Hidden invariance in Gurzadyan-Xue cosmological models

The dark energy formula derived by Gurzadyan and Xue which leads to a value fitting the SN data, provides a scaling relation between the physical constants and cosmological parameters and defines a set of cosmological models. In previous works we have considered several of those models and derived the cosmological equations for each case. In this letter, we present the phase portrait analysis of those models. Surprisingly we found, first, that the separatrix in the phase space which determines the character of solutions depends solely on the value of the current matter density. Namely, at $\Omega_m>2/3$ the equations describe Friedmannian Universe with the classical singularity at the beginning. While at $\Omega_m<2/3$ all solutions for all models start with zero density and non vanishing scale factor. Secondly, more remarkable, the value $\Omega_{sep}=2/3$ defining the separatrix is the same for all models, which reveales an underlying invariance hidden in the models, possibly, due to the basic nature of the GX-scaling.Comment: to appear in Physics Letters

Kolmogorov's Structure Functions and Model Selection

In 1974 Kolmogorov proposed a non-probabilistic approach to statistics and model selection. Let data be finite binary strings and models be finite sets of binary strings. Consider model classes consisting of models of given maximal (Kolmogorov) complexity. The ``structure function'' of the given data expresses the relation between the complexity level constraint on a model class and the least log-cardinality of a model in the class containing the data. We show that the structure function determines all stochastic properties of the data: for every constrained model class it determines the individual best-fitting model in the class irrespective of whether the ``true'' model is in the model class considered or not. In this setting, this happens {\em with certainty}, rather than with high probability as is in the classical case. We precisely quantify the goodness-of-fit of an individual model with respect to individual data. We show that--within the obvious constraints--every graph is realized by the structure function of some data. We determine the (un)computability properties of the various functions contemplated and of the ``algorithmic minimal sufficient statistic.''Comment: 25 pages LaTeX, 5 figures. In part in Proc 47th IEEE FOCS; this final version (more explanations, cosmetic modifications) to appear in IEEE Trans Inform T

Kinetic studies of oxidative coupling of methane on samarium oxide

Kinetic behaviour of three samples of samarium oxide (cubic (Sm-1 ), monoclinic (Sm-3) and mixed cubic-monoclinic (Sm 2) ) were studied in the oxidative coupling of methane using a gradientless flow circulation system. The specific rate of C2- product formation differed by a factor of 6-8 for Sm-1 and Sm-3. The specific activity for CO formation did not depend upon the crystal structure of samarium oxide while the rate of formation of CO2 was different for the samples studied. It is proposed that formation of CO and CO2 occurs via different reaction routes. The rate of CO2 formation at high CHJO2 ratio is limited by oxidant activation or surface CO2-complex decomposition

Game interpretation of Kolmogorov complexity

The Kolmogorov complexity function K can be relativized using any oracle A, and most properties of K remain true for relativized versions. In section 1 we provide an explanation for this observation by giving a game-theoretic interpretation and showing that all "natural" properties are either true for all sufficiently powerful oracles or false for all sufficiently powerful oracles. This result is a simple consequence of Martin's determinacy theorem, but its proof is instructive: it shows how one can prove statements about Kolmogorov complexity by constructing a special game and a winning strategy in this game. This technique is illustrated by several examples (total conditional complexity, bijection complexity, randomness extraction, contrasting plain and prefix complexities).Comment: 11 pages. Presented in 2009 at the conference on randomness in Madison