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12:16 AM
@ACuriousMind damn
so I need to run for mod and win to scratch that curiosity itch? I'm tempted to say it's worth it.
@Slereah boy, is that ever a snobbing of quantum gravity
didn't even get an actual equation
 
 
1 hour later…
vzn
1:26 AM
@EmilioPisanty agree with your sentiments on this but theres a Circle-the-Wagons mentality among the Powers That Be and (speaking from "experience learned the hard way") its not an area for open discussion despite some pretenses to the contrary.
 
@vzn I'm not sure you understand my sentiments on this
if you're basing your understanding of them based on the comments on this thread, there is simply not enough evidence for you to do so.
 
vzn
@EmilioPisanty sigh am just agreeing with your words, if they mean something different than what they say, then yes am not aware of that.
 
@vzn then say that you agree with the words and don't act as though you know the sentiment behind it
 
vzn
@EmilioPisanty ?!? dont know what youre talking about bye
 
particularly if you're going to go on to rail about "Circle-the-Wagons mentality" which is, frankly, entirely justified in this setting.
@vzn so shall we write this up as yet another missed opportunity to go for any meaningful debate, then? the old "turn tail as soon as anyone disagrees" routine?
 
vzn
1:32 AM
@EmilioPisanty ?!? you just said the suspension length was ridiculous, whatever (btw others eg bolbteppa have mentioned that iirc in the admin room, dont know if it got deleted though) chat.stackexchange.com/rooms/13775/physics-meta
@EmilioPisanty why do you want to debate when someone just said they agree with you? does that realliy say anything about me personally?
 
@vzn re-read your comments, and my initial response, carefully.
 
vzn
@EmilioPisanty maybe you know more about my feelings than me?
 
@vzn you said that you "agree with [my] sentiments". I simply cautioned you that you don't know what those are.
 
vzn
@EmilioPisanty agree with your stated sentiments
ps congrats on your reccent paper it looks cool :)
 
@vzn thanks
Don't distribute it too widely, though. It still needs to go through peer review.
 
vzn
1:37 AM
lol anyway hope it makes it into the glossy periodical as intended, that woiuld surely be a coup, not merely for you either :)
 
Indeed it would.
 
vzn
2:22 AM
lol more on groupthink & the "5 monkeys experiment" throwcase.com/2014/12/21/…
 
 
4 hours later…
6:23 AM
Anyone know a good book as an intro to mathematical logic?
Something focused pretty deeply on it
 
 
3 hours later…
9:48 AM
@SirCumference I've heard good things about Halmos' book, but never looked at it myself.
 
 
2 hours later…
11:48 AM
@Loong Try this: Hook a computer to the Predictor, if the solution to the halting problem is true, then press the button, if the solution to the halting problem is false, then don't press the button. Let's see how the Predictor will respond to that
then nature will have to make the choice of making us lose all freewill, or to become an oracle and answer every question we throw to it, including a question whose answer will delete reality from existence
either way, nature is screwed
 
bad video
he misses another winning move
"and then this is the key move here" NO
there are 2 winning moves, picking one is not "the key move"
 
@user54826 better video
 
thanks though im not exactly a fan of tic tac toe :)
i prefer crazyhouse
 
12:04 PM
what about checkers?
aka draughts
 
not a fan
 
In philosophy and mathematics, Newcomb's paradox, also referred to as Newcomb's problem, is a thought experiment involving a game between two players, one of whom purports to be able to predict the future. Whether the problem actually is a paradox is disputed. Newcomb's paradox was created by William Newcomb of the University of California's Lawrence Livermore Laboratory. However, it was first analyzed and was published in a philosophy paper spread to the philosophical community by Robert Nozick in 1969, and appeared in the March 1973 issue of Scientific American, in Martin Gardner's "Mathematical...
To outsmart a predictor: Force its condition where the button to be pressed will undermine its own existence
I am never short of elaborate ransom methods on reality itself
 
changing topic temporarily (I dont want to troll), did you know it's possible to measure the resistance of a wire without passing any voltage nor current
simply by measuring the voltage across it
 
but you cannot get voltage reading if there is no current passing through a voltmeter and there is no infinite resistors
Voltage as we measured experimentally is a function of current
 
you don't pass any current. this does not imply there is no current in it
there will be a current due to thermal agitation
so there's a non zero voltage thanks to temperature
there's no need to apply any current nor voltage
 
