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7:28 PM
oh yes i am really sorry i didn`t see the question thoroughly — Avnish Singh 1 min ago
::facepalm::
 
What is meant by "classical point of return" for a harmonic oscillator? :O
 
the turning point
 
Isn't that the amplitude?
 
it's also that, yes.
point is that, if you know the total energy of the harmonic oscillator, then it'll oscillate back and forth in [-A,A]
so x=A is where the classical trajectory stops and turns around
 
Well, in this case I only know $\omega$ and $m$ for a Weber rod and I am supposed to find the turning point in the ground state.
And iirc $\psi_0 = (\dfrac{m \omega}{\pi \hbar})^{1/4} e^{-\dfrac{m\omega}{2\hbar}x^2}$
Wait, I'm also given the length of the rod
So normalizing and finding the amplitude is probably a good idea
This is so wrong
 
7:49 PM
Sigh
At some point I just need a break
 
@Semiclassical If the oscillating rod is of length $L$, would solving $1=|A|^2 \int_0^{L} \psi_0 ^2 dx$ for $A$ give me the turning point?
 
No. You get the turning point from you knowledge of the total energy of the system
Which you definitely know for the ground state
 
Yeah $1/2 \hbar \omega$
 
The other thing that’s useful is that the probability density of finding the particle at a given spot should be inversely proportional to its velocity there
(the longer the particle stays in a given interval, the more likely you are to find it there)
One important point of comparison here is that the wavefunction is nonzero everywhere in space. In particular, it’s nonzero for x beyond the classical turning point
 
How can it be nonzero outside the rod?
 
8:00 PM
So there is a small but nonzero probability of finding it outside the range you’d expect for a classical harmonic oscillator
Because the harmonic oscillator you solve in QM doesn’t know about that length
if you wanted to enforce that, you’d need to set V=infinity for large enough x
But for the harmonic oscillator you only have V=infinity at x=infinity rather than any finite distance
As such, the ground state wavefunction is nonzero everywhere
It becomes exponentially small as you move away from the origin, so the probability gets small very fast
But there’s no sharp cutoff
 
Is there some online program where you can draw crappy pics and it will output the latex code, e.g. like you make your own inclined plane drawing with paint and it turns into code
 
Okay so should I try to find the expected value for the velocity? @Semiclassical
The revival time is $T=4mL^2/(\pi \hbar)$ so maybe if I multiply $T$ with $\dfrac{1}{m} \langle p \rangle$ I get something useful lol
 
oh my god there is
2
A: Online LaTeX diagram graphical editor

dacastrorA wonderful option is mathcha: www.mathcha.io just open the editor, you log in and you make your drawings and equations. this online program will create different export formats, including Likz, which is compatible with Latex.

can convert it to tikz
 
8:21 PM
@Lozansky no. You’re trying to work out the probability distribution for the positions of a classical harmonic oscillator, yes?
 
@Semiclassical I just want the magnitude of the classical turning point in the ground state :>
 
Ah. Again, though, that’s just the amplitude
Main thing is that the classical turning point doesn’t know about the wavefunction
Classically, what you’ve got is stuff like energy conservation
 
Yeah, but isn't the amplitude $|\psi_0|^2 = (\dfrac{m\omega}{\pi \hbar})^{1/2}e^{-m\omega/\hbar x^2}$?
 
No. That’s a probability density, for one thing
For another, when I say amplitude in reference to a classical harmonic oscillator I mean precisely “how far does it move from the origin before it stops and begins moving the other way”
 
Yeah, if the oscillations are given by $\psi_0 e^{-i\omega t/2}$ then $|\psi_0|^2$ would give me the amplitude, no?
 
8:29 PM
again, no
 
Hmm okay
 
A classical turning point is where v=0
 
Sure
 
All it knows about is the potential energy
It knows nothing about the wavefunction
Moreover, the units of what you’re saying don’t make sense either
A classical turning point is a location
 
Hi, everybody.
 
8:32 PM
Psi^2 is a probability density and has units of 1/length (in 1D)
 
Yeah
 
More accurately, Psi(x)^2 has dimensions of 1/length.
You can write wave functions of things other than position.
Important to keep in mind.
@EmilioPisanty Confirmed: It's now illegal to post answers on Stack Exchange unless they get a high score.
 
But I don't know anything about the potential, do I?
 
What, exactly, are you given then?
 
