Рассмотрены вопросы профессиональной направленности преподавания математических дисциплин в техническом вузе, актуальность которых обусловлена внедрением новых образовательных стандартов. Проведён анализ научных публикаций по данной тематике и обобщён педагогический опыт авторов. Обоснованы преимущества использования профессионально-ориентированных математических задач в сочетании с электронными образовательными ресурсами в учебном процессе. Приведены примеры задач профессионального характера по математическим дисциплинам и установлены межпредметные связи для ряда образовательных программ, реализуемых в Новосибирском государственном техническом университете. The issues of the applied orientation of teaching mathematical disciplines at university are considered, the relevance of which is due to the introduction of new educational standards. The analysis of scientific publications on this topic is carried out and the pedagogical experience of the authors is generalized. The advantages of using professionally oriented mathematical problems in combination with electronic educational resources in the educational process are substantiated. Examples of applied problems on mathematical subjects are given and interdisciplinary connections are established for a number of educational programs implemented at Novosibirsk State Technical University.

Introduction. Additional education for high school students is considered as an integral part of the sustainable education system. In addition, during the school years there is a period of professional self-determination, so students have an increased interest in classes, which allows them to decide on their future specialty and prepare for admission to university.Purpose setting. To explore the role of a technical university in additional education for high school students in mathematics and computer science and show the priority areas of work of the university with schoolchildren.Methodology and methods of the study. The scientific research is based on legislative and regulatory documents, publications of domestic specialists and personal experience of the authors on extended school studies at the Novosibirsk State Technical University.Results. Additional education is considered from the point of view of the implementation of career guidance for students by a technical university. It is shown that universities play an important role in the scientific and methodological support and staff maintenance of additional education of schoolchildren in the technical direction of training. Based on the results of the analysis of a survey of students and their parents is shown that the participation of students in additional classes increases their self-esteem, which is objectively manifested in the growth of motivation in the study of mathematics and computer science at school, in achievements at conferences of various levels, in the choice of technical specialties. All students attending classes as part of additional education entered technical universities, moreover, none of the guys changed their decision to become a technical specialist.Conclusions. The specificity of additional education allows a technical university toeffectively participate in the system of additional mathematical training of schoolchildren, identify talented students, increase their interest in technical sciences and engage them as applicants to technical universities.

The flow in time of an initial state ensemble in a multidimensional phase space, as a rule, models some dynamic process. Under what conditions is such a flow generated by a vector field in such a way that the given flow corresponds to the vector field in a unique way? A positive answer to this question is given by the classical uniqueness theorems for the solution of the initial value problem in the case of a regular vector field with the required properties of the modulus of continuity in space variables. In mathematical models of stochastic differential equations, in models of irregular hydrodynamic flows, and in a number of other cases when the flow is generated by a “bad” vector field that has a modulus of continuity in space variables that does not meet the conditions of the uniqueness theorem for solving the initial problem for a vector field, generating this flow, we cannot speak about the correctness of the initial problem for the vector field and, thus, about the correctness of finding the trajectories connecting the initial and actual states of the ensemble of particles in the phase space. In this case, the uniqueness of the flow generated by the vector field remains to be judged only by the properties of the flow itself. The only known result of this type is van Kampen's theorem, which states that the uniqueness of a flow generated by a vector field continuous in space variables is guaranteed by the properties of homeomorphism and the Lipschitz property of the flow in space variables. If the vector velocity field loses the property of continuity in space variables, then van Kampen's theorem does not work and some other properties of the flow are required to guarantee its uniqueness. In this paper, we establish such properties of a flow that guarantee its uniqueness even in the case of a violation of the continuity of the vector field that generates this flow. The conditions of van Kampen's theorem in a certain sense are a special case of the properties of the flow established in this paper, which guarantee its uniqueness as a solution to the initial problem for an irregular vector field. The general construction constructed here makes it possible to establish such properties of flows in various mathematical models that guarantee its uniqueness for a generating vector field.

The article is devoted to the problem of establishing the frames to use distant learning in higher education. The topic relevance is related to emerging a force majeure situation associated with a pandemic, which made it possible to carry out a unique experiment on such large-scale training applying. The paper analyzes some aspects of this problem concerning the mathematical discipline learning at the technical university junior courses. The study is based on the analysis of scientific publications by domestic and foreign authors devoted to the problems of mixed learning, distant learning, and peculiarities of teaching mathematical disciplines in universities. The author conclude that the main problems of introducing e-learning and distant learning technologies into the educational process is insufficient motivation of students pronounced especially in junior courses. They note that the effectiveness of using distant educational technologies in additional education is largely due to the good motivation of people who want to improve their professional level. The paper discusses results obtained during the forced transition to distant learning (March-July 2020), in particular, gives the rationale for a certain model of mixed learning. It emphasizes that, as the threat of the pandemic situation repetition, as well as the need for a new transition to e-learning are not excluded, the problem of motivating students should be given special attention at all educational process levels.

The paper presents the results of the experiment on the extracurricular individual students’ work based on web-support of learning activities. The issue relevance is contingent with reduction of the teaching load for regular students. The following methods of the academic research are applied: theoretical (analysis of scientific literature, generalisation of educational experience), empiric (lesson observation, a pedagogic experiment, questionnaires, tests, conversation) and statistical (determination of average values, scaling). A survey has been done among the first-year studentsbeing on the technical training programme at Novosibirsk State Technical University and studying mathematical analysis via the DiSpace 2.0 e-learning system. The use of the electronic educational-methodical complex for the support of extracurricular individual students’ learning activities has proved to increase motivation, interest in studies, and the level of acquired competence.