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## Behavior of the Shannon Function for Some Families of Classes of Three-Valued Logic Functions

An attempt to critically analyze the claims of the theory of self-organization of complex systems (synergetics) to the interdisciplinary generalizations and the universal efficacy of its models is made in the paper. The grounds of transfer of synergetic models to different disciplinary fields are under discussion. It is argued that synergetics is rather a mental scheme or a heuristic approach to exploring the complex behavior of systems, than a universal key to solving concrete scientific problems. Some prospects of development and the possible future of synergetics within the next decades are estimated.

We give a complete description of total equivalence systems (TES) for formulas based on closed classes of functions from the projection logic P3,2 with the property that the restrictions of its functions to the set {0,1} constitute a closed class of linear Boolean functions. For each such class, we find a total equivalence system, providing an algorithm for a transformation of an arbitrary formula to its canonical form.

Some peculiarities of the phenomenon of transdisciplinarity in the modern science, its differences from interdisciplinarity and multidisciplinarity, are under consideration in the article. The methodological principles of transdisciplinary studies and new possibilities of synthesis of scientific knowledge based on these principles are studied. The theory of complexity, futures studies, cognitive science, and the eco-evo-devo-perspective connected with cognitive biology are regarded as the most significant fields of the modern transdisciplinary researches. It is shown that transdisciplinary researches will, by all appearances, define the character of science in the medium-term future.

The author argues on expediency and mutual conditionality of evolutionary changes in the nature and in society. In the article three major factors of the evolution are allocated, namely: the accident, the factor of coincidence of circumstances and the factor of acceleration of social evolution.

The phenomenon of communication as a manifestation of complexity of interacting creatures. Communication is considered not as a privilege of a human being; it is shown that it is rooted in the world of living nature, it has an evolutionary origins. Communicative complexity is exposed by such concepts as flexibility, constructing, intersubjectivity, participatory sense-making, empathy, synergy, mutual incorporation and co-emergence of creatures which enter the process of communication. Understanding of communication from the position of the conception of enactivism allows disclosing some substantial aspects of the constructivist character of communicative interaction.

The conference *Philosophy, Mathematics, Linguistics: Aspects of Interaction 2014 (PhML-2014)* is a sequel in the series of conferences intended to provide a forum for philosophers, mathematicians, linguists, logicians, and computer scientists who share an interest in cross-disciplinary research. The conference PhML-2014 is endorsed by the American National Committee of the Division of Logic, Methodology and Philosophy of Science (DLMPS) of the International Union of the History and Philosophy of Science (IUHPS), the Japan Association for Philosophy of Science, the Swedish National Committee for Logic, Methodology and Philosophy of Science.

The monograph is devoted to the consideration of complex systems from the position of the end the 21st century. The considerable breakthrough in the understanding of complex systems is comprehensively analyzed. Such a breakthrough is connected with the use of the newest methods of nonlinear dynamics, of organization of the modern computational experiments. The book is meant for specialists in different fields of natural sciences and the humanities as well as for all readers who are interested in the recent advancements in science.

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.

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.