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2:30 AM
@Relativisticcucumber I'd make minor changes to straighten the logic flow. 1) After Einstein published 1905 SR pair of papers, Mink studied it and realised that Mink metric can summarise a lot of ingredients that would make SR tick. The symmetry group is Poincaré. 2) It is found out that if you simply start from a scalar action that is SR invariant, implying Poincaré invariant, then EL equations will spit out a theory for you that obeys SR. 3) These are subsets. SR actions are subset of Poincaré actions,
In particular, theories that arise from SR actions are subsets of all possible SR theories. But this guaranteed link means that if we are to search for SR theories to try, it is vastly easier to search from SR actions first. Other physicists will have a much less rejection rate because this path is so well-trotten.
@SillyGoose The actual answer to this is that, if you assert that you found an SR theory that you want other people to accept as being fundamental, and it is not found from an SR action, then you would have to have entire chapters worth of proving work to make others accept that your SR theory actually is SR.
@SillyGoose I think I just answered this. It is not some "finite list of assumptions". It is that everybody is going to be lazy. Students are lazy, teachers are lazy, the professors are the laziest of the entire lecture theatre... Like, even if you found an SR theory another way, e.g. Dirac guessing his equation first, he would just need to find an SR action that derives his equation, and viola, Dirac theory is necessarily SR. Note that it is an SR action, not the SR action.
Of course, Dirac also put some effort into proving directly that the equations and their solutions actually are SR, but that is beside the point.
 
 
2 hours later…
4:50 AM
@naturallyInconsistent someone should make an Iceberg meme about this. Step 1: "everything is a point dipole", step 5: "wtf is even macroscopic energy" step 10:"reject macroscopic, return to microscopic" xD
 
5:40 AM
Correct
 
 
1 hour later…
6:48 AM
@Slereah Some great answers on both sides of the fence in that post now, also he put it on arxiv
 
lots of answers for the Kerr part but what's going on with the Schwarzschild part
The Kerr part seems to be very extendible tho
 
7:17 AM
What's with the recent Kerr talk?
 
7:29 AM
17
Q: Why does Roy Kerr claim that the Kerr black hole does not contain a singularity?

noir1993In a preprint posted on the arXiv, Roy Kerr claims that there is a widespread misunderstanding related to the singularity inside the black hole that bears his name. Can anyone explain his argument in "simple terms"? Are there any pitfalls in his reasoning? Here's a link to the preprint: https...

 
 
1 hour later…
8:39 AM
Hi all. was just wondering why there's such a complex derivations of noether's theorem(such as this weizmann.ac.il/particle/perez/Courses/QMII19/…) . article assumes that there's a lagrangian symmetry. I mean, if there's a symmetry, euler lagrange equation's right side already is 0, which means LHS is conserved. What's the point of so complex derivations ?
 
In the reddit, ChatGPT is contradicting Kerr...
 
as another example, en.wikipedia.org/wiki/…, on the image, the author says that middle parts' actions are equal and only change contributing is buffering pieces, but on the buffering pieces, he assumes that only speed changes. why does he assume that ? i think he uses the fact that $\frac{\partial L}{\partial x_1} + \frac{\partial L}{\partial x_2} = 0$, if so, i don't see what's the point of this derivation as EL is simpler
 
Derivation 1 says if you plug a symmetry transformation into the Lagrangian directly and it changes to first order as a total derivative (4 - 6), then this has to be equal to what you get when you abstractly vary the Lagrangian and invoke the equations of motion (7 - 8), where the equality is in (9-10) implying (11-12).
The second way is a poor explanation for the 'Noether' method, which is basically just equation (9) or (14) integrated, where in (9) or (14) the parameter is constant as it was all through the derivation, and you now notice that if it was local and did IBP you'd end up with the conserved current attached to the derivative of the local parameter.
Thus this is another way to find the conserved current, vary a Lagrangian, throw away every total derivative you can until you're left with stuff attached to the derivative of the local parameter, and you call that stuff the conserved current
 
8:54 AM
@bolbteppa thank you <3. though, if lagrangian doesn't change when applying transformation, that this means: $\frac{\partial L}{\partial x_1} + \frac{\partial L}{\partial x_2} = 0$, which means LHS of lagrangian is conserved
this seems super simple. is this something wrong with this way ?
$\frac{d}{dt}(\frac{\partial L}{\partial \dot x_1}) = \frac{\partial L}{\partial x_1}$ <br>
$\frac{d}{dt}(\frac{\partial L}{\partial \dot x_2}) = \frac{\partial L}{\partial x_2}$ <br>

Let's multiply both sides by $\epsilon$(equation won't change for sure).

