Vol 30, No 4 (2022)

This editorial is based on Hadamard's own statements and various recollections of him.

The main purpose of this work is, first, a construction of the indirect Hamilton’s variational principle for the problem of motion of a pendulum with a vibration suspension with friction, oscillating along a straight line making a small angle with the vertical line. Second, the construction on its basis of the difference scheme. Third, to carry out its investigation by methods of numerical analysis. Methods. The problem of motion of the indicated pendulum is considering as a particular case of the given boundary problem for a nonlinear second order differential equations. For the solution of problem of its variational formulation there is used the criterion of potentiality of operators — the symmetry of the Gateaux derivative of nonlinear operator of the given problem. This criterion is also used for the construction of variational multiplier and the corresponding Hamilton’s variational principle. On its basis there is constructed and investigated a discrete analog of the given boundary problem and a problem of motion of the pendulum. Results. It is proved that the operator of the given boundary problem is not potential with respect to the classical bilinear form. There is found a variational multiplier and constructed the corresponding indirect Hamilton’s variational principle. On its basis there is obtained a discrete analog of the given boundary problem and its solution is found. As particular cases one can deduce from that the corresponding results for the problem of motion of the pendulum. There are performed numerical experiments, establishing the dependence of solutions of the problem of motion of the pendulum on the change of parameters. Conclusion. There is worked out a variational approach to the construction of two difference schemes for the problem of a pendulum with a suspension with friction, oscillating along a straight line making a small angle with the vertical line. There are presented results of numerical simulation under different parameters of the problem. Numerical results show that under sufficiently small amplitude and sufficiently big frequency of the oscillations of the point of suspension the pendulum realizes a periodical motion.

Purpose is to develop a mathematical model for the analysis of a variant in the development of a population process with a non-trivially regulated confrontation between an invading species and a biotic environment. Relevance. The situation we are studying arises in invasive processes, but is a previously unexplored special variant of their development. The task of modeling is to describe the transition to a deep ν-shaped crisis after intensive growth. The model is based on examples of the adaptive dynamics of a bacterial colony and the suppression of mollusk populations, carriers of dangerous parasitic diseases, after targeted anti-epidemic introduction of their antagonists. Methods. In our work equations with a retarded argument in the range of parameter values that have a biological interpretation were studied. The model uses a logarithmic form of species regulation, taking into account the theoretically permissible capacity of the medium. In the equation we included the function of external influence with flexible threshold regulation relative to the current and previous population size. Results. It is shown that the proposed form of impact regulation leads to the formation of a stable adapted population after the crisis, which does not have a destructive impact on the habitat. With an increase in the reproductive potential of an invasive species, a deep crisis becomes critically dangerous. The form of the crisis passage depends on the reproductive potential, on the size of the initial group of individuals, and also on the time of activation of the adaptive counteraction from the environment. It is established that at a sufficient level of resistance, a non destructive equilibrium is established. Conclusion. The actual scenario of sudden depression of an actively spreading population with a large reproductive r-parameter, which is caused by the delayed activity of its natural antagonists, has been studied. The threshold form of biotic regulation is characteristic of insects, the abundance of which is regulated by competing species of parasitic hymenoptera. The variant of rapid phase change considered by us in the model is relevant as a description of one of the forms of developing the body’s immune response to the development of an acute infection with a significant delay. If the immune response is prematurely inhibited by the body itself, then the chronic focus of the disease persists. Examples of the dynamics of two real biological processes in experiments with biological suppression methods are given, which correspond to the invasion scenario obtained in the new model.

Purpose of the work is to create a theoretical model of the thinking process, considered as a set of operations for the formation of cognitive generalizations of the level of categories (concepts). Method for creating a theoretical model is based on the approach used in natural sciences. It involves the selection of a small number of reliable facts, which are accepted as true on the basis of their evidence. On the basis of these facts, established in various scientific disciplines, the axioms of the proposed theory are formulated. Further, from the accepted axioms, they are logically deduced in the form of consequences: a) already known results that could be obtained in various fields of science, including those differing in the content of research, and therefore previously perceived as not related to each other; b) predictions of new connections and patterns in the study area. Results of the work are that it was possible to propose a version of the postulate dynamic theory of thinking, in which the main variables are the number of concepts formed, lost, realized and unconscious by the subject. The introduced postulates and variables made it possible to consider two types of models at the moment. Balanced integrodifferential models that describe the accumulation of the volume of conscious and unconscious concepts, as well as combinatorial models that describe the interactions of concepts. Conclusion. The proposed version of the dynamic thinking model made it possible to construct reasonable theoretical descriptions of the process of spontaneous language acquisition by bilingual children in a bilingual environment and a person’s ability to compare semantically heterogeneous objects with each other. The logical scheme of the approach and the concepts used in it made it possible to connect some facts known in psychology and in an explicitly compact formulation of the difference in the structure of scientific and artistic generalizations of the picture of the world.