Vol 24, No 125 (2019)
Articles
Enumeration problems associated with Donaghey’s transformation
Abstract
In this paper we consider enumeration problems associated with Donaghey’s transformation. We discuss two groups of questions. The first one is related to the enumeration of fragments of transformation orbits, which are referred to as the “arcs”. The second group of questions is concerned with finding the number of vertices in rotation graphs-a specific family of graphs that is by nature an approximation of Donaghey’s transformation. The basic results of this work are formulated in the form of generating functions and corresponding asymptotics.
Russian Universities Reports. Mathematics. 2019;24(125):5-25
5-25
On the existence of fixed points in completely continuous operators in F-space
Abstract
This work is dedicated to the development of the theory of fixed points of completely continuous operators. We prove existence of new theorems of fixed points of completely continuous operators in F -space (Frechet space). This class of spaces except Banach includes such important space as a countably normed space and Lp0
Russian Universities Reports. Mathematics. 2019;24(125):26-32
26-32
On exact triangle inequalities in (q1; q2) -quasimetric spaces
Abstract
For arbitrary ( q 1 ; q 2) -quasimetric space, it is proved that there exists a function f; such that f -triangle inequality is more exact than any ( q 1 ; q 2) -triangle inequality. It is shown that this function f is the least one in the set of all concave continuous functions g for which g -triangle inequality hold.
Russian Universities Reports. Mathematics. 2019;24(125):33-38
33-38
The set of regularity of a multivalued mapping in a space with a vector-valued metric
Abstract
We consider multivalued mappings acting in spaces with a vector-valued metric. A vector-valued metric is understood as a mapping satisfying the axioms “of an ordinary metric” with values in the cone of a linear normed space. The concept of the regularity set of a multivalued mapping is defined. A set of regularity is used in the study of inclusions in spaces with a vector-valued metric.
Russian Universities Reports. Mathematics. 2019;24(125):39-46
39-46
Construction of a fundamental solution for a one degenerating elliptic equation with a Bessel operator
Abstract
Degenerating elliptic equations containing the Bessel operator are mathematical models of axial and multi-axial symmetry of a wide variety of processes and phenomena of the surrounding world. Difficulties in the study of such equations are associated, inter alia, with the presence of singularities in the coefficients. This article considers a p -dimensional, p≥3 ; degenerating elliptic equation with a negative parameter, in which the Bessel operator acts on one of the variables. A fundamental solution of this equation is constructed and its properties are investigated, in particular, the behavior at infinity and at points of the coordinate planes xp -1 =0 , xp =0 : The results obtained will find application in the construction of solutions of boundary value problems, since on the basis of a fundamental solution, it is possible to choose the potential with which the singular problem is reduced to a regular system of integral equations.
Russian Universities Reports. Mathematics. 2019;24(125):47-59
47-59
Levenberg-Marquardt method for unconstrained optimization
Abstract
We propose and study the Levenberg-Marquardt method globalized by means of linesearch for unconstrained optimization problems with possibly nonisolated solutions. It is well-recognized that this method is an efficient tool for solving systems of nonlinear equations, especially in the presence of singular and even nonisolated solutions. Customary globalization strategies for the Levenberg-Marquardt method rely on linesearch for the squared Euclidean residual of the equation being solved. In case of unconstrained optimization problem, this equation is formed by putting the gradient of the objective function equal to zero, according to the Fermat principle. However, these globalization strategies are not very adequate in the context of optimization problems, as the corresponding algorithms do not have “preferences” for convergence to minimizers, maximizers, or any other stationary points. To that end, in this work we considers a different technique for globalizing convergence of the Levenberg-Marquardt method, employing linesearch for the objective function of the original problem. We demonstrate that the proposed algorithm possesses reasonable global convergence properties, and preserves high convergence rate of the Levenberg-Marquardt method under weak assumptions.
Russian Universities Reports. Mathematics. 2019;24(125):60-74
60-74
Symbolic integration algorithms in CAS MathPartner
Abstract
Risch theorem, published in 1969, gave beginning to creation of procedure library for symbolic integration. But such library, for past almost 50 years, still not been created. Some attempts of creation such libraries is known, but not one of them not finished. In computer algebra system MathPartner a new procedure library for symbolic integration, based on Risch theorem, is creating. We give detailed description of basic procedures contained in this library, and role of each procedure in symbolic integration algorithm. We represent procedural block diagram of whole algorithm and examples of computed integrals.
Russian Universities Reports. Mathematics. 2019;24(125):75-89
75-89
On differential-operator partial differential equations in locally convex spaces
Abstract
Рассматривается дифференциально-операторное уравнение первого порядка в частных производных относительно векторнозначной аналитической вектор-функции двух переменных со значениями в локально-выпуклом пространстве. Актуальность исследования обусловливается сложностью, а порой и невозможностью перенесения существующих методов исследования дифференциально-операторных уравнений в частных производных с нормированных пространств на локально выпуклые пространства. В работе сформулирована и доказана теорема о существовании и единственности решения дифференциально-операторного уравнения первого порядка в частных производных. В этом утверждении существенно используются предложенные и исследованные В.П. Громовым понятия порядка и типа оператора. На основе полученных результатов получены решения двух конкретных дифференциально-операторных уравнений.
Russian Universities Reports. Mathematics. 2019;24(125):90-98
90-98
99-111
112-118
Asymptotic expansion of a solution for one singularly perturbed optimal control problem with a convex integral quality index depends on slow variables and smooth control constraints
Abstract
The paper deals with the problem of optimal control with a convex integral quality index depends on slow variables for a linear steady-state control system with a fast and slow variables in the class of piecewise continuous controls with a smooth control constraints x ε = A 11 x ε + A 12 y ε + B 1 u, εy ε = A 21 x ε + A 22 y ε + B 2 u, J ε u ≔φ x ε T + 0 T u(t) 2 dt→ min, t∈ 0, T , x ε0 = x 0 ,u ≤1, y ε0 = y 0 , where x ε ∈Rn , y ε ∈Rm , u∈Rr ; A ij , B i , i, j =1,2, - are constant matrices of the corresponding dimension, and φ(·) - is the strictly convex and cofinite function that is continuously differentiable in Rn in the sense of convex analysis. In the general case, Pontryagin’s maximum principle is a necessary and sufficient optimum condition for the optimization of a such a problem. The initial vector of the conjugate state l ε is the unique vector, thus determining the optimal control. It is proven that in the case of a finite number of control switching points, the asymptotics of the vector l ε has the character of a power series.
Russian Universities Reports. Mathematics. 2019;24(125):119-136
119-136