$\epsilon(\frac{d}{dt}(\frac{\partial L}{\partial \dot x_1})) = \epsilon(\frac{\partial L}{\partial x_1})$ <br>
$\epsilon(\frac{d}{dt}(\frac{\partial L}{\partial \dot x_2})) = \epsilon(\frac{\partial L}{\partial x_2})$ <br>

$\epsilon(\frac{d}{dt}(\frac{\partial L}{\partial \dot x_1} + \frac{\partial L}{\partial \dot x_2})) = \epsilon(\
in the last equation, symmetry of lagrangian implies RHS is 0, which means left side is conserved
If you tell me that this way is not good because it doesn't help me find conserved quantity in all cases of problems and tell me that on the article I shared, that way helps to find conserved quantity in all cases, then sure, I would understand
 
Not sure what you're doing, but $L$ depends on more variables than $x_1$ and $x_2$, when you added those two they don't sum to zero
 
well, as many as it depends, my case can be used for as many as you want
 
The Lagrangian can change, in (4) for example it changes, if it changes it just needs to change up to a total derivative
 
if you understand how i got to: $\epsilon(\frac{d}{dt}(\frac{\partial L}{\partial \dot x_1} + \frac{\partial L}{\partial \dot x_2})) = \epsilon(\frac{\partial L}{\partial x_1} + \frac{\partial L}{\partial x_2})$, then what I am saying from here is: RHS basically says how much L changes when you change both $x_1$ and $x_2$ by $\epsilon$ and if you got symmetry, RHS is 0, so LHS is conserved
I think, my proof is for the cases, when Lagrangian doesn't change at all.
@bolbteppa on the image(en.wikipedia.org/wiki/…), can you tell me what logic he uses such as on the buffering pieces, he assumes only speed changes ?
 
Symmetry of the Lagrangian does not imply anything here necessarily has to vanish, if $L$ does not depend on $x_1$ and $x_2$ then the RHS will vanish
 
9:02 AM
yes, i was referring to the cases when $L$ doesn't depend on position variables or if it does, it depends on them such as $U(x_1 - x_2)$ in which case, my proof is solid. but for the cases when $L$ changes by total time derivative of some function, my proof is not covering that case
 
That picture is just trying to describe the case where $L = L(x,\dot{x}) = L(\dot{x})$ does not depend on $x$, so that $\delta S = \int L_{\dot{x}} \delta \dot{x} dt \approx L_{\dot{x}} \delta x |^2_1 = 0$ tells us that $p = L_{\dot{x}} = \partial L/\partial \dot{x}$ is conserved throughout the motion. In this case there is no 'total derivative' term in the variation of the action, i.e. the $J$ term is zero in (5) and (9),
we are intuitively showing the momentum at the end of (10) is conserved directly (i.e. I just integrated across assuming $L_{\dot{x}}$ is approximately constant), you need to argue like (10) to justify it properly over the whole region
You don't need to add your two equations in this case, they independently vanish, so the LHS involves two separately conserved quantities, whose sum is also conserved
 
so if the picture is only describing the scenario when $L$ doesn't depend on $x$, that case is like super simple one
@bolbteppa how did you figure out that the picture uses the scenario of $L(\dot x)$ and not $L(\dot x, x)$ ?
maybe they use $L(\dot x, x)$, but assumes that since symmetry happens, $\frac{\partial L}{\partial x_1} + \frac{\partial L}{\partial x_2 = 0$
 
The language is confusing but in the picture they say $\Delta S = \int \Delta L = \int \frac{\partial L}{\partial \dot{q}} \Delta q$ which is only possible if $L$ only depends on $\dot{x}$, otherwise they'd have to write $\Delta L = (L_x \delta x + L_{\dot{x}} \delta \dot{x} + L_t \delta t)$ in general for $L = L(t,x,\dot{x})$
 
exactly. that's what was confusing me and I thought they skipped the part where they had to say: $\frac{\partial L}{\partial x_1} + \frac{\partial L}{\partial x_2} = 0$, but seems like that's not the case and it's just $L = L(\dot x)$
@bolbteppa so the proof on the image is pretty weak since it uses the super simple case :( would the same type of proof be possible(as on wikipedia picture) for $L(\dot x, x)$ ? because if I try that, I fail because to succeed and end up with conservation, I need to use $\frac{\partial L}{\partial x_1} + \frac{\partial L}{\partial x_2} = 0$, and if I use this, what's the point of such proof lol i could just directly use EL to show conservation.
 
