A new simulation of the EPR-Bohm correlations

Foundations of physics and/or philosophy of physics, and in particular, posts on unresolved or controversial issues

Re: A new simulation of the EPR-Bohm correlations

Postby Schmelzer » Mon Jun 08, 2015 1:30 pm

Joy Christian wrote:I have made no claims that are even slightly wrong.


Amen :D
Schmelzer
 
Posts: 123
Joined: Mon May 25, 2015 1:44 am

Re: A new simulation of the EPR-Bohm correlations

Postby Joy Christian » Wed Jun 10, 2015 11:20 pm

Here is a nice article from which the flatlanders can learn a thing or two:

https://johncarlosbaez.wordpress.com/20 ... dimension/

Image

See also: http://challengingbell.blogspot.co.uk/2 ... tians.html.
Joy Christian
Research Physicist
 
Posts: 1578
Joined: Wed Feb 05, 2014 3:49 am
Location: Oxford, United Kingdom

Re: A new simulation of the EPR-Bohm correlations

Postby Q-reeus » Thu Jun 11, 2015 4:06 am

According to a one Gavin Wince, we need not 1, not 2, but 3 dimensions of time to 'really make sense of it all'. His 'Existics' website: http://existics101.com/
And here's a lengthy YouTube vid setting it all out: https://www.youtube.com/watch?v=V7bbYNCdqak
Could not watch it all the way through - 'confusing' best sums it up for me. Then again, maybe others have looked at his stuff and found it stimulating or enlightening or even exhilarating, or whatever. Any thoughts?
Q-reeus
 
Posts: 259
Joined: Sat Jun 07, 2014 11:18 pm

Re: A new simulation of the EPR-Bohm correlations

Postby Ben6993 » Thu Jun 11, 2015 5:36 am

Q-Reeus,
I couldn't watch it all the way through, though it was entertaining, as the style of delivery put me off.

Armin, who used to post in the old s.p.f used to have a model where the electron had its own internal time dimension.
I believe the same, too. Black holes have their own internal time dimensions?
I may have oversimplified it but the s time scale of Greg Egan, in one of the links that Joy has just above posted, is the time in the n+1 dimensions which runs differently to the projected time onto the n dimensions. But t and s are not really fundamentally different, it is just that t is a projection of s? This reminds me of a cubist comment on Joy's new graphic where. ... No.... it is easier to see on http://challengingbell.blogspot.co.uk/2 ... tians.html where the electrons move in the red dimensions but flatland is the table top. The projection of the redland is an ellipse on the table top (which is equivalent to the 'good' data in the old R simulation). The cubist comment is equivalent to saying that the simulations must be run in the circle on the table top where the green area contains the disputed, discarded data. Does that green data exist in real experiments? No, not if real experiment can give the cos curve. Does that green data exist in Joy's model? No, except as trimmed junk. Does that green data exist in flatland simulations? Should it? Impasse again.

There is a Leonard Susskind lecture at:
http://www.youtube.com/watch?v=yK4rhkcsQDc
The black-hole information paradox, complementarity, and firewalls by Leonard Susskind
(I hope this is the right one!)
In it he mentions five main aspects for quantum gravity which from memory are:
QM, GR, BHs, entropy/thermodynamics, entanglement.
(It is interesting in this quantum gravity connection that JAY has newly reentered thermodynamic territory)
Susskind seems to link BHs to the entanglement. If you have two perfectly entangled BHs, and if Alice and Bob are in two different regions which are beyond each others horizons, then if Alice fell into her BH and Bob fell into his BH then they could eventually meet in the 'middle', despite originally being in regions of space which are disconnected. (Wormhole?) But it requires perfect entanglement. The less perfect the entanglement the less chance Alice and Bob have of meeting up.

I have a paper at http://wp.me/p18gTT-26 where the conclusion is "This paper shows that a Rasch analysis compresses its location parameter space according to the level of uncertainty in making judgements within that space. The more uncertain the judgements, the more compressed are the points on the scale." This seems to fit in with Suuskind's claim about entangled BHs leading to close proximity (in some dimensions or other) in that entanglement maximises the chance element, as seen in the 50% chance of A being -1 or +1. However since the A outcome in Joy's model is deterministic, and the chance only arises because of the counfounding of the double cover. I am not sure where that leaves entanglement supposedly being a fifth component to quantum gravity work.

