Oh yeah... do notice how the data bits x per message is always an integer, and that y and r are not necessarily so unless the number of messages is a power of two and all messages are equiprobable.
On Whether or Not Non-Gravitational Interaction Can Occur in the Absence of Gravity by Shawn Halayka
Notice that when you analyze the classical binary messages that the mean radial distance increases, but the standard deviation decreases.
It does kind of seem, at first glance, like a "spherization" of the positions in the message (state) space.
The C++ code is attached.
In the following list, the "max message size" is the number of bits per message. So the first in the list analyzes just the two 1-bit messages, the second in the list analyzes the four 2-bit messages, etc, etc. I have to redo the code so that it does not store the radii in an array, since it inevitably runs out of memory on this system when the "max message size" gets to about 25. I can just analyze the radii twice; the first time to get the mean, the second time to get the standard deviation.
The n-bit messages of course "live" in a discrete n-dimensional space that has only two positions (0, 1) for each dimension.
max message size: 1
min radius: 0
max radius: 1
mean radius: 0.5 -/+ 0.5
max message size: 2
min radius: 0
max radius: 1.41421
mean radius: 0.853553 -/+ 0.521005
max message size: 3
min radius: 0
max radius: 1.73205
mean radius: 1.12184 -/+ 0.491409
max message size: 4
min radius: 0
max radius: 2
mean radius: 1.33834 -/+ 0.456989
max message size: 5
min radius: 0
max radius: 2.23607
mean radius: 1.52183 -/+ 0.428974
max message size: 6
min radius: 0
max radius: 2.44949
mean radius: 1.68313 -/+ 0.408759
max message size: 7
min radius: 0
max radius: 2.64575
mean radius: 1.82867 -/+ 0.394921
max message size: 8
min radius: 0
max radius: 2.82843
mean radius: 1.96247 -/+ 0.385622
max message size: 9
min radius: 0
max radius: 3
mean radius: 2.08713 -/+ 0.379348
max message size: 10
min radius: 0
max radius: 3.16228
mean radius: 2.20439 -/+ 0.37503
max message size: 11
min radius: 0
max radius: 3.31662
mean radius: 2.31552 -/+ 0.371968
max message size: 12
min radius: 0
max radius: 3.4641
mean radius: 2.42143 -/+ 0.369719
max message size: 13
min radius: 0
max radius: 3.60555
mean radius: 2.52281 -/+ 0.368006
max message size: 14
min radius: 0
max radius: 3.74166
mean radius: 2.62022 -/+ 0.366657
max message size: 15
min radius: 0
max radius: 3.87298
mean radius: 2.7141 -/+ 0.365562
max message size: 16
min radius: 0
max radius: 4
mean radius: 2.80482 -/+ 0.364652
max message size: 17
min radius: 0
max radius: 4.12311
mean radius: 2.89268 -/+ 0.36388
max message size: 18
min radius: 0
max radius: 4.24264
mean radius: 2.97793 -/+ 0.363215
max message size: 19
min radius: 0
max radius: 4.3589
mean radius: 3.0608 -/+ 0.362634
max message size: 20
min radius: 0
max radius: 4.47214
mean radius: 3.14148 -/+ 0.362121
max message size: 21
min radius: 0
max radius: 4.58258
mean radius: 3.22012 -/+ 0.361665
max message size: 22
min radius: 0
max radius: 4.69042
mean radius: 3.29689 -/+ 0.361256
max message size: 23
min radius: 0
max radius: 4.79583
mean radius: 3.37191 -/+ 0.360886
max message size: 24
min radius: 0
max radius: 4.89898
mean radius: 3.44529 -/+ 0.360552
max message size: 25
min radius: 0
max radius: 5
mean radius: 3.51713 -/+ 0.360246Attachment #1: radius.txt
Attached is an image of the radii for different spaces of n bits, where n = 10, 18, 26. The radii are normalized by sqrt(n), binned, and then drawn. The lighter coloured lines (bins) have more messages in them.
The radii slowly creep together as n increases.Attachment #1: shell.jpg
... This is all equivalent to saying:
The set of n-bit messages contains 2^n messages total. The bits in each message are Cartesian coordinates in the message space, and the radial distance squared of each message (the count of the bits with the value 1 in the message) can be any one of the integer values 0 through n.
