@Mithrandir24601 What type of noise are they talking about when they say "Unfortunately for analog computation it turns out that when realistic assumptions about the presence of noise in analog computers are made, their power disappears in all known instances; they cannot efficiently solve problems which are not solvable on a Turing machine." (in Nielsen & Chuang)? Looks like they're speaking of thermal noise, but I'm not sure
Wanted to switch things up a bit. Building an interpreter(a basic one ).
Then might return to some physics study, then programming project
In other news, I think I now understand what Taylor expansions are. I mean I've been doing them for years, but someone told me something about someone who did something awesome that changed physics and now I think I know what a Taylor expansion is about lolz :P
ok ok back to my super simple intepreter. Need to tokenize now or something like that , then label stuff
The switch from $$\frac{d}{dt} \frac{\partial L}{\partial \dot q^n} - \frac{\partial L}{\partial q} = 0$$ to $$\ddot q^{n'} \frac{\partial^2 L}{\partial \dot q^{n'} \partial \dot q^n} = \frac{\partial L}{\partial q^n} - \dot q^{n'} \frac{\partial^2 L}{\partial q^{n'} \partial \dot q^n}$$ involves the use of the total derivative formula and all
But how do you do it for field theories
Where it's all partial derivatives
Do we have $$\partial_\mu F(\phi, \partial_\mu \phi) = (\partial_\mu \phi) \frac{\partial F}{\partial \phi} + (\partial_\mu \partial_\nu \phi) \frac{\partial F}{\partial \partial_\nu \phi}$$
@Blue I don't think they're talking about any single specific type of noise here, but yeah, it would most likely be thermal noise, (although this is actually very general - you couple the system to a thermal bath, then the type of interaction determines the specific type of noise, to my knowledge - this is known as 'non-Markovian' noise). The problem in QC is that the above noise generally don't have any nice sort of scaling with the size of the thing
If you ever feel the urge to read a paper on interpretations of QM first you should do a few precautionary checks first. e.g. do your toenails need cutting? Have you hoovered the living room?
@JohnRennie yep :) I need to make breakfast, finish a PhD in about 3 years-ish, then I've got ~250 books to read. At that point, assuming I don't have anything else to do, I'll have enough time to read it
Hmm..., if I understood correctly, the complementary view of a stationary observer seeing a stretched horizon while an inflating observer does not see anything weird at the horizon, and unitarity is preserved by black holes by having a global wavefunction consists of a superposition of black hole geometries, detector/observer states and the degrees of freedom inside and outside of the black hole.
In particular, the inflating observer inside the black hole and the stationary observer outside the black hole are located at different branches of the global wavefunction, where the former will see the inflating information get scrambled by the black hole, while the latter see nothing strange going on.
The hawking radiation emitted then serves to decohere these branches, thus this increase in the number of black hole and exterior environment spacetime geometries in the product state will then manifest itself as the black hole entropy
Or in other words, the black hole spacetime geometry, observers and any inflating information together forms a superposition, which is then decohere by the hawking radiation became entangled with the exterior environment as it escapes from the black hole
However I have two questions:
1. What about an infalling observer that is outside the black hole. Will the branches between it and the stationary observer be compatible (e.g. both seeing the state by using the information from their detectors as a nontrivial transformation of the interior state of the black hole which is the state of the stretched horizon)?
2. The models and the maths supported that by modelling the evaporating black hole geometry as a superposition of semiclassical spacetime geometries and then invoking decoherence, black hole complementarity, unitarity and the black hole entropy (in terms of bounds) are recovered, suggesting that the model indeed works. But suppose we managed to flew a space probe near a black hole, how can we test that it is indeed consists of decohered branches of a superposition of spacetime geometries,
that is, how can we distinguish this from other information paradox resolution models that does not invoke the global wavefuction (or an ensemble of spacetime geometries)?
The above are my thoughts after reading that paper for you guys to ponder about...