By analogy one could argue that if it were possible to fall into an electron, like into a BH, one would exit our time dimension and enter the electron's internal time dimension. If Alice and Bob did this in their entangled particles, they could meet in the middle? Treating our universe like a particle, which I am in favour of, then as all particles collapse wavefunction on interaction, which signifies the end of the electron's internal time dimension, our universe could end its time at an interaction cf Penrose's CCC. Penrose's ends of cycle ought, in this logic, to correspond to particle interactions, where the universe is a particle.

Sorry to rabbit on but I am between projects.
Ben6993
 
Posts: 287
Joined: Sun Feb 09, 2014 11:53 am

Re: A new simulation of the EPR-Bohm correlations

Postby Yablon » Thu Jun 11, 2015 5:47 am

Q-reeus wrote:According to a one Gavin Wince, we need not 1, not 2, but 3 dimensions of time to 'really make sense of it all'. His 'Existics' website: http://existics101.com/
And here's a lengthy YouTube vid setting it all out: https://www.youtube.com/watch?v=V7bbYNCdqak
Could not watch it all the way through - 'confusing' best sums it up for me. Then again, maybe others have looked at his stuff and found it stimulating or enlightening or even exhilarating, or whatever. Any thoughts?

Please look at my new post on Euclidean space time transformations and quantum gravity. I do indeed have three time dimension for which the observed time is an invariant, up to its Lorentz transformation with space. Jay
Yablon
Independent Physics Researcher
 
Posts: 233
Joined: Tue Feb 04, 2014 9:39 pm
Location: New York

Re: A new simulation of the EPR-Bohm correlations

Postby Joy Christian » Thu Jun 11, 2015 10:44 am

Something I wrote on Albert Jan's blog page about the latest simulation that is worth reproducing here: http://challengingbell.blogspot.co.uk/2 ... tians.html:

Joy Christian wrote:To be sure, a pre-ensemble M of the pre-states u has been used to build this simulation, but such scaffolding mathematical structures are routinely used in modern physics --- for example, in gauge field theories. The scaffolding structures are eventually "gauged out" to extract the correct physics out of the auxiliary mathematics. Similarly, the true ensemble N of the true states (u, s) has been extracted in my simulation from the pre-ensemble M of the pre-states u, thereby reducing the auxiliary pre-pair {M, u} to the true pair {N, (u, s)}, just as in any modern gauge theory.
Joy Christian
Research Physicist
 
Posts: 1578
Joined: Wed Feb 05, 2014 3:49 am
Location: Oxford, United Kingdom

Re: A new simulation of the EPR-Bohm correlations

Postby AnotherGuest » Fri Jun 12, 2015 1:14 am

Joy Christian wrote:Something I wrote on Albert Jan's blog page about the latest simulation that is worth reproducing here: http://challengingbell.blogspot.co.uk/2 ... tians.html:

Joy Christian wrote:To be sure, a pre-ensemble M of the pre-states u has been used to build this simulation, but such scaffolding mathematical structures are routinely used in modern physics --- for example, in gauge field theories. The scaffolding structures are eventually "gauged out" to extract the correct physics out of the auxiliary mathematics. Similarly, the true ensemble N of the true states (u, s) has been extracted in my simulation from the pre-ensemble M of the pre-states u, thereby reducing the auxiliary pre-pair {M, u} to the true pair {N, (u, s)}, just as in any modern gauge theory.


There is also a suggestion at http://challengingbell.blogspot.co.uk/2015/06/a-flatlanders-view-on-joy-christians.html that the "true ensemble of true states" is different for each different pair of settings. Inspection of the R code http://rpubs.com/jjc/84238 confirms that this could well be the case: the function g depends on the setting and it can deliver outcomes 0 as well as -1 and +1; then the 0's are discarded from the correlation calculation.

The published output of the simulation http://rpubs.com/jjc/84238 reveals that the size of the "true ensemble" is different for each pair of settings: N = 52585, 52101, 52291, and 51915 for just four particular setting pairs. It seems to me that this contradicts the theoretical claim, that the selection of the "true ensemble" from the "pre ensemble" does not depend on the actual settings.
AnotherGuest
 

Re: A new simulation of the EPR-Bohm correlations

Postby Heinera » Fri Jun 12, 2015 10:56 am

AnotherGuest wrote:The published output of the simulation http://rpubs.com/jjc/84238 reveals that the size of the "true ensemble" is different for each pair of settings: N = 52585, 52101, 52291, and 51915 for just four particular setting pairs. It seems to me that this contradicts the theoretical claim, that the selection of the "true ensemble" from the "pre ensemble" does not depend on the actual settings.

