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## On trees of bounded degree with maximal number of greatest independent sets

For each n and d, we describe the structure of trees with the maximal possible number of greatest independent sets in the class of n-vertex trees of vertex degree at most d. We show that for all even n an extremal tree is unique but uniqueness may fail for odd n; moreover, for d = 3 and every odd n>6, there are exactly ceil{(n-3)/4} + 1 extremal trees. In the paper, the problem of searching for extremal (n; d)-trees is also considered for 2-caterpillars, i.e., trees in which every vertex lies at distance at most two from some simple path. For each n and d=3,4, we completely reveal all extremal 2-caterpillars on n vertices each of which has degree at most d.

For any *n*, in the set of *n*-vertex trees such that any two leaves have no common adjacent vertex, we describe the trees with the smallest number of maximal independent sets.

The book contains the necessary information from the algorithm theory, graph theory, combinatorics. It is considered partially recursive functions, Turing machines, some versions of the algorithms (associative calculus, the system of substitutions, grammars, Post's productions, Marcov's normal algorithms, operator algorithms). The main types of graphs are described (multigraphs, pseudographs, Eulerian graphs, Hamiltonian graphs, trees, bipartite graphs, matchings, Petri nets, planar graphs, transport nets). Some algorithms often used in practice on graphs are given. It is considered classical combinatorial configurations and their generating functions, recurrent sequences. It is put in a basis of the book long-term experience of teaching by authors the discipline «Discrete mathematics» at the business informatics faculty, at the computer science faculty* *of National Research University Higher School of Economics, and at the automatics and computer technique faculty of National research university Moscow power engineering institute. The book is intended for the students of a bachelor degree, trained at the computer science faculties in the directions 09.03.01 Informatics and computational technique, 09.03.02 Informational systems and technologies, 09.03.03 Applied informatics, 09.03.04 Software Engineering, and also for IT experts and developers of software products.

We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.

We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.

We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.

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.