9:19 AM
No, if you try it with $L(x,\dot{x})$ it's going to completely fail, you can't fix it by adding those terms, also in the picture there is only an $x_1$ no $x_2$
 
it's not going to fail only if $L$ is symmetric without even time derivative difference. if that's the case, the proof succeeds, but the proof becomes pointless as to use that proof, as i said, i need to use $\frac{\partial L}{\partial x_1} + \frac{\partial L}{\partial x_2} = 0$ and if i use this, proof is pointless at all. makes sense ? @bolbteppa
I think I have to follow the proof on the first article I shared
 
$\Delta S = \int \Delta L = \int (L_x \Delta x + L_{\dot{x}} \frac{\Delta x}{\Delta t} ) \Delta t = \int (L_x \Delta x \Delta t + L_{\dot{x}} \Delta x)$, what the hell is the first term supposed to mean,
 
@bolbteppa $\delta S = \int_{t_0}^{t_1} \epsilon (\frac{\partial L}{\partial \dot x_1} \dot f(t) + \frac{\partial L}{\partial \dot x_2} \dot f(t) + \frac{\partial L}{\partial x_1} f(t) + \frac{\partial L}{\partial x_2}f(t))dt = 0$
 
No it doesn't make sense, there is no $x_2$ anywhere, and even if we include an $x_2$ the Lagrangian now depends on $x_1$ and $x_2$ so they don't even add to give zero nor separately give zero
 
$\delta S = \int_{t_0}^{t_1} \epsilon (\frac{\partial L}{\partial \dot x_1} \dot f(t) + \frac{\partial L}{\partial \dot x_2} \dot f(t) + \frac{\partial L}{\partial x_1} f(t) + \frac{\partial L}{\partial x_2}f(t))dt = 0$ if lagrangian is symmetric under $x_1(t) + \epsilon$ and $x_2(t) + \epsilon$ (symmetric such as it exactly stays the same, then 3rd and 4th member in the interal add up to 0, i.e only 1st and 2nd stay in the integral and you can get conserved quantity from that @bolbteppa
 
9:25 AM
You can't use the same function $f$ for $x_1$ and $x_2$
I'm not sure what you're saying
 
why not ? if you got 2 paths($x_1$ and $x_2$), we can choose f such as $f(t_1) = 0$ and $f(t_2) = 0$. this works for both paths.
 
In $L = L(x_1,x_2;\dot{x}_1,\dot{x}_2)$, you vary $x_1$ and $x_2$ independently of one another in general, also, what have you got other than a big mess right now that doesn't mean anything
 
I wouldn't agree. I can just vary them with the same function. @bolbteppa see here(physics.stackexchange.com/a/19865/366605)
 
It's not this hard, these kinds of posts just confuse and mislead people
 
my point was that there's no constraint why i can't vary $x_1$ and $x_2$ by the same $\epsilon f(t)$
 
9:32 AM
Fine vary then by the same function, now what, you still have a big mess
 
I don't in the case when Lagrangian is symmetric without total time derivative(i.e L = L'), because if so, $\delta S = \int_{t_0}^{t_1} \epsilon (\frac{\partial L}{\partial \dot x_1} \dot f(t) + \frac{\partial L}{\partial \dot x_2} \dot f(t) + \frac{\partial L}{\partial x_1} f(t) + \frac{\partial L}{\partial x_2}f(t))dt = 0$ (3rd and 4th members add up to 0) and I successfully get conserved quantity @bolbteppa
 
