The Universe as a Nonuniform Lattice in the Finite-Dimensional Hypercube II.Simple Cases of Symmetry Breakdown and Restoration

This paper continues a study of field theories specified for the nonuniform lattice in the finite-dimensional hypercube with the use of the earlier described deformation parameters. The paper is devoted to spontaneous breakdown and restoration of symmetry in simple quantum-field theories with scalar fields. It is demonstrated that an appropriate deformation opens up new possibilities for symmetry breakdown and restoration. To illustrate, at low energies it offers high-accuracy reproducibility of the same results as with a nondeformed theory. In case of transition from low to higher energies and vice versa it gives description for new types of symmetry breakdown and restoration depending on the rate of the deformation parameter variation in time, and indicates the critical points of the previously described lattice associated with a symmetry restoration. Besides, such a deformation enables one to find important constraints on the initial model parameters having an explicit physical meaning.Comment: 9 pages,Revte

Deformed Density Matrix and Generalized Uncertainty Relation in Thermodynamics

A generalization of the thermodynamic uncertainty relations is proposed. It is done by introducing of an additional term proportional to the interior energy into the standard thermodynamic uncertainty relation that leads to existence of the lower limit of inverse temperature. The authors are of the opinion that the approach proposed may lead to proof of these relations. To this end, the statistical mechanics deformation at Planck scale. The statistical mechanics deformation is constructed by analogy to the earlier quantum mechanical results. As previously, the primary object is a density matrix, but now the statistical one. The obtained deformed object is referred to as a statistical density pro-matrix. This object is explicitly described, and it is demonstrated that there is a complete analogy in the construction and properties of quantum mechanics and statistical density matrices at Plank scale (i.e. density pro-matrices). It is shown that an ordinary statistical density matrix occurs in the low-temperature limit at temperatures much lower than the Plank's. The associated deformation of a canonical Gibbs distribution is given explicitly.Comment: 15 pages,no figure

Pure States, Mixed States and Hawking Problem in Generalized Quantum Mechanics

This paper is the continuation of a study into the information paradox problem started by the author in his earlier works. As previously, the key instrument is a deformed density matrix in quantum mechanics of the early universe. It is assumed that the latter represents quantum mechanics with fundamental length. It is demonstrated that the obtained results agree well with the canonical viewpoint that in the processes involving black holes pure states go to the mixed ones in the assumption that all measurements are performed by the observer in a well-known quantum mechanics. Also it is shown that high entropy for Planck remnants of black holes appearing in the assumption of the Generalized Uncertainty Relations may be explained within the scope of the density matrix entropy introduced by the author previously. It is noted that the suggested paradigm is consistent with the Holographic Principle. Because of this, a conjecture is made about the possibility for obtaining the Generalized Uncertainty Relations from the covariant entropy bound at high energies in the same way as R. Bousso has derived Heisenberg uncertainty principle for the flat space.Comment: 12 pages,no figures,some corrections,new reference

Quantum Mechanics at Planck's scale and Density Matrix

In this paper Quantum Mechanics with Fundamental Length is chosen as Quantum Mechanics at Planck's scale. This is possible due to the presence in the theory of General Uncertainty Relations. Here Quantum Mechanics with Fundamental Length is obtained as a deformation of Quantum Mechanics. The distinguishing feature of the proposed approach in comparison with previous ones, lies on the fact that here density matrix subjects to deformation whereas so far commutators have been deformed. The density matrix obtained by deformation of quantum-mechanical density one is named throughout this paper density pro-matrix. Within our approach two main features of Quantum Mechanics are conserved: the probabilistic interpretation of the theory and the well-known measuring procedure corresponding to that interpretation. The proposed approach allows to describe dynamics. In particular, the explicit form of deformed Liouville's equation and the deformed Shr\"odinger's picture are given. Some implications of obtained results are discussed. In particular, the problem of singularity, the hypothesis of cosmic censorship, a possible improvement of the definition of statistical entropy and the problem of information loss in black holes are considered. It is shown that obtained results allow to deduce in a simple and natural way the Bekenstein-Hawking's formula for black hole entropy in semiclassical approximation.Comment: 18 pages,Latex,new reference

Underlying Challenges for Russian Venture Industry Development and Methods for Their Solution

The authors of this article set a goal to identify the most relevant obstacles for venture capital development in Russia. In order to achieve this goal, statistical analysis was carried out as well as valuation of different quantitative and qualitative information, including primary sources (interviews with venture industry experts) was conducted. Russian and foreign literature was exploited during this research. The analysis of the Russian venture capital development dynamics was carried out, as well as the major regulatory aspects of the industry were examined. Based on the results of the research, certain recommendations were provided, which, according to the authors, are capable of supporting venture investments in the long term and accelerating the volume growth of capital raising by domestic startups

Chow's theorem and universal holonomic quantum computation

A theorem from control theory relating the Lie algebra generated by vector fields on a manifold to the controllability of the dynamical system is shown to apply to Holonomic Quantum Computation. Conditions for deriving the holonomy algebra are presented by taking covariant derivatives of the curvature associated to a non-Abelian gauge connection. When applied to the Optical Holonomic Computer, these conditions determine that the holonomy group of the two-qubit interaction model contains $SU(2) \times SU(2)$. In particular, a universal two-qubit logic gate is attainable for this model.Comment: 13 page

Nonlinear difference approximations for evolutionary PDEs

The authors describe a procedure to improve both the accuracy and computational efficiency of finite difference schemes used to simulate nonlinear PDEs. The underlying idea is that of enslaving, which is the estimation of the small unresolved scales in terms of the larger resolved scales. They discuss details of the procedure and illustrate them in the context of the forced Burgers` equation in one dimension. They present computational examples that demonstrate the predicted increases in accuracy and efficiency

On distributions of functionals of anomalous diffusion paths

Functionals of Brownian motion have diverse applications in physics, mathematics, and other fields. The probability density function (PDF) of Brownian functionals satisfies the Feynman-Kac formula, which is a Schrodinger equation in imaginary time. In recent years there is a growing interest in particular functionals of non-Brownian motion, or anomalous diffusion, but no equation existed for their PDF. Here, we derive a fractional generalization of the Feynman-Kac equation for functionals of anomalous paths based on sub-diffusive continuous-time random walk. We also derive a backward equation and a generalization to Levy flights. Solutions are presented for a wide number of applications including the occupation time in half space and in an interval, the first passage time, the maximal displacement, and the hitting probability. We briefly discuss other fractional Schrodinger equations that recently appeared in the literature.Comment: 25 pages, 4 figure