On the Motion of a Solid Body on Spherical Surfaces
A motion problem for material points embedded in a standard three-dimensional sphere $S^3$ is considered in terms of classical mechanics. In particular, spherical analogs of Newton’s laws are discussed.
A mathematical model is proposed to analyze the spinal strain-deformation condition resulting from axial and lateral g-loads that are generated by changes in the gravity field and/or pilot’s actions during high-performance aircraft maneuvering under flight overload conditions. An algorithm of solution has been developed, which takes into account changes in the intervertebral disk pressure and the fibrous ring shape at the time when loading reaches close-to-critical g-values. Calculation of the spinal-strain deformation condition was implemented using the SPLEN computer system (KOMMEK, Russia). Analysis of the spinal straindeformation condition was made for two types of external loads: normal load and unilateral load with the bending moment. Maximum permissible loads on the spinal segment were evaluated, and a pattern of distribution of strain intensity and mean strains, spinal deformation, and the destruction field was described. The developed computer models can be used as a basis for developing a technique of predicting characteristic spinal injuries due to different extreme loads and pathologies.
This book aims to present the newest research in the fields of mathematics and mechanics. Theories of mathematics and mechanics are the basis of modern technological improvements and, therefore, interest towards them are increasingly important.
A method based on the spectral analysis of thermowave oscillations formed under the effect of radiation of lasers operated in a periodic pulsed mode is developed for investigating the state of the interface of multilayered systems. The method is based on high sensitivity of the shape of the oscillating component of the pyrometric signal to adhesion characteristics of the phase interface. The shape of the signal is quantitatively estimated using the correlation coefficient (for a film–interface system) and the transfer function (for multilayered specimens).
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.