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The problem of the tunnel-effect in the three-dimensional Lobachevsky space is formulated and solved. It is shown that the tunneling probability essentially decreases when radius of the space curvature is of the same order as linear sizes of the well in which a particle is locked.

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In this paper we have studied upper bound of time derivative of information entropy for colored cross-correlated noise driven open systems. The upper bound is calculated based on the Fokker–Planck equation and the Schwartz inequality principle. Our results consider the effect of the noise correlation strength and correlation time due to the correlation between additive and multiplicative white noises on the upper bound as well as relaxation time. The interplay of deterministic and random forces reveals extremal nature of the upper bound and its deviation from the time derivative of information entropy.

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We investigate some cylindrically symmetric nonstationary and nonstatic solutions of Einstein field equations. We first study some physical properties of a solution which can be considered as Kasner generalization of static Levi-Civita vacuum solution. Then we generalize this metric to include a solution where a space-time is filled with null dust or a stiff fluid.

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The purpose of these lectures is to review some of the recent work devoted to understanding the microscopic foundations of irreversible behavior in fluid systems. We begin by considering the properties of systems whose microscopic dynamics is chaotic. Our goal is to show that, for simple model systems, one can understand the approach of a sufficiently smooth initial phase space distribution to an equilibrium state, or more properly, to a local equilibrium state, without the need to introduce stochastic elements into the description of the system’s dynamics. To follow this argument one needs to understand some of the basic ideas of dynamical systems theory as applied to chaotic systems. We first consider the notions of ergodicity and mixing, and discuss the extent to which these ideas really might be applied to systems of large numbers of particles. Then we broaden the discussion to describe the behavior of hyperbolic dynamical systems with exponential separation of infinitesimally close phase space trajectories. These systems are characterized by stable and unstable manifolds, and nonzero Lyapunov exponents. We will briefly touch upon such topics as SRB measures, entropy production and the relations between transport coefficients and properties of the underlying microscopic chaotic behavior of the phase space trajectories of the system. We illustrate these ideas as well as their application to transport theory with several simple models, among them the baker and multi-baker maps. In the second part of these lectures we consider the fact that microscopic chaos is neither necessary nor sufficient for good transport properties. This will lead to a brief discussion of classical systems that are pseudochaotic. These are systems with zero Lyapunov exponents, but with some microscopic properties that are similar to those of chaotic systems, including the separation, in time, of nearby phase space trajectories. However for pseudochaotic systems the separation is proportional to some power of time rather than exponential. The third and final part of these lectures is devoted to a consideration of transport in a quantum version of the simple model discussed in the first part, the multi-baker map. The quantum version shows quite different behavior and we conclude the lectures with a brief description of the transition from quantum to classical behavior in the semi-classical limit.

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The purpose of these lecture notes is twofold. First we aim at introducing the reader to the basic concepts pertaining to Casimir forces, starting from the seminal work of Casimir. In a broader sense, we also review some aspects of dispersion forces, in particular the status of van der Waals forces in vacuum as well as in a finite density and non zero temperature medium. The Lifshitz theory of forces between dielectric bodies is briefly described. In the second place, the course deals with a more recent analysis of the Casimir force for metals based on an exact microscopic statistical mechanical treatment of matter and field fluctuations. It reveals that charges fluctuations inside the conductors cannot be ignored (as is done in the conventional Casimir calculation). It also helps clarifying present day controversies about the contribution of thermal fluctuations to the force and the proper way to recover the metallic case in the framework of the Lifshitz theory. Finally the occurrence of Casimir forces in critical phenomena is illustrated in the case of the Bose–Einstein condensation in a free Bose gas. The Casimir effect in other contexts (general quantum field theory, particle physics, cosmology, … ) is not considered here.

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We consider the motion of a particle subjected to the constant gravitational field and scattered inelastically by oscillating boundaries which possess the shape of parabola, wedge, and hyperbola. The linear dependence of the restitution coefficient on the particle velocity is assumed. We demonstrate that this dynamical system can be either regular or chaotic, which depends on the billiard shape and the oscillation frequency. The trajectory calculations are compared with the experimental data; a good agreement has been achieved. Moreover, the properties of the system has been studied by means of the Liapunov exponents and the Kaplan–Yorke dimension. The period-doubling bifurcation route to chaos has been found. Chaotic and nonuniform patterns visible in the experimental data are interpreted as a result of large embedding dimension.

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Initial data for boosted Kerr black hole are constructed in an axially symmetric case. Momentum and Hamiltonian constraints are solved numerically using finite element method (FEM) algorithms. Both Bowen–York and puncture boundary conditions are adopted and appropriate results are compared. Past and future apparent horizons are also found numerically and the Penrose inequality is tested in detail.

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We investigate the fermion creation in quantum kinetic theory by applying “oscillator representation” approach, which was earlier developed for bosonic systems. We show that in some particular cases (Yukawa-like interaction, fixed direction of external vector field) resulting Kinetic Equation (KE) reduces to KE obtained by time-dependent Bogoliubov transformation method. We conclude “oscillator representation” approach to be more universal for the derivation of quantum transport equations in strong space-homogeneous time-dependent fields. We discuss some possible applications of obtained KE to cosmology and particle production in strong laser fields.

