Abelian Equations and Rank Problems for Planar Webs

We find an invariant characterization of planar webs of maximum rank. For 4-webs, we prove that a planar 4-web is of maximum rank three if and only if it is linearizable and its curvature vanishes. This result leads to the direct web-theoretical proof of the Poincar\'{e}'s theorem: a planar 4-web of maximum rank is linearizable. We also find an invariant intrinsic characterization of planar 4-webs of rank two and one and prove that in general such webs are not linearizable. This solves the Blaschke problem ``to find invariant conditions for a planar 4-web to be of rank 1 or 2 or 3''. Finally, we find invariant characterization of planar 5-webs of maximum rank and prove than in general such webs are not linearizable.Comment: 43 page

On a class of linearizable planar geodesic webs

We present a complete description of a class of linearizable planar geodesic webs which contain a parallelizable 3-subweb.Comment: 7 page

Quantum Versus Classical Decay Laws in Open Chaotic Systems

We study analytically the time evolution in decaying chaotic systems and discuss in detail the hierarchy of characteristic time scales that appeared in the quasiclassical region. There exist two quantum time scales: the Heisenberg time t_H and the time t_q=t_H/\sqrt{\kappa T} (with \kappa >> 1 and T being the degree of resonance overlapping and the transmission coefficient respectively) associated with the decay. If t_q < t_H the quantum deviation from the classical decay law starts at the time t_q and are due to the openness of the system. Under the opposite condition quantum effects in intrinsic evolution begin to influence the decay at the time t_H. In this case we establish the connection between quantities which describe the time evolution in an open system and their closed counterparts.Comment: 3 pages, REVTeX, no figures, replaced with the published version (misprints corrected, references updated

Proof of the Variational Principle for a Pair Hamiltonian Boson Model

We give a two parameter variational formula for the grand-canonical pressure of the Pair Boson Hamiltonian model. By using the Approximating Hamiltonian Method we provide a rigorous proof of this variational principle

Dynamics of a coefficient of friction during non-stationary sliding of a parabolic indenter on visco-elastic foundation

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in AIP Conference Proceedings 1783, 020041 (2016) and may be found at https://doi.org/10.1063/1.4966334.We have studied the coefficient of friction between a rigid parabolic indenter and a visco-elastic Kelvin foundation under step-wise change of sliding velocity. We have obtained analytical estimations for normal and tangential forces in a contact and their limiting values during transition process that occurs after a jump of sliding velocity. The results of numerical simulation are in good agreement with analytical estimates

A model of fretting wear in the contact of an axisymmetric indenter and a visco-elastic half-space

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in AIP Conference Proceedings 1683, 020040 (2015) and may be found at https://doi.org/10.1063/1.4932730.We propose a simple and efficient model of wear of axially symmetric bodies in contact with a visco-elastic foundation based on the method of dimensionality reduction. The results of simulation of wear of a parabolic indenter have been demonstrated. It has been shown that dissipation due to viscosity of a material leads to increase the size of the worn region of an indenter. The noted effect is conditioned with an increase of effective shear modulus of visco-elastic material under sufficiently high velocities of tangential loading. The model can be generalized to a wide range of materials with complex visco-elastic properties