I mean $1/2 \hbar \omega$ is the energy of the system?
 
8:35 PM
The potential energy of a harmonic oscillator is 1/2 m omega^2 x^2
So if you know the mass of your particle and you know omega, you know the potential
 
Hm OK, and so we solve for $x$ in $1/2 \hbar \omega = 1/2 m \omega^2 x^2$ (all energy is potential)?
 
At the turning point, yes
 
@Semiclassical So the length of the rod doesn't affect the turning point?
So in effect, I could have a rod smaller than the turning point...
 
To the extent that you take the length of the rod as being independent of the frequency, yes
On the other hand, you could take the ends of the rod to be the classical turning points, and infer the frequency from that
So in that setup they’d coincide.
 
8:55 PM
@SirCumference play the drums
 
@DanielSank how so?
Ah. I missed the pointer.
@DanielSank you mean good answers?
 
Okay so the rod absorbs some energy
And jumps from energy state $E_n$ to $E_{n+1}$
Then the change in amplitude should be $\Delta x = \sqrt{\dfrac{2\hbar}{m\omega}}(\sqrt{n+3/2}-\sqrt{n+1/2})$
How can I rewrite this in terms of $\Delta x = C /\sqrt{n}$? :O
 
Take n to be large probably
 
Yeah, that'd do it I guess =)
 
@bolbteppa Me too and big time!
 
9:03 PM
Which is really just the correspondence principle in action
 
9:14 PM
@BalarkaSen ok revisiting the issue
are you here
 
no
i am over there
 
@BalarkaSen Let $M$ be oriented, and let $D$ be the PD map $H^k\to H_{n-k}$, ok
 
Is PD nice in the sense that $D(\alpha)\frown\beta =D(\alpha\smile\beta)$?
 
PD = Poincare dual ? (Just curious)
 
9:17 PM
yeah
 
@0celo7 what's \frown?
 
cap product
 
In algebraic topology the cap product is a method of adjoining a chain of degree p with a cochain of degree q, such that q ≤ p, to form a composite chain of degree p − q. It was introduced by Eduard Čech in 1936, and independently by Hassler Whitney in 1938. == Definition == Let X be a topological space and R a coefficient ring. The cap product is a bilinear map on singular homology and cohomology ⌢ : H p ( X ; R ) × H ...
 
9:18 PM
@0celo7 oh god that's horrifying
 
I think this follows from $D(\alpha) = \alpha \frown [M]$ and how cap product interacts with cup product.
 
Does that imply a certain asymmetry between A and B, or am I reading too much into that
 
@BalarkaSen Right I don't know 100% how they interact
hence the question
 
I know that at the chain/cochain level, $\psi(\varphi \frown \sigma) = (\varphi \smile \psi)(\sigma)$
Hm.
 
If D(a)-cap-b = a-cap-D(b) then my remark is superfluous
 
9:19 PM
@BalarkaSen that's the definition, no?
@Semiclassical I'm being sloppy with the order
 
There are many definitions
 
what I said is probably only true mod 2 as written
i think I have to switch some things around to get the right sign
 
Confusing
 
so
is this an HNQ in the making?
1
Q: Why are the $\mathbf E$ and $\mathbf B$ fields of an electromagnetic wave mutually perpendicular?

SealWhy are the wave number $\mathbf k$ and the electric and magnetic fields $\mathbf E$ and $\mathbf B$ are perpendicular to each other? I know it but I haven't thought about it deeply. How can I prove this conclusion mathematically?

 
9:24 PM
Would circularly polarized light be a counterexample? I’m forgetting
 
@EmilioPisanty ?
@Semiclassical no
 
@G.Bergeron Hot Network Question
 
Fair enough
 
the ones that get pushed on the sidebar on the right across the whole of SE
sometimes known as the Stack-Exchange-Wide Advertisement for General Enlightenment list
 
Aka “protect question incoming” :p
 
9:25 PM
@EmilioPisanty Seems a bit over the top for the question, no?
 
@G.Bergeron it's more of a systemic, software issue
 
@0celo7 The right order should be $\beta \frown D(\alpha) = D(\alpha \smile \beta)$, by checking for $\alpha = pt \in H^0$.
 
Well, 5 answer in 40 minutes
 
the most cogently I've managed to articulate my views on the matter is here.
 