Indeed.
Heinera
 
Posts: 378
Joined: Thu Feb 06, 2014 12:50 am

Re: A new simulation of the EPR-Bohm correlations

Postby Joy Christian » Fri Jun 12, 2015 11:22 am

I am going to ignore the comments made by AnotherGuest and Heinera, because the issues they are raising are already addressed by me in my previous posts.

Instead, for those of you who have followed my work and understand my 3-sphere model, I have cleaned up the simulation to make the main ideas clearer:

http://rpubs.com/jjc/84238.
Joy Christian
Research Physicist
 
Posts: 1578
Joined: Wed Feb 05, 2014 3:49 am
Location: Oxford, United Kingdom

Re: A new simulation of the EPR-Bohm correlations

Postby Schmelzer » Fri Jun 12, 2015 1:50 pm

Joy Christian wrote:I am going to ignore the comments made by AnotherGuest and Heinera, because the issues they are raising are already addressed by me in my previous posts.

Instead, for those of you who have followed my work and understand my 3-sphere model, I have cleaned up the simulation to make the main ideas clearer:

http://rpubs.com/jjc/84238.


I think, the claim
http://rpubs.com/jjc/84238 wrote:# This ratio does not depend on the settings a and b, as
# proved by the graph below: N/L = 1. Consequently, the probability density
# rho(u, s) for the (u, s) also remains independent of the settings a and b.

can be already classified as deception. As N, as L depends on a and b, in the same way, as can be easily seen from the formulas, and we have told this to Joy many times. Giving these terms names which suggest that there is no such dependence does not change the fact that there is such a dependence.

Ok, Joy may disagree, but even in this case it would be appropriate to inform the readers that several people have argued that the expression depends on a and b. But, given that this is hidden from the reader, I think the accusation of deception is not too strong.
Schmelzer
 
Posts: 123
Joined: Mon May 25, 2015 1:44 am

Re: A new simulation of the EPR-Bohm correlations

Postby Joy Christian » Fri Jun 12, 2015 2:10 pm

So the flatlanders think that they are being deceived. They certainly are --- by their inability to liberate themselves from the flatland, and by their inability to read:

http://rpubs.com/jjc/84238

Joy Christian wrote:To be sure, a pre-ensemble M of the pre-states u has been used to build this simulation, but such scaffolding mathematical structures are routinely used in modern physics --- for example, in gauge field theories. The scaffolding structures are eventually "gauged out" to extract the correct physics out of the auxiliary mathematics. Similarly, the true ensemble N of the true states (u, s) has been extracted in my simulation from the pre-ensemble M of the pre-states u, thereby reducing the auxiliary pre-pair {M, u} to the true pair {N, (u, s)}, just as in any modern gauge theory.


PS: I have added calculations of the numbers N and L explicitly in the four Bell-test cases so that no doubt remains. Not surprisingly, N = L in each of the four cases.
Joy Christian
Research Physicist
 
Posts: 1578
Joined: Wed Feb 05, 2014 3:49 am
Location: Oxford, United Kingdom

Re: A new simulation of the EPR-Bohm correlations

Postby Schmelzer » Fri Jun 12, 2015 3:11 pm

Joy Christian wrote:So the flatlanders think that they are being deceived. They certainly are --- by their inability to liberate themselves from the flat land, and by their inability to read:
http://rpubs.com/jjc/84238
Joy Christian wrote:To be sure, a pre-ensemble M of the pre-states u has been used to build this simulation, but such scaffolding mathematical structures are routinely used in modern physics --- for example, in gauge field theories. The scaffolding structures are eventually "gauged out" to extract the correct physics out of the auxiliary mathematics. Similarly, the true ensemble N of the true states (u, s) has been extracted in my simulation from the pre-ensemble M of the pre-states u, thereby reducing the auxiliary pre-pair {M, u} to the true pair {N, (u, s)}, just as in any modern gauge theory.


PS: I have added calculations of the numbers N and L explicitly in the four Bell-test cases so that no doubt remains. Note surprisingly, N = L in each of the four cases.


