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## Brown's theorem and its application for enumeration of dissections and planar trees

We present new functional equations connecting the counting series of plane and planar (in the sense of Harary and Palmer) dissections. Simple rigorous expressions for counting symmetric rr-dissections of polygons and planar SS-dissections are obtained.

Currently, the effective use of all available geographical information in Earth Sciences worldwide is associated with problems of their processing and effective application, although more recently, the problems of storing large amounts of data have been added to this. In modern conditions, the spatial data Fund is a complex and extensive information field containing the most heterogeneous data in terms of spatial coverage and resolution. This is due to an increase in the volume of information, and the ways to choose the means and methods of processing. In the research work of scientists and the routine work of managers, only knowledge-intensive information is needed, which has specifics due to the specialization, complexity and strong connectivity of data.

This book describes modern tools and methods of geoinformation mapping, remote monitoring in geographical research for complex spatial analysis of natural and socio-economic processes. The synthesis of industry knowledge is also necessary for studying different aspects of nature and society, for establishing patterns and deepening geographical knowledge, and for making forecasts. Modern geoinformation technologies, remote sensing of the Earth, and cartographic works based on them most fully meet such complex requests.

Observers are able to extract summary statistical properties, such as numerosity or the average, from spatially overlapping subsets of visuals objects. However, this ability is limited to about two subsets at a time, which may be primarily caused by the limited capacity of parallel representation of those subsets. In our study, we addressed several issues regarding subset representation. In four experiments, we presented observers with arrays of dots of one to six colors and instructed them to judge the number of colors. We measured both speed and accuracy of those judgments. Following standard criteria used for the interpretation of object enumeration data, we recognized two modes of subset representation: a) parallel, effortless and strategy-independent representation of no more than two subsets, and b) serial representation modulated by different attentional strategies and a working memory template. We also found an advantage of large sets over small ones, demonstrating that subset representation can be formed based on some statistical accumulation of information from individual objects.

We enumerate chord diagrams without loops and without both loops and parallel chords. For labelled diagrams we obtain generating functions, for unlabelled ones we derive recurrence relations.

This book constitutes the proceedings of the 15th International Computer Science Symposium in Russia, CSR 2020, held in Yekaterinburg, Russia, in June 2020.

The 25 full papers and 6 invited papers were carefully reviewed and selected from 49 submissions. The papers cover a broad range of topics, such as: algorithms and data structures; computational complexity, including hardness of approximation and parameterized complexity; randomness in computing, approximation algorithms, fixed-parameter algorithms; combinatorial optimization, constraint satisfaction, operations research; computational geometry; string algorithms; formal languages and automata, including applications to computational linguistics; codes and cryptography; combinatorics in computer science; computational biology; applications of logic to computer science, proof complexity; database theory; distributed computing; fundamentals of machine learning, including learning theory, grammatical inference and neural computing; computational social choice; quantum computing and quantum cryptography; theoretical aspects of big data.

The conference was cancelled as a live conference due to the corona pandemic.

All investigations of the scheme defined in its name are fulfiled by the direct enumeration of its outcomes, namely: the number of its outcomes of the scheme and its probability distribution are defined, the numbering problem for outcomes of the scheme is solved, this gives the possibility of a quick modeling of its possible values and approximate calculation of the number of outcomes of the scheme by the method of proportion.

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.