Good piece [LWN.net]

Good piece

Posted Nov 29, 2012 17:47 UTC (Thu) by davidescott (guest, #58580)
In reply to: Good piece by wolfgang
Parent article: LCE: Don't play dice with random numbers

Two comments:
a) Classical Mechanics (as described by Newton) is NOT deterministic. See "Space Invaders" or for an example without infinities (but a non-differentiable surface) John D. Norton's work. http://plato.stanford.edu/entries/determinism-causal

b) When you claim that QM is non-deterministic you appear to be assuming the Copenhagen interpretation. From a Bohmian perspective QM may still be deterministic, but forever beyond our powers to make a deterministic prediction of its behavior, because of its disturbing non-locality.

Saying that QM processes provide a true source of randomness is really the "Church's thesis" of cryptography. Its hard to see (given our current knowledge of physics) how we could ever arrive at a useful definition of "randomness" that was in any way stronger than what we get by saying that "QM processes are random". In other words when you say that a QM process is truly random you are defining "randomness" not ascribing a property to QM processes.

to post comments

Good piece

Posted Nov 29, 2012 18:33 UTC (Thu) by Cyberax (✭ supporter ✭, #52523) [Link] (12 responses)

I don't know what they're smoking but I definitely want some.

1) "Space invaders" scenario is perfectly deterministic, they just have a problem with incorrect abstraction. In particular, treating gravity as infinitely fast. If we instead treat gravity as a classical field with finite interaction propagation speed, then this paradox disappears. Ditto for special relativity (it has its own problems with singularities, though).

2) Even if we allow the "space invaders" it still does NOT mean the system is not deterministic. If we know all the details about the system then we can predict its behavior with arbitrary precision. That knowledge, of course, must include all the infinitely fast objects. Laplace demon is still able to predict everything with arbitrary precision.

Good piece

Posted Nov 29, 2012 20:45 UTC (Thu) by man_ls (guest, #15091) [Link] (4 responses)

You are aware that Laplace's demon (as Maxwell's) is nothing short of God-like in nature. And that its very nature was refuted (as seen in Wikipedia) by the second law of Thermodynamics:

According to chemical engineer Robert Ulanowicz, in his 1986 book Growth and Development, Laplace's demon met its end with early 19th century developments of the concepts of irreversibility, entropy, and the second law of thermodynamics. In other words, Laplace's demon was based on the premise of reversibility and classical mechanics; however, under current theory, thermodynamics (i.e. real processes) are thought to be irreversible in practical terms (compared to the age of the universe, for instance).

Apparently this last part has not been stated convincingly enough since I brought it up several comments up, and it is indeed hard to grasp. The consequence is that no entity can predict what will happen in a system with absolute precision, not even a theoretical demon, not even in theory. In theory the amount of information that can be stored is always limited, so that (to state it in modern terms) below the noise level the signal is unpredictable. And the noise level cannot be made arbitrarily small.

Good piece

Posted Dec 4, 2012 22:31 UTC (Tue) by chithanh (guest, #52801) [Link] (3 responses)

> You are aware that Laplace's demon (as Maxwell's) is nothing short of God-like in nature. And that its very nature was refuted (as seen in Wikipedia) by the second law of Thermodynamics:

And the second law of thermodynamics has been refuted by Poincaré recurrence.

http://en.wikipedia.org/wiki/Poincaré_recurrence_theorem

Good piece

Posted Dec 5, 2012 8:35 UTC (Wed) by man_ls (guest, #15091) [Link] (2 responses)

Hardly a rebuttal since this theorem and the second law operate on different time scales. Boltzmann had a similar argument for the whole universe; it only made the second law stronger.

Whenever something contradicts the second law, that something loses. But if you feel better with a recurrence every 10^10^100... seconds or so, then so be it.

Good piece

Posted Dec 5, 2012 21:25 UTC (Wed) by mathstuf (subscriber, #69389) [Link] (1 responses)

> recurrence every 10^10^100

And don't fret, those 10's are rounded up from e, so it's not as long as it might seem (though I think you're missing few stacks and should be 10^10^10^10^10^1.1).

Good piece

Posted Dec 5, 2012 21:38 UTC (Wed) by man_ls (guest, #15091) [Link]

You are right, I added the ellipsis on a Googolplex because I could not find any good estimate at the moment. Now I have: in the Wikipedia no less, and it is as you say. Saying it is bigger than the age of the universe is a bit of an understatement. So don't hold your breath for a recurrence.

