### Article

## Equitable two-colorings of uniform hypergraphs

An equitable two-coloring of a hypergraph $H=(V,E)$ is a proper vertex two-coloring such that the cardinalities of color classes differ by at most one. In connection with the property B problem Radhakrishnan and Srinivasan proved that if $H$ is a $k$-uniform hypergraph with maximum vertex degree $\Delta(H)$ satisfying

$$

\Delta(H)\leqslant c\,\frac {2^{k-1}}{\sqrt{k\,\ln k}}

$$

for some absolute constant $c>0$, then $H$ is 2-colorable. By using the Lov\'asz Local Lemma for negatively correlated events and the random recoloring method we prove that if $H$ either is a simple hypergraph or has a lot of vertices, then under the same condition on the maximum vertex degree it has an equitable coloring with two colors. We also obtain a general result for equitable colorings of partial Steiner systems.

The paper deals with the classical extremal problem concerning colorings of hypergraphs. The problem is to find the value m(n,r), equal to the minimum number of edges in a n-uniform hypergraph with chromatic number greater than r. We obtain new upper and lower bounds for m(n,r) in the case when the parameter r is very large in comparison with n.

Most publications on the theory and practice of creating control systems are devoted to deterministic problems, when the input quantities and output characteristics have deterministic values or functions. In the article, the authors draw attention to the fact that all input and output signals, as well as the internal parameters of control systems, always have random variations. Technological random deviations have parameters for construction materials and electronic components and devices. In addition, the ambient temperature randomly influences the control systems, mechanical effects from the installation object, as well as temporary aging, wear and other factors. The authors draw attention to the insufficient number of publications devoted to the system discussion of probabilistic methods for studying control systems. In this article, the authors not only systematized the methods for studying random variations in the parameters of control systems, but also set forth their unified approach to solving the most frequently encountered problems of studying accuracy, stability, serial seriality and parametric reliability in the case of gradual failure of control systems. The necessary mathematical apparatus is given.

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 paper deals with a combinatorial problem concerning colourings of uniform hypergraphs with large girth. We prove a new lower bound for the maximum edge degree for an n-uniform non-r-colourable simple hypergraph. As an application of our probabilistic technique we establish a lower bound for the classical van der Waerden number W(n, r), the minimum natural N such that in an arbitrary colouring of the set of integers {1, . . . ,N} with r colours there exists a monochromatic arithmetic progression of length n.

We propose a new machine learning concept called Randomized Machine Learning, in which model parameters are assumed random and data are assumed to contain random errors. Distinction of this approach from “classical” machine learning is that optimal estimation deals with the probability density functions of random parameters and the “worst” probability density of random data errors. As the optimality criterion of estimation, randomized machine learning employs the generalized information entropy maximized on a set described by the system of empirical balances. We apply this approach to text classification and dynamic regression problems. The results illustrate capabilities of the approach.

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.