Symmetry-preserving discrete schemes for some heat transfer equations

Lie group analysis of differential equations is a generally recognized method, which provides invariant solutions, integrability, conservation laws etc. In this paper we present three characteristic examples of the construction of invariant difference equations and meshes, where the original continuous symmetries are preserved in discrete models. Conservation of symmetries in difference modeling helps to retain qualitative properties of the differential equations in their difference counterparts.Comment: 21 pages, 4 ps figure

Hierarchy of Conservation Laws of Diffusion--Convection Equations

We introduce notions of equivalence of conservation laws with respect to Lie symmetry groups for fixed systems of differential equations and with respect to equivalence groups or sets of admissible transformations for classes of such systems. We also revise the notion of linear dependence of conservation laws and define the notion of local dependence of potentials. To construct conservation laws, we develop and apply the most direct method which is effective to use in the case of two independent variables. Admitting possibility of dependence of conserved vectors on a number of potentials, we generalize the iteration procedure proposed by Bluman and Doran-Wu for finding nonlocal (potential) conservation laws. As an example, we completely classify potential conservation laws (including arbitrary order local ones) of diffusion--convection equations with respect to the equivalence group and construct an exhaustive list of locally inequivalent potential systems corresponding to these equations.Comment: 24 page

Sociological analysis of the problem of unemployment among Ukrainian youth

У статті здійснено соціологічний аналіз проблеми безробіття серед сучасної української молоді. Незважаючи на невисокий рівень безробіття в Україні порівняно із загальноєвропейським рівнем, молодь являє особливу категорію, в якій проблема зайнятості має яскраво виражений характер і вимагає перманентного державного моніторингу та розробки політики молодіжної зайнятості як частини загальної політики зайнятості в країні.За допомогою різних соціологічних досліджень детально розглянута структура безробіття серед української молоді. За даними 2015 року лише 9% молоді безробітні. Великою проблемою визнається низька можливість працевлаштування за обраною спеціальністю.Оцінено вплив на структуру молодіжного безробіття таких факторів як вікові групи та розподіл за видами діяльності. Встановлено, що найбільш уразлива до безробіття є вікова група 15-24 роки. Велика увага приділена відповідності рівня освіти молоді та вимог роботодавців до своїх працівників. Виявлено, що є необхідним корекція роботи всієї системи освіти.Запропоновано в якості важливого кроку для регулювання проблеми безробіття серед молоді поліпшення роботи Державної служби зайнятості, якій варто звернутися до міжнародного досвіду та здійснити реформування механізму державної підтримки підприємницької діяльності шляхом надання пільг, дотацій або державної допомоги для розвитку бізнесу тощо. Саме молодь є найбільш психологічно гнучкою до сприйняття нових сучасних технологій підприємницької діяльності.В статье проведен социологический анализ проблемы безработицы среди современной украинской молодежи. Несмотря на относительно низкий уровень безработицы в Украине по сравнению с общеевропейским уровнем, молодежь относится к особой категории, в которой проблема занятости имеет ярко выраженный характер и требует перманентного государственного мониторинга и разработки политики молодежной занятости как части общей политики занятости в стране.В свете различных социологических исследований подробно рассмотрена структура безработицы среди украинской молодежи. По данным 2015 года только 9% молодежи являются безработные. Большой проблемой признается  малая вероятность трудоустройства по избранной специальности.Оценено влияние на структуру молодежной безработицы таких факторов как возрастные группы и распределение по видам деятельности. Установлено, что наиболее предрасположена к безработице возрастная группа 15-24 года. Большое внимание уделено соответствию уровня образования молодежи и требований работодателей к своим работникам. Определена необходимость коррекции работы всей системы образования.Предложено в качестве важного шага для регулирования проблемы безработицы среди молодежи – улучшение работы Государственной службы занятости, которой следует обратиться к международному опыту и осуществить реформирование механизма государственной поддержки предпринимательской деятельности путем предоставления льгот, дотаций или государственной помощи для развития бизнеса и тому подобное. Именно молодежь является наиболее психологически гибкой к восприятию новых современных технологий предпринимательской деятельности.This article provides a sociological analysis of the problem of unemployment among modern Ukrainian youth. Despite the low unemployment rate in Ukraine compared with European levels, youth is a special category in which the problem of employment has a distinct character and needs permanent state monitoring and policy youth employment as part of total employment policies in the country. Continuously declining proportion of young people to the total population. In Ukraine, for the last ten years the number dropped from her 22 to 20%. According to all forecasts, this trend will continue. With various sociological studies reviewed in detail the structure of unemployment among Ukrainian youth. According to 2015 only 9% of young people are unemployed. The big problem is recognized low employment opportunities in the chosen specialty. A major problem completely for all graduates, regardless of their theoretical training, enterprises consider isolation of knowledge from practice unprepared to work in real business and a lack of understanding of how the business works. Thus it is necessary to orient the student during the training in higher education with the requirements of the employer. The analysis of the youth labor market in Ukraine showed that the highest levels of unemployment  youthful equip individuals 24 years of age, it gives reason to include young people in this age groupe highest risk for difficulties first search of employment are not only with general economic and political crises, but also to the volatility of attitudes of young people, too ambitious position of young workers from the essence of practical skills and experience and, consequently, low professional status.The effect on the structure of youth unemployment factors such as age groups and distribution activities. Found that the most vulnerable to unemployment is the age group 15-24 years. Much attention is given to matching the level of education of young people and requirements of employers to their employees. Revealed that the necessary correction of the entire education system. Thus, the main conclusion is significant opportunity to improve the efficiency of the State Employment Center as one of the basic tools of the modern state youth policy. Proposed as an important step to regulate the problem of youth unemployment improvement of the State Employment Service, which should appeal to the international experience and to reform the mechanism of state support for entrepreneurship by providing incentives, subsidies or state aid to business expansion and others. That young people are the most responsive to the psychological perception of new technologies entrepreneurship

