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## Одномерные текстуры биаксиальных жидких кристаллов

We have derived the equations for 1D textures of biaxial nematics and showed that they are based on the struc

ture of the group SL(3;R) of real 3×3 matrices of determinent +1. To describe the conformations of the above textures

we introduce the curvature vector Ω that corresponds to a skew-symmetric matrix decribing infitesimal rotations

of the order parameter while moving along the axis of the texture. The construction is helpful for the analysis

of data provided by the numerical modelling. Thus we have obtained the conditional periodic textures of the biaxial

nematics. Our results may be applied for studying the optical phenomenae in the biaxial nematics.

Textures (i.e., smooth space nonuniform distributions of the order parameter) in biaxial nematics turned out to be much more complex and interesting than expected. Scanning the literature we find only a very few publications on this topic. Thus, the immediate motivation of the present paper is to develop a systematic procedure to study, classify, and visualize possible textures in biaxial nematics. Based on the elastic energy of a biaxial nematic (written in the most simple form that involves the least number of phenomenological parameters) we derive and solve numerically the Lagrange equations of the first kind. It allows one to visualize the solutions and offers a deep insight into their geometrical and topological features. Performing Fourier analysis we find some particular textures possessing two or more characteristic space periods (we term such solutions quasiperiodic ones because the periods are not necessarily commensurate). The problem is not only of intellectual interest but also of relevance to optical characteristics of the liquid-crystalline textures.

Much of our understanding (and applications) of biaxial nematic liquid crystals requires the study of the textural transformations in external electric or magnetic fields. To that end, one should employ theoretical approaches which could have bearing on the minimization problem of the multi-parametric free energy. The immediate shortcoming of the direct free-energy minimization (widely used for uniaxial nematics) is the need to resolve several non-linear constraints. To overcome this difficulty, in what follows we shall use the “angular velocity”, which describes space rotations of the order parameter, and is in fact a vector internal curvature of the texture. This method provides a means to resolve the constraints imposed on the order parameter. Thus, we have obtained the set of equations to find all possible one-dimensional textures of biaxial nematics in the external field. To illustrate our method, we calculate the critical fields corresponding to some basic configurations for textural transitions in the biaxial nematics. We feel that this result could be useful to determine the intrinsic degree of biaxiality for liquid crystalline materials.

We consider textures of biaxial nematics confined between two parallel plates. The boundary conformations at the bordering plates are supposed to be identical, the gradients of the order parameter being generally nonzero. We claim that for any texture (including stable uniform order parameter alignment) there exists its counterpart texture which is also a minimum of the gradient elastic energy. Our arguments are based on the topological analysis of the conformation of the order parameter.

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.