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## Collineation group as a subgroup of the symmetric group

Let Ψ be the projectivization (i.e., the set of one-dimensional vector subspaces) of a vector space of dimension ≥ 3 over a field. Let *H* be a closed (in the pointwise convergence topology) subgroup of the permutation group GΨ of the set Ψ. Suppose that *H *contains the projective group and an arbitrary self-bijection of Ψ transforming a triple of collinear points to a non-collinear triple. It is well-known from [9] that if Ψ is finite then *H* contains the alternating subgroup AΨ of GΨ.

We show in Theorem 3.1 below that *H *= GΨ, if Ψ is infinite.

We classify all finite simple subgroups of the Cremona group Cr3(C).

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.

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.

The collection represents proceedings of the nineth international conference "Discrete Models in Control Systems Theory" that is held by Lomonosov Moscow State Uneversity and is dedicated in 90th anniversary of Sergey Vsevolodovich Yablonsky's birth. The conference subject are includes: discrete functional systems; discrete functions properties; control systems synthesis, complexity, reliability, and diagnostics; automata; graph theory; combinatorics; coding theory; mathematical methods of information security; theory of pattern recognition; mathematical theory of intellegence systems; applied mathematical logic. The conference is sponsored by Russian Foundation for Basic Research (project N 15-01-20193-г).

Conference covers both fundamental problems ofthe theory, and application to research of complex organizational and technical systems.

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.

A form for an unbiased estimate of the coefficient of determination of a linear regression model is obtained. It is calculated by using a sample from a multivariate normal distribution. This estimate is proposed as an alternative criterion for a choice of regression factors.