Indeed, no doubt remains:
http://rpubs.com/jjc/84238 wrote:N = n((A * B), a, u, b, u) # The total number of simultaneous +/- events
Ns[i, j] = N # Total number of simultaneous events observed by Alice and Bob
Ls[i, j] = n(s, a, u, b, u) # The number of initial states (u, s) within S^3

As N, as L depends on a and b. That N=L is not surprising at all, given that above depend on a and b.
Schmelzer
 
Posts: 123
Joined: Mon May 25, 2015 1:44 am

Re: A new simulation of the EPR-Bohm correlations

Postby Joy Christian » Fri Jun 12, 2015 3:29 pm

I see no point in keep responding to the foolish argument being insisted upon by the above flatlander, who has never bothered to read either the simulation or the theoretical paper on which it is based. Like minkwe did, it is best to put him on ignore list. But I hope that other readers are not hoodwinked by his foolish argument.
Joy Christian
Research Physicist
 
Posts: 1578
Joined: Wed Feb 05, 2014 3:49 am
Location: Oxford, United Kingdom

Re: A new simulation of the EPR-Bohm correlations

Postby Schmelzer » Fri Jun 12, 2015 8:49 pm

Here we have not only a defamation, but a really interesting one:
Joy Christian wrote:I see no point in keep responding to the foolish argument being insisted upon by the above flatlander, who has never bothered to read either the simulation or the theoretical paper on which it is based. Like minkwe did, it is best to put him on ignore list. But I hope that other readers are not hoodwinked by his foolish argument.


Ok, a cheap accusation. But, think about it: It follows a posting this contains the evidence which proves it wrong, namely, an explicit quote of text from the simulation, with a link to the simulation, and not some arbitrary quote, but a very special one, necessary to make the argument. Moreover, he knows that I will read this, and that I can, yet, respond, thus, make this point and prove him wrong.

What's this? Can anybody explain me the purpose of such behaviour? A defamation which is demonstrably wrong, with all the evidence in the posting before, and knowing the one who is defamed is not yet banned and can answer? Not a "too stupid to understand", which would be defamation too, but hard to prove that it is defamation, but never bothered to read where it is obvious that I have read it given that I have quoted it.

A complete mystery for me. The only theory where this makes at least sense is the following: In an earlier posting, he has used "true believer" for name-calling and supported this with a link to a book of Eric Hoffer "The true believer". Indeed, a remarkable book, I also recommend you to read it. It describes the mind-set of "true believers", and part of it is the thesis that these true believers will be completely disconnected from reality, they simply refuse to question anything what is claimed by the Guru. So, may be this is a sort of sociological experiment, and a test if his supporters are already "true believers" of his theory, so that they do not even care about such obviously wrong, demonstrative defamations?
Schmelzer
 
Posts: 123
Joined: Mon May 25, 2015 1:44 am

Re: A new simulation of the EPR-Bohm correlations

Postby Ben6993 » Fri Jun 12, 2015 9:39 pm

Hi Joy

I have long ago said that it would be impasse because the 3spherists' geometry confounds with cubist non-locality. Can you explain a little more the significance of the flat plane for N/L, please? You earlier agreed that counterfactual realism returns if the data are restricted to the 'good' set, which is excellent. But if the 'good' dataset contains a different allowable range of b values for any given a and theta, then the cubists will see that as non-local information. Don't you need to see the 'good' dataset in the four dimensions (as a sphere, with one centre) to avoid the apparent non-locality (on the ellipsoid where there are two focii leading to flatland non-locality, when the sphere is projected on to flatland)?
Ben6993
 
Posts: 287
Joined: Sun Feb 09, 2014 11:53 am

Re: A new simulation of the EPR-Bohm correlations

Postby Q-reeus » Sat Jun 13, 2015 2:16 am

Ben6993 wrote:Q-Reeus,
I couldn't watch it all the way through, though it was entertaining, as the style of delivery put me off.

Likewise - Gavin's full-on use of 'immersive multimedia experience' needs toning down. :D (also - pardon please my late response)
Armin, who used to post in the old s.p.f used to have a model where the electron had its own internal time dimension.
I believe the same, too. Black holes have their own internal time dimensions?