It would be interesting to somehow be able to keep a black hole in a superposition, then we can use diffraction experiments to show the existence of a superposition. alternately, we need to prepare many copies of such black hole with the same initial state, and then the information of our detectors should reveal some probability distribution of observables related to the nearby spacetime geometry that suggests the spacetime was once in a superposition before being decohered by hawking radiation
But either way, I like this idea of the superposition of spacetime geometries used in this paper
The mathematics of pendulums are in general quite complicated. Simplifying assumptions can be made, which in the case of a simple pendulum allow the equations of motion to be solved analytically for small-angle oscillations.
== Simple gravity pendulum ==
A so-called "simple pendulum" is an idealization of a "real pendulum" but in an isolated system using the following assumptions:
The rod or cord on which the bob swings is massless, inextensible and always remains taut;
The bob is a point mass;
Motion occurs only in two dimensions, i.e. the bob does not trace an ellipse but an arc.
The motion...
As a workaround while this request is pending, there exist several client-side workarounds that can be used to enable LaTeX rendering in chat, including:
ChatJax, a set of bookmarklets by robjohn to enable dynamic MathJax support in chat. Commonly used in the Mathematics chat room.
An altern...
@Slereah so in the image which you sent is k the maximum potence of sin? You said that I'm in $$T_2$$ and in that formula there is $$sin^2$$. And in $$T_8$$ there is $$sin^8$$
apparently if you write out the GR action relative to an ADM splitting, it's identically zero for any field config
that's much better than saying Gauss-Bonnet and hoping no mathematician is listening
Anonymous
@Mithrandir24601 I was wondering if they were referring to noise is a more generalized setting. Like even if we get rid of the conventional noise - like industrial noise, vibrational noise, thermal noise (or reduce them to negligible levels), still noise could refer to the uncertainties in amplitude, phase, etc, which arise due to the underlying quantum mechanical nature of the system.
Anonymous
The Turing machine concept doesn't take into account this type of noise, which is impossible to get rid of, in any practical scenario. But then again, we can't even totally get rid of the classical sources of noise - like thermal, vibrational, etc in any practical scenario.
Anonymous
However, I think I more or less get it now. Thanks
I got hit with "isn't your work just a derivative version of instantons" on PhD Student Presentation Day on my second year, while presenting imaginary-time approaches to tunnel ionization (as described e.g. here)
in the end, it isn't, but I'd never heard of the things
@Curio for $\theta > 90º$ the equation of motion will change again because the string will go slack. But yes the equation does have an infinite number of terms and yes with an infinite number of terms it describes the motion correctly for $-90º \le \theta \le 90º$.
@JohnRennie wait, what do you mean with -90°? Consider the pendulum motionless , I mean it doesn't move. The wire of the pendulum forms two 180° angles, one right to the wire and one left. I mean, if the angle is 200°, it doesn't make really sense because truly it's 160°
@CaptainBohemian that's a trick question because depending on your definition of "soliton", instantons may just be a specific kind of solutions. Or they might not, I've never seen a technical definition of solitons that everyone could agree on to cover everything off handedly referred to as solitons
@Curio you could use a massless rigid rod in place of the string (at least in principle) and the equation of motion would be unchanged for $-\pi/2 \le \theta \le \pi/2$. I'm not sure if the equation would apply for angles outside this range.
OK, I worked this out on a spreadsheet and it seems to give the desired result (a formula for the distance between two points on this surface). Let me know if there's something I got wrong:$$ds^2(t)=dt^2\phi^2(dt-2t)^2-dx^2-dy^2-dz^2$$I don't know if the notiation is right, but I know the metric...
wha...?
how does one do differential geometry on Excel?
(I mean, apart from the obvious answer of "wrongly")
@EmilioPisanty given that OP's question as written is non-sensical to begin with, I don't think using a spreadsheet to answer it makes anything worse :P
@Curio yes. If you imagine replacing the wire with a rigid rod, then in the range -90 to 90 degrees the rod is in tension. At exactly -90 and 90 degrees the force in the rod is zero, and for larger angles the rod is in compression.
