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## Математика. Конспект лекций

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

We calculate characteristic polynomials of operators explicitly represented as polynomials of rank $1$ operators. Applications of the results obtained include a generalization of the Forman--Kenyon's formula for a determinant of the graph Laplacian and also provide its level $2$ analog involving summation over triangulated nodal surfaces with boundary.

Reinterpretation of approaches of John Locke, Immanuel Kant and Charles Peirce helps to single out three integral organons of cognition. One is mathematics, or cognition of measure and art of all kind of measurement. Another is morphology or cognition of forms and art of arranging shapes and configurations. One more is semiotics or cognition of meanings and art of their transfer. It is demonstrated that all three organons vary and provide specific fields of knowledge and areas of research. Current versions and varieties of disciplinary manifestations of organons are reviewed. Mathematics is the most developed complex of scientific disciplines. Morphology is a constellation of a number of assorted and fully independent disciplines. Semiotics is rifted by a gap between rough outline of general or «pure» semiotics (Morris) and a nebula of unevenly elaborated semiologies of various sorts – that of languages, literatures, cinema, heraldry, race discrimination or ideological manipulations. Analysis of political discourses and speech acts can contribute to integration of common area of semiotic research.

The introductory article clarifies the title of the current issue of «METHOD» and explicates the purpose of the entire publication. It explains slight but telling differences between the Russian, English and German phrasings that expound the meaning of the title and purpose of the yearbook. Subtle but indicative differences between languages and modes of speech and thought highlight a major issue of knowledge transfer. The yearbook departs from knowledge transfer to a more incentive issue of convergence and divergence of cognitive skills. Introduction focuses on transdisciplinary organons. They derive from our basic cognitive abilities. The initial one is the faculty to tell relative degrees of our sensations (bigger - smaller, warmer - colder etc.) and then to rate sizes of things and intensity of processes. The following one is pattern recognition or our ability to single out certain ‘rated’ entities from their environment. The subsequent one is our capacity to assign meaning to the ‘recognized’ figures and forms of the world around. It further supplements with the gift to use words and images to grasp sense and to convey it. Each of the three fundamental cognitive abilities diverge into further generations of abundant skills and proficiencies. Elaborate methods of scientific research outreach to thresholds of our knowledge. Right there they intertwine with each other. Interdisciplinary linkages develop. Transdisciplinary prospects loom. We conceive imminent convergence of our methodological skills into three transdisciplinary organons congenial to the three cognitive abilities. The first one is metretics or the higher technique of measurement and calculus. It resides in mathematical and statistical studies. The next one is morphetics or the expertise of exploring forms, shapes and figures. It resides in all kinds of morphological, comparative, configurative and evolutionary research. The last one is semiotics or the art of processing sense and reference. It resides in still budding semiologies, cognitive arts and still rudimentary humanities.

The paper discusses in detail the scale of translation of primary points scored by school graduates in the unified state exam in mathematics, used from 2013 to the present time. Based on the analysis of the dynamics of these scales, a conclusion is made about the annual increase in the "average" 100-point result, as well as the presence of a significant increase in the final grade compared with the linear scale. Additionally, the authors describe the effect of reducing the value of primary points as they approach the maximum.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.