Sci.Physics.Foundations

[quote=gill1109 post_id=313 time=1637898047 user_id=60]
More maths: https://ieeexplore.ieee.org/document/9622238 [/quote]
You really should stop writing nonsense papers and posting nonsense to the forum. :mrgreen: :mrgreen: :mrgreen:
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More maths: https://ieeexplore.ieee.org/document/9622238

@gill1109 None of this really matters any more. New CHSH 10 run average,

[b]CHSH = 2.81796![/b] :mrgreen: :mrgreen: :mrgreen:
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Mathematica's warning message is misleading. We are working over the real numbers, when we try to solve that equation. The inverse of the function "exp" is "log". These are strictly monotone, continuous, real functions. The equation is easily seen to have exactly one solution. You can easily write down the formula for it. Mathematica probably can too, but it apparently needs to be given some help.

Actually, we want to find the smallest positive integer which satisfies an *inequality*, not an "equality*.

Mathematica programming challenge: see if you can tell Mathematica to do what we actually want it to do!

> 8 * exp(-171126 / 160^2)
[1] 0.01000002
> 8 * exp(-171127 / 160^2)
[1] 0.009999633

[quote=FrediFizzx post_id=308 time=1637832701 user_id=58]
[quote=gill1109 post_id=305 time=1637815457 user_id=60]
Fred, that was not the smartest way to solve that equation
I recommend you start by taking logs
log_e(0.1 / 8) = - N (eta/16)^2
N = log_e(800) x 160^2

About 172 thousand

You should have done Solve, not Reduce. [/quote]

[img]https://sciphysicsfoundations.com/download/gill_formula2.png[/img]

Solve doesn't find all the solutions. Not good.
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[/quote]
A little thought will show that there is exactly one real solution to that equation. Solve found it for you, up to whatever numerical accuracy you asked for. The smallest integer at least as large as the unique real solution is what you really want. Well done.

The answer is N = 171127 or larger.

[code]> 8 * exp(-171126 / 160^2)
[1] 0.01000002
> 8 * exp(-171127 / 160^2)
[1] 0.009999633[/code]

[quote=gill1109 post_id=305 time=1637815457 user_id=60]
Fred, that was not the smartest way to solve that equation
I recommend you start by taking logs
log_e(0.1 / 8) = - N (eta/16)^2
N = log_e(800) x 160^2

About 172 thousand

You should have done Solve, not Reduce. [/quote]

[img]https://sciphysicsfoundations.com/download/gill_formula2.png[/img]

Solve doesn't find all the solutions. Not good.
.

Enjoy: https://rpubs.com/gill1109/theorem

Notice: my theorem states a bound on a probability. It does not state that that bound is the best bound. I can give you much sharper but rather more complicated bounds.

I can also give you approximations. Good ones. With much smaller N, the chance that the CHSH-like quantity I defined in the theorem would be bigger than 2.1 is already very small indeed.

You could easily do simulations to verify my claims. That is also a good way to check that you understand the statement of the theorem.

Fred, that was not the smartest way to solve that equation
I recommend you start by taking logs
log_e(0.1 / 8) = - N (eta/16)^2
N = log_e(800) x 160^2

About 172 thousand

You should have done Solve, not Reduce

[quote=gill1109 post_id=302 time=1637775072 user_id=60]
How to use the theorem. For instance you want to know how large N should be such that the chance that the CHSH-like quantity in the theorem is smaller than 2.1 with probability 99%. Then find the value of N such that, with eta = 0.1, 8 exp( - N (eta/16)^2 ) = 0.01 [/quote]
So, like this?

[img]https://sciphysicsfoundations.com/download/gill_formula.png[/img]

Then N = 171126 minus some imaginary number? What the heck does that tell us?

But it matters not because this kills it all.

https://www.wolframcloud.com/obj/fredifizzx/Published/CS-39-S3quat-3D-new-prodcalc-forum.nb

Like I was saying, you got NO proof anymore! :mrgreen: :mrgreen: :mrgreen:
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PS. Which means you can stop posting your nonsense any time now! :D
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How to use the theorem. For instance you want to know how large N should be such that the chance that the CHSH-like quantity in the theorem is smaller than 2.1 with probability 99%. Then find the value of N such that, with eta = 0.1, 8 exp( - N (eta/16)^2 ) = 0.01

Ha, got it.