12:15 PM
but that will only cause the zero value to oscillate, thus is part of background noise, thus that voltage is not the voltage measured in your wire?
 
you measure the noise voltage. this noise contains the information of the resistance of the wire
in general, noise contain a lot of information, this is known since at least 70 years
before that, people thought it was useless
I thought the same, until yesterday when I read up about that
 
How much % of information can we recover from that thermal noise, can we get that more than 80%?
 
depends. you need to know the temperature, so you need to perform the measuremnet of temperature and of the voltage itself
but in principle the noise contains R perfectly
 
Johnson–Nyquist noise (thermal noise, Johnson noise, or Nyquist noise) is the electronic noise generated by the thermal agitation of the charge carriers (usually the electrons) inside an electrical conductor at equilibrium, which happens regardless of any applied voltage. The generic, statistical physical derivation of this noise is called the fluctuation-dissipation theorem, where generalized impedance or generalized susceptibility is used to characterize the medium. Thermal noise in an ideal resistor is approximately white, meaning that the power spectral density is nearly constant throughout...
hmm...
 
the 1st page, Introduction
it looks like fascinating, we can extract a shitton of info from the noise
 
12:22 PM
Ah I see
Well, if I recall, they are now controlling and directing noise to where it is needed in some quantum and thermoelectric devices, let me see if I can dig out those links
 
I'd be interested with regards to the thermoelectric stuff ^^
 
Here's a quantum example, where interference is used to direct thermal noise to certain regions. It is still theory though
and the arxiv
 
i basically have 0 knowledge in noise. i ordered an old book (but looks promising) on it, yesterday
 
I knew little about noise other than the different "colors" and flucturation theorem
 
12:57 PM
Anyone knows what happened here?
 
@TheDarkSide What do you mean?
 
@ACuriousMind Hi. Long time. What I meant was this post appears to have some historical significance, but this gibberish answer bumps it to the front page. It is interesting since it doesn't violate any rule I know :D
Yet it is gibberish.
 
@TheDarkSide I don't understand what you mean. The entire post is a sandbox, it is precisely of no significance at all.
 
> You can use this question as a formatting sandbox (if you can edit CW questions), and you can post answers if you want to test out formatting there as well.
 
Do we need this?
 
Anonymous
1:03 PM
That thread shouldn't have been bumped to the front page since it has a score less than -8
 
Anonymous
"It's hard not to like this meta post but please keep it downvoted to -8 or less (rather than upvoting it) so that it doesn't get bumped to the front meta page every time someone is experimenting in the sandbox, which usually is not interesting to anyone else."
 
Anonymous
But is that true any longer?
 
I don't see it on the front page.
 
Anonymous
 
Anonymous
@Loong I see it on the active page
 
1:05 PM
@Blue The front page is not the same as the questions/active tab :P
 
@Loong Active tab.
 
Anonymous
 
Anonymous
Oh, this is what you call front page? ^
 
Anonymous
Hmm, makes sense
 
Anonymous
I hardly ever check that "Top Questions" page
 
1:07 PM
@ACuriousMind So, this is meant for people who are learning how to post and want to test out the formatting help provided?
 
Anonymous
Although new users who come from search engines are directed to that page I suppose
 
Wouldn't it be better if they delete their posts after testing this out?
Just saying.
 
Deleting answers would not solve the problem for users with 10 000 reputation. And a collection of examples could actually be useful for other users.
 
Anonymous
I wonder how the Physics site looks for 10k+ users. Probably a mess on the more popular threads.
 
1:17 PM
@Loong Alright, if it is alright. Though I didn't get the premise of retaining such posts, individual "confusions" would all differ, and it is more likely people will create a new post to suit their interests rather than skim though a glossary of earlier posts.
But alright. If it alright.
Second question (on visiting this site after a long time): Over here, is "92" referring to 2092, or is this some bug that (somehow) refers to 1992.
 
