Том 187, № 1 (2016)

We suggest using the Hall–Littlewood version of the Rosso–Jones formula to define the germs of p-adic HOMFLY-PT polynomials for torus knots [m, n] as coefficients of superpolynomials in a q-expansion. In this form, they have at least the [m, n] ↔ [n, m] topological invariance. This opens a new possibility to interpret superpolynomials as p-adic deformations of HOMFLY polynomials and poses a question of generalizing to other knot families, which is a substantial problem for several branches of modern theory.

In the framework of the sine-Gordon integrable model for spiral magnetic structures, we investigate the behavior at large times of a weakly nonlinear dispersive wave field generated by a spatially local initial excitation of the structure. The method used is based on a direct asymptotic analysis of the corresponding matrix of the Riemann problem on the torus.

We establish some properties of the zeros of sums and differences of contiguous Bessel functions of the first kind. As a by-product, we also prove that the zeros of the derivatives of Bessel functions of the first kind of different orders are interlaced the same way as the zeros of the Bessel functions themselves. As a physical motivation, we consider gated graphene nanodots subject to Berry–Mondragon boundary conditions. We determine the allowed energy levels and calculate the mean charge at zero temperature. We discuss its dependence on the gate (chemical) potential in detail and also comment on the effect of temperature.

We propose a general approach for solving the basic magnetostatic equation in the case of nonhomogeneous magnetic solids when their magnetic permeability depends on the coordinates. Seeking the strength of the resulting field reduces to finding two scalar functions satisfying a system of one integral and one differential equation with certain boundary conditions. Using this approach, we obtain an analytic solution of the magnetostatic equation for a ball with a model permeability in an arbitrary external field.

We investigate the dynamics of collapse in massive and massless relativistic theories of gravity for different equations of state for matter numerically and analytically. This allows clarifying the character of the collapse dynamics in the massive relativistic theory of gravity; in particular, we establish the graviton-mass dependence of the time of reaching the turning point (i.e., the point of transition from contraction to expansion). For the massless relativistic theory of gravity, we clarify the relation between the known general relativity solution for cold dust and the corresponding solution in the relativistic theory of gravity. We show that the harmonic time is singular, including the case of a smooth distribution of matter corresponding to a compact object with a strongly diffused boundary, which means that the Oppenheimer–Snyder solution cannot be fully embedded into the Minkowski space. We in addition investigate the effect of a nonzero pressure on the collapse dynamics.

About half a century ago, A. M. Molchanov conjectured the existence of a resonance structure of the solar system, stating that as the solar system evolves, its planets inevitably drift to orbits for which the ratios of their orbital periods become close to ratios of relatively small natural numbers. In Molchanov’s opinion, this resonance property is ensured by small dissipative forces. We offer another possible explanation of this phenomenon without taking dissipation into account.

We present a new multifactorial global search algorithm (MGSA) and check the operability of the algorithm on the Michalewicz and Rastrigin functions. We discuss the choice of an objective function and additional search criteria in the context of the problem of reactive force field (ReaxFF) optimization and study the ranking of the ReaxFF parameters together with their impact on the objective function.