Vol 38, No 6 (2017)
- Year: 2017
- Articles: 17
- URL: https://journals.rcsi.science/1995-0802/issue/view/12493
Article
2-colored diagrams of knots
Abstract
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.
Local structure of Gromov–Hausdorff space around generic finite metric spaces
Abstract
We investigate the local structure of the space M consisting of isometry classes of compact metric spaces, endowed with the Gromov–Hausdorff metric. We consider finite metric spaces of the same cardinality and suppose that these spaces are in general position, i.e., all nonzero distances in each of the spaces are distinct, and all triangle inequalities are strict. We show that sufficiently small balls in M centered at these spaces and having the same radii are isometric. As consequences, we prove that the cones over such spaces (with the vertices at the single-point space) are isometric; the isometry group of each sufficiently small ball centered at a general position n points space, n ≥ 3, contains a subgroup isomorphic to the symmetric group Sn.
Tippe-top on visco-elastic plane: steady-state motions, generalized smale diagrams and overturns
Abstract
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.
Integrable systems with linear periodic integral for the Lie algebra e(3)
Abstract
Integrable systems with a linear periodic integral for the Lie algebra e(3) are considered. One investigates singularities of the Liouville foliation, bifurcation diagram of the momentum mapping, transformations of Liouville tori, topology of isoenergy surfaces and other topological properties of such systems.
Continuous orbital invariants of integrable Hamiltonian systems
Abstract
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.
Three-dimensional non-reductive homogeneous spaces of solvable groups Lie, admitting affine connections
Abstract
When a homogeneous space admits an invariant affine connection? If there exists at least one invariant connection then the space is isotropy-faithful, but the isotropy-faithfulness is not sufficient for the space in order to have invariant connections. If a homogeneous space is reductive, then the space admits an invariant connection. The purpose of the work is the classification of three-dimensional non-reductive homogeneous spaces, admitting invariant affine connections. We concerned only case, when Lie group is solvable. The local classification of homogeneous spaces is equivalent to the description of effective pairs of Lie algebras. The peculiarity of techniques presented in the work is the application of purely algebraic approach, the compound of different methods of differential geometry, theory of Lie groups, Lie algebras and homogeneous spaces.
Topological classification of the Goryachev integrable systems in the rigid body dynamics: non-compact case
Abstract
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.
On convergence of combinatorial Ricci flow on surfaces with negative weights
Abstract
Chow and Lou in 2003 had shown that the analogue of the Hamilton Ricci flow on surfaces in the combinatorial setting converges to the Thruston’s circle packing metric. The combinatorial setting includes weights defined for edges of a triangulation. Crucial assumption in the paper of Chow and Lou was that the weights are nonnegative. We show that the same results on convergence of Ricci flow can be proved under weaker condition: some weights can be negative and should satisfy certain inequalities. As a consequence we obtain theorem of existence of Thurston’s circle packing metric for a wider range of weights.
Fixed points of mappings on ordered sets
Abstract
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.
An algorithm for construction of commuting ordinary differential operators by geometric data
Abstract
We give a description and program realisation of an algorithm which produces explicit examples of commuting ordinary differential operators with rational coefficients by geometric data consisting of a rational curve and a torsion free sheaf with some trivialisation.
Deformation quantization and the action of Poisson vector fields
Abstract
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.
Topological classification of integrable geodesic flows in a potential field on the torus of revolution
Abstract
A Liouville classification of integrable Hamiltonian systems which are the geodesic flows on 2-dimensional torus of revolution in a invariant potential field in the case of linear integral is obtained. This classification is obtained using the Fomenko–Zieschang invariant (marked molecules) of investigated systems. All types of bifurcation curves are described. Also a classification of singularities of the system solutions is obtained.
Approximate factorization for a class of second-order matrix functions
Abstract
The notion of approximate factorization of a Hölder matrix function on a simple smooth closed contour is introduced. The elements of a (2 × 2)-matrix function of a special form are approximated by polynomials in z and 1/z. Sufficient conditions for the vanishing of the partial indices of the approximating matrix function are found. Factorization of the approximating matrix function is explicitly constructed.
Clarkson’s inequalities for periodic Sobolev space
Abstract
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.
On the explicit scheme with variable time steps for solving the parabolic optimal control problem
Abstract
The optimal control problem including a linear parabolic equation as the state problem is considered. Pointwise constraints are imposed on the control function. The objective functional contains a given observation function on the entire domain at each moment of time. The optimal control problem is approximated by a finite-dimensional problem with grid approximation of the state equation by using an explicit scheme with variable time steps. The existence and uniqueness of solutions for the continuous and grid optimal control problems are proved. The finite-dimensional optimal control problemis solved by the Udzawa iterationmethod. Results of numerical experiments are presented.
Modeling of radio-frequency capacitive discharge under atmospheric pressure in Argon
Abstract
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.