What's conserved, all you have is some integral of complicated functions over the whole path ends up being zero
 
no, we got $\int_{t_0}^{t_1} \epsilon (\frac{\partial L}{\partial \dot x_1} \dot f(t) + \frac{\partial L}{\partial \dot x_2} dt = 0$ .
you now split this in 3 integral parts for each interval(as on the wikipedia page) and you will notice that 2nd integral will be 0, i.e you're left with: $\delta S = \underbrace{\epsilon (\frac{\partial L}{\partial \dot x_1} + \frac{\partial L}{\partial \dot x_2}) f(t)\Bigr|_{t_0}^{t_0+\tau}}_{\text{At the beginning}} + \underbrace{\epsilon (\frac{\partial L}{\partial \dot x_1} + \frac{\partial L}{\partial \dot x_2}) f(t)\Bigr|_{t_1 - \tau}^{t_1}}_\text{At the end} = 0$
$\delta S = \epsilon (\frac{\partial L}{\partial \dot x_1} + \frac{\partial L}{\partial \dot x_2})(1-0) + \epsilon (\frac{\partial L}{\partial \dot x_1} + \frac{\partial L}{\partial \dot x_2})(0-1) = 0$
and you got conserved quantity :P
the proof works, but i need to use $\frac{\partial L}{\partial x_1} + \frac{\partial L}{\partial x_2} = 0$
which i don't like because if i use that, it's the same as if i had directly used chat.stackexchange.com/transcript/message/64827821#64827821 which is simpler
@RyderRude hey mate. what's your thoughts ? basically, I wanted to use the same technique as in here(en.wikipedia.org/wiki/…), but for $L(x_1, x_2, \dot x_1, \dot x_2)$ and if I do, only way I succeed is by using $\frac{\partial L}{\partial x_1} + \frac{\partial L}{\partial x_2} = 0$ and if i use that, the proof loses its point, because I could directly just use EL to show this without such complications.
 
9:52 AM
I'm just not sure what you're trying to do, maybe it makes sense
 
@GiorgiLagidze i dont understand which proof on wiki u r following
 
all good. @bolbteppa thanks for your time and listening and helping ^_^
@RyderRude en.wikipedia.org/wiki/… (this should directly get you to the image)
 
oh it's the one u linked
 
it kind of uses feynman's technique
but the situation is for $L(\dot x)$ and not for $L(x, \dot x)$
 
this seems identical to Feynman's proof
 
9:56 AM
yes, but they discuss the case when $L = L(\dot x)$.
if you now use the same technique for $L = L(x, \dot x)$, you will realize that you need to use the fact that $\frac{\partial L}{\partial x_1} + \frac{\partial L}{\partial x_2} = 0$, otherwise, you can't prove. and if you use this fact, then what was the point of using this feynman technique in the first place ? if $\frac{\partial L}{\partial x_1} + \frac{\partial L}{\partial x_2} = 0$, you could just directly use EL ( chat.stackexchange.com/transcript/message/64827821#64827821)
 
i dont think Feynman's technique is a general proof. it only works for the conservation of canonical momentum, which is the simplest case of Noether's theorem
 
yes, i agree, but can you make it work for $L(x, \dot x)$ without using $\frac{\partial L}{\partial x_1} + \frac{\partial L}{\partial x_2} = 0$ ?
 
Maybe this will help, if $L = \frac{1}{2}(m_1 \dot{x}_1^2 + m_2 \dot{x}_2^2 - k_1 x_1^2 - k_2 x_2^2)$, from from $\frac{d}{dt} (m_1 \dot{x}_1) = k_1 x_1$ and $\frac{d}{dt}(m_2 \dot{x}_2) = k_2 x_2$, you're saying in the special case when $\frac{d}{dt}(m_1 \dot{x}_1 + m_2 \dot{x}_2) = k_1 x_1 + k_2 x_2 = 0$ i.e. when $x_2 = - (k_1/k_2) x_1$, then $m_1 \dot{x}_1 + m_2 \dot{x}_2$ is conserved. Okay, but what about when $x_2 \neq - (k_2/k_1) x_1$? There is no reason this has to be true in general
 
@GiorgiLagidze do u mean for L(x1, v1,x2,v2)?
 
@RyderRude yes
@bolbteppa yes, true. it's just I wanted to use feynman's technique without using $\frac{\partial L}{\partial x_1} + \frac{\partial L}{\partial x_2} = 0$
it seems impossible to me :P
 
10:05 AM
the symmetry transform here is L(x1,v1,x2,v2)---> L(x1+a, v1, x2+a, v2)
for the conservation of momentum, the case where Feynman's proof works
 
@RyderRude and how do you prove with feynman's technique ?
 