@BalarkaSen I hate that I can't run mathjax properly at the office
It's seems like a very ''dramatic'' part of math from that non-mathjax vantage point... :p
 
9:27 PM
@G.Bergeron what?
 
its annoying to not being able to render mathjax
 
are they blocking the mathjax CDN?
 
@BalarkaSen So you think it's true in general?
 
you should probably be able to provide the script locally
 
@EmilioPisanty I don't have admin privileges, it's an old browser on centos...
 
9:28 PM
If not, I can say that $\alpha$ is a cup product of 1-chains
 
@G.Bergeron yeah, figures
 
and $\beta$ another 1-chain
 
on chat you may be able to change where the bookmarklet pulls the script
on main that's probably harder
 
@0celo7 Give me a minute to do the computation. I think $\alpha \frown (\beta \frown \gamma) = (\alpha \smile \beta) \frown \gamma$.
Let me see if I can do this.
 
@EmilioPisanty similarly, the tikz package for commutative diagram (tikz-cd) is not available. Trying to simply put in the same directory led to a several hours long dependancy chase only to realize that the version of pgf on our tex build is not compatible with tikz-cd
-_-
@EmilioPisanty Could probably make it work, there is a fair amount of lazyness at work here...
 
9:31 PM
@G.Bergeron yeah, if it's a system you control and a known fix then that's one thing
but figuring out how to get an old browser do things it wasn't really built to do on a machine with no sudo, definitely another order of magnitude
 
@EmilioPisanty Yes, but more simply, I could probably just run the script on the javascript console
 
I'm at a similar sudo-less desktop at work and that can indeed be a pain
 
@EmilioPisanty Yes, and wait for it.... WE ONLY HAVE 500 MB 8(
 
@BalarkaSen Oh, Hatcher has the formula way back in the back of the chapter
 
thankfully it's a mostly fully-kitted-out texlive (I think bundled with ubuntu 16.04?) but at some point IT just utterly failed at installing a font I needed wanted because texlive just doesn't have it.
 
9:34 PM
That, coupled with the fact the every damn unix applications wants to put their stuff on home is a major PIA
Like my dropbox!!!
 
oh, and also
booom! rep-cap
cc @ACuriousMind
 
Then you have to mess around with symlinks and stuff which dropbox definitively does not like
@EmilioPisanty rep-cap? Reputation is capped?
 
@BalarkaSen I have another question for you
 
... unless somebody unupvotes-then-upvotes a post of mine a minute before and a minute after GMT midnight
@G.Bergeron daily rep is capped
 
@BalarkaSen actually 2 more
 
9:36 PM
@0celo7 Where in Hatcher
What are the questions
 
200 max, with accepts and bounties exempt
you get a badge if you hit it
 
259
 
and then a gold badge if you hit it 150 times
 
@Semiclassical btw, I loved the greek quote you posted! Especially in the context it was posted! I was becoming quite exasperated before I just quit! :P
 
lol
I wondered who all would get that :P
 
9:38 PM
i dont have the hard copy w/ me rip
 
(I'm personally fonder of the bit of Kant I quoted here, though it's not nearly as pithy)
 
@BalarkaSen Suppose I have $\alpha_1,\dotsc,\alpha_{n-1}$ classes in H^1
denote the total cup by $A$
 
what does it mean for D(A) to be nonzero
 
@Semiclassical I already rage-quit by that time. Just saw it 5 minutes ago.
 
9:38 PM
geometrically
 
@Semiclassical there's this thing called "google"
it's really quite good
 
lol, true
 
and why does, $M\# T^n$ always have this nonzero
 
I guess I should more say: I wondered who all would bother to look that up :P
 
or does it?
 
9:39 PM
@EmilioPisanty Yeah, but had I not starred it at that point, would you have noticed it?
 
@G.Bergeron nope
 
or maybe can you find classes for which that's true
I'm not sure what the right phrasing is
 
@Semiclassical c'mon, Greek on the star board?
 
@EmilioPisanty Let's just say it was really on topic.
 