Besides, it would only happen if our universe is an energy-conserving system; probably just changing size would break the recurrence.

Finally, even if a recurrence was possible in an expanding universe, it would just leave us at the starting point; not just diminish the entropy.

Good piece

Posted Nov 29, 2012 21:13 UTC (Thu) by davidescott (guest, #58580) [Link] (4 responses)

> In particular, treating gravity as infinitely fast. If we instead treat gravity as a classical field with finite interaction propagation speed, then this paradox disappears.

You are moving the bar and no longer talking about classical mechanics.

We know classical mechanics is not the correct theory, the point is that one of the ways it is incorrect is that it allows for mathematical singularities in its solutions. If you start slapping restrictions on the theory: "finite force propagation, energy and speed limits, regularity of solutions, continuity of fields, etc...." to prevent those singularities from appearing then it is no longer classical mechanics. It is something else.

> If we know all the details about the system then we can predict its behavior with arbitrary precision. That knowledge, of course, must include all the infinitely fast objects.

Again you aren't talking about Classical Mechanics. Your Demon wants to predict what will happen to the ball at time t_2, and wants to make that prediction based on the state of the universe at time t_0. Suppose that the correct prediction is "the ball does not exist at time t_2, because an Invader appears at time t_1 \in (t_0, t_2) and vaporizes the ball before t_2." If your Demon is capable of making such a prediction then he/she must be able to:

Express in the notation of classical Newtonian mechanics the position and velocity of the Space Invader at time t_0, so that I can deterministically show that the invader MUST begin to slow at time t_1 and destroy the ball prior to time t_2.

You CANNOT do so because v=infinity and x=everywhere (or x=emptyset) is not a valid expression of position and velocity of a body in Newtonian Mechanics. In Newton's formulas, the infinitely fast object DOES NOT EXIST within the classical universe, but his formulas allow it to be the finite-time limit of a classical process.

Good piece

Posted Nov 29, 2012 21:36 UTC (Thu) by Cyberax (✭ supporter ✭, #52523) [Link] (3 responses)

>Again you aren't talking about Classical Mechanics. Your Demon wants to predict what will happen to the ball at time t_2, and wants to make that prediction based on the state of the universe at time t_0.
Yup.

>Suppose that the correct prediction is "the ball does not exist at time t_2, because an Invader appears at time t_1 \in (t_0, t_2) and vaporizes the ball before t_2." If your Demon is capable of making such a prediction then he/she must be able to
It must be able to integrate equations of motion. That's all.

>Express in the notation of classical Newtonian mechanics the position and velocity of the Space Invader at time t_0, so that I can deterministically show that the invader MUST begin to slow at time t_1 and destroy the ball prior to time t_2.
Wrong. "Space invader" exists only at ONE point - it has infinite speed and can't slow down.

>You CANNOT do so because v=infinity and x=everywhere (or x=emptyset) is not a valid expression of position and velocity of a body in Newtonian Mechanics. In Newton's formulas, the infinitely fast object DOES NOT EXIST within the classical universe, but his formulas allow it to be the finite-time limit of a classical process.
Newtonian mechanics has no problems with infinitely fast objects, as long as you don't collide them with something else.

That makes them a little bit like black holes - they are singularities, but they are fairly well-behaved

Good piece

Posted Nov 29, 2012 23:21 UTC (Thu) by davidescott (guest, #58580) [Link] (2 responses)

> Newtonian mechanics has no problems with infinitely fast objects, as long as you don't collide them with something else.

WHAT?

If you think that is the case solve the following single particle 1-dimensional, force-less system:
t=0: the particle is "at" x=0 and has dx/dt=\infty and d^2x/dt^2=0.
Solve for t=1 to get x_1,v_1,a_1

Now solve the following systems for t=1 and t=-1:
t=0: x=x_1, dx/dt=-v_1, d^2x/dt^2=a_1
t=0: x=2*x_1, dx/dt=-v_1, d^2x/dt^2=a_1
t=0: x=x_1, dx/dt=-2*v_1, d^2x/dt^2=a_1
t=0: x=2*x_1, dx/dt=-2*v_1, d^2x/dt^2=a_1

Either you cannot do this, or something will be contradictory.