Multiple Hamiltonian structure of Bogoyavlensky-Toda lattices

This paper is mainly a review of the multi--Hamiltonian nature of Toda and generalized Toda lattices corresponding to the classical simple Lie groups but it includes also some new results. The areas investigated include master symmetries, recursion operators, higher Poisson brackets, invariants and group symmetries for the systems. In addition to the positive hierarchy we also consider the negative hierarchy which is crucial in establishing the bi--Hamiltonian structure for each particular simple Lie group. Finally, we include some results on point and Noether symmetries and an interesting connection with the exponents of simple Lie groups. The case of exceptional simple Lie groups is still an open problem.Comment: 65 pages, 67 reference

Symmetries of the near horizon of a Black Hole by Group Theoretic methods

We use group theoretic methods to obtain the extended Lie point symmetries of the quantum dynamics of a scalar particle probing the near horizon structure of a black hole. Symmetries of the classical equations of motion for a charged particle in the field of an inverse square potential and a monopole, in the presence of certain model magnetic fields and potentials are also studied. Our analysis gives the generators and Lie algebras generating the inherent symmetries.Comment: To appear in Int. J. Mod. Phys.

Lie symmetry analysis and exact solutions of the quasi-geostrophic two-layer problem

The quasi-geostrophic two-layer model is of superior interest in dynamic meteorology since it is one of the easiest ways to study baroclinic processes in geophysical fluid dynamics. The complete set of point symmetries of the two-layer equations is determined. An optimal set of one- and two-dimensional inequivalent subalgebras of the maximal Lie invariance algebra is constructed. On the basis of these subalgebras we exhaustively carry out group-invariant reduction and compute various classes of exact solutions. Where possible, reference to the physical meaning of the exact solutions is given. In particular, the well-known baroclinic Rossby wave solutions in the two-layer model are rediscovered.Comment: Extended version, 24 pages, 1 figur

New results on group classification of nonlinear diffusion-convection equations

Using a new method and additional (conditional and partial) equivalence transformations, we performed group classification in a class of variable coefficient $(1+1)$-dimensional nonlinear diffusion-convection equations of the general form $f(x)u_t=(D(u)u_x)_x+K(u)u_x.$ We obtain new interesting cases of such equations with the density $f$ localized in space, which have large invariance algebra. Exact solutions of these equations are constructed. We also consider the problem of investigation of the possible local trasformations for an arbitrary pair of equations from the class under consideration, i.e. of describing all the possible partial equivalence transformations in this class.Comment: LaTeX2e, 19 page

Symmetries of Differential Equations via Cartan's Method of Equivalence

We formulate a method of computing invariant 1-forms and structure equations of symmetry pseudo-groups of differential equations based on Cartan's method of equivalence and the moving coframe method introduced by Fels and Olver. Our apparoach does not require a preliminary computation of infinitesimal defining systems, their analysis and integration, and uses differentiation and linear algebra operations only. Examples of its applications are given.Comment: 15 pages, LaTeX 2.0

Reduction Operators of Linear Second-Order Parabolic Equations

The reduction operators, i.e., the operators of nonclassical (conditional) symmetry, of (1+1)-dimensional second order linear parabolic partial differential equations and all the possible reductions of these equations to ordinary differential ones are exhaustively described. This problem proves to be equivalent, in some sense, to solving the initial equations. The ``no-go'' result is extended to the investigation of point transformations (admissible transformations, equivalence transformations, Lie symmetries) and Lie reductions of the determining equations for the nonclassical symmetries. Transformations linearizing the determining equations are obtained in the general case and under different additional constraints. A nontrivial example illustrating applications of reduction operators to finding exact solutions of equations from the class under consideration is presented. An observed connection between reduction operators and Darboux transformations is discussed.Comment: 31 pages, minor misprints are correcte

Nonlinear Evolution Equations Invariant Under Schroedinger Group in three-dimensional Space-time

A classification of all possible realizations of the Galilei, Galilei-similitude and Schroedinger Lie algebras in three-dimensional space-time in terms of vector fields under the action of the group of local diffeomorphisms of the space \R^3\times\C is presented. Using this result a variety of general second order evolution equations invariant under the corresponding groups are constructed and their physical significance are discussed