[tex]\Pr( \langle AB \rangle_{obs}+\langle AB' \rangle_{obs}+\langle A'B \rangle_{obs} - \langle A'B' \rangle_{obs}\le 2 + \eta ) \ge 1- 8 e^{- N (\frac{\eta}{16})^2 }[/tex]

Here, <AB>_{obs} stands for the average of the products of what is in columns A and B, over just those rows where A and B were both selected for observation. In each row, either A or A' is selected, and B or B' is selected, each with probability a half. All 2N selections independent of one another.

[quote=FrediFizzx post_id=298 time=1637766620 user_id=58]
[quote=gill1109 post_id=285 time=1637645213 user_id=60]
I presented a mathematical theorem. It uses the standard notation and concepts of probability theory. If you don’t know them, you might have difficulties reading what I wrote. Ask questions. Let me know what you don’t understand. [/quote]
Now that is really funny because I don't think I have ever seen you do,
Theorem 1: Blah, Blah, Blah, ....
IOW, I don't think I have ever seen you make a real theorem statement like they do in most math documents. Send me a link if you have.
[/quote]
Take a look at my [b]Theorem 1[/b] in [url]https://arxiv.org/abs/1207.5103[/url]
[i]Statistics, Causality and Bell's Theorem[/i]

Abstract: Bell's [Physics 1 (1964) 195-200] theorem is popularly supposed to establish the nonlocality of quantum physics. Violation of Bell's inequality in experiments such as that of Aspect, Dalibard and Roger [Phys. Rev. Lett. 49 (1982) 1804-1807] provides empirical proof of nonlocality in the real world. This paper reviews recent work on Bell's theorem, linking it to issues in causality as understood by statisticians. The paper starts with a proof of a strong, finite sample, version of Bell's inequality and thereby also of Bell's theorem, which states that quantum theory is incompatible with the conjunction of three formerly uncontroversial physical principles, here referred to as locality, realism and freedom. Locality is the principle that the direction of causality matches the direction of time, and that causal influences need time to propagate spatially. Realism and freedom are directly connected to statistical thinking on causality: they relate to counterfactual reasoning, and to randomisation, respectively. Experimental loopholes in state-of-the-art Bell type experiments are related to statistical issues of post-selection in observational studies, and the missing at random assumption. They can be avoided by properly matching the statistical analysis to the actual experimental design, instead of by making untestable assumptions of independence between observed and unobserved variables. Methodological and statistical issues in the design of quantum Randi challenges (QRC) are discussed. The paper argues that Bell's theorem (and its experimental confirmation) should lead us to relinquish not locality, but realism.

Statistical Science 2014, Vol. 29, No. 4, 512-528
DOI: 10.1214/14-STS490

Theorem 1. Given an N × 4 spreadsheet of numbers ±1 with columns A, A′, B and B′, suppose that, completely at random, just one of A and A′ is observed and just one of B and B′ are observed in every row. Then, for any η ≥ 0, (3) [displayed equation].

The various objects defined in equation (3) are defined earlier in the paper. I will see if I can put the equation into this thread later on, as well as the needed definitions.

The proof is given in full later in the paper.

[quote=gill1109 post_id=285 time=1637645213 user_id=60]
I presented a mathematical theorem. It uses the standard notation and concepts of probability theory. If you don’t know them, you might have difficulties reading what I wrote. Ask questions. Let me know what you don’t understand. [/quote]
Now that is really funny because I don't think I have ever seen you do,

Theorem 1: Blah, Blah, Blah, ....

IOW, I don't think I have ever seen you make a real theorem statement like they do in most math documents. Send me a link if you have.
.

I presented a mathematical theorem. It uses the standard notation and concepts of probability theory. If you don’t know them, you might have difficulties reading what I wrote. Ask questions. Let me know what you don’t understand.