9:40 PM
I mean, if one wants to poke fun at certain parts of modern physics with regards to how speculative and out there it can be, I can't say I entirely object
e.g. the struggle of string theory / SUSY to find something that can actually be empirically validated
 
@Semiclassical What pissed me off was the dismissive and quite condescending attitude AFTER I took the time to seriously read and analyze the paper
 
yeah
why I like the Kant quote is that it captures how important it is to have stuff to push against when it comes to scientific questions
 
@0celo7 For the second question, I think the PD of the generating classes of the meridians of $T^n$ have that.
$H_*(T^n)$ is an exterior algebra on those
 
without that, you're more likely to convince yourself that you're making progress than to actually do so
 
@0celo7 I need to get hands-on practice with those cup and cap products... They're pointing their noses left and right in a few talks I attend.
@Semiclassical Indeed
 
9:45 PM
I'm fine with being poetic. But one shouldn't conflate that with the actual labor of mathematical/scientific investigation
2
 
@BalarkaSen but even when you connect sum with M?
 
@0celo7 The ring structure doesn't change. There's a map $M \# T^n \to M \vee T^n$ given by punching the sphere along which you took the connected sum. This gives a map $H^*(M \vee T^n) \to H^*(M \# T^n)$
 
Should I put my face in that pink square beside my name?
 
what is * in M*T^n
 
Wedge sum
 
9:47 PM
why not \vee
whatever
is that map an isomorphism?
 
cuz I forgot notation whoops
 
@BalarkaSen I don't get the point of this comment
 
Anybody have a clue on how to generalize the coordinate ring (seen as a formal dual) of Lie group to the Weyl group?
From what I tried, it seems to lead to something infinitely generated, which in that form, is not that useful...
 
@BalarkaSen I think the (co)homology in this paper is done really shittily and not correctly
 
@G.Bergeron that's unfortunately par for the course with that user
 
@EmilioPisanty Yeah well I learn my lesson and will never again engage seriously... -_-
learned?
 
So, you know, engage at your peril. Can be worth it if the paper is good, but unfortunately you can't count on a proper engagement in return if you do.
@Semiclassical there was an excellent bit on this subject in A serious man
 
@Semiclassical Yeah, but poor debate is different than calling every physicists clueless and telling me I have low self-esteem. Like, ok, I was just trying to help.
 
@0celo7 Sorry I ran away for a while. That map is an isomorphism in grade less than $\dim M + n$, me thinks
 
I'm fine with people making portentous statements tbh so long as they acknowledge that they're not a demonstration
 
10:02 PM
Roughly saying "the math is what counts, the stories are just stories"
 
I don't have a quick argument though
 
@BalarkaSen I'm going to restart this conversation when I get through the rest of the paper
 
@EmilioPisanty :)
 
Unlikely to be youtubeable though
 
just looking at the rest of the proof, it seems very shady
no need spending time on that now
 
10:03 PM
Mmmk
 
It's a great film and well worth watching in any case
 
@EmilioPisanty about?
 
@G.Bergeron hard to define
Life?
I guess
 
It's not a movie about a specific plot from what I know about it
I mean, there are certainly events and themes
 
10:05 PM
It's a Coen Brothers movie, for reference
 
I'll go check the trailer, anyway I've slept too little to be really efficient in my work now. Damn those early dentist appointment...
 
@G.Bergeron I don't know if the trailer will do it justice
 
Was the greek quote conversation with le epic troller and paradigm shifter
 
The chalkboard might
 
@BalarkaSen probably?
 
10:07 PM
That's one annoying fellow
 
who is le epic troller
 
@EmilioPisanty Well, there was a dentist in it! :p But, yeah, it's not one of those crazy post-modern movie like that one ''Pi''?
 
@0celo7 the Very Ziggy Nun
 
Milo?
I don't get it
 
I wouldn't say it's crazy post-modern from what I know of it
 
10:09 PM
you're thinking too hard
vzn
 
though I think there is a certain magical realism to it
 
I don't think he's a troll...
 
Eh. I find him more frustrating than trolly
 
I think he's a crank but that's not Nice
 
@0celo7 Doubt it
 
10:10 PM
(mr last letter of the alphabet over in the math chat, by contrast, is one of the few people I've actually Ignored in a while)
 
I think Balarka is projecting tbh o_O
 
@Semiclassical Ok ok
 
Accurate impression
@Semiclassical I kinda like that guy
 
I very much don't.
 
please delete that
 
10:11 PM
@BalarkaSen that's also not nice
 
@DavidZ What if I haven't specified who is an idiot
 
@BalarkaSen Are you the kind of person going through youtubes comments to find the worse? :P
 
That guy over there is an idiot
Is that not nice?
@G.Bergeron Lol that's kind of spot on
 
@BalarkaSen it doesn't really matter. Though of course none of this is as bad as saying "you're an idiot" to someone in the chat.
 
