Vol 38, No 6 (2017)

It is well known that there are many presentations of a knot and each presentation has both its own advantages and disadvantages. In the paper we present another presentation of a knot. For the new presentation we define moves and rewrite the criterion of realizability.

We consider the dynamics of a tippe-top on a visco-elastic plane with dry friction. Tippetop is a rigid dynamically symmetric sphere with the center of mass that lies on the dynamical axis of symmetry but does not coincide with the center of the sphere. It was shown earlier [1] that the full mechanical energy conserves only on the steady-state motions of the tippe-top while the linear on the pseudovelocities Jellet integral remains constant for a quite general model of friction, for example, viscous friction. It allows studying the stability of the steady-state motions and constructing the generalized Smale diagrams [1]. Here we consider another model of friction—so-called distributed dry friction proposed in [2] that is more suitable for the convex bodies on a rough plane. In this case, both energy and Jellet function change their value on solutions. The earlier investigated case is considered as generating. Steady state motions transform in pseudo-steady motions with slow changing parameters. Calculations illustrate the analytical investigation.

We study integrable Hamiltonian systems with 2 degrees of freedom on regular compact isoenergy 3-manifolds. Such a system is given by a pair (B,F) of a closed 2-form B without zeros and a Bott function F (called the first integral) with dF ∧ B = 0 on a compact 3-manifold Q endowed with a volume form. We prove that, under some additional assumptions, any continuous orbital invariant of integrable systems is “trivial”, i.e. it can be expressed in terms of local extremes of rotation functions on one-parameter families of invariant tori, provided that the systems admit a cross-section of genus 0. We also show which of nontrivial orbital invariants are continuous in the genus 1 case.

We study the topology of the Liouville foliation of the Goryachev integrable case in the rigid body dynamics which is a one-parameter family of completely integrable Hamiltonian systems with two degrees of freedom. For this problem P. E. Ryabov has found a real separation of variables with the aid of which he studied the phase topology of the Goryachev systems for positive values of the parameter. We solve the similar problem for negative values of the parameter. This case is of special interest because all the leaves of the Liouville foliation and the surfaces of constant energy turn out to be non-compact. The results are presented in the form of Fomenko invariants for all regular energy levels.

In this paper sufficient conditions are presented for the least element existence in the common fixed point set of a family of multivalued mappings on ordered sets. The problem of the iterative search for common fixed point of a family of multivalued mappings on ordered sets is considered as well. In addition, we suggest conditions to guarantee invariance of the fixed point property of mappings under an order-homotopy.

As one knows, for every Poisson manifold M there exists a formal noncommutative deformation of the algebra of functions on it; it is determined in a unique way (up to an equivalence relation) by the given Poisson bivector. Let a Lie algebra g act by derivations on the functions on M. The main question, which we shall address in this paper is whether it is possible to lift this action to the derivations on the deformed algebra. It is easy to see, that when dimension of g is 1, the only necessary and sufficient condition for this is that the given action is by Poisson vector fields. However, when dimension of g is greater than 1, the previous methods do not work. In this paper we show how one can obtain a series of homological obstructions for this problem, which vanish if there exists the necessary extension.

We calculate Jordan–Kronecker invariants for semi-direct sums of Lie algebras so(n) and sp(n) with several copies of Rⁿ with respect to their standard representation.

The validity of Clarkson’s inequalities for periodic functions in the Sobolev space normed without the use of pseudodifferential operators is proved. The norm of a function is defined by using integrals over a fundamental domain of the function and its generalized partial derivatives of all intermediate orders. It is preliminarily shown that Clarkson’s inequalities hold for periodic functions integrable to some power p over a cube of unit measure with identified opposite faces. The work is motivated by the necessity of developing foundations for the functional-analytic approach to evaluating approximation methods.

We consider a one-dimensional self-consistentmathematical model of capacitive radiofrequency discharge in Argon between symmetrical electrodes at atmospheric pressure in the local approximation. The model incorporates electrons, atomic and molecular ions, metastable atoms, Argon dimers, and ground-state atoms. The numerical algorithm for the model is based on a finitedimensional approximation of the problem using difference schemes with subsequent iterations. A software package in the MatLab environment has been developed to implement the numerical algorithm. Using this software for a model problem, we have obtained the characteristics of a radiofrequency discharge in a plasmatron with interelectrode distances of 0.2 and 2 cm at atmospheric pressure. The results of numerical calculations are in good agreement with data known from literature of field experiments and calculations.