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by **gill1109** » Thu Feb 27, 2014 12:32 am

Fred, have you even looked at the latest simulations by Joy and Chantal?

They have incorporated a large number of my ideas. They are even using R. I am very proud of their achievements.

But you can always see if you can do better still with Mathematica: good luck.

FrediFizzx wrote:
gill1109 wrote:Fred, you are a bit behind the game. Joy has already accepted the fact that what you say here is *not* true. You seem not to have noticed that Joy's changed his simulation model. He's also working with R, nowadays. I am not going to waste my time writing out a mathematical proof of something that Joy and I both agree on.

That is patently false that Joy agrees with you. So stop saying stuff like that since it is pure propaganda.

I didn't think you would be able to disprove what I am saying about the limits of going to infinity for the number of trials and infinitesimal degree increments. But you are right that it is a waste of time since I am 100 percent correct. We think it is you that is behind the game completely.

by **FrediFizzx** » Wed Feb 26, 2014 2:43 pm

gill1109 wrote:Fred, you are a bit behind the game. Joy has already accepted the fact that what you say here is *not* true. You seem not to have noticed that Joy's changed his simulation model. He's also working with R, nowadays. I am not going to waste my time writing out a mathematical proof of something that Joy and I both agree on.

That is patently false that Joy agrees with you. So stop saying stuff like that since it is pure propaganda.

I didn't think you would be able to disprove what I am saying about the limits of going to infinity for the number of trials and infinitesimal degree increments. But you are right that it is a waste of time since I am 100 percent correct. We think it is you that is behind the game completely.

by **gill1109** » Wed Feb 26, 2014 10:17 am

Ben6993 wrote:Richard: "breaking computation into pieces" ...

I remember doing that in the 1970s with Fortran66 using the "equivalence" statement so my graphs and tables could use the same memory space within a program. And later, the IBM S370(?) DOS system used 'paging' to do the same sort of thing. And if your variable arrays were stored columnwise, you hoped the paging wasn't being done rowwise!

Me too! Seems we are of similar vintage.

by **Ben6993** » Wed Feb 26, 2014 10:09 am

Richard: "breaking computation into pieces" ...

I remember doing that in the 1970s with Fortran66 using the "equivalence" statement so my graphs and tables could use the same memory space within a program. And later, the IBM S370(?) DOS system used 'paging' to do the same sort of thing. And if your variable arrays were stored columnwise, you hoped the paging wasn't being done rowwise!

by **gill1109** » Wed Feb 26, 2014 9:52 am

Ben6993 wrote:I tried your code {http://rpubs.com/gill1109/ChaoticUnsharpBall1} as it stood but received error warnings about size constraints on my PC,"Error: cannot allocate vector of size 76.3 Mb"
...
I have also had about five tries with gamma at different values in the range 0.455 to 0.465 without finding improvement.

Splendid. Sorry about the memory allocation! I have a 64 bit and rather new laptop with maxed out memory, processor speed and so on... some of my scripts are approaching my limits for "fast" computation. If numbers were to get bigger still I would have to break the computation into pieces so that everything is not all in memory at the same time, and then speed would suffer.

by **Ben6993** » Wed Feb 26, 2014 9:44 am

Hi Richard

I tried your code {http://rpubs.com/gill1109/ChaoticUnsharpBall1} as it stood but received error warnings about size constraints on my PC,"Error: cannot allocate vector of size 76.3 Mb".

So I reduced to 10^6 data pairs rather than 10^7 and it seemed to work despite "error warnings" about size [required more than 1021M bytes]

OUTPUT:
> angles[66] * 180 / pi
[1] 65
> corrs[66]
[1] 0.4204177
> cos(angles[66])
[1] 0.4226183
> plot(angles * 180/pi, corrs - cos(angles), type = "l")
> 1/sqrt(Ns[66])
[1] 0.001304742
>

My output graph for {plot(angles * 180/pi, corrs - cos(angles), type = "l")} was similar to yours but not as smooth, which is as one would expect with a smaller run. And, as in your run, most deviant from zero at 30-40 degrees and 70 degrees.

I have also had about five tries with gamma at different values in the range 0.455 to 0.465 without finding improvement.

by **gill1109** » Wed Feb 26, 2014 9:41 am

Joy Christian wrote:OK, we are done. I have revised my simulation: http://rpubs.com/chenopodium/13653.

