Book
Математический анализ для менеджеров
Theorems of existence and uniqueness of Cauchy’s problem solution for systems of nonlinear functional and differential equations are proved. During the proof of the theorems the positivity of the Cauchy’s matrix corresponding linear system is used essentially.
This journal publishes the mathematics section of Series I of the Vestnik (Herald) of St. Petersburg University, and is one of the oldest Russian mathematics journals in English translation. Articles cover the major areas of pure and applied mathematics.
Many famous mathematicians are associated with the Faculty of Mathematics and Mechanics at St. Petersburg University, including Chebyshev, Lyapunov, Alexandrov, Smirnov, Kantorovich, to name a few. Today, mathematics/mechanics faculty members continue the excellent tradition in mathematics associated with the University and the Vestnik is the prime outlet for their research results.
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 paper addresses the on-line teaching of Calculus using webMathematica interactive electronic tutorials developed by the author. The tutorials are available on the web site http://wm.iedu.ru. It is obvious that e-learning technologies need new pedagogy. It is usually called e-pedagogy. We share and realize the main pedagogical principle of webMathematica based learning. The principle is laid out as follows. To teach mathematics not calculation or math not equal calculating.
In order to reduce the Gibbs phenomenon exhibited by the partial Fourier sums of a periodic function, discontinuous in 0, Driscoll and Fornberg considered so-called singular Fourier-Padé approximants constructed from the Hermite-Padé approximants. Convincing numerical experiments have been obtained by these authors, but no error estimates have been proven so far. In the present paper we study the special case of Nikishin systems and their Hermite-Padé approximants, both theoretically and numerically. We obtain rates of convergence by using orthogonality properties of the functions involved along with results from logarithmic potential theory. In particular, we address the question of how to choose the degrees of the approximants, by considering diagonal and row sequences, as well as linear Hermite-Padé approximants. Our theoretical findings and numerical experiments confirm that these Hermite-Padé approximants are more efficient than the more elementary Padé approximants, particularly around the discontinuity of the goal function.
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.