Good piece

Posted Nov 30, 2012 0:33 UTC (Fri) by Cyberax (✭ supporter ✭, #52523) [Link] (1 responses)

That's just an artifact of a chosen coordinate system. If you really want to solve it - write down Lagrangian of a system and see what happens.

Good piece

Posted Nov 30, 2012 2:18 UTC (Fri) by davidescott (guest, #58580) [Link]

I'm telling you I can't solve that. I don't know how. If you think it is so easily solved I would love to see your solution.

Good piece

Posted Nov 30, 2012 7:07 UTC (Fri) by apoelstra (subscriber, #75205) [Link] (1 responses)

> 1) "Space invaders" scenario is perfectly deterministic, they just have a problem with incorrect abstraction. In particular, treating gravity as infinitely fast. If we instead treat gravity as a classical field with finite interaction propagation speed, then this paradox disappears. Ditto for special relativity (it has its own problems with singularities, though).

This isn't the problem with space invaders, though. The problem is that you have an object that disappears to infinity, where it remains for all time. But since classical mechanics is time-reversible, it could just-as-legitimately said that the object "has been at infinity since eternity, then moves to a finite position at time t".

But nothing in classical mechanics predict when "time t" is, hence the indeterminacy.

As you have pointed out, special relativity wrecks up this pathology (though general relativity introduces many more, much worse, ones). But that's irrelevant to whether the claim "classical mechanics is deterministic" is true.

Good piece

Posted Nov 30, 2012 14:07 UTC (Fri) by davidescott (guest, #58580) [Link]

As an aside. SR eliminates infinite velocity singularities in classical mechanics, but there are other non-regular solutions in Classical Mechanics which are finite velocity. To eliminate those you would have to enforce C^2 on all constraint surfaces (and hope that covers all possible singularities). In many ways these surface is C^1 but not C^2 problems are easier to understand than space invaders because you don't have any messy infinities to deal with. So go read Norton's paper.

Good piece

Posted Nov 29, 2012 20:55 UTC (Thu) by man_ls (guest, #15091) [Link] (10 responses)

Its hard to see (given our current knowledge of physics) how we could ever arrive at a useful definition of "randomness" that was in any way stronger than what we get by saying that "QM processes are random".

I disagree. Information theory can indeed provide a definition of randomness: a signal with maximum entropy, where each bit of the signal carries one bit of information. In a normal text different bits are correlated and therefore the information content is always less than one bit per bit of signal.

Going from there to quantum processes is not easy, but I would say that it equates to "quantum events are not correlated in time with previous events, only with states". This means that the probability of emitting a photon at any given moment is the same, regardless of the time that the photon was absorbed; only the state of the particle is relevant, not its history. Since the time when the photon is emitted is not correlated with the moment that it was absorbed, this time is random and therefore impossible to predict.

Good piece

Posted Nov 29, 2012 22:24 UTC (Thu) by davidescott (guest, #58580) [Link] (9 responses)

This is a philosophical point so bear with me a second.

I'm someone who things it very strange when people talk about wave-form collapse, and even stranger when some physicists talk about many worlds (in the universes are duplicated to realize all possible outcomes sense). As if the way to deal with our limited ability to measure the universe around us is by assuming it to be random, and then even more perversely to correct this "God does not play dice" problem by presuming that all outcomes occur in the correct probabilities.

To me it makes much more sense to say. The Universe IS. There is no such thing as "wave-form collapse" or even "wave function evolution." There is no such thing as "time." A excitation exists in some 4 dimensional space. It has a particular shape and you can take cross-sections of it along various planes and compute quantities such as Energy or Charge that happen to be conserved across that class of cross-sections. There are even descriptions of the dynamics of excitation as you continuously vary the cross-section taken. We happen to call one of these local descriptions the Copenhagen interpretation of QM, it is probabilistic and non-deterministic. There is non-local description of the same dynamics, Bohmian QM, it is deterministic.

We use the Copenhagen interpretation because we are interested in the predictive capability of the theory, which Bohmian QM just can't do. If Bohmian QM predicts spin up and its spin down the scientist says "there was a non-local reaction with something outside the experimental apparatus that affected the hidden variable which I can never measure directly." You can start playing games with things and saying I have 99% confidence the hidden variable value is up, and a 99.999% confidence that no non-local interaction will affect that, but its no advantage over Copenhagen, and just saying the outcome is UP 98.999% of the time.