If $L$ depends on $x$, or $x_1$ and $x_2$, then at best you can only use it in some special directions, but you can't use it in general because $x_1$ and $x_2$ are in general more independent than to be constrained such that $L_{x_1} + L_{x_2} = 0$ has to hold. We're trying to find the completely general solution and general conservation laws not special cases
 
@bolbteppa he wants to assume invariance of the Lagrangian under the transformation i mentioned, and only then prove conservation
he is not trying to prove conservation for a generic Lagrangian
but he also wants to use Feynman's technique
 
The problem is, you have two separate equations, $\frac{d}{dt} L_{\dot{x}_1} = L_{x_1}$ and $\frac{d}{dt} L_{\dot{x}_2} = L_{x_2}$, when you add them you are restricting the set of possible solutions and thus considering only a special case, not the most general solutions
 
10:09 AM
@ACuriousMind Well (1.12) defines the measure $\delta_{H(q,p)-E}$, as the text says it is defined through expectations of observables $\varphi$. I am just confused why you can straight away define it as the uniform measure over the level set $\{(q,p)~:~H(q,p)=E\}$ of the Hamiltonian. To me thats what it looks like they are trying to do.
 
@RyderRude here(physics.stackexchange.com/a/19865/366605), why do you think that Ron only assumes that only speed changes on those paths ? :P Definitely, he uses the logic that $\frac{\partial L}{\partial x_1} + \frac{\partial L}{\partial x_2} = 0$, but if you use that logic, there's no point to even start feynman's proof because EL simply is enough.
 
@GiorgiLagidze Feynman's proof is only for conservation of momentum. that's y the term $dL/dx$ isnt there, but this term is there in the general proof of Noether's theorem
 
you can still get conservation of momentum even if dL/dx is there
 
u can see the diagram Feynman draws. it only involves an unchanging x and an infinite v
it is not the case of a general symmetry
 
I think the wikipedia page doesn't use feynman's technique.
 
10:12 AM
as wiki notes, this proof is only for illustration. it's not a general proof
 
forget feynman at all. my point is that, on wiki, can you use the same proof for $L(x, \dot x)$ ?
no right ?
 
wiki also doesnt consider the term dL/dx
it's the same proof
 
exactly :P
if you don't have $x$ in the L, fine.
 
it's only an illustration of conservation of momentum
 
and why can't i have illustration for $L(x, \dot x)$ case ?
 
10:15 AM
Noether's theorem considers all the terms in the variation
but it loses this illustration
y cant u just accept the mathematical proof :P
 
yes. let's forget this conversation ever happened.
 
@GiorgiLagidze ok wiki says this : Roughly speaking, they contribute mostly through their "slanting"
so they r ignoring the dL/dx contribution
 
@Monty Ah, that's indeed because of the zero Lebesgue measure of $S(E)$ - you want to express the uniform measure on $S(E)$ as an integral w.r.t. the Lebesgue measure $\mathrm{d}q\mathrm{d}p$ in the ambient phase space
of course you could just abstractly say "let's take the uniform measure on $S(E)$", but as a physicist you want this integral representation so you can do (more or less justified :P) symbolic manipulations of it
 
If you include $x$, then the wiki proof now goes as $\Delta S = \int \Delta L = \int (L_x \Delta x + L_{\dot{x}} \frac{\Delta x}{\Delta t} ) \Delta t = \int (L_x \Delta x \Delta t + L_{\dot{x}} \Delta x)$ where you now have this first term to deal with as well, if you just assume it's constant over some small interval and integrate then you get an extra $L_x \Delta x t$ which obviously quickly gives the wrong answer as it only depends on $t$ and linearly at that
and it's not conserved over the interval either it's some function of $t$ which is different at different times
 
@ACuriousMind gotcha
 
10:23 AM
in Neother's theorem proof, this dL/dx gets cancelled out due to the EL eqn
 
@bolbteppa that's not the case. because you use $\epsilon f(t)$ . there's no way you end up with $t$ at all
the proof works as I said, but assumes that dL/dx1 + dL/dx2 = 0
anyways, gotta go. thanks
 
@GiorgiLagidze note that Ron is ignoring this term because the change is x is an infinitesimal while the change is v is infinite, so the former is negligible
 
yes, Ron's proof is better than wikipedia
that was my point
 
I explained why this only works for a special case where the variables are related in a special way, it does not hold in general so it's not a general conservation law
 
I can't understand fully Ron's proof, so I gave up on it
 
10:31 AM
but this argument only works for translational symmetry. Noether's theorem is always more general
wiki is using the same proof but they also want to consider a general symmetry. so they just handwave and say "Roughly speaking, they contribute mostly through their "slanting"
this allows them to throw away the dL/dx term
but they onlh claim it to be an illustration, rather than a proof
 
yeah, I don't get why they throw dL/dx out. is it because of translation invariance ?
 
