Integer triangles, Pell’s equation and Chebyshev polynomials

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.

On the existence of fixed points in completely continuous operators in F -space

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 Lp(0 < p < 1), lp(0 < p < 1).

A class of strongly stable approximation for unbounded operators

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.

Decay of the solutions of the generalized Korteweg–de Vries equation at large times

In this paper the existence of weak solutions of the nonlinear generalized KdV equation is shown and conditions for which weak solutions decay to zero at large times are obtained.

Symbolic integration algorithms in CAS MathPartner

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.

Enumeration problems associated with Donaghey’s transformation

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.

About the general solution of a linear homogeneous differential equation in a Banach space in the case of complex characteristic operators

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

About a complex operator exponential function of a complex operator argument main property

Operator functions e^A, sin B, cos B of the operator argument from the Banach algebra of bounded linear operators acting from E to E are considered in the Banach space E . For trigonometric operator functions sin B, cos B, formulas for the sine and cosine of the sum of the arguments are derived that are similar to the scalar case. In the proof of these formulas, the composition of ranges with operator terms in the form of Cauchy is used. The basic operator trigonometric identity is given. For a complex operator exponential function e^Z of an operator argument Z from the Banach algebra of complex operators, using the formulas for the cosine and sine of the sum, the main property of the exponential function is proved. Operator functions e^At , sin Bt, cos Bt, e^Zt of a real argument t∈(-∞;∞) are considered. The facts stated for the operator functions of the operator argument are transferred to these functions. In particular, the group property of the operator exponent e^Zt is given. The rule of differentiation of the function e^Zt is indicated. It is noted that the operator functions of the real argument t listed above are used in constructing a general solution of a linear n th order differential equation with constant bounded operator coefficients in a Banach space.

Bergman–Hartogs domains and their automorphisms

For Cartan–Hartogs domains and also for Bergman–Hartogs domains, the determination of their automorphism groups is given for the cases when the base is any bounded symmetric domain and a general bounded homogeneous domain respectively.

The harmonic balance method for finding approximate periodic solutions of the Lorenz system

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.