But being predictive IS NOT the same as being descriptive. Don't tell me something *IS* (in the descriptive sense) random just because you can't predict the next number. That's not randomness that just limited intelligence, mathematical ability, and computational capability. Pick an arbitrary sequence from OEIS.org or some sample from the digits of PI and ask people if it is random. NOBODY will say: Oh that's PI beginning at digit 288342341234, but that doesn't make it random. That makes us dumb.

------------------------------------------------------------------

So to the particulars of your comments:
> a signal with maximum entropy, where each bit of the signal carries one bit of information.

"1" was that random or not? Can't say... I need to talk about a sequence of digits.

"3.14159"? -- not random that is "PI."
I don't know what PI is can you explain it to me in less than 6 digits. If you cannot then you cannot compress the stream. If I gave you a thousand digits you could encode a C program that computes PI, but I don't know what C is so now you have to express that. What prevents me from playing the "what is a?" game all day long and forcing you to give me a complete description of every particle in a universe containing a computer that calculates "PI."

Finally we get to the indeterminate sequence of digits generated by your overvolted transistor. You claim that is random because the numbers generated by it cannot be compressed because the underlying process is quantum mechanical and thereby random.
I disagree. If you get to reference something outside the sequence like "PI" then so should I. I will thereby reference the output of your transistor in my deterministic non-local Bohmian description of the dynamics of our static universe. DONE. Its compressed. (self-referential, but compressed)

In other words I reject your statement that "quantum events are not correlated in time with previous events, only with states." I believe they are correlated because the universe has deterministic non-local dynamics, and hidden variables (if we could ever know them) would allow us to correlate the QM events over time.

Another way to put this is to say that you need multiple *INDEPENDENT* outputs from the process in order to calculate its entropy. I REJECT your ability to do so. There is no amount of time-space separation within this universe that you can put between multiple RNGs that will ever make them independent. You need multiple universes to get a correct calculation of entropy. You can only approximate entropy within a single universe.

That said when we calculate this approximate entropy, QM processes seem to have maximal entropy. That does not prove them to be random (descriptive) rather we predict them to be random. If the LHC starts pumping out particles that are all spin up, we would call that new physics and would have to adjust to a world where QM processes are no longer random.

Also the next digit in my sequence was 8, so it wasn't PI anyways.

Good piece

Posted Nov 30, 2012 0:30 UTC (Fri) by nybble41 (subscriber, #55106) [Link] (8 responses)

> To me it makes much more sense to say. The Universe IS. ... There is no such thing as "time."

I think I see your problem. Without time, randomness is indeed impossible. There is no point in talking about random events when you can directly observe the entirety of space-time; from that perspective, there are no probabilities, only facts. This discussion, fortunately, was about a more human perspective, in which the past may be known but knowledge of the future is beyond our reach.

> If you get to reference something outside the sequence like "PI" then so should I. I will thereby reference the output of your transistor in my deterministic non-local Bohmian description of the dynamics of our static universe.

You can reference things outside of the sequence, but only if they are knowable before the sequence is generated. PI is known, being a fundamental mathematical constant, but the output of the transistor is not.

However, I would say that it is meaningless to talk about whether a specific _sequence_ is random; what matters is whether the _process_ of producing the sequence introduced new information into the system. (The same process must, of course, destroy some existing information in order that the entropy of the system as a whole remains monotonically increasing.)

If one can predict the result of a measurement with certainty using only information knowable beforehand, then the measurement itself introduces no new information, and thus is not random. Otherwise, the result of the measurement is random, with entropy greater than zero and less than or equal to the number of significant bits measured.

> Don't tell me something *IS* (in the descriptive sense) random just because you can't predict the next number.

It's not a question of whether you or I could predict the next number; the question is whether the rules of the system permit _anyone_ to predict the next number with 100% certainty, given all the information which it is possible for them to know. Even in a completely deterministic system it may not be possible, even in theory, for any one person to know all the initial conditions which can affect the result.

Good piece

Posted Nov 30, 2012 4:03 UTC (Fri) by davidescott (guest, #58580) [Link] (7 responses)

> You can reference things outside of the sequence, but only if they are knowable before the sequence is generated. PI is known, being a fundamental mathematical constant, but the output of the transistor is not.