@EmilioPisanty only if we assume that OP is asking for the induced metric from embedding the surface in standard euclidean space, which has not been clarified yet
I'm not a fan of guessing what askers mean - on the one hand it teaches them they can get away with asking ill-defined questions, and on the other the answerer's effort is potentially wasted if their guess is wrong
@EmilioPisanty You don't. Which the asker would have realised if people had forced them to clarify the question instead of answering as if OP understood the topic at hand :P
Aha! I VTC'd thinking I didn't have a dupehammer for the string-theory tag, but Emilio had just that moment edited in the tags I do have a hammer for :-)
@ACuriousMind do you mean ''instantons may just be a specific kind of 'solitons' " rather than'' instantons may just be a specific kind of 'solutions' "?
@0celo7 The beauty is that Tao's post shows that the regulator (what you probably mean by "convergence method") does not actually matter - all regularizations that give an answer at all must agree on the "finite part" of the divergent series.
@Curio I'm not sure what you mean by $F_p$. If you're just expressing the total force on the bob as the sum of horizontal and vertical components then of course that is fine.
@0celo7 you also have the hunger trouble during bad weather! I just came back from hunting for food in the rain. I really hate hunting in the rain, but hadn't eaten a whole day long. The rain has persisted for 2 whole days.
The tangential component creates a torque, and the equation of motion is then $$T = I\ddot \theta$$ where $T$ is the torque and $I$ is the moment of inertia.
The torque and the moment of inertia are both proportional to the mass, so the mass factors out of the equation.
@0celo7 Yes - instead of partial sums truncated at $N$, he uses infinite series $\sum_n a_n \eta(n/N)$ suppressed by a "nice" regulator function $\eta$ and then shows that their asymptotic finite part - the term independent of $N$ - does not depend on $\eta$ - that term is the "value" of the divergent series.
@Curio If you draw a free body diagram you'll find the torque is always proportional to the mass. So let's write it as $T = T'm$. The moment of inertia is just $I = m\ell^2$. So our equation becomes: $$T'm = m\ell^2\ddot\theta$$ and we can divide both sides by $m$ to get an equation in which the mass does not appear.
@ACuriousMind your argument has considerable merit :-)
@0celo7 I doubt you found a single actual error, and since we're describing the natural world, it's the math we're using which has the problem not reality apparently...
Not the fault of physics people want their nice functions yet are forced into distributions etc
Sure, you can't show "from first principles" that you should use the finite part of divergent sums, but there's nothing wrong with simply adding that as a first principle.
@DavidZ Thanks for clarifying the point about flag warnings! Anyway, the particular question I'd flagged was "How much force/TNT does it take to blow up the average suburban house…?" and the response I'd gotten to my custom flag was "declined - Using standard flags helps us prioritize problems and resolve them faster. Please familiarize yourself with the list of standard flags: see What is Flagging?".
@DavidZ So, I was basically just trying to figure out which flag may've fit.
For context, the question was asked April 1<sup>st</sup>, 2017, so April Fool's day. It was also deleted pretty promptly, so presumably there was agreement that the post needed to be addressed.
@0celo7 I don't understand Rourke & Sanderson's proof. Pls help
I have been stuck on it for a week
(Nah, I can prolly figure it out)
I need to see what a submanifold $M$ of $Q \times \Bbb R^k$ behaves like if it's sitting in $Q \times B_\epsilon(0)$ where $B_\epsilon(0)$ is an $\epsilon$-ball around $0$, and the induced Riemannian metric on $M$ is $1/\epsilon$-times large compared to $Q \times \Bbb R^k$
That's what I want to do
The approximate picture is, "lotta wiggles"
I think the tangent bundle $TM$, appropriately extended beyond $M$ to a distribution on $Q \times \Bbb R^k$, has the property that it cannot be an element of a small neighborhood (in the appropriate Grassmannian) of any subdistribution of the horizontal distrubution $\pi_1^* Q$ on $Q \times \Bbb R^k$