### Working paper

## Interpolation Macdonald polynomials and Cauchy-type identities

Moore's generalization of the game of Nim is played as follows. Given two integer parameters $n, k$ such that $1 \leq k \leq n$, and $n$ piles of tokens. Two players take turns. By one move a player reduces at least one and at most $k$ piles. The player who makes the last move wins. The P-positions of this game were characterized by Moore in 1910 and an explicit formula for its Sprague-Grundy function was given by Jenkyns and Mayberry in 1980, for the case $n = k+1$ only. We modify Moore's game and introduce Exact $k$-Nim in which each move reduces exactly $k$ piles. We give a simple polynomial algorithm computing the Sprague-Grundy function of Exact $k$-Nim in case $2k > n$ and an explicit formula for it in case $2k = n$. The last case shows a surprising similarity to Jenkyns and Mayberry's solution even though the meaning of some of the expressions are quite different. On the Sprague-Grundy function of Exact $k$-Nim. Available from: https://www.researchgate.net/publication/281144667_On_the_Sprague-Grundy_function_of_Exact_k-Nim [accessed Oct 26 2017].

This article discusses the planning of electric rolling stock (ERS) maintenance: to implement all maintenance requirements under limited resources. In this case, several effective criteria can be considered for the planning of turnover schedule:

- satisfaction of traffic safety requirements, provided by correcting the estimated train movement time which should not exceed limited time between maintenances;

- uniformity of maintenance.

The solution by using graph theory allows to get the whole set of reasonable maintenance schedule and to choose which maintenance corresponds to the train movement schedule and minimum differs from the optimal by selected criteria. It takes a significant amount of time.

The discussed task can be solved quickly by using genetic algorithm. New criterion – total exceeded time over permissible interval time between maintenances – allows to get a solution for any initial data. It is not always available when using the criterion of uniform maintenance.

In this article developed mathematical tool is based on combinatorics, graph theory, and genetic algorithms.

The authors executed the adaptation of crossover algorithms and mutations, implemented in the framework of the genetic algorithm, with the features of the agent involved in the solution. The authors investigated the possibility of using various crossover types, mutations and the parameters’ influence of the genetic algorithm on the obtained results for turnover schedule planning.

The obtained results have been tested for the conditions of the Moscow subway.

The collection represents proceedings of the XVIII international conference “Problems of Theoretical Cybernetics” (Penza, 19–23 June, 2017), that is sponsored by Russian Foundation for Basic Research (project N 17-01-20217-г). The conference subject area includes: control systems synthesis, complexity, reliability, and diagnostics; automata; computer languages and programming; graph theory; combinatorics; coding theory; theory of pattern recognition; mathematical programming and operations research, mathematical theory of intelligence systems; applied mathematical logic; functional systems theory; optimal control theory; applications of cybernetics in natural science and technology.

This volume contains a selection of contributions from the "First International Conference in Network Analysis," held at the University of Florida, Gainesville, on December 14-16, 2011. The remarkable diversity of fields that take advantage of Network Analysis makes the endeavor of gathering up-to-date material in a single compilation a useful, yet very difficult, task. The purpose of this volume is to overcome this difficulty by collecting the major results found by the participants and combining them in one easily accessible compilation.

Full papers (articles) of 2nd Stochastic Modeling Techniques and Data Analysis (SMTDA-2012) International Conference are represented in the proceedings. This conference took place from 5 June by 8 June 2012 in Chania, Crete, Greece.

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.

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.

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.