Let me remind again that the initial state of the system is still (e_o, theta_o), as derived in http://libertesphilosophica.info/blog/w ... mplete.pdf, but the choice of the initial function f(theta_o) is now different:

f(theta_o) = (1/2.47) sin(theta_o)^{1.61}.

It is also important to note that the Monte Carlo accuracy of the simulation is about 0.0001, but any remaining wrinkles in the correlation function are much smaller than 0.0001.

The difference between the theoretical and the simulated values is at its most about 0.02, very much larger than the simulation error. Have you looked at the bottom graph?

I have got the difference down to 0.002 in my latest.

http://rpubs.com/gill1109/ChaoticUnsharpBall1

The only difference with yours is a slightly different formulat for f(theta_0).

Compare the bottom graphs on both pages. It will not be difficult to get even better by new small tweaks. Chantal will want to get back ahead of me again, I am sure she will help you!

by **gill1109** » Wed Feb 26, 2014 9:32 am

Joy Christian wrote:
gill1109 wrote:Here's the best simulation of Joy's model ever!

http://rpubs.com/gill1109/ChaoticUnsharpBall1

That is not a simulation of my model, regardless of its quality.

This is a simulation of my model: http://rpubs.com/chenopodium/joychristian.

Dear Joy

There is a mathematical equivalence between the abstract structure of a simulation of Caroline Thompson's model, and the abstract structure of your simulation of your model. One can translate between one and the other. You have theta_0. She has R. One can be expressed in terms of the other, the formula is simple, you can look it up, if you are beginning to be able to read R now. She has measurement directions a and b. You have measurement directions a and b. She has two opposite random points on the sphere called "N" and "S" (N for North, S for South, determining a direction in space). You have a uniform random point on S^2 called e_0.

She derives her model from consideration of a ball spinning in space chaotically... you derive your model from consideration of S^3 etc etc ... But when we get down to a computer simulation in R, Python, Java or whatever, we simulate mathematical things like: "pick a random point on S^2". Pick a random number from some interval with some distribution ...

Recognising the same abstract mathematical structure in completely different physical systems is what mathematics is all about, what gives it its power! I am sure you recognise that.

You can ask Chantal to "translate" my latest R code into a language which you prefer. I think you'll be very pleased with the result. We are getting closer all the time, and gaining insights all round.

by **Joy Christian** » Wed Feb 26, 2014 9:07 am

gill1109 wrote:Here's the best simulation of Joy's model ever!

http://rpubs.com/gill1109/ChaoticUnsharpBall1

That is not a simulation of my model, regardless of its quality.

This is a simulation of my model: http://rpubs.com/chenopodium/joychristian.

by **Joy Christian** » Wed Feb 26, 2014 8:49 am

OK, we are done. I have revised my simulation: http://rpubs.com/chenopodium/13653.

Let me remind again that the initial state of the system is still (e_o, theta_o), as derived in http://libertesphilosophica.info/blog/w ... mplete.pdf, but the choice of the initial function f(theta_o) is now different:

f(theta_o) = (1/2.47) sin(theta_o)^{1.61}.

It is also important to note that the Monte Carlo accuracy of the simulation is about 0.0001, but any remaining wrinkles in the correlation function are much smaller than 0.0001.

by **gill1109** » Wed Feb 26, 2014 8:34 am

Here's the best simulation of Joy's model ever!

http://rpubs.com/gill1109/ChaoticUnsharpBall1

Unfortunately it is still not spot-on, the maximal deviation between the curves (0.002) is somewhat larger than the statistical error (0.0004). The statistical error is much larger than any numerical error (numerical precision / rounding errors) so we can safely ignore purely numerical error. [For Fred: this is what I mean by a computational/Monte Carlo proof that some sequence of simulations is *not* going to converge to the cosine. It is not a question of knowing something about calculus. It's a question of knowing something about Monte Carlo simulation, statistics, and numerical precision.]

I thought I had come up with a simple exact analytical solution, R = (1 + sqrt U) * 45 degrees, U ~ Unif(0,1), but it turned out that the 0.46th power fitted better than the 1/2 power, and neither fits exactly.

R is the radius of the circular cap in the Chaotic ball model which is mathematically equivalent to the Minkwe simulation in 3-D. It is related in a simple way to the theta_o of the latest Christian-Roth simulation.