14 hours ago, by ACuriousMind
@EmilioPisanty I've asked and it's indeed simply "as permanent as the system currently allows". As to what triggered the lengthening all I can tell you is that it wasn't a random act.
Why are y'all compulsively checking Ron's profile, anyway? :P
 
@ACuriousMind Ahh. That's been discussed !!
@ACuriousMind I think we have already established that we have a lot of fan-girls on this site :P
Jokes aside, for me, the progression was (One of his answers) -> (Wow again) -> (Check how many of his great answers can I understand now) -> (Profile visit) -> (Good Lord, what did he do now?)
And of course, as Emilio brought up:
16 hours ago, by Emilio Pisanty
I'd like to understand how network-wide suspensions get their term extended from ten years into longer timespans while still in the middle of the suspension
It is strange how everyone has the same question:
16 hours ago, by Emilio Pisanty
Does that '92 mean 2092? as in seventy-four-years-from-now 2092?
 
someone has to check the logs of the server hosting PSE
who modified the date and why and how and when
changing topic (I dont want to troll by randomly picking a new topic).... for those who write papers
do you use websites like sharelatex or overleaf?
if not, how do you collaborate with people with the writing of the doc?
 
Anonymous
1:39 PM
@user54826 Yeah, I like ShareLaTeX more than Overleaf. It just seems more "steady" than Overleaf.
 
what do you mean by steady?
 
Anonymous
Almost everyone I know in our group uses ShareLaTeX. But professors tend to be more old school and use software like TeXMaker.
 
btw witih overleaf (the free version), there is no history so we cannot revert to older versions. but it's easy to sync with github, so this limitation is like... stupid
 
Anonymous
@user54826 Better autocomplete, overall layout, file uploading seems more intuitive, etc
 
Anonymous
Anyhow, Overleaf bought ShareLaTeX
 
Anonymous
1:47 PM
So in the future we'll probably have the best of both
 
Anonymous
Also, you have got 4 different compilers on ShareLaTeX and a chat on the side. Moreover, you can see all the edits your collaborators are making in real time and scold them via chat if they're writing nonsense :P
 
i didnt know that...
i went for overleaf but i really didnt know which to pick.
 
it's a bit outdated
now overleaf has a v2 thing
 
Anonymous
v2 is in beta though
 
Anonymous
1:52 PM
But yeah
 
Anonymous
I really want to buy the paid version, but it's expensive :/
 
Anonymous
If it were a "one-time" thing I'd be fine. But monthly is a bit too much.
 
that sucks yeah. personally I would never pay for any software
but im a special case, i've been lobotomized by richard stallman ^^
 
Anonymous
Nah, we've had more extreme cases of you on this chat. You'd know if you spoke to Bernardo!
 
never heard of him
 
Anonymous
2:01 PM
Dec 29 '17 at 17:27, by Bernardo Meurer
Your universe can't attack me, I'm protected by 4 layers of FREE SOFTWARE VMs
 
ah a paranoid :)
 
that's quite a conversation
 
3:04 PM
@user1732 QED?
 
Anonymous
@SirCumference ?
 
Jesus I thought you were linking a quantum electrodynamics book for some reason
 
Anonymous
@SirCumference Tao writing a quantum electrodynamics book....well not that improbable either :P
 
@SirCumference 'QED' stands for quod erat demonstrandum (roughly "that which was to show") and was the traditional way of closing proofs before the little box gained popularity.
 
@ACuriousMind Yeah I know, I'm just a physicist before a mathematician :P
Although I am so used to the tombstone I forgot about QED
 
3:17 PM
Hello everybuddy!
 
Howdy
 
Anonymous
Good evening
 
Anonymous
More like good night actually
 
It's morning here :P
 
@Blue it's good night for the weak
;)
 
Anonymous
3:20 PM
@AvnishKabaj Umm?
 
@user1732 The website is confusing me ;-;
 
Just watched Doctor Strange for the 5th time. I don't know but I always relate the stuff he does to physics and weird quantum mechanics.
 
@ACuriousMind You mean Algebraic Logic?
 
4:13 PM
@Blue I meant it was just 9 back then
 
Anonymous
@AvnishKabaj Ah. My "good night'' 's don't necessarily imply "I'm going to sleep now" ;)
 
Anonymous
(although my "good morning" 's might imply that often)
 
my good mornings are usually more "hmgmrg"
 
Quick question. Why in a microwave, microwaves are used to heat the food? I mean, why can't waves with other frequency be used? Such as radio waves or uv?
 
Anonymous
4:29 PM
Because they aren't readily absorbed?
 
Anonymous
Microwaves
 
@IceInkberry Why?
 
Anonymous
11
A: Why do we use microwaves in microwave oven?

