On the directional derivative of the Hausdorff dimension of quadratic polynomial Julia sets at 1/4

Let $d(\varepsilon)$ and $\mathcal D(\delta)$ denote the Hausdorff dimension of the Julia sets of the polynomials $p_\varepsilon(z)=z^2+1/4+\varepsilon$ and $f_\delta(z)=(1+\delta)z+z^2$ respectively. In this paper we will study the directional derivative of the functions $d(\varepsilon)$ and $\mathcal D(\delta)$ along directions landing at the parameter $0$, which corresponds to $1/4$ in the case of family $z^2+c$. We will consider all directions, except the one $\varepsilon\in\mathbb{R}^+$ (or two imaginary directions in the $\delta$ parametrization) which is outside the Mandelbrot set and is related to the parabolic implosion phenomenon. We prove that for directions in the closed left half-plane the derivative of $d$ is negative. Computer calculations show that it is negative except a cone (with opening angle approximately $150^\circ$) around $\mathbb{R}^+$

NA61/SHINE experiment - programme beyond 2020

The fixed-target NA61/SHINE experiment (SPS CERN) looks for the critical point (CR) of strongly interacting matter and the properties of the onset of deconfinement. It is a scan of measurements of particle spectra and fluctuations in proton-proton, proton-nucleus and nucleus-nucleus interactions as a function of collision energy and system size, corresponding to a two dimensional phase diagram (T-$\mu_B$). New measurements and their objectives, related to the third stage of the experiment after 2020 are presented and discussed here.Comment: 7 pages, 6 figures, 27 references, Lecture given at the Colloqium on Nonequilibrium Phenomena in Strongly Correlated Systems, Dubna, 18 - 19 April, 201

Nonlinear Constraints from Non-Abelian Internal Symmetries

Symmetry invariant local interaction of a many body system leads to global constraints. We obtain explicit forms of the global macroscopic condition assuring that at the microscopic level the evolution respects the overall symmetry.Comment: 4 pages, to appear in Nuclear Physics Proceedings Supplement: STATISTICAL QCD, held at Bielefeld August 200

Fluctuations, correlations and finite volume effects in heavy ion collision

Finite volume corrections to higher moments are important observable quantities. They make possible to differentiate between different statistical ensembles even in the thermodynamic limit. It is shown that this property is a universal one. The classical grand canonical distribution is compared to the canonical distribution in the rigorous procedure of approaching the thermodynamic limit.Comment: 5 pages. To appear in Int. J. Mod. Phys E, QM2006/Shanghai proceedings - poster sessio

Finite Size Universe or Perfect Squash Problem v2

We give a physical notion to all self-adjoint extensions of the operator $id/dx$ in the finite interval. It appears that these extensions realize different non-unitary equivalent representations of CCR and are related to the momentum operator viewed from different inertial systems. This leads to the generalization of Galilei equivalence principle and gives a new insight into quantum correspondence rule. It is possible to get transformation laws of wave function under Galilei transformation for any scalar potential. This generalizes mass superselection rule. There is also given a new and general interpretation of a momentum representation of wave function. It appears that consistent treatment of this problem leads to the time-dependent interactions and to the abrupt switching-off of the interaction.Comment: 15 pages, Accepted for publication in Journal of Mathematical Physic