Vol 24, No 126 (2019)
Articles
Existence of inverse function in a neighbourhood of a critical value
Abstract
The classical inverse function theorems guarantee the existence of an inverse function in a neighborhood of the value of a given point if the regularity condition is satisfied at this point, that is, the first derivative at a given point is nondegenerate. A more general condition for the existence of an implicit function is the 2-regularity condition. It holds, for example, for many quadratic mappings at zero. It is known that under natural smoothness assumptions, the existence of a continuous inverse function follows from a 2-regularity of a map at a point in a certain direction. In this paper, it is shown that, in the known statements guaranteeing the existence of an inverse function when the 2-regularity condition is satisfied, we can weaken the smoothness assumptions. However, the inverse function may not be continuous.
Russian Universities Reports. Mathematics. 2019;24(126):141-149
141-149
Hybrid globalization of convergence of subspace-stabilized sequential quadratic programming method
Abstract
Local superlinear convergence of the stabilized sequential quadratic programming method is established under very weak assumptions not involving any constraint qualification conditions. However, all attempts to globalize convergence of this method inevitably face principal difficulties related to the behavior of this method when the iterates are still relatively far from solutions. Specifically, the stabilized sequential quadratic programming method has a tendency to generate long sequences of short steps before its superlinear convergence shows up. To that end, the so-called subspace-stabilized sequential quadratic programming method has been proposed, demonstrating better “semi-local” behavior, and hence, more suitable for development of practical algorithms on its basis. In this work we propose two techniques for hybrid globalization of convergence of this method: algorithm with backups, and algorithm with records.We provide theoretical results on convergence and rate of convergence of these algorithms, as well as some results of their numerical testing.
Russian Universities Reports. Mathematics. 2019;24(126):150-165
150-165
Procedural interpretation of symbolic integration algorithms in MathPartner system
Abstract
The work is devoted to the development of a procedure library for the computer algebra system MathPartner. A software implementation of symbolic integration algorithms is being developed. The solution of the problem of symbolic integration is divided into three stages. At the first stage, the integrand is reduced to the form necessary for applying the Rish algorithm. A description is given of the corresponding procedures that reduce the integrand to an expression containing a finite set of arithmetic operations and compositions of logarithmic functions and exponentials, and also make a set of regular monomials. At the second stage, the integration of the fractional part of the integrand is performed. A description is given of the procedures that reduce the fractional part to the form required for the application of the integration algorithm. At the third stage, the polynomial part of the integrand is integrated. Procedures are obtained that allow, depending on the type of the integrand, to apply the appropriate integration algorithms. The appendix contains a description of the user’s language commands of the MathPartner system, which are designed to calculate integrals in symbolic form.
Russian Universities Reports. Mathematics. 2019;24(126):166-178
166-178
Integer triangles, Pell’s equation and Chebyshev polynomials
Abstract
In this paper we consider some types of integer triangles: “almost equilateral”, rectangular “almost isosceles”, rectangular “whose angle is almost 300”. The description is reduced to Pell’s equation. We state the theory of Pell’s equation on the basis of an “iterated matrix”. Powers of this matrix are expressed in terms of Chebyshev polynomials.
Russian Universities Reports. Mathematics. 2019;24(126):179-186
179-186
The harmonic balance method for finding approximate periodic solutions of the Lorenz system
Abstract
We consider the harmonic balance method for finding approximate periodic solutions of the Lorenz system. When developing software that implements the described method, the math package Maxima was chosen. The drawbacks of symbolic calculations for obtaining a system of nonlinear algebraic equations with respect to the cyclic frequency, free terms and amplitudes of the harmonics, that make up the desired solution, are shown. To speed up the calculations, this system was obtained in a general form for the first time. The results of the computational experiment are given: the coefficients of trigonometric polynomials approximating the found periodic solution, the initial condition, and the cycle period. The results obtained were verified using a high-precision method of numerical integration based on the power series method and described earlier in the articles of the authors.
Russian Universities Reports. Mathematics. 2019;24(126):187-203
187-203
Projective congruent symmetric matrices enumeration
Abstract
Projective spaces over local ring R = 2R with principal maximal ideal J; 1+J ⊆ R*2 have been investigated. Quadratic forms and corresponding symmetric matrices A and B are projectively congruent if kA = UBU T for a matrix U ∈ GL(n;R) and for some k ∈ R * : In the case of k = 1 quadratic forms (corresponding symmetric matrices) are called congruent. The problem of enumerating congruent and projective congruent quadratic forms is based on the identification of the (unique) normal form of the corresponding symmetric matrices and is related to the theory of quadratic form schemes. Over the local ring R on conditions R * =R *2 ={1;-1; p;-p} and D(1; 1)=D(1; p)={1; p}; D(1;-1)=D(1;-p)={1;-1; p;-p} (unique) normal form of congruent symmetric matrices over ring R is detected. Quantities of congruent and projective congruent symmetric matrix classes is found when maximal ideal is nilpotent.
Russian Universities Reports. Mathematics. 2019;24(126):204-210
204-210
About the general solution of a linear homogeneous differential equation in a Banach space in the case of complex characteristic operators
Abstract
A linear inhomogeneous differential equation (LIDE) of the n th order with constant bounded operator coefficients is studied in Banach space. Finding a general solution of LIDE is reduced to the construction of a general solution to the corresponding linear homogeneous differential equation (LHDE). Characteristic operator equation for LHDE is considered in the Banach algebra of complex operators. In the general case, when both real and complex operator roots are among the roots of the characteristic operator equation, the n -parametric family of solutions to LHDE is indicated. Operator functions eAt ; sinBt ; cosBt of real argument t ∈ [0;∞) are used when building this family. The conditions under which this family of solutions form a general solution to LHDE are clarified. In the case when the characteristic operator equation has simple real operator roots and simple pure imaginary operator roots, a specific form of such conditions is indicated. In particular, these roots must commute with LHDE operator coefficients. In addition, they must commute with each other. In proving the corresponding assertion, the Cramer operator-vector rule for solving systems of linear vector equations in a Banach space is applied
Russian Universities Reports. Mathematics. 2019;24(126):211-217
211-217
A class of strongly stable approximation for unbounded operators
Abstract
We derive new sufficient conditions to solve the spectral pollution problem by using the generalized spectrum method. This problem arises in the spectral approximation when the approximate matrix may possess eigenvalues which are unrelated to any spectral properties of the original unbounded operator. We develop the theoretical background of the generalized spectrum method as well as illustrate its effectiveness with the spectral pollution. As a numerical application, we will treat the Schr¨odinger’s operator where the discretization process based upon the Kantorovich’s projection.
Russian Universities Reports. Mathematics. 2019;24(126):218-234
218-234