97. M. Biesiada, Z. Golda and M. Szydłowski
On some group properties of Newtonian static star structure equations
J. Phys. A: Math. Gen., vol. 20, pp. 1313-1321 (1987).
[abstract] [journal]

Abstract:
By using Lie group theory, symmetries of the system of equations describing Newtonian static stars in radiative equilibrium are investigated. It turns out that the most general symmetries are those resulting from quasi-homologous transformations. These symmetries enforce a corresponding equation of state. Stromgren's homologous stars are a special case of this, more general, class of solutions.

98. M. Demiański, Z. Golda, L. M. Sokołowski, M. Szydłowski, P. Turkowski
The group-theoretical classification of the 11-dimensional classical homogeneous Kaluza-Klein cosmologies
J. Math. Phys. , vol. 28, pp. 171-173 (1987).
[abstract] [journal]

Abstract:
In the context of the classical Kaluza-Klein cosmology the genalized Bianchi models in 11 dimensions are considered. These are space-times whose spacelike ten-dimensional sections are the hypersurfaces of transitivity for a ten-dimensional isometry group of the total space-time. Such a space-time is a trivial principal fiber bundle $P(M,G_7)$, where $M$ is four-dimensional physical space-time with an isometry group $G_3$ (of a Bianchi type) and $G_7$ is a compact isometry group of the compact isometry group of the compact internal space. The isometry group of $P$ is $G_{10} = G_3 \otimes G_7$, hence all the generalized Bianchi models are classified by enumerating the relevant groups $G_7$. Due to the compactness of $G_7$ the result is astonishingly simple: there are three distinct homogeneous internal spaces in addition to the 11 ordinary Bianchi types for $M$.

99. Zdzislaw A. Golda, Marek Szydlowski, Michal Heller
Generic and nongeneric world models
Gen. Rel. Grav., vol. 19, pp. 707-718 (1987).
[abstract] [journal]

Abstract:
Catastrophe theory methods are employed to obtain a new classification of those world models which can be presented in the form of gradient dynamical systems. Generic sets and structural stability of models in the potential space are strictly defined. It is shown that if a cosmological model is required to be Friedman and generic, it must be flat.

100. M. Demianski, Z. A. Golda, M. Heller and M. Szydlowski
The dimensional reduction in a multi-dimensional cosmology
Class. Quantum Grav. , vol. 3, pp. 1199-1205 (1986).
[abstract] [journal]

Abstract:
Einstein's field equation are solved for the case of the eleven-dimensional vacuum spacetime which is the product $R\times\mbox{Bianchi}(V)\times T^7$, where $T^7$ is a seven-dimensional torus. Among all possible solutions the authors identify those in which the macroscopic space expands and the microscopic space contracts to a finite size. The solutions with this property are `typical' within the considered class. They implement the idea of a purely dynamical dimensional reduction.

101. M. Szydłowski, M. Heller, Z. Golda
Stochastic Time Scale for the Universe
Acta Phys. Pol. , vol. B17, pp. 19-24 (1986).
[abstract]

Abstract:
An intrinsic time scale is naturally defined within stochastic gradient dynamical systems. It should be interpreted as a ``relaxation time'' to a local potential minimum after the system bas been randomly perturbed. It is shown that for a flat Friedman-like cosmological model this time scale is of order of the age of the Universe.

102. M. Szydłowski, M. Heller, Z. Golda
Stochastic Properties of the Friedman Dynamical System
Acta Phys. Pol. , vol. B16, pp. 791-798 (1985).
[abstract]

Abstract:
Some mathematical aspects of the stochastic cosmology are discussed in its relationship to the corresponding ordinary Friedman world models. In particular, it is shown that if the strong and Lorentz energy conditions are known, or the potential function is given, or a stochastic measure is suitable defined then the structure of the phase plane of the Friedman dynamical system is determined.