Vestnik St. Petersburg University: Mathematics 

About: Vestnik St. Petersburg University: Mathematics is an academic journal published by Springer Science+Business Media. The journal publishes majorly in the area(s): Nonlinear system & Random variable. It has an ISSN identifier of 1063-4541. Over the lifetime, 711 publications have been published receiving 1640 citations. The journal is also known as: Vestnik Sankt-Peterburgskogo Universiteta. & Mathematics.

On the spectrum of the Steklov problem in a domain with a peak

Abstract: We consider the spectral Steklov problem in a domain with a peak on the boundary. It is shown that the spectrum on the real nonnegative semi-axis can be either discrete or continuous depending on the sharpness of the exponent.

Solution of generalized linear vector equations in idempotent algebra

Abstract: The problem on the solutions of homogeneous and nonhomogeneous generalized linear vector equations in idempotent algebra is considered. For the study of equations, an idempotent analog of matrix determinant is introduced and its properties are investigated. In the case of irreducible matrix, existence conditions are found and the general solutions of equations are obtained. The results are extended to the case of arbitrary matrix. As a consequence the solutions of homogeneous and nonhomogeneous inequalities are presented. 1. Introduction. For analysis of different technical, economical, and engineering systems the problems are often occurred which require the solution of vector equations linear in a certain idempotent algebra [1–5]. As a basic object of idempotent algebra one usually regards a commutative semiring with an idempotent summation, a zero, and a unity. At once many practical problems give rise to idempotent semiring, in which any nonzero (in the sense of idempotent algebra) element has the inverse one by multiplication. Taking into account a group property of multiplications, such a semiring are called sometimes idempotent semifield. Note that in passing from idempotent semrings to semifields, the idempotent algebra takes up an important common property with a usual linear algebra. In this case it is naturally expected that the solution of certain problems of idempotent algebra can be obtained by a more simple way and in a more conventional form, in particular, due to the applications of idempotent analogs of notions and results of usual algebra. Consider, for example, the problem on the solution with respect to the unknown vector x the equation A ⊗x ⊕b = x, where A is a certain matrix, b is a vector, ⊕ and ⊗ are the signs of operations of summation and multiplication of algebra. Different approaches to the solution of this equation were happily developed in the work [3–7] and the others. However many of these works consider a general case of idempotent semiring and, therefore, the represented in them results have often too general theoretical nature and are not always convenient for practical application. In a number of works it is mainly considered existence conditions of solution of equations and only some its partial (for example, minimal) solution is suggested in explicit form. In the present work a new method for the solution of linear equations in the case of idempotent semiring with the inverse one by multiplication (a semifield) is suggested which can be used for obtaining the results in compact form convenient for as their realization in the form of computational procedures as a formal analysis. For the proof of certain assertions the approaches, developed in [1, 2, 4], are used. In the work there is given first a short review of certain basic notions of idempotent algebra [2, 4, 5, 8], involving the generalized linear vector spaces and the elements of matrix calculus, and a number of auxiliary

Standard commutator formula

Abstract: Let R be a commutative ring with 1, A, B ⊴ R be its ideals, GL(n, R, A) be the principal congruence subgroup of level A in GL(n, A), and E(n, R, A) be the relative elementary subgroup of level A. We prove the following commutator formula $$ [E(n,R,A),GL(n,R,B)] = [E(n,R,A),E(n,R,B)], $$ which generalizes known results. The proof is yet another variation on the theme of decomposition of unipotents.

Cycles of two-dimensional systems: Computer calculations, proofs, and experiments

Abstract: One of the central problems in studying small cycles in the neighborhood of equilibrium involves computation of Lyapunov’s quantities. While Lyapunov’s first and second quantities were computed in the general form in the 1940s–1950s, Lyapunov’s third quantity was calculated only for certain special cases. In the present work, we present general formulas for calculation of Lyapunov’s third quantity. Together with the classical Lyapunov method for calculation of Lyapunov’s quantities, which is based on passing to the polar coordinates, we suggest a method developed for the Euclidian coordinates and for the time domain. The calculation of Lyapunov’s quantities by two different analytic methods involving modern software tools for symbolic computing enables us to justify the formulas obtained for Lyapunov’s third quantity. For quadratic systems in which Lyapunov’s first and second quantities vanish, while the third one does not, large cycles were calculated. In the calculations, the quadratic system was reduced to the Lienard equation, which was used to evaluate the domain of parameters corresponding to the existence of four cycles (three “small” cycles and a “large” one). This domain extends the region of parameters obtained by S.L. Shi in 1980 for a quadratic system with four limit cycles.

The curvature tensor and the einstein equations for a four-dimensional nonholonomic distribution

Abstract: The space of possible particle velocities is a four-dimensional nonholonomic distribution on a manifold of higher dimension, say, M 4 × ℝ1. This distribution is determined by the 4-potential of the electromagnetic field. The equations of admissible (horizontal) geodesics for this distribution are the same as those of the motion of a charged particle in general relativity theory. On the distribution, a metric tensor with Lorentzian signature (+, −, −, −) is defined, which gives rise to the causal structure, as in general relativity theory. Covariant differentiation (a linear connection) and the curvature tensor for this distribution are introduced. The Einstein equations are obtained from the variational principle for the scalar curvature of the distribution. It is proved that the Dirac operator for the four-dimensional distribution can be extended to functions defined on the manifold M 4 × S 1, where S 1 is the circle. For such functions, electric charges are topologically quantized.