You might as well assume $x_1$ and $x_2$ are constants, then you get another useless special case conservation law
 
in case of translational symmetry, delta x is negligible because delta v is infinite in the diagram
 
@RyderRude even on wikipedia ?
 
no, on wikipedia, they consider a general symmetry
so they dont have a rigorous reason to throw away that term
 
10:34 AM
yes ! that's why they don't consider $L(x, \dot x)$ in the first place, but only $L(\dot x)$ :D
 
the diagram they've drawn is not a mere translation of the trajectory
it's not that they dont consider the dependence of L on x
 
if they consider, why do they throw $dL/dx$ ?
 
it's that they assume delta x due to the infinitesimal symmetry transformation is negligible
the full change in the lagrangian wud hav been dL/dx times delta x
just like it's dL/dv times delta v
 
okay, they assume delta x is negligible
 
yes.
 
10:37 AM
why don't they assume that delta v is also negliglbe :D
 
they write : Roughly speaking, they contribute mostly through their "slanting"
@GiorgiLagidze idk :P
it's a bad proof
 
well, it's horrible.
yeah, seems like this noether proof, I have to forget about illustrations because every single time, I saw anyone's illustration proof, it sucks unfortunatelly
the only way is pure mathematics and that's it i guess :P
 
yes, the original proof is a very short manipulation
stick to it until Hamiltonian mechanics proves it in one line
 
yes, better to forget it now ! thanks so much. have to go for a couple of hours. have a good day
 
ok :)
 
 
1 hour later…
11:57 AM
Hello, does anyone know the name of this change of variables?
$b_i^\dagger\to e^{i\varphi_i}$
 
12:35 PM
@Mr.Feynman isn't this Jordan-Wigner
 
1:20 PM
@ACuriousMind The context is that of Jordan Wigner, but the definition I have is a bit different. I'll check with some algebra after my siesta
By the way, the definition I have is that of the inverse transformation $$\sigma_i^x=(a_i^\dagger+a_i)\prod_{j=1}^{i-1}(a_j^\dagger+a_j)$$, something similar in for $y$ matrix and $z$ matrix is the same as wiki
 
that was a short siesta :P
 
Lol, a couple of minutes and I'll go non-interacting
Oh ok, wiki writes it for the ladder operators, but still in the picture above there is something different: they are mapping $\b^\dagger$ to a "phase" operator
It would make sense if it were some kind of polar decomposition but it isn't
(And now I'll go non-interacting)
 
5 mins of seista can feel like hours of sleep
 
1:39 PM
I just learnt that siesta is the human norm throughout history. We are working way too much and getting paid way too little and paying too much and the almost doubling the amount of work done by basically everybody in the world only improved actual productivity by 30%, i.e. overworked for nothing much.
And there are just the usual devils to blame.
Absolutely psychotic
 
i wish superheros existed to sort this out
why on earth do we need so much productivity
the world shud work the same with same pay on half the productivity
one could have expected that sort of a world back when desk jobs started getting big
but government and rich people made it awful for everyone else anyway
 
 
7 hours later…
8:52 PM
The day of honking has arrived
Honk
So do you all think that physics will forever remain in pieces that each describe their respective regimes precisely
 
 
1 hour later…
10:13 PM
@naturallyInconsistent i see -- thanks !
@SillyGoose going ryder rude style today
 
10:44 PM
@Relativisticcucumber the impression is spot on, that question isn't receiving any feedback (sorry Ryder, I needed to make this joke :P)
 
11:16 PM
Consider a Cauchy foliation $\Omega_s(x)$ of a Lorentzian (1+1) manifold for space variable, $x$ and time variable $s$. Does anyone know how to interpret $\Omega_s(x)$ given that it satisfies a heat/diffusion equation, with Cauchy data being $\Omega_s(x)|_{s=0}$?
 

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