirely axiomatized like that

No manifold or anything

0celo7

I'm down

Do we derive the manifold somehow?

Slereah

I guess technically that's kind of the causal set axiomatization of special relativity

0celo7

Link?

Slereah

Just defining causal and chronological relations

It's referenced in Penrose's bibliography

Some 60's paper

0celo7

how do you get the linear structure of Minkowski space?

it's pretty important

Slereah

03:04

It's 4 AM so I don't have much ideas right now

I should go to bed

0celo7

nooo

I don't want to do my lab report

ZeroTheHero

03:20

To any moderator: please check physics.stackexchange.com/questions/316673/… and possibly edit out "because I am a fu***** idi** " as offensive..

0celo7

@ZeroTheHero Better?

ZeroTheHero

yeah... thanks.

0celo7

I also took out his line at the end about being stupid or whatever

Although personally I agree with him

He's at least unstable (this is not an official medical diagnosis)

ZeroTheHero

I reviewed the post and was a bit disconcerted by the tone...

0celo7

@Slereah are you still here

ZeroTheHero

03:24

but yeah... thanks for the quick reaction.

0celo7

$\epsilon_n=\epsilon_n^{(0)}+\lambda V_{nn}+\lambda^2\sum_{m\ne n}\frac{|V_{mn}|^2}{\epsilon_n-\epsilon_m}$

0celo7

03:39

there is so much wrong with this picture

@dmckee I've been looking at dissertations from people in my department any they're all strangely formatted and the committees all have engineers on them

I doubt any electrical engineering prof cares about curvature flow or whatever

What's up with that?

These things are all off-center

wtf

My linear algebra TA was working on some cool stuff

Sir Cumference

04:07

wtf, i went to my TA with questions about astronomy

she said "actually, this is what I'm learning right now..."

0celo7

you're doing PhD level stuff, wow!

Sir Cumference

@0celo7 i wouldn't say that

but it was certainly weird that she couldn't help much

she was certainly surprised

0celo7

@ACuriousMind This book of Kato is neat. He seems to treat linear algebra via functional analysis

much more natural than algebra, imo

Secret

0

Q: How real is belief and their impacts, and how exactly they are limited by an objective reality?

Some definitions and notes: A worldview of an entity is an unifying collection that summarises the way it perceives, understands and modelling reality An entity is any existence that has the ability to generate a worldview. This includes conscious beings like humans and gods, as well other type...

epistemology metaphysics belief

The ultimate belief question

ZeroTheHero

04:31

@0celo7 sorry to bother you again. Check this guy: physics.stackexchange.com/users/147611/d-winstein 4 homework questions in the last 4 hours with 0 attempts at anything... He is just hoping someone will do his entire homework for him.

Secret

Actually, I don't study physics, so that's why there's a lack of effort ^^'. I am trying to help my friend who has struggled with certain questions, I should have asked him how he tried to solve the question. — D.Winstein 43 secs ago

What, that is NOT ALLOWED

That's even worse than getting someone to do his homework

2

0celo7

@ZeroTheHero I'm not a mod, I can't do anything about it

ZeroTheHero

@0celo7 :( thanks for looking it up anyways... strange.

(I mean: this poster is strange)

Secret

This guy makes me so trigger happy: This is the first time I made 4 downvotes within a day

I cannot emphasize it enough

$$\Huge{\text{That's even worse than getting someone to do his homework!!}}$$

0celo7

lol

ZeroTheHero

04:41

yeah. I deleted an answer I had posted to one question. I don't mind helping people out with trickier stuff if there is a greater benefit but I find this case to be a little extreme.

well... good day/good night gentlemen.

David Z

Cast flags or VTC next time, rather than complaining in chat. It might have taken a long time for these questions to be closed just based on your chat messages, if I hadn't happened to be here.

Secret

05:29

[Super random question] Can we have a blend theory that includes LQG and string theory...?

Secret

05:58

newscientist.com/article/…

loltospoon

Friends, where can I take a very introverted girl on a date that's both fun, creative, and brings out the fire within her???

By that last part, I mean in it a non-sexual way. I've heard her genuine laugh and I've fallen in love with it. I want her to do it more. Any ideas?

Secret

> Then again, if the speed of light were infinite, massless particles and the information they carry would move from A to B instantaneously, cause would sit on top of effect and everything would happen at once. The universe would have no history and no future, and time as we understand it would disappear

It might be more interesting if not the entire universe behave like so, then we have interesting systems where it is instantenous only for some bounded region and that will give interesting dynamics

@loltospoon While I have no experience on dating, if I were you, logically I will try to understand what makes her happy, and then I will do such thing if it is possible after taking account of a variety of factors

loltospoon

@Secret yup, already working on learning as much as I can about her. Now I'm looking for ways to surprise her :3

Secret

The way I handle A and B that has instantaneous interactions is the same as A and B are inseparable and cannot be described as two subsystems, kinda a generalisation of the concept of correlations

> Imagine a world where, if you and I had once met, my missing the bus to work would automatically make you late too. Or where, if I put on odd socks, yours would be odd too. A great excuse, maybe – but also deeply weird.

That is not just the usual classical correlation, and obviously not entanglement (because it is not the wavefunction that becomes non factorisable). This is a deep correlation that given two systems, once interacted, result in all their future evolution to be correlated, even if they are separated from each other.

Esoterics love to talk about this kind of correlation in the context of how the mind affect the physical world

I might model this some time later, should not take too much adjustment on lagrangians and so and so...Should make an interesting worldbuilding setting

aip.scitation.org/doi/abs/10.1063/1.4773174

Leonhard Euler

06:47

Hello @JohnRennie

John Rennie

@NickAlexeev The reflectivity and transmission are calculated using the Fresnel equations. These are straightforward but messy, and note that the different polarisations of the light have different reflectivities and transmission coefficients.

@LeonhardEuler Hi

Leonhard Euler

@JohnRennie, how are you/

John Rennie

I'm fine. And you?

Leonhard Euler

@JohnRennie, I am also fine.

@JohnRennie, what are the chemical properties of salts?

John Rennie

How old are you, i.e. what level are you at in school? I ask because that affects the level of explanation that is appropriate.

Leonhard Euler

06:54

@JohnRennie, I study in grade 10

John Rennie

What age is that? In the UK we use a different way of describing school so I don't know what grade 10 means.

Leonhard Euler

@JohnRennie, 16-17

Years

John Rennie

OK, so that would be A levels in the UK.

Leonhard Euler

@JohnRennie, Ya

John Rennie

When you ask about the chemical properties of salts that's a rather broad question. Did you have anything specific in mind?

Leonhard Euler

06:59

@JohnRennie, Actually this question was asked in a test. I tried searching in textbooks, but they only have the physical properties

Anonymous

@LeonhardEuler Salts have thousands of properties! What test is this ?

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