Featured post

# On Kolmogorov Complexity of Random Very Long Braided Words

http://arxiv.org/abs/1308.0211v2

Any positive word comprised of random sequence of tokens form a finite alphabet can be reduced (without change of length) using an appropriate size Braid group relationships. Surprisignly the Braid relations dramatically reduce the Kolmogorov Complexity of the original random word and do so in distinct bands of (rate of change) values with gaps in between. Distribution of these bands are estimated and empirical statistics collected by actually coding approximations to the Kolmogorov Complexity (in Mathematica 9.0). Lempel-Ziv-Welch lossless compression algorithm techniques used to estimate the distribution for gaped bands. Evidence provided that such distributions of reduction in Kolmogorov Complexity based upon Braid groups are universal i.e. they can model more general algebraic structures other than Braid groups.

This is the distribution for the 1-pass reduction of Braid relations:

This is the 100-pass case i.e. the Braid relations were applied up to 100 times to reduce the long random word, and the curve for rate of change of Kolmogorov Complexity behaves like a Gamma distribution:

Basically the Braid relationships can be envisaged as the twists in the rubber band and propeller releases the additional complexity or entropy or energy but does so in integral quantized manner.

This is what the author has coined as Topological Oscillator vs. the usual primitive wave equations in complex numbers.

These braided forms are seen every day of our lives on surface of sun:

The white curves are indeed magnetic braids suspected to be the cause of immense heat on the surface of the sun, which is much hotter than its internals:

http://www.space.com/19394-magnetic-braids-sun-corona-nasa.html

“A small NASA space telescope has revealed surprising magnetic braids of super-hot matter in the sun’s outer atmosphere, a find that may explain the star’s mysteriously hot corona, researchers say.”

Featured post

# 14-Sets Algebra

A novel 14-D algebra based upon Kuratowski Monoid’s Closure and Complement operators is uncovered. 14×14 Matrix Representation, Conditions for inverse elements, 0-Commutator and Coproduct computed using Mathematica 9.0 . Almost Random Almost Incompressible large-length coefficients are found to have inverse. Matrix representation has the form of a Stochastic Matrix, and its infinite power has a limit with three 1-valued eigenvalues with determinant 0. As a consequence of this algebra, a topological (Twist on Z2xZ2xZ2) construction of the Octonions developed. An approximate Logarithmic algorithm devised for mapping Kuratowski’s topological operators to 182 dimensional Lie Algebra of 14×14 matrices with columns sum up to 0, thus a concept of pre-geometry is brought forth. De Novo semantics attributed to these 14-D vectors described, in categoreal form, borrowing from the seminal work of Alfred North Whitehead’s Process and Reality.

Algebra only all code removed:

http://files.lossofgenerality.com/14Sets_algebra.pdf

With code:

http://files.lossofgenerality.com/14sets_lowres.pdf

http://files.lossofgenerality.com/14sets.pdf

http://files.lossofgenerality.com/14sets.nb

# The forces dark tunnel or light magic of Mach’s Principle

By Alexander IIin

According to the Principle of Mach inert properties of the physical body are determined by all the physical bodies in the universe, which means that among all the bodies in the universe there is an instant and continuous exchange of information about their condition and body influence each other’s instant action.

The complex objection about the impossibility of superluminal communication usually involves a number of concepts and case studies, namely, the phase velocity, the ray of sunlight, shadows, scissors, and a number of others…

http://files.lossofgenerality.com/For_Text_The%20_forces_dark_tunnel.pdf

# YBE R-matrices over Grassmann Algebra, 2-4 Generators

Experiment on solving the constant R-matrices for Grassmann Algebra with 2-4 generators:

http://files.lossofgenerality.com/YBE_Grassmann.pdf

Also computed the solutions modulus 5:

All solutions obtained symbolically and numerically using Mathematica 9.0 . No pencil and paper computations.