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## О новом типе безмассовых дираковских фермионов в кристаллических топологических изоляторах

A new type of massless Dirac fermions in *crystalline *three_dimensional topological insulators (three_dimen_sional two_dimensional situation) has been predicted. The spectrum has fourfold degeneracy at the top of the two_dimensional Brillouin zone (M point) and twofold degeneracy near the M point. Crystal symmetry along with the time reversal invariance in three_dimensional topological insulators allows fourfold degenerate Dirac cones, which are absent in the classification of topological features in R._J. Slager et al., Nat. Phys. **9**, 98 (2013). The Hamiltonian in the cited work does not contain Dirac singularities with more than twofold degeneracy. For this reason, the corresponding topological classification is incomplete. The longitudinal magnetic field in the spinless case holds the massless dispersion law of fermions and does not lift fourfold degeneracy. In the spinor case, the magnetic field lifts fourfold degeneracy, holding only twofold degeneracy, and results in the appearance of a band gap in the spectrum of fermions.

The research is carried out in accord with the main principles of communicative and paradigmatic linguistics. It is devoted to the study of lingual ways of expressing harmony as an ethetic category. Rhythm as a regular repetition of similar and commensurable units of language is considered to be the main mechanism of the harmonious organization of belle-lettres style texts. The high efficiency of rhythm as a stylistic device creates the expressiveness of the text at all the levels of language, beginning with the phonemic level and finishing with the dictemic level. The principle of repeatability of elements and their relations within the system of semantic and syntactic ties manifests internal symmetry; and the symmetry of text elements is the main indicator of its harmony.

Recently much attention has been devoted to the optimization of transportation networks in a given geographic area. One assumes the distributions of population and of services/workplaces (i.e. the network's sources and sinks) are known, as well as the costs of movement with/without the network, and the cost of constructing/maintaining it. Both the long-term optimization and the short-term, "who goes where" optimization are considered. These models can also be adapted for the optimization of other types of networks, such as telecommunications, pipeline or drainage networks. In the monograph we study the most general problem settings, namely, when neither the shape nor even the topology of the network to be constructed is known *a priori*.

The book contains the reports of the member of the congress from the different countres. They consider the idea of the symmety in the science and in the art.

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.

The problem from the area of Personology of Self, being the most important part of general personology, is studied in this paper. The paper offers the model of basic features and abilities of Self. The role of ability of Self to be its own ability, the ability which integrates many intentions of “I can” type depending on life tasks faced by the personality, is stressed. Existential value of the task to be effective in dialog based on the intention “I express myself” is substantiated. The model of narration, the model of author’s reflection in the moment of narration and three-dimensional topological model of ability of Self to express itself in the dialog are offered.

The symmetry nature of the appearance of specific surface (edge) states at the boundaries of low_dimension structures with the symmetry of ribbons (borders) invariant with respect to time reversal is discussed. Symmetry reasons for the stability of such states against the elastic scattering from nonmagnetic impurities have been revealed.

The paper examines the significance of Platonic Solids in different parts of modern science.

We propose a model that evaluates how much a network has changed over time in terms of its structure and a set of central elements. The difference of structure is evaluated in terms of node-to-node influence using known nodes correspondence models. To analyze the changes in nodes centralities we adapt an idea of interval orders to the network theory. Our approach can be used to investigate dynamic changes in temporal networks and to identify suspicious or abnormal effects in terms of the topology and its critical members. We can also transform the stability measure to the similarity measure in order to cluster the network in some homogeneous periods. To test our model, we consider the international migration network from 1970 to 2015 and attempt to analyze main changes in migration patterns.

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.

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.