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## Mathematical model of an active biological continuous medium with account for the deformations and rearrangements of the cells

A continuum model of the embryonic epithelial tissue with account for the active deformations and rearrangements of the cells is proposed. The stress tensor is represented as the sum of the stresses undergone by the cell directly and the tensor of active stresses that arise owing to contracting cellular protrusions anchored on the surface of neighboring cells and developing in response to cell reshaping (deformation). The strain rate tensor includes three components: elastic and two inelastic related to the active deformation of the cells and their rearrangement. The first of these components depends on the stresses in the cells and the reached cellular deformation level, whereas the second is determined by the active stresses. The problem of reaction of a thin sheet to a rapid stretching is solved and agreement with experimental data is obtained.

The problem of deformation of a planar embryonic epithelium layer that is unloaded after a short period of uniaxial stretching with subsequent fixation in the stretched state for different periods of time is solved. The initial conditions for solving this problem are derived from the previously discussed problem of the uniform stretching of a tissue fragment (explant) with subsequent fixation of the obtained length. In this study we used the previously developed continuum model that describes the stress–strain state of epithelial tissue taking the parameters that characterize the shape of the cells and their stress state into account, as well as the active stresses they exert when they interact with each other. The experimentally observed continuation of the deformation of a stretched tissue after the external force has ceased to act is described theoretically as a result of active cell reactions to mechanical stress. The duration of explant fixation is shown to have a strong effect on its further elongation and on the pattern of cell activity.

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.

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.