Тематические тесты для систематизации знаний по математике. Часть 1.
The article discusses the classifications of traditional sciences (vidyā, śāstra) in the Sanskrit texts of the Upanishads (Chāndogya and Muṇḍaka), in the Manusmṛti, Kauṭilya’s Arthaśāstra, Lalitavistara, Vatsyayāna’s Kāmasūtra et al. N. Kanaeva demonstrates that the authors of these classifications were brāhmans whereas the non-brāhmanical systems of science classifications did not introduce anything new into them because they had inherited them along with the traditions of brāhmanical educational system. Brāhmanical classifications of systems of knowledge were built according to a pragmatic criterion as lists of types of knowledge employed in the social practices of the higher varṇas: brāhmans, kṣatriyas and vaiśyas. In the Middle Ages another criterion for classification of sciences emerges — orientation to tradition (traditionalism) resulting in complicated lists combining theoretical and practical knowledge (“sciences” and “arts”).
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 article considers the questions of the use of educational tests for the formation of the skills, systematization of knowledge and monitoring the results of teaching mathematical disciplines at National Research University Higher School of Economics – Perm branch. Presents the technology of the use of educational and methodical complex consisting of multilevel thematic tests according to the main sections of the course of the higher mathematics. The author shares the experience of using tests in teaching of a course of the higher mathematics in an economic university. It is proposed to use the educational tests not only for monitoring the results, but as a training tool for systematization of knowledge and organization of independent activity of students, including their out-of-class work.
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.