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

Features of Technological Approaches to Teaching

У статті розглянуто поняття «технології» та «навчання», описано технологічні підходи закордонних науковців до навчання у вищій освіті, проаналізовано історичний аспект. Представлено характеристику технологічних підходів закордонних науковців до навчання. Встановлено, що до основних ознак технології відносять проектованість, керованість, коректованість, результативність, діагностичну доцільність, аналіз діючих факторів, ефективність методів, системність і цілісність. Зазначено, що до провідних чинників технологічного підходу відноситься мотивація самостійної роботи з придбання знань, джерела якої знаходяться як у контексті практичної, так і в навчальній діяльності.The article deals with the concepts of technology and teaching, technological approaches of foreign scholars to the teaching in higher education; the historical aspect is also analyzed. The article highlights the main features of the technology: projecting, performance, manageability, results, diagnostic expediency, analysis of existing factors, effectiveness of methods, consistency and integrity. It is indicated that the leading factors of technological approach include motivation of individual work for gaining knowledge; the sources of this work are in the context of practical activity and studying. In Ukraine the increase of necessity in highly qualified specialists is caused by technological progress in the society. Modern experts must be efficiently and professionally prepared. The problem of teaching efficiency has always been important. Pedagogical technology gives a description, project of the process of student’s individuality formation and it must include diagnostic purposes and content of education, didactic processes and organizational forms of teaching. The components of the didactic process are: motivation as a forming of stable student’s interest to studying activity and transformation of external purposes into internal needs; cognitive activity of the student, the result of which is knowledge comprehension; teacher management depending on the purpose of study. Educational technologies should contain conceptuality – didactic, psychological, philosophical and social pedagogical argumentation of educational purposes; consistency – logics and interconnection of all parts and aspects of the process, the integrity of the designing and implementation of the educational process; the management – variation of teaching methods and tools for correction of results which are revealed in the process of diagnostics; the effectiveness of results and economic parameters; reproducibility by other teachers and departments. One of the leading factors of technological approach is motivation of individual work with the obtaining of knowledge. Its sources are in the context of practice and lerning activity. The other important factor is flexibility of the technology and its ability to adapt quickly and flexibly to the learning environment, to the students, to time budget and to other circumstances. The cognitive teaching technology is also effective, it is based on the subjective quality of erudition and problem-modular studying and activity-based approach

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