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A specific universal shape of empirical mass formula is proposed for all leptons \(\nu _1, \nu _2 , \nu _3\) and \(e^-,\,\mu ^-,\,\tau ^- \) as well as all quarks \(u,\,c,\,t\) and \(d,\,s,\,b\) of three generations, parametrized by three free constants \( \mu , \varepsilon , \xi \) assuming four different triplets of values. Four such triplets of parameter values are determined or estimated from the present data. Mass spectra in the four cases are related to each other by shifting the triplet of parameters \( \mu , \varepsilon , \xi \). For charged leptons \(\xi \simeq 0\) (but probably \(\xi \neq 0\)). If for them \(\xi \) is put to be exactly 0, then \(m_\tau = 1776.80\) MeV is predicted after the input of experimental \(m_e\) and \(m_\mu \) (the central value of experimental \(m_\tau = 1776.99^{+0.29}_{-0.26}\) MeV corresponds to \( \xi =1.8\times 10^{-3}\neq 0\)). For neutrinos \(1\)/\(\xi \simeq 0\) (but \(1\)/\(\xi \neq 0\) in the case of normal hierarchy \(m^2_{\nu _1} \ll m^2_{\nu _2} \ll m^2_{\nu _3}\)). If for neutrinos \(1\)/\(\xi \) is conjectured to be exactly 0, then \((m_{\nu _1}, m_{\nu _2}, m_{\nu _3}) \sim (1.5, 1.2, 5.1)\times 10^{-2}\) eV are predicted after the input of experimental estimates \(|m^2_{\nu _2} - m^2_{\nu _1}| \sim 8.0\times 10^{-5}\;{\rm eV}^2\) and \(|m^2_{\nu _3} - m^2_{\nu _2}| \sim 2.4\times 10^{-3}\;{\rm eV}^2\). Thus, the mass ordering of neutrino states 1 and 2 is then inverted, while the position of state 3 is normal.

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The scattering amplitude of hadrons in high energy Regge limit can be rewritten in terms of reggeized gluons, i.e. Reggeons. We consider three-Reggeon states that possess either \(C=+1\) or \(C=-1\) parity. In this work using Janik–Wosiek method the spectrum of conformal charges is calculated for states with conformal Lorentz spin \(n_h=0,1,2,3,\ldots \,\). Moreover, corrections to WKB approximation are computed.

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\(N\)-reggeized gluon states in Quantum Chromodynamics are described by BKP equation. In order to solve this equation for \(N\gt 3\) particles the \(Q\)-Baxter operator method is used. Spectrum of the integrals of motion of the system exhibits a complicated structure. In this work we consider the case with \(N=4\) Reggeons where complicated relations between \(q_3\)-spectrum and \(q_4\)-spectrum are analysed. Moreover, corrections to WKB approximation for \(N=4\) and \(q_3=0\) are computed.

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A multisource ideal gas model is used to give an uniform description of the angular distributions of different kinds of target fragments produced in nucleus–nucleus collisions at high energies. The theoretical results calculated by the Monte Carlo method are qualitatively in agreement with the experimental angular distributions of target black, grey, and heavy fragments produced in \(^{84}\)Kr-Emulsion (Em) collisions at 1.7 \(A\)GeV and \(^{16}\)O-Em collisions at 3.7 and 60 \(A\)GeV.

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Collective \(1^+\) scissors mode states of \(^{162,164}\)Dy isotopes are investigated in the framework of the rotational-invariant QRPA model. Results are compared with the ones from both the nuclear resonance fluorescence (NRF) and inelastic neutron scattering experiments. The \(M1\) strengths as well as the total \(M1\) in these well deformed isotopes in 2–4 MeV are calculated to be in a general agreement with especially the NRF values.

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Recently suggested method of measuring Renyi entropies of multiparticle systems produced in high-energy collisions is presented in the form of a “do-list”, explaining explicitely how to perform the measurement and suggesting improvements in the treatment of the data.

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During the last centuries of human history, many questions was repeated in connection with the great problems of the existence and origin of human beings, and also of the Universe. The old questions of common sense and philosophy have not been solved in spite of the indisputable results of modern natural sciences. Recently the so-called anthropic principles show that these questions are still present. We investigated some important results of the modern cosmology and their consequences with respect to the corresponding questions of philosophy and logic. After a short conceptual introduction there are two baselines. It is shown first how Goedel’s theorem affects the foundation of anthropic principles. Our train of thought shows that Goedel’s incompleteness theorem may deny some efforts claiming that anthropic principles can be ruled out. After this in the Appendix we touch the branch of questions that are connected with the philosophical aspects of anthropic principles and the multiple-world hypothesis. Here we investigated those formulae of quantum theory, which are supposed to be the ground for the theory of many worlds-hypothesis. Although our method is based partly on philosophy and logic, it is mainly grounded in the results and methods of natural sciences. So we need both physics and philosophy to go in our way.