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## A lower bound on the size of an absorbing set in an arc-coloured tournament

Bousquet, Lochet and Thomassé recently gave an elegant proof that for any integer n, there is a least integer f(n) such that any tournament whose arcs are coloured with n colours contains a subset S of vertices of size f(n) with the property that any vertex not in S admits a monochromatic path to some vertex of S. In this note we provide a lower bound on the value f(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.

The work deals with a generalization of Erdos-Lovasz problem concerning colorings of non-uniform hypergraphs. We establish a new sufficient condition for r-colorability of a non-unifrom hypergraph with large edge sizes and girth at leats 4 in terms of expectation of the number of monochromatic edges in a random coloring.

The work deals with combinatorial problems concerning colorings of non-uniform hypergraphs. Let $H=(V,E)$ be a hypergraph with minimum edge-cardinality $n$. We show that if $H$ is a simple hypergraph (i.e. every two distinct edges have at most one common vertex) and $$ \sum_{e\in E}r^{1-|e|}\leqslant c\sqrt{n}, $$ for some absolute constant $c>0$, then $H$ is $r$-colorable. We also obtain a stronger result for triangle-free simple hypergraphs by proving that if $H$ is a simple triangle-free hypergraph and $$ \sum_{e\in E}r^{1-|e|}\leqslant c\cdot n, $$ for some absolute constant $c>0$, then $H$ is $r$-colorable.

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.

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.

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.

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.