Do Pursuit-Evasion Games Hold the Key to Understanding Physics and Nature?

[Al.Rivero@gmail.com]

I am sure most of you have noticed how convenient the English keyboard
becomes when using it to label 26 and 10 dimensional spaces: there is
26 alphabetical characters, and 10 numerical ones.

I feel ready to discard the ten numerical chars as mere coincidence,
but I am not so sure about the alphabetical. Let me to tell you why:
the dimensions of the bosonic string are related to the 24 dimensional
Leech lattice and its 25+1 dimensional companion (of signature
25-1=24-0). On the other hand, the 24 dimensional unimodular lattices,
particularly Leech's, are useful in coding theory, because they build
the densest packings we know. Golay's codes and a whole of
error-correcting industry come from these lattices, and it is only for
a small miss that it can not be claimed that error correction is a
byproduct of string theory... Conway's pals come there before, using
their own branch of unorthodox mathematics.

Now I wonder if the existence of Golay codes or Leech packagings has
been speculated to appear in natural recognition system (say, our
neural networks processing sound) or used to justify the survival of
our alphabetical notation system (3300 years old!). Far fetched, but it
could explain the coincidence on the basis of a common mathematical
structure.

Alejandro

 

[Dcase]

Hi Al.Rivero@gmail.com:

Thanks for this post.

I was unaware of "... Golay's codes and a whole of error-correcting industry ..."

I had only a vague awareness of "... Leech packagings ..."

I have been reading about both.

From my perspective the "neural networks processing sound" [or other neuro-sensory modality] probably involves decision analysis of information theory which appears to be mathematical game theory [related to but not necessarily cellular automata]. Game theory saddle points may be related to the mathematics used in string / loop physics.
http://www.math.uri.edu/~kulenm/mth381pr/GAMETH/gametheory.html

There may be a relationship to the complex-24D + string-D + time-D of the Monster of Conway and Borcherds.

I think that I saw on Tony Smith website that he was able to correlate the Monster with either a 27-Cliiford or 27-Jordan algebra.
I am not able to currently find this information at his site.
http://www.valdostamuseum.org/hamsmith/BlackHole.html

However, this certainly seems possible for MOD[26] in these three forms:
a - clock-face numbering 1-26
b - classic MOD numbering 0-25
c - helical numbering 0-26.

Also, for packing consider:
Science 13 February 2004:
Vol. 303. no. 5660, pp. 990 - 993
DOI: 10.1126/science.1093010
Reports
Improving the Density of Jammed Disordered Packings Using Ellipsoids
Aleksandar Donev,1,4 Ibrahim Cisse,2,5 David Sachs,2 Evan A. Variano,2,6 Frank H. Stillinger,3 Robert Connelly,7 Salvatore Torquato,1,3,4* P. M. Chaikin2,4
Abstract:
Packing problems, such as how densely objects can fill a volume, are among the most ancient and persistent problems in mathematics and science. For equal spheres, it has only recently been proved that the face-centered cubic lattice has the highest possible packing fraction . It is also well known that certain random (amorphous) jammed packings have 0.64. Here, we show experimentally and with a new simulation algorithm that ellipsoids can randomly pack more densely—up to = 0.68 to 0.71for spheroids with an aspect ratio close to that of M&M's Candies—and even approach 0.74 for ellipsoids with other aspect ratios. We suggest that the higher density is directly related to the higher number of degrees of freedom per particle and thus the larger number of particle contacts required to mechanically stabilize the packing. We measured the number of contacts per particle Z 10 for our spheroids, as compared to Z 6 for spheres. Our results have implications for a broad range of scientific disciplines, including the properties of granular media and ceramics, glass formation, and discrete geometry.

 

[Al.Rivero@gmail.com]

Al.Rivero@gmail.com wrote:

> byproduct of string theory... Conway's pals come there before, using
> their own branch of unorthodox mathematics.[/color]

Sorry I have been misleading here; from an answer to my post in the
physics forums interface, it seems that the mention of Conway
mistakenly addressed the game of Life. No, Conway's pals does not refer
to celular automata nor to the game of Life, nor to "winning ways for
your mathematical plays". It refers to a serious book from Conway on
lattices and codes and subsequent work by Borcherds on the fake monster
algebra.

 

[Dcase]

Hi Al.Rivero:

I have been reading more Game Theory [GT] literature, especially dynamic noncooperative with static, discete and continuous time; noting GT semantics in a prestigious journal.

NATURE - Current issue: Volume 445 Number 7126 pp339-458

1 - Fish can infer social rank by observation alone
Logan Grosenick, Tricia S. Clement and Russell D. Fernald
doi:10.1038/nature05511

2 - 'Infotaxis' as a strategy for searching without gradients p406
Massimo Vergassola, Emmanuel Villermaux and Boris I Shraiman
doi:10.1038/nature05464

3 - Comparison of the Hanbury Brown–Twiss effect [HBTE] for bosons and fermions p402
T Jeltes, ..., CI Westbrook, et al
doi:10.1038/nature05513

Please read editor's summary first, then article, if desired.

The fish behavior and "Infotaxis" [robotics] appear to be consistent with pursuit-evasion [P-E] games.
This is related to biophysics.

HBTE discusses the social life of atoms: HE-3 fermions and HE-4 bosons display bunching and anti-bunching [or attractor and dissipator] behavior.
Perhaps this type of high energy physics [HEP] may be analyzed via P-E games.

Perhaps before there can be a grand unified theory of physics [both in mechanics and nature], there might be required a grand unified theory of mathematics.

GT appears to possibly encompass all branches of mathematics.

P-E may result in escape, equilibria or capture.

In nuclear physics, P-E may help explain:
escape: radioactice half-life
equilibria: stability of various electron shells about various nuclei
capture: k-capture.

GT may allow for more diverse algorithms than cellular automata.

An idea of GT mathematics for dynamic noncooperative type can be found in the following slide lecture by Tamer Basar [U-IL@UC].
Especially interesting is slide 23 / 83, various lines of singular surfaces, which may allow for exploration and analysis of singularities such as dipoles, sinks and sources. The "transition line" or the "switching envelope" may allow for the transition from sink to source.
http://decision.csl.uiuc.edu/~tbasar/Delft05-slides.pdf