John RennieIn microwave ovens what matters is how much energy the radiation carries and how that energy is absorbed by the food. Visible light and IR are rapidly absorbed by most foods, so they would only heat the outer layer of the food. You'd get food with the outside carbonised and the inside raw. Micro...

 
Ah... I should learn to google stuff :D
 
@Blue o.O
@IceInkberry you googled that dint you
 
4:42 PM
Why are microwaves harded to absorb by foods? Is it because their wavelength matches something?
 
Anonymous
24
A: How do microwaves heat moisture-free items?

GeoffreyMicrowave heating is largely caused by the changing electric and magnetic fields (i.e. the "microwaves") which are emitted by your microwave oven affecting polar molecules. As the direction of the electric field changes over time, the polar molecules (often, of water) attempt to follow the field ...

 
Anonymous
1
A: Why does the food in the microwave heat up but the bowl doesn't?

user108787The microwaves are primarily designed to vibrate/heat the water molecules in the food, as a way of ensuring that the foods gets cooked evenly. An aid to this process is the rotating plate within the machine. Microwaves that cook your food pass through plastics, glass, and ceramics, with mimima...

 
@Blue You are my personal googling machine xD (joking)
 
Anonymous
Heh XD
 
Anonymous
PSE has a lot of good answers on these type of general questions
 
5:26 PM
Hmm, I'm learning about lenses and is this what happens in reality? (My drawing, not very good)
I tried to apply the refraction car anology in my head.
If there is anything wrong, please correct me.
Also I think it's really cool how the concave/diverging lense has a virtual focal point.
 
lenses are a bit of a weird topic in physics, insofar as they're quite evidently applications of physics principles
in that sense they're more of an engineering topic.
 
Cool, @Semiclassical is my drawing right?
 
(there are other examples of that in intro physics of course, but lenses are a particularly stark example)
they look right. of course, no ray-tracing diagram is ever going to be truly 'realistic': you do these drawings under certain approximations
 
Sorry, what do you mean?
 
well, consider the difference between a circular mirror and a parabolic one
strictly speaking, the idea of a curved mirror having a focus only applies to the latter
in other words, if you shoot multiple parallel rays at a curved mirror and see how they reflect, they in general won't cross perfectly at a common focus
that's only strictly true for a parabolic mirror
 
5:40 PM
Why? Because the mirror won't be the perfect shape?
 
yeah.
 
Well, if the mirror is engineered perfectly enough, what's the problem?
 
well, that depends. if your mirror is engineered as a parabola, then sure
but if it's engineered as a circle---and that's a lot simpler to do in practice---then it's simply not going to be true that it'll have a single focus
 
What do you mean 'as a circle'?
 
as in, a circular cross-section
well, semicircle
like how you'd draw a curved mirror in a ray-tracing diagram
if you draw it as a circular arc, then strictly speaking the rays won't all land at a focus
the geometry of a circular arc simply isn't consistent with that.
however, if you're dealing with a mirror with fairly low curvature, this isn't really a big deal since the difference between a parabolic and a circular arc is pretty small
so if you've got a mirror with low curvature, you can approximate it as a parabolic mirror and therefore get a single focus
 
5:45 PM
Do you mean that type of mirror?
Or a semi concave one?
 
that first image isn't loading for me. (i'm behind an annoying firewall atm, so that might be why)
 
Ok, sorry, I don't understand when (at what type of mirror) there won't be a perfect focal point?
 
basically, there's no perfect focal point unless you're dealing with a parabolic mirror
 
Anonymous
Moreover your ray diagrams are valid only with the paraxial ray approximation
 
yah, that too
that in particular means that rays drawn near the edge of a lens will be the least realistic.
since that's where the angle of the ray to the surface normal of the lens is largest
 
5:51 PM
But the ray would still go through the focal point?
What do you mean, "least realistic"?
 
in the paraxial approximation, yes. in reality, probably not
 
What is paraxial approximation in short?
 
suppose you draw a horizontal ray that passes just above the lens
 
in that case, the ray never actually enters the lens and therefore will not be bent
 
Anonymous
5:52 PM
 
Anonymous
For reference ^
 
now suppose you move the ray down slightly. subject to the usual assumptions of ray-tracing, this ray would enter the lens at its very edge and therefore come out bent in order to pass through the focus
so you'd have the ray changing direction sharply based on that small change in height
but that doesn't make a lot of sense: it only passes through a very small portion of the lens, so that much bending doesn't seem realistic
 
Hmm, I can't imagine it. In the pic that @Blue sent, the paraxial ray looks just like marginal ray.
(but for some reason doesn't land on the focal point)
 
Anonymous
Paraxial = Close to Principal Axis
 
Anonymous
Marginal = Away from Principal Axis
 
5:57 PM
I get what that diagram means, but I think it by itself isn't very clear
 
Why would a horizontal ray closer to the axis of thecenter of the lense (principal axis) not land on the focal point?
 
