79. M. Heller, Z. Odrzygóźdź, L. Pysiak, W. Sasin
Quantum Groupoids of the Final Type and Quantization on Orbit Spaces
Demonstratio Mathematica , vol. 37, pp. 671-678 (2004).

Abstract:

80. Marek Szydlowski, Wojciech Czaja
Particle-Like Description in Quintessential Cosmology
Phys. Rev., vol. D69, p. 083518 (2004).
[abstract] [preprint] [journal]

Abstract:
Assuming equation of state for quintessential matter: $p=w(z)\rho$, we analyse dynamical behaviour of the scale factor in FRW cosmologies. It is shown that its dynamics is formally equivalent to that of a classical particle under the action of 1D potential $V(a)$. It is shown that Hamiltonian method can be easily implemented to obtain a classification of all cosmological solutions in the phase space as well as in the configurational space. Examples taken from modern cosmology illustrate the effectiveness of the presented approach. Advantages of representing dynamics as a 1D Hamiltonian flow, in the analysis of acceleration and horizon problems, are presented. The inverse problem of reconstructing the Hamiltonian dynamics (i.e. potential function) from the luminosity distance function $d_{L}(z)$ for supernovae is also considered.

81. Marek Szydlowski, Wojciech Czaja
Stability of FRW cosmology with a generalized Chaplygin gas
Phys. Rev., vol. D69, p. 023506 (2004).
[abstract] [preprint] [journal]

Abstract:
We apply methods of dynamical systems to study the behavior of a universe dominated by the generalized Chaplygin gas. We reduce the dynamics to a two-dimensional Hamiltonian system and study its behavior for various ranges of parameters. The dynamics is studied on the phase plane by using methods of a qualitative analysis of differential equations. The behavior of trajectories at infinity is studied in some convenient coordinates introduced on the phase plane. Hence we show that the Friedmann-Robertson-Walker model with the generalized Chaplygin gas is structurally stable. We clearly find the domains of cosmic acceleration as well as conditions for which the horizon problem is solved. We also define some general class of fluids which generalize the Chaplygin gas. The dynamics of such models in terms of energy conditions is also discussed.

82. Marek Szydlowski, Wojciech Czaja
Toward reconstruction of the dynamics of the Universe from distant type Ia supernovae
Phys. Rev., vol. D69, p. 083507 (2004).
[abstract] [preprint] [journal]

Abstract:
We demonstrate a model-independent method of estimating qualitative dynamics of an accelerating universe from observations of distant type Ia supernovae. Our method is based on the luminosity-distance function, optimized to fit observed distances of supernovae, and the Hamiltonian representation of dynamics for the quintessential universe with a general form of equation of state $p=w(a(z))\rho$. Because of the Hamiltonian structure of FRW dynamics with the equation of state $p = w(a(z)) \rho$, the dynamics is uniquelly determined by the potential function $V(a)$ of the system. The effectiveness of this method in discrimination of model parameters of Cardassian evolution scenario is also given. Our main result is the following, restricting to the flat model with the current value of $\Omega_{m,0}=0.3$, the constraints at $2\sigma$ confidence level to the presence of $\rho^{n}$ modification of the FRW models are $-0.50 \lesssim n \lesssim 0.36$.

83. P. Bizoń, S. J. Szybka, A. Wassserman
Periodic self-similar wave maps coupled to gravity
Phys. Rev. D, vol. 69, p. 064014 (2004).
[abstract] [preprint] [journal]

Abstract:
We continue our studies of spherically symmetric self-similar solutions in the SU(2) sigma model coupled to gravity. Using mixed numerical and analytical methods we show existence of an unstable periodic solution lying at the boundary between the basins of two generic attractors.

84. S. J. Szybka
Chaotic self-similar wave maps coupled to gravity
Phys. Rev. D, vol. 69, p. 084014 (2004).
[abstract] [preprint] [journal]

Abstract:
We continue our studies of spherically symmetric self-similar solutions in the SU(2) sigma model coupled to gravity. For some values of the coupling constant we present numerical evidence for the chaotic solution and the fractal threshold behavior. We explain this phenomenon in terms of horseshoe-like dynamics and heteroclinic intersections.