On the Lagrangian and Hamiltonian aspects of infinite -dimensional dynamical systems and their finite-dimensional reductions

A description of Lagrangian and Hamiltonian formalisms naturally arisen from the invariance structure of given nonlinear dynamical systems on the infinite--dimensional functional manifold is presented. The basic ideas used to formulate the canonical symplectic structure are borrowed from the Cartan's theory of differential systems on associated jet--manifolds. The symmetry structure reduced on the invariant submanifolds of critical points of some nonlocal Euler--Lagrange functional is described thoroughly for both differential and differential discrete dynamical systems. The Hamiltonian representation for a hierarchy of Lax type equations on a dual space to the Lie algebra of integral-differential operators with matrix coefficients, extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems, is obtained via some special Backlund transformation. The connection of this hierarchy with integrable by Lax spatially two-dimensional systems is studied.Comment: 30 page

The general differential-geometric structure of multidimensional Delsarte transmutation operators in parametric functional spaces and their applications in soliton theory. Part 2

The structure properties of multidimensional Delsarte transmutation operators in parametirc functional spaces are studied by means of differential-geometric tools. It is shown that kernels of the corresponding integral operator expressions depend on the topological structure of related homological cycles in the coordinate space. As a natural realization of the construction presented we build pairs of Lax type commutive differential operator expressions related via a Darboux-Backlund transformation having a lot of applications in solition theory. Some results are also sketched concerning theory of Delsarte transmutation operators for affine polynomial pencils of multidimensional differential operators.Comment: 10 page

Frequency locking of modulated waves

We consider the behavior of a modulated wave solution to an $\mathbb{S}^1$-equivariant autonomous system of differential equations under an external forcing of modulated wave type. The modulation frequency of the forcing is assumed to be close to the modulation frequency of the modulated wave solution, while the wave frequency of the forcing is supposed to be far from that of the modulated wave solution. We describe the domain in the three-dimensional control parameter space (of frequencies and amplitude of the forcing) where stable locking of the modulation frequencies of the forcing and the modulated wave solution occurs. Our system is a simplest case scenario for the behavior of self-pulsating lasers under the influence of external periodically modulated optical signals

Тенденції розвитку міжнародного туризму

The sphere of international tourism for the period 2000–2018 has been explored and analyzed in the article. The dynamics of the world tourist flows development and income from international tourism are considered, the determinants of development are derived, the regional structure of the world market of tourist services is given. The development of the tourism industry in the world is analyzed by indicators: the number of tourist arrivals, tourism revenues at current prices, total contribution of tourism to GDP. The main tourist regions are analyzed in the context of the study, they are compared both in terms of the number of tourist visits, and a list of the TOP 10 countries by volume of foreign exchange earnings from tourism in recent years. It is concluded that in recent years, there is a continuing upward trend in demand for emerging markets, with old markets such as the European market, partially losing its popularity. In the process of research, it is determined that in the conditions of development of the world economy, expansion of volumes of international trade, improvement of means of transport, communication, increase of internationalization and globalization of the world economy the demand for tourist services increases. One of the peculiarities of the development of the international tourist services market is the globalization of the market, which has reached all sectors of the tourism industry. It is concluded that nowadays international tourism is an important area of international economic activity, a complex sphere that significantly affects both the structure and the situation in the world economy as well as the economies of most individual countries and regions of the world. Tourism development has a stimulating effect on such sectors of the economy as transport, communications, trade, construction, agriculture, production of consumer goods, and іt is one of the most promising areas of structural restructuring of the economy, confirming the amount of foreign exchange earnings from tourism in the world in general and in individual countries. It is the increasing demand for tourism services and the volume of foreign exchange earnings from tourism in the world makes the topic relevant for further research

Weak ferromagnetism and glassy state in kappa-(BEDT-TTF)2Hg(SCN)2Br

Since the first observation of weak ferromagnetism in the charge-transfer salt kappa-(BEDT-TTF)2-Cu[N(CN)2]Cl [U. Welp et al., Phys. Rev. Lett. 69, 840 (1992)], no further evidence of ferromagnetism in this class of organic materials has been reported. Here we present static and dynamic spin susceptibility measurements on kappa-(BEDT-TTF)2Hg(SCN)2Br revealing weak ferromagnetism below about TWF = 20 K. We suggest that frustrated spins in the molecular dimers suppress long-range order, forming a spin-glass ground state in the insulating phase