It would---or, at least, it'd land sufficiently close to the focal point as to be indistinguishable. The point is that rays which are farther from the principal axis will cross farther from the focal point.
I guess the point is this. The action of a lens is ultimately based off of Snell's law, since the light rays will refract at both sides of the lens
However, actually using Snell's law in such scenarios is a big pain
it's quite tedious
 
I don't see any difference between these two except the weirdness about the rays not landing on the focal point in the second picture.
 
(this firewall isn't letting me see anything hosted on non-US sites, which is...really dumb)
 
As @Blue said, a paraxial ray is a ray that is closer to the center of a convex lens. Why would such a ray not land exactly on the focal point?
 
Anonymous
6:06 PM
@NovaliumCompany Counter question: Can you derive $f=R/2$ and state the approximations used to arrive at that equation?
 
You want me to find the derivative of it?
 
Derive = deduce
@NovaliumCompany one thing to notice in that first drawing: where are the rays bending?
 
On the convex example?
 
6:08 PM
yes. in the first diagram you linked (with blue lens) where do the rays actually change direction?
 
Well, in my opinion they should change direction at the left surface of the lens, but for some reason they change when there are in the middle of the lense.
 
right. that means that this picture isn't going to be truly realistic
 
Yep, it isn't mine so.
I thought to derive something meant to find the derivative?
 
by contrast, the second picture correctly has it be the case that the rays change direction twice
Not in this context. All it means is "can you deduce this equation from first principles"
So derive in the common sense of extract/obtain rather than the sense of calculus
 
Ok, so what am I supposed to do with he equation @Blue gave me?
 
6:13 PM
Let me ask it a little differently. How do you know that, if you draw a horizontal ray through a convex lens, that it'll pass through the focal point?
 
Well, I thought if the lens has the right shape, it would?
 
How would you show it?
 
Well, I can't if I have to be honest ;\
 
A lens is just a piece of glass whose edges are taken to be circular arcs
so one should be able to validate a given ray using snell's law
 
Yep, and If I apply refraction and snell's law, I can see if the rays match up to the focal point?
 
6:17 PM
Well, there's the thing: If you apply refraction and snell's law, you'll actually get a picture like the second one
 
Hmm, seriously? Why?
 
Because that's what the lens geometry forces, basically.
The point is that the statement "the transmitted rays of a convex lens pass through a common focal point" is not a principle of nature. rather, it's an approximate statement
 
Hmm, so the first pic with the blue lens is not right?
 
Correct. It's wrong on two accounts. One is that the internal rays will be straight inside the lens, i.e. the bend occurs at the edges of the lens not at its center
the other is that the rays which exit the lens do not actually all land nicely on the focal point.
 
Yep, and the rays closer to the principal axis won't perfectly align with the focal point?
 
6:21 PM
no. the rays closer to the principal axis will land closer to the focal point than the rays which are farther from the principal axis
the marginal focal point is not the paraxial focal point, but it's the paraxial focal point which you use in ray-tracing
in that black-white diagram, it's point B which is the 'usual' focal point
point A is where the marginal rays will focus.
 
So the marginal rays (the rays away from the principal axis) are the ones actually away from the focal point (usual, paraxial focal point)?
 
right
 
Cool, I'm wearing contact lenses in the eyes at the moment. If the rays don't land on one point, doesn't this mess up my vision?
 
in principal, yes.
however, if your lens isn't highly curved, then the amount of difference between A and B isn't very big
 
Hmm, ok. Is it possible to engineer a lense where all rays would land exactly on one focal point?
 