My code is taken copy-paste from the Christian-Roth simulation model, which was the previous front runner.

by **gill1109** » Wed Feb 26, 2014 2:56 am

Here is computational proof that it can be done *exactly*: clearly, a continuous convex combination of the blue curves, which are Christian-Roth models with fixed theta_0, and simultaneously Caroline Thompson models with fixed R, can be found which yields the black curve (cosine), exactly.

http://rpubs.com/gill1109/ChaoticUnsharpBall3

Fred: maybe you can use Mathematica to find out what convex combination is needed????

by **gill1109** » Wed Feb 26, 2014 1:18 am

Fred, you are a bit behind the game. Joy has already accepted the fact that what you say here is *not* true. You seem not to have noticed that Joy's changed his simulation model. He's also working with R, nowadays. I am not going to waste my time writing out a mathematical proof of something that Joy and I both agree on.

I suggest you install and learn R yourself. Both are pretty easy jobs.

Actually we are still only very close, still not yet spot-on. The problem is, what distribution to take for theta_0? At present Chantal Roth and Joy Christian together have homed in empirically on a pretty good choice, which in terms of Caroline Thompson's equivalent ball model, is circular caps of radius

R = acos((sin(Phi)^1.32)/3.16), Phi ~ Unif(0, pi/2).

Minkwe (lifted to 3-D - another of my innovations which Joy has taken on board) had

R = acos((sin(Phi)^2)/2), Phi ~ Unif(0, pi/2).

I've written an R script so one can experiment with various different probability distributions for R. And in particular, try various *constant* values of R. It is pretty clear that as you vary R between pi/4 and pi/2 you get various curves, none of which look much like a cosine, but such that a continuous convex linear combination of them can be found, which is exactly equal to the cosine! (OK: this is not a theorem, it is a conjecture, but it is a bloody good one, pardon my French!).

See:

http://rpubs.com/gill1109/ChaoticUnsharpBall2

The code is taken by copy-paste from Chantal and Joy's latest production http://rpubs.com/chenopodium/joychristian. The only thing I have changed is the probability distribution of theta_0

by **FrediFizzx** » Wed Feb 26, 2014 12:32 am

This is how you do it.

Those small deviations around the peaks will disappear in the limit when the number of trials goes to infinity and the degree increments are infinitesimal.

by **FrediFizzx** » Wed Feb 26, 2014 12:13 am

gill1109 wrote: I guess the word of thanks is the closest we are going to get to an acknowledgement by Christian that my criticism of earlier simulation models was justified. I wonder if Fred will also admit that he was wrong, too.

Not from that for sure.

Fred Diether replied on Feb. 24, 2014 @ 00:01 GMT Richard said, "It can be mathematically proven that we *don't* go from Bell's straight line to the cosine curve." in relation to what I said.

OK, go for it. Let's see the rigorous mathematical proof that what I am saying above is false. I would advise you not to waste your time on it since Joy has already extremely mathematically and rigorously proven that it is true here and with this paper here from a different direction.

From the FQXi blog. I am still waiting for you to show your mathematical proof to what I said here.

Fred Diether wrote on Feb. 23, 2014 @ 20:22 GMT In addition to what Joy posted, it is very clear to see what is going on with the Minkwe simulation via John Reed's Mathematica translation. Take a look at the 6 graphs as theta_0 goes from 0 to pi/2 with 5 million trials and 1 degree resolution. We go from Bell's straight line to the cosine curve. Anyone that knows calculus can see that as the number of trials goes to the infinity limit and as the degree resolution goes to the infinitesimal limit, that the perfect cosine curve will be achieved. No doubt this could be mathematically proven should there be an enterprising soul out there. It is easy to see though so not sure if anyone should waste their time on it.

You need to do the EPR Challenge here.
viewtopic.php?f=22&t=22#p448

by **gill1109** » Tue Feb 25, 2014 9:05 pm

Joy Christian wrote:I am grateful to Richard Gill for insisting on greater precision for the simulation, which helped me discover a more accurate choice for the initial state (e_o, theta_o) in the EPR-Bohm case. I am also grateful to Chantal Roth for encouraging me to learn R and thus investigate Richard Gill's simulation myself. The result of my investigations is not devoid of beauty.

Indeed not devoid of beauty, and getting very close, but not quite there yet. I have been discussing this with Chantal. What to tweak next? My suggestion is the lower bound to the radii of the caps. It's strange that it's such a funny angle, and I think this sharp cut-off explains the straightness of the correlation between 60 and 90 degrees.

I guess the word of thanks is the closest we are going to get to an acknowledgement by Christian that my criticism of earlier simulation models was justified. I wonder if Fred will also admit that he was wrong, too.

by **Joy Christian** » Tue Feb 25, 2014 12:16 pm

So let us put some perspective on my version of Richard Gill's S^2 version of Michel Fodje's original simulation of my 3-sphere, or SU(2), model for the EPR-Bohm correlation.