@BalarkaSen I do that sometimes. I call it ''ethnological field trip''
 
10:13 PM
@DavidZ I was joking around a little. I deleted it.
 
I think that's a pretty silly standard, tbh. But it's one which is consistent with what's been said before.
 
@DavidZ Are there written rules to this chat somewhere?
 
LOL
 
I don't mind people who I disagree with, but I do mind people who (intentionally or otherwise) invalidate my or other people's experiences
 
Well I think it's mostly a sign of shallow thinking skills to have radical views about most things
 
10:14 PM
@BalarkaSen Cool. This one wasn't such a big deal though, because of its abstractness... I think deleting it is good choice, but not strictly necessary.
 
The person in question doing so was what moved him from "tolerate" to the "ignore and forget about" category
 
@G.Bergeron good question :P
 
Which is what mr last letter does
 
@BalarkaSen Given $\alpha\in H^1(M,Z)$, can one always find a smooth hypersurface in the class of $D(\alpha)$? I was under the impression this is not possible with integer coefficients.
 
@DavidZ I am sometime not sure where is the line, not necessarily in cases as above, but also, say, about acceptable topics, etc.
 
10:16 PM
@DavidZ Yup, I understood :) I intended to delete it in a minute or so actually
 
In general the expectations for people's conduct are pretty similar to what is in the help center, e.g. don't be rude, do be accommodating of other people's views. But it's not as clearly specified as on the main and meta sites.
 
@BalarkaSen From what I'm currently listening: ''Dude, your mixes are out of this world, I feel like I'm travelling through a robots intestines.''
 
@G.Bergeron As far as topics, in general anything is fine to discuss here, though it's best to stay away from certain topics that are widely considered not suitable for public conversation (e.g. sex, that's come up in the past I think).
 
@DavidZ So ''PG13''?
 
Something like that
 
10:20 PM
@BalarkaSen My only regret from my last interaction with them was not linking this: penny-arcade.com/comic/2006/04/10/i-hope-you-like-text
 
@BalarkaSen please, the suspense is killing me
 
@0celo7 I am multi-talking!!!
 
I need to go eat, but have to know the answer to this first
 
Ok so $D(\alpha)$ is an element of $H_{n-1}(M; \Bbb Z)$
 
yes
 
10:22 PM
I believe it's true that every integral homology class in codimension 1 can be represented by a submanifold
 
oh really
amazing
oh yeah
 
I don't like homology in Z...
 
36
Q: When is a Homology Class Represented by a Submanifold?

Steve Possible Duplicate: Cohomology and fundamental classes Given an oriented manifold $M$ and an oriented submanifold $\phi:N\to M$ we can obtain a homology class $\phi_*[N]\in H_*(M)$ where $[N]$ is the fundamental class of $N$. In general, it is not true that every hom...

 
the canonical MO answer says that
 
yup
 
10:24 PM
not that I understand the proof
Yau lives on for another day
 
Lol
It takes more work to unwrap the proof given
I tried once
 
I need to understand it
it must go in the thesis
chapter 6 of the thesis is alg.top $\cap$ GMT
the horror
 
I can recall and shoot the proof at you some time after
@G.Bergeron Amazing
 
@BalarkaSen Not that it is particularly surprising, it's a psytrance mix.
 
Wanna link me?
 
10:26 PM
@EmilioPisanty It sure sounds like it, but I don't think it is an actual German word.
 
1667
Astronomyastronomy.stackexchange.com

Beta Q&A site for astronomers and astrophysicists.

Currently in public beta.

Jesus
Just 19 visits a day?
 
@SirCumference most people dislike astronomy, no?
 
@BalarkaSen Doubt you'll like... but ok youtube.com/watch?v=1cyVgOdXVPc
 
not bashing it, most people dislike my field too
not that I blame them
 
@0celo7 If you mean the general public, I'd imagine they are far more interested with the stars than mathematics
I mean no one writes pop-math books like they do pop-science
 
10:28 PM
@0celo7 algebraic topology? If yes, then I don't see people disliking it!
 