The number of n-bit messages with the radial distance squared x is
f(n, x) = n! / [x! (n - x)!]
which is a way of counting the combinations (see binomial coefficient).
Summing the various f(n, x) for x from 0 through n, the grand total is 2^n.
5 days later
[deleted]
Hello,
Thanks for your comment. Have a good day.
- Shawn
After studying about 250 essays in this contest, I realize now, how can I assess the level of each submitted work. Accordingly, I rated some essays, including yours.
Cood luck.
[deleted]
Hi Sergey,
Thanks for your rating, whatever it may have been. Good luck in the contest as well.
- Shawn
If you do not understand why your rating dropped down. As I found ratings in the contest are calculated in the next way. Suppose your rating is [math]R_1 [/math] and [math]N_1 [/math] was the quantity of people which gave you ratings. Then you have [math]S_1=R_1 N_1 [/math] of points. After it anyone give you [math]dS [/math] of points so you have [math]S_2=S_1+ dS [/math] of points and [math]N_2=N_1+1 [/math] is the common quantity of the people which gave you ratings. At the same time you will have [math]S_2=R_2 N_2 [/math] of points. From here, if you want to be R2 > R1 there must be: [math]S_2/ N_2>S_1/ N_1 [/math] or [math] (S_1+ dS) / (N_1+1) >S_1/ N_1 [/math] or [math] dS >S_1/ N_1 =R_1[/math] In other words if you want to increase rating of anyone you must give him more points [math]dS [/math] then the participant`s rating [math]R_1 [/math] was at the moment you rated him. From here it is seen that in the contest are special rules for ratings. And from here there are misunderstanding of some participants what is happened with their ratings. Moreover since community ratings are hided some participants do not sure how increase ratings of others and gives them maximum 10 points. But in the case the scale from 1 to 10 of points do not work, and some essays are overestimated and some essays are drop down. In my opinion it is a bad problem with this Contest rating process. I hope the FQXI community will change the rating process.
[deleted]
Okay!
Dimensionology. See attached (some references still missing).Attachment #1: shannon.pdf
"Van der Pauw sheet resistance and the Schwarzschild black hole"
The entropy of the Schwarzschild black hole is considered in terms of Shannon's
mathematical theory of communication and van der Pauw's theory of sheet resistance.Attachment #1: 1210.0021v1.pdf
[deleted]
Could very well be "sheet conductance", if you want to think of a black hole with zero temperature as a superconductor... Which might be more reasonable than taking the backwards approach in the paper. Will add this in for v2.0, with references re: extremal black holes.
5 days later
"Van der Pauw sheet resistance-conductance and the Schwarzschild black hole"
v3
With section on black hole interior volume / volume derivative.Attachment #1: 2_shannon.pdf
5 days later
v4
Added intro, conclusion. Expanded sections on binary discrete signals and black hole interior volume / volume derivative.
Made it clear that it's not the theory of everything handed down to me by alien space gods...Attachment #1: 3_shannon.pdf
Oh yeah, moved the mention of the factorial / Stirling's approximation to the section on binary discrete signals. ;)
[deleted]
Perhaps it is not obvious that getting the "root" signals (a set of points on an (n-1)D shell) from all of the signals (a set of points on an nD ball) is a matter of ignoring noise and phase distortion. Came up with a toy "wavelet" reconstruction of the continuous signal to show how this "dimensional reduction" is straightforward and natural (even for toy models). Will add it in soon. I got a job in construction! Should be indoors. Bonus.
[deleted]
Looking around at all the comments from the past two weeks... Like I kind of hinted at a month or two ago: if this particular essay contest was not a psychology experiment, it should have been.
[deleted]
LOL. Outdoors!
[deleted]
Some discussion about the black hole entropy and gauge redundancy:
`Introduction to Quantum Fields in Curved Spacetime and the Hawking Effect' by Ted Jacobson
`How empty is the black hole interior?' by Lubos Motl
`String Theory returns to symmetry' by Philip Gibbs (see comment about "gauge redundancy" of signal-states).
`What Would Weyl Do?' by Blake Stephen Pollard
I only include the following link because it talks about the "happyon". Uh huh: