Generalization of the Grad method in plasma physics

The Grad method is generalized based on the Bogolyubov idea of the functional hypothesis for states at the end of relaxation processes in a system. The Grad problem (i.e., description of the Maxwell relaxation) for a completely ionized spatially uniform two-component electron-ion plasma is investigated using the Landau kinetic equation. The component distribution functions and time evolution equations for parameters describing the state of a system are calculated, and corrections are obtained to the known results in a perturbation theory in a small electron-to-ion mass ratio.Comment: 10 pages, 2 table

On relaxation phenomena in a two-component plasma

The relaxation of temperatures and velocities of the components of a quasi-equilibrium two-component homogeneous completely ionized plasma is investigated on the basis of a generalization of the Chapman-Enskog method applied to the Landau kinetic equation. The generalization is based on the functional hypothesis in order to account for the presence of kinetic modes of the system. In the approximation of a small difference of the component temperatures and velocities, it is shown that relaxation really exists (the relaxation rates are positive). The proof is based on the arguments that are valid for an arbitrary two-component system. The equations describing the temperature and velocity kinetic modes of the system are investigated in a perturbation theory in the square root of the small electron-to-ion mass ratio. The equations of each order of this perturbation theory are solved with the help of the Sonine polynomial expansion. Corrections to the known Landau results related to the distribution functions of the plasma and relaxation rates are obtained. The hydrodynamic theory based on these results should take into account a violation of local equilibrium in the presence of relaxation processes.Comment: 18 page

Introducing learning creative mathematical activity for students in extra mathematics teaching

The objective of the research is determined by the need to introduce creative learning mathematics activities for school students, because that is one of the ways of ensuring the effective acquisition of knowledge at the inter-subject level and further successful adaptation while choosing a future career. Moreover, the resources of extra mathematics teaching at a general secondary school can be widely used. Thus, the purpose of the research is a) studying the theoretical basis for stimulating creative learning mathematics activities for the students and b) developing the teaching techniques of this stimulation in extra mathematics teaching at general secondary schools. The leading methods are a) modeling the task systems ensuring the development of the five types of learning activities: reproductive, productive, research, project, and project-research and b) the system analysis of the selections of experimental data based on estimating three criteria: fluency - the ability to generate a lot of ideas; flexibility - the ability to produce different ideas; ingenuity - the ability to react unconventionally. The experimental research has been carried out since 2001 and the problem systems characteristic of the five types of learning activities have been used. Subsequently, methodological approaches in extra mathematics teaching have been developed introducing creative mathematics activities for students

Development of meta-subject competencies of the 7-9 grades basic school students through the implementation of interdisciplinary mathematical courses

© Authors. The article is aimed at describing one of the possible interdisciplinary courses for students of the 7-9 classes of the basic school connecting mathematics with natural sciences and the study of such courses role in the formation and development of meta-subject competencies of students. The leading method for this is the modeling of interdisciplinary courses, the main tools for the development of meta-subject competence in which are the interdisciplinary character of the content, including open and partially open type tasks in the course structure and the organization of project activities of schoolchildren in the field of studying mathematics and related disciplines. As a result of the research conducted by the authors they worked out several author's interdisciplinary courses, among them a special place is taken by the course "Mathematics in Natural sciences" for the classes of the natural-science profile. It is suggested in it to consider mathematical questions when applying them to the material of related disciplines both through the study of theoretical material and in the process of solving problems, including the open type, and involving schoolchildren in the project of interdisciplinary activity. Practical use of the interdisciplinary courses allows to see the achievements of meta-subject results by schoolchildren and the successes of students both in mathematical preparation, expressed by high marks on the subject, and related disciplines, which indicates the improvement of mathematical education quality of students who received training using the proposed methodology

The leading-order hydrodynamics of the Landau-Vlasov kinectic equation with the nonlocal collision integral

This paper is devoted to the hydrodynamics of a one-component gas with small potential interaction. The basis of investigation is the kinetic equation in case of small potential interaction which contains general nonlocal collision integral [1] and describes arbitrary non-uniform states. In the local approximation this equation coincides with the well-known Landau–Vlasov kinetic equation. In hydrodynamics the system is supposed to be weakly non-uniform

THE LEADING–ORDER HYDRODYNAMICS OF THE LANDAU-VLASOV KINECTIC EQUATION WITH THE NONLOCAL COLLISION INTEGRAL

This paper is devoted to the hydrodynamics of a one-component gas with small potential interaction. The basis of investigation is the kinetic equation in case of small potential interaction which contains general nonlocal collision integral [1] and describes arbitrary non-uniform states. In the local approximation this equation coincides with the well-known Landau–Vlasov kinetic equation. In hydrodynamics the system is supposed to be weakly non-uniform

Thermal expansion and polarization of (1-x)PNNxPT solid solutions

The paper presents the results of detailed studies of the thermal expansion of (1-x)PbNi1/3Nb2/3O3-xPbTiO3 solid solutions with x¼0- 0.8. The anomalous and lattice contributions to deformation and the thermal expansion coefficient are analyzed and the mean square polarization Pd is determined. The results obtained are discussed within the framework of the thermodynamic theory and the Landau 2-4-6 coefficients for solid solutions are estimate

Open type tasks in mathematics as a tool for students’ meta-subject results assessment

© 2015 by iSER, International Society of Educational Research. The relevance of the present study is due to the tasks of assessing the quality of secondary education, which determine students’ subject knowledge along with their personal achievements, and where meta-subject skills define not only the level of educational results achieved, but also the prospects for the future high-quality training for a school graduate at any age. Thus, the purpose of the study conducted is to create approaches for making the system of tasks together with the criteria of their assessment to determine the level of students’ meta-subject results. The leading methods applied are modeling systems of open type tasks, with mathematical content and systematic analysis of large samples of experimental data, based on the evaluation of the two-point scale four parameters: the optimality of students’ proposed ideas, efficiency of reasoning, originality of their answer and development degree of their solutions. A pilot study conducted since 2008 has applied open tasks with mathematical content and the criteria for their assessment formed the approach to determine the level of high school students’ meta-subject results, expressed quantitatively in the form of an integrated assessment of the relative character – meta-subject intelligence quotient. Practical use of the intelligence quotient enables the making of accurate calculations for each age group in order to show the level of students’ meta-subject results. This in turn may determine the future direction of students’ individual development, and to ensure their transition to a higher level of meta-subject skills and, consequently, higher quality of education

Surface transportation engineering technology: prerequisite for accident-free traffic at signal-controlled intersections

Introduction: Improving intersection capacity will not be possible without accounting for traffic safety. Purpose of the study: We aim to determine the prerequisite for accident-free traffic at signal-controlled intersections with turning traffic flows. Methods: In our study, we used the methods of observation, comparison, and mathematical analysis. Results: We have carried out a field observation of traffic intensity at signal-controlled intersections in the city of Yekaterinburg, focusing on vehicles that moved when the green light was on. We have also analyzed traffic flow moving in three directions in the same lanes. We have discovered that traffic accident likelihood is the highest (54%) at four-way intersections. Three-way intersections account for 44% of traffic accidents, while the remaining 2% of accidents occur at multi-way intersections. Furthermore, we have determined the additional factors that impact safety in turning traffic flows at intersections. Our study demonstrates that in order to ensure maximum intersection capacity, the duration of the traffic signal cycle must be adjusted with the minimum safe distance between vehicles in mind. © St. Petersburg State University of Architecture and Civil Engineering