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Vol 227, No 4 (2017)

Article

pages 387-394 views

Chromatic Number with Several Forbidden Distances in the Space with the q-Metric

Berdnikov A.V.

Abstract

We study the chromatic number \( \overline{\chi}\left(X;\rho; k\right) \) of a metric space X with a metric ρ and k forbidden distances. We obtain an estimate of the form \( \overline{\chi}\left({\mathbb{R}}^n;\rho; k\right)\ge {(Bk)}^{Cn} \) for cases where the metric ρ on the set n is generated by the q-norm.

Journal of Mathematical Sciences. 2017;227(4):395-401
pages 395-401 views

Fast Matrix Multiplication by Using Color Algebras

Aidagulov R.R., Shamolin M.V.

Abstract

A method for calculation of matrix product based on color algebras is proposed.

Journal of Mathematical Sciences. 2017;227(4):402-406
pages 402-406 views

Diameters of Random Distance Graphs

Iskhakov L., Mironov M.

Abstract

This paper contains an almost exhaustive description of all cases where a distance random graph asymptotically almost surely has diameter 1, 2, or greater than 2.

Journal of Mathematical Sciences. 2017;227(4):407-418
pages 407-418 views

Integrable Motions of a Pendulum in a Two-Dimensional Plane

Shamolin M.V.

Abstract

In this paper, we examine new cases of integrability of dynamical systems on the tangent bundle to a low-dimensional sphere, including flat dynamical systems that describe a rigid body in a nonconservative force field. The problems studied are described by dynamical systems with variable dissipation with zero mean. We detect cases of integrability of equations of motion in transcendental functions (in terms of classification of singularity) that are expressed through finite combinations of elementary functions.

Journal of Mathematical Sciences. 2017;227(4):419-441
pages 419-441 views

Transcendental First Integrals of Dynamical Systems on the Tangent Bundle to the Sphere

Shamolin M.V.

Abstract

In this paper, we examine the existence of transcendental first integrals for some classes of systems with symmetries. We obtain sufficient conditions of existence of first integrals of second-order nonautonomous homogeneous systems that are transcendental functions (in the sense of the theory of elementary functions and in the sense of complex analysis) expressed as finite combinations of elementary functions.

Journal of Mathematical Sciences. 2017;227(4):442-460
pages 442-460 views

Pendulum Systems with Dynamical Symmetry

Lokshin B.Y., Samsonov V.A., Shamolin M.V.

Abstract

In this paper, we consider a class of oscillatory mechanical systems described by nonlinear second-order differential equations that contain parameters with variable dissipation and possess the property of dynamic symmetry. We study properties of mathematical models of such systems depending on parameters. Special attention is paid to bifurcations and reconstructions of phase portraits and the appearance and disappearance of cycles that envelope the phase cylinder or lie entirely on its covering. We present a complete parametric analysis of system with position-viscous friction and four parameters, where the force action linearly depends on the speed. The space of parameters is split into domains in which the topological behavior of the system is preserved. For some classes of pendulum systems whose right-hand sides depend on smooth functions, a qualitative analysis is performed, i.e., the space of systems considered is split into domains of different behavior of trajectories on the phase cylinder of quasi-velocities. We also perform a systematic analysis of the problem on an aerodynamic pendulum and detect periodic modes unknown earlier; these modes correspond to cycles in the mathematical model.

Journal of Mathematical Sciences. 2017;227(4):461-519
pages 461-519 views

Asymptotics of Motions of Viscous Incompressible Fluids with Large Viscosity

Khatskevich V.L.

Abstract

We examine a nonstationary initial-boundary-value problem on the motion of a viscous incompressible fluid with large viscosity. We obtain estimates of the convergence of solutions of this problem to solutions of the corresponding linearized problems as the viscosity tends to infinity. We also consider the case of the problem periodic in time.

Journal of Mathematical Sciences. 2017;227(4):520-530
pages 520-530 views