6:26 PM
yeah. it'd be the lens equivalent of a parabolic mirror
note, though, that this issue (spherical aberration) is one which can develop naturally in the eye
as such, one can actually use the spherical aberration of a contact lens so as to counteract this effect
so spherical aberration can itself be a tool
this starts to get into serious engineering territory, though
 
So the natural lens of an eye actually has the margin, paraxial ray focal point difference and the contact lenses can be used to fix that? (as well as the vision)
 
nvm
Do you see this?
 
that's neat
 
There is no margin, paraxial ray focal point difference I guess?
 
6:30 PM
right.
 
Cool.
I'll be going for now. Thanks @Semiclassical and @Blue for the help, it's always pleasure chatting with you :3
 
I suspect it's not quite perfect either, since a fresnel lens (in that picture) is built out of smaller pieces whereas the parabolic reflector is one smooth surface
but it's definitely a far closer approximation than a standard spherical lens would be
(this is altogether why I tend to dislike lenses as a subject: it's closer to an engineering topic than a physics topic imo)
 
Anonymous
There are several "more accurate" models of ray optics with lesser approximations but those tend to be extremely hard and very geometrical. I don't think people study those anymore though
 
yeah. i mean, i'm sure there are still some places where it's relevant
 
Anonymous
Maybe in the lens and optics industry.....haven't met anyone like that though :P
 
6:41 PM
in particular, if you do computer modelling of lenses you probably wouldn't use the paraxial approximation
but I'm not sure you'd use a ray treatment in any case. might as well just go over to a wave-based approach
 
Anonymous
True
 
i mean, the physics of light in that regime is really wave optics rather than ray optics
ray optics is a convenient approximation, but it's not fundamental
 
Anonymous
Yes, at that scale indeed wave optics would work better. But in the pre-wave optics age, Newtonian ray optics was sort of the norm and it is sort of amazing how far ray optics (by the way, I'm referring to the particle model of light) can go without invoking any wave optics.
 
ya
hard to deal with interference phenomena using only particles, tho
 
Anonymous
This seems like an interesting read:
 
Anonymous
6:56 PM
1
A: How did Newton explain his interference rings without wave optics?

Conifold"Newton rings" in thin plates were discovered not by Newton but by Hooke, and not in 1717 but in 1664 (Boyle described a similar phenomenon in soap bubbles at about the same time). Hooke reported his experiments in Micrographia (1665), from which Newton learned about them, "in which book he hath ...

 
Anonymous
Albeit wrong, the brilliant attempts at explaining the typically wave optics phenomena using particles only, are commendable. :P
 
the interesting point in there was that "[Newton's] considered position was actually a hybrid of corpuscular and wave theories"
so "particle + wave" rather than "particle" or "wave"
which is sorta amusing given how much one uses 'Newtonian' as a shorthand for 'corpuscular'
 
Anonymous
@Semiclassical Yeah, but I guess Newton gradually shited to that stance in the later stages of his career (?). I thought that he very strongly opposed all wave notions initially.
 
"history is complicated"
 
Anonymous
7:02 PM
Heh
 
"physics attributed to Newton" is not equivalent to "all physics as envisioned by Newton"
 
7:27 PM
are the mathematicians around?
I need some help halp
say that you have a manifold $M$ of dimension $n$
is it possible to have $m>n$ functions $f_i:M\to \mathbb R$ which are functionally independent? i.e. such that none of the $f_i$ is a function of the others
and if so, doesn't that contradict the fact that the manifold has dimension $n$?
 
@EmilioPisanty What is your exact definition of "functionally independent" here?
 
@ACuriousMind I'm reading arxiv.org/abs/1803.01269
as a follow-up to doi.org/10.1063/1.4895466
 
That's not really an answer to my question ;)
 
> Similarly, a set of isotropic invariants $f_1, \ldots, f_m$ of A is said to be functionally irreducible if none of them can be a function of the others. An integrity basis of A is said to be a minimal integrity basis of A if it is polynomially irreducible, and a function basis of A is said to be an irreducible function basis of A if it is functionally irreducible
in Olive & Auffray's terms,
> A functional basis is said to be irreducible if none of its elements can be expressed as a function of the others. It is worth noting that this definition does not preclude that some functional relations between generators exist
so Chen et al. show that there is an eleven-member irreducible function[al] basis for third-order symmetric tensors in three dimensions
but that space has dimension ten
and, in addition, those functions are SO(3) invariant, which by manifold-dimension counting should knock out three dimensions, leaving only room for seven independent 'shape' parameters
so it feels like there's something elementary that I missed somewhere
 
I think you are confusing the invariants with coordinate functions. No one demands that the invariants be invertible.
You're right that if these invariants were diffeomorphisms $V\to \mathbb{R}$, then you could build a chart $V\to\mathbb{R}^{11}$, indeed producing a contradiction. But they're not diffeomorphisms, so all is well
 
7:40 PM
@ACuriousMind what do you mean by that the invariants are not diffeomorphisms?
 