@gill1109 You've got no theorem. All you have is a junk math theory. Stop spewing nonsense like the above!
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[quote=FrediFizzx post_id=245 time=1637421029 user_id=58]
@gill1109 "Obviously, there are very many ways to do this!" ??? Then what you are claiming as a theorem is not a theorem but just a theory.
[/quote]
It's a mathematical theorem. I say that if A, B, C, D and E all hold, then F is true as well. I simply do not discuss the many ways you can avoid the conclusions of the theorem. The theorem has a number of assumptions. You can avoid the conclusions by violating some or all of the assumptions.

Joy Christian has tried several different ways in his papers.

You could:

Relax the assumption that outcomes are binary
Alter your definition of correlation
Violate locality
Violate no-conspiracy (statistical independence)
Violate realism

I do not say that A, B, C, D and E are true [b]if and only if[/b] F is true.

Fine has a theorem giving necessary and sufficient conditions for the 8 one-sided CHSH inequalities (but still assuming some background assumptions); Boole did the same (in retrospect) for the 6 one-sided original Bell inequalities.

@gill1109 "Obviously, there are very many ways to do this!" ??? Then what you are claiming as a theorem is not a theorem but just a theory.
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Since so far *nobody* has actually responded to the original post, I will repeat it here.

It's about a little exercise in elementary probability theory which I published in several papers more than 20 years ago, and recently, again, in my published "Comments" on some papers by Joy Christian in IEEE Access and in RSOS.

So far, only @minkwe entered the mathematics discussion, but he did this by talking about a different proof of a quite different claim.

Here it goes again:

[quote]Let[tex] X_1, X_2, Y_1, Y_2[/tex] be four random variables taking the values [tex]+/-1[/tex]
Let [tex]A, B[/tex] be two random variables taking the values [tex]1, 2[/tex]
Suppose [tex](A, B)[/tex] is statistically independent of [tex](X_1, X_2, Y_1, Y_2)[/tex]
Define [tex]X = X_A[/tex] and [tex]Y = Y_B[/tex]
In other words, [tex]X = X_1[/tex] if [tex]A =1[/tex], [tex]X = X_2[/tex] if [tex]A = 2[/tex]; [tex]Y = Y_1[/tex] if [tex]B = 1[/tex], [tex]Y = Y_2[/tex] if [tex]B = 2[/tex]
Define [tex]\rho_{ab} = E(XY | A = a, B = b)[/tex]

Theorem: [tex]\rho_{11} - \rho_{12} - \rho_{21} - \rho_{22}[/tex] lies in the interval [tex][-2 , +2][/tex].

Proof:

[tex]X_1 Y_1 - X_1 Y_2 - X_2 Y_1 - X_2 Y_2 = X_1 (Y_1 - Y_2) - X_2 (Y_1 + Y_2)[/tex] can only take the values [tex]+/-2[/tex]

Hence
[tex]E(X_1 Y_1) - E(X_1 Y_2) - E(X_2 Y_2) - E(X_2 Y_2)[/tex] lies in [tex][-2 , +2][/tex].

But [tex]E(X_a Y_b) = E(X_a Y_b | A = a, B = b)[/tex] by statistical independence of [tex](X_1, X_2, Y_1, Y_2)[/tex] from [tex](A, B)[/tex]
[tex]= E(X Y | A=a, B =b)[/tex] because [tex]X Y = X_a Y_b[/tex] when [tex]A = a, B = b[/tex]
[tex]= \rho_{ab}[/tex].

QED

[/quote]

I claim that this theorem is useful in the analysis of classical computer simulation of quantum correlations. It tells us that in order to achieve your aim, you will have to violate the conditions of the theorem. Obviously, there are very many ways to do this!

[quote=FrediFizzx post_id=228 time=1637294919 user_id=58]
[quote=gill1109 post_id=227 time=1637294341 user_id=60]
My problem was the physicists’ attitude “shut up and calculate”. Don’t ask questions. Just watch how we do it, and copy that. The mathematicians’ attitude was: find out *why* a certain weird looking calculation does work. Keep on asking questions. Find deep connections between seemingly unrelated parts of mathematics. Always question the assumptions which you make without thinking, for convenience’ sake and because everyone else does. [/quote]
Yeah, yeah, yeah, blame it on somebody. :mrgreen:
[/quote]
I'm not blaming anything on anybody. I think that in science, we need mathematicians and physicists. Nobody should underestimate what the other way of thinking has to offer. One must learn to communicate and to appreciate.