PI is just the limit of a sequence. It happens to be particularly useful in Mathematical theorems, but is it any less random because of that? Which of the following is not a random number, but a useful mathematical constant?

0.1039089930584351
0.2519318504932542
0.5772156649015328
0.6394164761104365
0.7741085613465057
0.9458817764941707

The number being recognizable is not a sufficient definition. Its like the old joke about the mathematician and the license plate. "KZJ-1249" what are the chances of seeing that! Every real number is special in its own way, but I agree the process is more important than the actual output. One wants to compare multiple instances of the same RNG process, recognizing that a true RNG almost never (but not never) outputs a sequence that is "obviously not random."

> I think I see your problem. Without time, randomness is indeed impossible.

In a some fundamental sense that is true. The larger point is that any deterministic system is "without time" in this sense, and that contrary to commonly held views Classical Mechanics is not deterministic, and QM can be deterministic.

So when people say QM is random, mechanics is not, they are usually assuming a non-deterministic interpretation of QM, which is not required by experimental evidence (only non-locality is required).

> the question is whether the rules of the system permit _anyone_ to predict the next number with 100% certainty, given all the information which it is possible for them to know

This is perhaps the closest to a meaningful definition of "random," but it depends upon the validity of the uncertainty principle, and who qualifies as _anyone_. It is in fact what I said earlier, that QM is how we define "random", not that QM *is* random.

In particular since physical law can never be proven complete and correct, "random" can never be proven to be correctly defined. QM is our best understanding of the rules of the system, and thus our current best standard for "randomness." If we find the rules to be wrong, and that the uncertainty principle is not fundamental to future models of physics then our definition of "random" will change, and something else might be brought forward as the standard for "randomness."

In this respect saying "QM is random" is vacuous, and is the same as saying "QM is our best understanding of physics." Before we had QM we had Brownian motion which could be have been appropriately called "random" although we now call it merely "chaotic."

Good piece

Posted Nov 30, 2012 8:47 UTC (Fri) by man_ls (guest, #15091) [Link] (1 responses)

This is perhaps the closest to a meaningful definition of "random," but it depends upon the validity of the uncertainty principle, and who qualifies as _anyone_.

Why does this definition involve the uncertainty principle, at all? It depends on information theory alone: that the information held by any party is limited and can be computed.

Again, we go back to the beginning: the next number can be decided by a chaotic system (such as a coin toss), a statistical system (such as thermal noise), a truly quantum system (such as the spin of a particle in a pair) or just a clever adversary (as in cryptography). As long as the next number cannot be guessed by someone, it appears random. If the next number cannot be guessed by anyone (i.e. is not correlated to previous values), it is truly random. The difference is subtle but essential.

With this latest rehashing of the old deterministic conundrum I will leave the discussion, if you don't mind.

Good piece

Posted Nov 30, 2012 14:00 UTC (Fri) by davidescott (guest, #58580) [Link]

> the information held by any party is limited

by the uncertainty principle. Uncertainty is why information is limited.

Uncertainty is the only reason (currently provided by physics) why we cannot propose the existence of an advanced species on another planet capable of mapping the positions and momenta of all particles in the universe.

If such a species existed then they would be able to correlate all outputs and nothing would be truly random.

Yes its determinism, but I don't see how it is a conundrum. Its all fairly straightforward. If there is determinism, and if there are no limits on information that can be gained about a system, then there are no "physical secrets" and no true randomness.

-----------------------------------------------------

My only point in bringing all this up was to correct wolfgang's false statements concerning the determinism of classical mechanics, the non-determinism of QM, and the supposed relationship between randomness and this non-determinism.

a) Classical Mechanics has no limits on information, but contrary to what many seem to believe it is not deterministic.

If Classical Mechanics were a true theory for the universe we could still define "truly random" values, by say counting the number of space invaders passing through a detector over a fixed time interval. This would be randomness deriving from the non-determinism of the universe.

Classical Mechanics is a provably false theory, and thankfully no space invaders have ever been seen.

b) QM has limits on information, and that alone is enough to define some notion of randomness. Contrary to what many seem to believe, QM is not non-deterministic. The Copenhagen interpretation is non-deterministic and very popular, but it is not the only interpretation. The Bohm interpretation is deterministic.

c) Your subtle differences are not lost on me. YOUR definition of random (random := next number cannot be guessed by anyone) is a definition based on the limits of information. It is crucial what "anyone" can do. Those limits are derived from the limits on what can be learned about other systems in the universe. It has absolutely nothing to do with determinism.