Can't speak re electrons, but one basic reason I do not believe in BH's is precisely the temporal disconnect predicted between any (static) R > 2M, and R < 2M. Mathematically formal swapping of t and r Schwarzschild coordinates inside of 'EH' makes no physical sense imo. Resorting to e.g. Kruskal-Szekeres 'rain' coordinates seems like a fix but conveniently ignores 'minor issues' such as infinite time passing for exterior universe once infaller passes R = 2M.
I may have oversimplified it but the s time scale of Greg Egan, in one of the links that Joy has just above posted, is the time in the n+1 dimensions which runs differently to the projected time onto the n dimensions. But t and s are not really fundamentally different, it is just that t is a projection of s? This reminds me of a cubist comment on Joy's new graphic where. ... No.... it is easier to see on http://challengingbell.blogspot.co.uk/2 ... tians.html where the electrons move in the red dimensions but flatland is the table top. The projection of the redland is an ellipse on the table top (which is equivalent to the 'good' data in the old R simulation). The cubist comment is equivalent to saying that the simulations must be run in the circle on the table top where the green area contains the disputed, discarded data. Does that green data exist in real experiments? No, not if real experiment can give the cos curve. Does that green data exist in Joy's model? No, except as trimmed junk. Does that green data exist in flatland simulations? Should it? Impasse again.

Can't say much there. Other than that the blog-spot article's reference to a second time dimension was the impetus for my last post here.
There is a Leonard Susskind lecture at:
http://www.youtube.com/watch?v=yK4rhkcsQDc
The black-hole information paradox, complementarity, and firewalls by Leonard Susskind
(I hope this is the right one!)
In it he mentions five main aspects for quantum gravity which from memory are:
QM, GR, BHs, entropy/thermodynamics, entanglement.
(It is interesting in this quantum gravity connection that JAY has newly reentered thermodynamic territory)
Susskind seems to link BHs to the entanglement. If you have two perfectly entangled BHs, and if Alice and Bob are in two different regions which are beyond each others horizons, then if Alice fell into her BH and Bob fell into his BH then they could eventually meet in the 'middle', despite originally being in regions of space which are disconnected. (Wormhole?) But it requires perfect entanglement. The less perfect the entanglement the less chance Alice and Bob have of meeting up.

More avante garde notions built imo on some shaky foundations.
I have a paper at http://wp.me/p18gTT-26 where the conclusion is "This paper shows that a Rasch analysis compresses its location parameter space according to the level of uncertainty in making judgements within that space. The more uncertain the judgements, the more compressed are the points on the scale." This seems to fit in with Suuskind's claim about entangled BHs leading to close proximity (in some dimensions or other) in that entanglement maximises the chance element, as seen in the 50% chance of A being -1 or +1. However since the A outcome in Joy's model is deterministic, and the chance only arises because of the counfounding of the double cover. I am not sure where that leaves entanglement supposedly being a fifth component to quantum gravity work.

Outside of anything I can meaningfully comment on, other than to say; fascinating ideas!
By analogy one could argue that if it were possible to fall into an electron, like into a BH, one would exit our time dimension and enter the electron's internal time dimension. If Alice and Bob did this in their entangled particles, they could meet in the middle? Treating our universe like a particle, which I am in favour of, then as all particles collapse wavefunction on interaction, which signifies the end of the electron's internal time dimension, our universe could end its time at an interaction cf Penrose's CCC. Penrose's ends of cycle ought, in this logic, to correspond to particle interactions, where the universe is a particle.

I like that Penrose fully acknowledges and offers a possible resolution to the huge issue of initial state re entropy in particular. Not sure what the latest is on his joint-effort search for 'imprints from previous aeon'.
Last edited by Q-reeus on Sat Jun 13, 2015 2:26 am, edited 1 time in total.
Q-reeus
 
Posts: 259
Joined: Sat Jun 07, 2014 11:18 pm

Re: A new simulation of the EPR-Bohm correlations

Postby Q-reeus » Sat Jun 13, 2015 2:20 am

[quote="Yablon"
Please look at my new post on Euclidean space time transformations and quantum gravity. I do indeed have three time dimension for which the observed time is an invariant, up to its Lorentz transformation with space. Jay[/quote]
Interesting! I'd imagine your 3 time dimensions are conceptually different to that of Gavin Wince. I was immediately put off with his seeming identifications based on 'point time', 'rate time', and 'duration time'. Seemed odd if not deranged to put it mildly. Maybe reading everything of his through might start to make sense - big maybe. :D
Q-reeus
 