I mean something entirely different, sorry :D
Of course they're not diffeomorphisms
 
I mean, diffeomorphisms $V\to\mathbb R$ doesn't make sense to begin with
 
Is each $f$ in the paper not $f : \mathbb{R} \to \mathbb{R}$ and simply acting on the value of the $A_{i_1 i_2 \dots}$ tensors?
 
@bolbteppa no, the domain is the tensor space itself
they're each a different polynomial on the components of the tensor
 
@EmilioPisanty What I really mean is that the map $V\to\mathbb{R}^{11}$ where you map a tensor to its 11 invariants is not guaranteed to be bijective.
You get your contradiction only if it is a bijection and the inverse is continuous.
 
7:44 PM
@ACuriousMind so you're saying it fails to be injective, or does it fail to cover the domain?
I imagine the latter, right?
that would gel with
> It is worth noting that this definition does not preclude that some functional relations between generators exist
 
@EmilioPisanty Failure to be surjective would not be necessarily bad (charts are allowed to map only to open subsets), but I'd rather bet that it is not injective.
 
@ACuriousMind I'm thinking more in terms of the image of the chart having an empty interior
 
On the other hand, maybe it is injective and maps to a 10-dimensional subset of $\mathbb{R}^{11}$, i.e. it's an embedding of the symmetric tensors in $\mathbb{R}^{11}$.
 
but on the other hand, the dimension counting would imply that it can occupy at most a seven-dimensional submanifold of $\mathbb R^{11}$
 
@EmilioPisanty Yeah, it's not injective. By the very definition of the invariants, they map a tensor and its transform by any orthogonal matrix to the same thing.
 
7:48 PM
@ACuriousMind exactly
 
So...there's no problem here, right?
 
well, so here's the thing
I'm trying to split up the space of tensors into independent "shape" and "orientation" parameters
the idea being that you'll have three Euler-angle-like orientation parameters, and if you keep the shape parameters constant and range over the orientation, you'll get the full orbit
so presumably there'll be seven independent shape parameters
I would like to get those seven parameters explicitly
.... but along come Chen et al. to say that it's impossible to have fewer than eleven such parameters?
I figure I'm almost certainly misinterpreting that result, but I can't see how.
 
YES! Finally after 6 hours of nonstop coding, the 2D scanner is ready. Now I can finally RIP
 
@EmilioPisanty That would only be true if the result of quotienting out the orbits of $\mathrm{SO}(3)$ from the space would result in a space that nice in any way.
 
Get ready for a 2D Potential energy surface scan that will all finish in minutes!
 
7:53 PM
But the result is certainly not a manifold, and probably not even an orbifold.
So there's no guarantee it has nice "parameters" at all.
 
(and also, I imagine that I will also run into some of that orbifold trouble we talked about earlier, 'cause some of the orbits are going to be more symmetric than others. But one step at a time, I figure.)
@ACuriousMind yeah, I imagine
but still, my bet is that there will be places where things get "kinks" of some form or other, but that there will be large stretches of 'bulk' of the tensor space where some decomposition of that form needs to hold
 
All the invariants show is that there is a map from this space to $\mathbb{R}^{11}$, and that they're a functional basis means it's not possible to embed it in any lower dimension
But that doesn't mean there aren't local charts to $\mathbb{R}^7$, just that there's no global such chart.
 
ah, gotcha
so if I want a chart I just take those eleven invariants, look at the Jacobian, and trim it down until I get a nonsingular square matrix?
 
Ok they are functions on a space of tensors, ref. 17 is clearer, I don't think you are allowed to use calculus, ref. 17 clearly frames it in Hilbert basis theorem language
 
and the functional-basis irreducibility tells me that I will be unable to provide a single trimming set that works globally? i.e. that any such trimming will have some part of space where the Jacobian becomes singular?
@bolbteppa ref. 17 in olive & auffray or in chen et al?
 
7:59 PM
This stuff clearly has to work for modules over rings where you can't differentiate
 
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