As we can see from the details discussed in the above paper, my analytical model predicts the (negative) cosine correlation exactly. What is more, in Bell's local-realistic framework (discussed in his 1964 paper) we are completely free to choose whatever initial or complete state of the system we like. In the present representation of the 3-sphere this initial state is represented by the pair (e_o, theta_o), which are simply four numbers. These numbers depend on the system under consideration. They depend on the symmetry of the physical situation. For example, for the GHZ or Hardy case the initial state would be nothing like these four numbers.

Now have a look at the derivation of this state within my 3-sphere model. As we can see from just above the box of eq.(10), I had made a simple choice for the initial state by choosing the function f(theta_o) defined in eq.(7). This choice simply specifies the magnitude of the sum of the two initial quaternions, p_o and q_o, and thereby also specifies the initial state (e_o, theta_o). In Michel Fodje's simulation the choice I made for f(theta_o) seemed necessary and sufficient to produce the correct correlation. It now appears that when one zooms-in with greater precision, the initial state is in fact what has now been chosen in the latest simulation.

I am grateful to Richard Gill for insisting on greater precision for the simulation, which helped me discover a more accurate choice for the initial state (e_o, theta_o) in the EPR-Bohm case. I am also grateful to Chantal Roth for encouraging me to learn R and thus investigate Richard Gill's simulation myself. The result of my investigations is not devoid of beauty.

by **gill1109** » Tue Feb 25, 2014 9:29 am

Yes it is really beautiful! What I see:

You "create" two free parameters and by choosing them cleverly you get much closer to what you want. Joy/Chantal write

s <- (sin(theta)^1.32)/3.16

I had

s <- (sin(theta)^2)/2

Note: the formula I used was given to us by Joy Christian, I believe. Seems he was wrong, that time.

It's indeed a fun game. I hope Christian is now very glad that he is now very very close to the cosine.

It will be hard for anyone to prove that these two new numbers 1.32 and 3.16 are somehow universal properties of nature which follow from the S^3 theory. It will be hard for anyone to prove that the final curve definitely is not *exactly* the cosine, but obviously it cannot be, however "accurately" one determines 1.32 and 3.16 (though Joy will say, obviously it must be, because of his S^3 model).

I hope Joy now admits that all previous simulation models were somehow "wrong" and that Richard Gill and Chantal Roth together helped him to fix them. Using the very advanced programming language R.

by **Joy Christian** » Tue Feb 25, 2014 7:50 am

Here, finally, is my version of Richard Gill's S^2 version of Michel Fodje's classic simulation:

http://rpubs.com/chenopodium/joychristian (with thanks to Chantal Roth).

Do scroll all the way down on the page to check out the second plot. What do you see?

by **gill1109** » Tue Feb 25, 2014 12:05 am

Dear Minkwe,

Seems we have a different sense of humour!

It was not obvious to me whether you were talking about words written by Gisin and Gisin, about their mathematics, or about my code. Three different things!

But good: you were criticizing what Gisin and Gisin wrote about their model.

Now please tell me whether you think my simulation is non local! I admit that it was in some way inspired by the Gisin Gisin model, but I do not have to take on board any of their ideas about what it could mean. Where is the non-locality in my code?

If you prefer I can write out the simple basic formulas which the code implements, and you can tell me where is the non-locality in my formulas?

Richard

minkwe wrote:
gill1109 wrote:I am waiting for Minkwe to tell us where precisely he sees a difference in angle between Bob and Alice being used to define a vector of Alice's. Where he sees it in my code. He does not need to await instructions from The Big Boss. He's an independent, thinking, scientist.

Richard,
What's all this talk about The Big Boss, is it coming from the same place as your earlier claims without evidence that I had been paid to write code?

As concerns your demands that I make a precise statement about your code, it is obvious that I made a statement about the Gisin model, not your code. If you want to see the part of the paper about which I was speaking, you can check for yourself at the top of page 2:

http://arxiv.org/pdf/quant-ph/9905018.pdf

Where they derive the -cos(alpha) relationship, you can see that they do not use separate angles on each side. Rather, they define the vectors by using the difference of the angles. Now I'm not saying it is not possible to redo it using different angles, all I'm saying is they define b = (0,0,1) and a = (sin(alpha), 0, cos(alpha)), which clearly shows that we are using information about Bob's angle choice to define Alice's angle choice.

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