I think the average person is concerned with the waste of money that telescopes are
@G.Bergeron I'm no algebraic topologist
I am just dealing with homological variational theory right now
 
C'mon, astronomy is probably the most interesting branch of physics to the general public. The vast majority of pop science books are based on it
 
I think the average person is more interested in astrology than astronomy unfortunately
2
 
@SirCumference Probably, though it wasn't the case for me.
@Semiclassical LOL
 
@Semiclassical Hah I saw that comic from earlier
It's 100/10
 
10:29 PM
I am going to have a 2-page definition. Ugh.
 
@Semiclassical "There's a difference?"
 
@SirCumference .…
 
@Semiclassical Note the quotation marks :P
 
Ow
Yeah, fair
I think astronomy has a better shot at public consciousness than a lot of other disciplines though
 
Tbh when I mention studying astronomy, the most common questions are "can you read my fortune?", "so you must be a fan of Neil deGrasse Tyson, right?" and "what's the seventh planet in the solar system? durr durr durr"
 
10:32 PM
@G.Bergeron This sounds like some cyberpunko-futuristic shit
 
@BalarkaSen Is it simpler if one considers $H_k(M;\mathbb{R})$?
@BalarkaSen It's called psytrance, short for psychedelic trance. You know dirty hippies on a beach in Goa... ore rather what evolved from that.
 
Lol
 
@SirCumference What about Planet X or nibiru stuff?
 
@G.Bergeron Most people have never heard of those hypotheses
 
@SirCumference Or where was the moon landing film after all?
 
10:34 PM
I have heard some trance music of various sorts, so I could catch that this falls into the genre
 
@SirCumference I though it was all the rage with that 21 December 2012 end of mayan calendar.
 
@G.Bergeron Well you get people like this
 
@SirCumference It's a new language base around ''not'' ''very'' and ''good''. It turned out to be not very good.
2
 
Yep sounds right
 
@G.Bergeron This has some of my favorite comments of all time
"When I grow up, I want to be a stuffed capsicum"
 
10:39 PM
@SirCumference On topic: When we were little, my father showed us a documentary about the fake moon landing that was well made and all. The goal of the entire exercise was to show us what bullshit looks like with a nice coating of paint.
 
vzn
@0celo7 lol thx for the ringing endorsement :P
 
"I am a zygote and i love this kind of music. I don't understand why my fellow pre-embryonic eukariote colleagues only listen to chart music but i've been told i'm an old soul"
 
@BalarkaSen Poetic fail to my ears...
 
@G.Bergeron to clarify: the moon landing was BS, or a documentary trying to debunk it was BS?
 
@EmilioPisanty Yes.
 
10:42 PM
@Semiclassical Don't worry, the documentary was. :p
 
Oh good
 
vzn
@G.Bergeron many thx for reading the paper & the analysis/ discussion (on the other hand, its not mine...). fyi the snarky comments were performance art, sorry you took them seriously/ personally
 
“We are what we pretend to be, so we must be careful about what we pretend to be.”
 
vzn
@Semiclassical yeah and theres also this thing called projection identification in psychology... o_O
 
@vzn Look, I don't hold a grudge. It's just the entire exercise seemed like a waste of time in the end.
In any case, I have to leave for home
See you people
 
vzn
10:49 PM
@G.Bergeron am sorry to hear/ disappointed you think it was a "waste". not sure what you expected. sorry the discussion and/ or paper did not meet your expectations. enjoyed it myself at time but will have to reevaluate that feeling now.
 
theres also this thing called nAmEdRopPing in conversations OoOooo_oooOOo
 
vzn
yeah lots of "droppings" on my name in here lately :(
 
11:07 PM
@vzn you're the trump of the hbar
 
@vzn maybe you should evaluate whether that's because you treat interactions with others as "performance art" (?) instead of actual scientific discussions
 
@BalarkaSen Is this MO thing really that hard to make precise?
I guess it's not clear why the preimage represents the homology class?
 
170
A: I believe I have solved a famous open problem. How do I convince people in the field that I am not a crank?

KavehFirst, make sure you are not really a crank before trying to convince others. Read these common characteristics of cranks. If they apply to you then get professional help. For the rest of the answer I will assume that you have really solved a famous open problem. In the following "he" refers...

Cf. in particular point 4 in this answer
> He does not understand it is not a puzzle.
We do care about physics.
 
11:22 PM
@BalarkaSen this is Theorem VI.11.16 in Bredon
good enough for me
 
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