This definition is only correct if the uncertainty principle is not violated, which is why I say that "QM defines random" and not "QM is random". A Bohmian sees no randomness in the outcome of the quantum process, it is all predetermined. A Bohmian defines random as outcomes which are unpredictable due to the limits on information, which is again your definition.

--------------------------------------------------

Final remark. You state:
> If the next number cannot be guessed by anyone (i.e. is not correlated to previous values), it is truly random. The difference is subtle but essential.

I'm going to pick on your parenthetical remark. The definition proposed by the parenthetical "anything not correlated to previous values is truly random" is either false, or a bad definition (depending on exactly how it is read).

Bohmian QM is a system where outcomes can be unpredictable, but they are not in a philosophical sense "uncorrelated" because everything is mediated through hidden variables and is fully deterministic. This is nonsense to a Bohmian.

Or maybe this is a practical definition for a Copenhagenist (who believes in intrinsic randomness). Take the outputs and run them through an auto-correlation test. If they fail they are non-random, if they pass they are random. The problem here is that any auto-correlation test has a confidence interval, and has a false positive and false negative rate. Just because a RNG passes the auto-correlation test doesn't mean it isn't just psuedo-random with a long period so you got a false-negative, and just because an over-volted transistor fails a test doesn't mean it wasn't obeying QM, its just a false-positive.

Good piece

Posted Nov 30, 2012 17:00 UTC (Fri) by nybble41 (subscriber, #55106) [Link] (1 responses)

> Which of the following is not a random number, but a useful mathematical constant?

I just said that it's not the numbers which are random, but rather the process which produced them. Any of those numbers could be produced by a random process, or a universal constant useful in some context. The output of a random process could (with infinitesimal probability) be equal to PI to an arbitrary number of digits, and still be random; the output of a deterministic process with known initial conditions could be completely unrecognizable without actually repeating the process, and pass all the statistical heuristics for randomness, and yet the process would still not be random.

> The larger point is that any deterministic system is "without time" in this sense, and that contrary to commonly held views Classical Mechanics is not deterministic, and QM can be deterministic.

I'm not arguing with that. True randomness does not require non-determinism, provided no one can know all the initial conditions. The difference is that, given non-determinism, a process can be random even _with_ complete knowledge of the initial conditions.

>> the question is whether the rules of the system permit _anyone_ to predict the next number with 100% certainty, given all the information which it is possible for them to know
> This is perhaps the closest to a meaningful definition of "random," but it depends upon the validity of the uncertainty principle, and who qualifies as _anyone_. It is in fact what I said earlier, that QM is how we define "random", not that QM *is* random.

First, it doesn't depend on QM at all, or the uncertainty principle. QM (whether non-local or non-deterministic) is one system which defines cases where the number cannot be predicted, but it is hardly the only one. Classical mechanics also works, if the initial conditions are unknowable. Second, "anyone" simply means "anyone"; there are no qualifications.

> In particular since physical law can never be proven complete and correct, "random" can never be proven to be correctly defined.

Whether a physical process is random does depend on the physics of the process. If we're wrong about the physical laws, we may also be wrong about whether the process is random.

That doesn't really affect the definition of "random" so much as the system to which the definition is applied: what the system permits one to know, and whether that knowledge is sufficient to predict the next result given the system's rules.

Good piece

Posted Nov 30, 2012 17:30 UTC (Fri) by davidescott (guest, #58580) [Link]

Agreed. I don't think you and I disagree on anything.

Good piece

Posted Nov 21, 2013 14:24 UTC (Thu) by mirabilos (subscriber, #84359) [Link] (2 responses)

> KZJ-1249

Care to explain to those non-USA people among us?

Good piece

Posted Nov 21, 2013 14:42 UTC (Thu) by cladisch (✭ supporter ✭, #50193) [Link] (1 responses)

The point of the joke is that the license plate is completely random without any obvious pattern.

Good piece

Posted Nov 21, 2013 14:48 UTC (Thu) by mirabilos (subscriber, #84359) [Link]

Ah, thanks.