Posts: 259
Joined: Sat Jun 07, 2014 11:18 pm

Re: A new simulation of the EPR-Bohm correlations

Postby Joy Christian » Sat Jun 13, 2015 3:49 am

Ben6993 wrote:Hi Joy

I have long ago said that it would be impasse because the 3spherists' geometry confounds with cubist non-locality. Can you explain a little more the significance of the flat plane for N/L, please? You earlier agreed that counterfactual realism returns if the data are restricted to the 'good' set, which is excellent. But if the 'good' dataset contains a different allowable range of b values for any given a and theta, then the cubists will see that as non-local information. Don't you need to see the 'good' dataset in the four dimensions (as a sphere, with one centre) to avoid the apparent non-locality (on the ellipsoid where there are two focii leading to flatland non-locality, when the sphere is projected on to flatland)?

I will answer your question mathematically so that no ambiguity remains. Not withstanding the argument minkwe is making elsewhere, there are only three traditional ways that non-locality can enter into the model (or the simulation) under discussion. The two obvious ways non-locality can enter the discussion is through the results of Alice and Bob. But it is easy to see from the simulation that Alice's result A does not depend on either Bob's choice of setting b or Bob's result B, and likewise Bob's result B does not depend on either Alice's choice of setting a or Alice's result A:

A = +sign(g(a, u, s))

B = -sign(g(b, u, s)),

where the pair (u, s) is the initial state. Now the only remaining way non-locality can enter the discussion is through a possible dependence of the probability density (u, s) on the settings a and b, in which case it should be expressed as (a, b, u, s). But I have explicitly demonstrated in the simulation that (u, s) is not a function of a and b --- i.e., it is independent of a and b (see the first graph in the simulation: http://rpubs.com/jjc/84238).

Thus the 'good' dataset does not contain a different allowable range of b values for any given a value, and there is no question of any non-local information being exchanged between Alice and Bob.

The flatlanders keep insisting otherwise because they are unable to refrain from referring to the non-existing or auxiliary physical objects like the pre-ensemble M.

With respect to such flatland objects the world does look distorted. But from within S^3, the world is perfectly local-realistic, with no undue dependence on a and b.
Joy Christian
Research Physicist
 
Posts: 1578
Joined: Wed Feb 05, 2014 3:49 am
Location: Oxford, United Kingdom

Re: A new simulation of the EPR-Bohm correlations

Postby Ben6993 » Sat Jun 13, 2015 6:56 am

Joy

Thanks for the clarification.
Joy wrote:
Thus the 'good' dataset does not contain a different allowable range of b values for any given a value ...
Thank you, that is what I wanted to be sure of.

Using your graphic here for a reduced dimensions analogy:
http://challengingbell.blogspot.co.uk/2015/06/a-flatlanders-view-on-joy-christians.html
This seems to me to illustrate what you have written. The complete spherical symmetry in the red circle, or real world, shows that there is no special dependence of any one a with any other b. (Though not exactly sure where the a, b, A, B and theta are in the red circle.) The projection of the red circle onto the flatland gives an ellipse. As the ellipse it is not spherical, there can appear to be asymmetry wrt a and b? Is there a different ellipse for each (a,b) pair? Which is what makes the snipping away of the redundant green data seem biased as that snipping depends on particular values of a and b when viewed from flatland, but not biased when viewed in red S^3.
Ben6993
 
Posts: 287
Joined: Sun Feb 09, 2014 11:53 am

Re: A new simulation of the EPR-Bohm correlations

Postby Schmelzer » Sat Jun 13, 2015 8:43 am

Joy Christian wrote:But I have explicitly demonstrated in the simulation that (u, s) is not a function of a and b --- i.e., it is independent of a and b (see the first graph in the simulation: http://rpubs.com/jjc/84238).


I would suggest a simplification: Prove that L(s,a,u,s,b,u,s) does not depend on a and b simply by plotting L(s,a,u,s,b,u,s)/L(s,a,u,s,b,u,s).
Schmelzer
 
Posts: 123
Joined: Mon May 25, 2015 1:44 am

PreviousNext

Return to Sci.Physics.Foundations

Who is online

Users browsing this forum: No registered users and 1 guest

cron
CodeCogs - An Open Source Scientific Library