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No 2 (2025)
COMPUTER ALGEBRA
3-5
STATISTICS OF THE DISTRIBUTION OF FAMILIES OF PERIODIC SOLUTIONS TO HILL’S PROBLEM
Abstract
All generating solutions of families of periodic orbits of the planar circular Hill problem of the second type can be described in terms of limiting arc solutions of the integrable Henon problem. Each generating solution is a finite sequence composed, according to certain rules, of a countable set of arcs of two types connected at the origin by a hyperbolic conic. Each generating solution determines the symmetry type, the global orbit multiplicity, and other characteristics of the corresponding periodic solutions to the generated family. The symbolic dynamics on a finite subset of arc solutions, which is used to calculate the statistics of the distribution of generated families by symmetry types is studied. For this purpose, a class hierarchy is implemented using the Python ecosystem, and simulation for three sets of arcs is carried out performed.
Programmirovanie. 2025;(2):6-19
6-19
ON INTEGRABILITY OF TWO- AND THREE-DIMENSIONAL DYNAMICAL SYSTEMS WITH A QUADRATIC RIGHT-HAND SIDE
Abstract
A heuristic method that allows us to determine in advance the cases of integrability of autonomous dynamical systems with a polynomial right-hand side is used. The capabilities of this method are demonstrated using examples of two- and three-dimensional dynamical systems with a quadratic nonlinearity. A significant achievement compared to previous works is the ability to study systems of a general type without resonances in the linear parts, which is achieved by generalizing the results of resonance cases. Thus, it becomes possible to use the obtained results when working with dynamical models of real systems.
Programmirovanie. 2025;(2):20-26
20-26
PROJECTIVE GEOMETRIC ALGEBRA IN PLANE AND ITS IMPLEMENTATION IN THE LIBRARY GANJA.JS
Abstract
Geometric algebra is currently considered as a universal mathematical apparatus of computer graphics. Active research, both academic and applied, is being conducted in this area. Due to the applied nature of the research, many results are immediately implemented in the form of computer codes and libraries. One of such libraries is Ganja.js. The aim of this paper is to review some capabilities of Ganja.js using the example of projective geometric algebra 𝒞2,0,1(R) in its dual version. The paperuses the apparatus of linear algebra, elements of projective geometry, and geometric algebra (Clifford and Grassmann algebras). The software used is JavaScript. Ganja.js implements a mathematical syntax that allows you to define various Clifford algebras, manipulate their elements using algebraic operations, and visualize algebra elements as geometric objects. The created visualizations can be interactive and animated. Even though JavaScript is a completely non-standard language for academic mathematical research, Ganja.js can be a useful tool for computation, visualization, and research in geometric algebra.
Programmirovanie. 2025;(2):27-42
27-42
ON THE CALCULATION OF THE NUMBER OF REAL ROOTS OF A SYSTEM OF NONALGEBRAIC EQUATIONS USING COMPUTER ALGEBRA
Abstract
Systems of nonalgebraic equations containing entire functions of several complex variables are considered. The number of real zeros of such systems is studied using computer algebra methods. For this purpose, a computer implementation of Newton’s recurrences and formulas for the resultant of the functions under study in Maple is proposed. The relevance of this problem is due to the fact that in applied problems, e.g., in the equations of chemical kinetics, it is necessary to determine the number of stationary states of the system.
Programmirovanie. 2025;(2):43-48
43-48
49-54
STRONGLY CYCLIC VECTORS
Abstract
Systems of form y(x)′ = A(x)y(x) are considered, with matrix elements being initial segments of unknown infinite power series. The concept of a cyclic vector is generalized to the case of these systems by introducing the concept of a strongly cyclic vector. A method for checking the strong cyclicity of a vector is discussed. A sufficient condition that allows one to check the strong cyclicity of a vector by the form of the matrix of derivatives with respect to the system is derived.
Programmirovanie. 2025;(2):55-63
55-63
64-72
ON BINARY SOLUTIONS TO A SYSTEM OF LINEAR EQUATIONS MODULO THREE
Abstract
We consider the problem of finding a binary solution to a system of linear equations modulo three. In case the number of equations is less than a sufficiently slowly growing function of the number of variables, a new polynomial-time algorithm is proposed to recognize the existence of a binary solution to such a system. The algorithm is based on the note that if the coefficient matrix contains non-zero columns proportional to each other, then the elimination of the corresponding variables preserves the property of having no binary solution to the system. In particular, every system of two equations in five variables allows the elimination of some variables that preserves the property of having no binary solution to the system. Based on these results, we propose an errorless heuristic algorithm, which is implemented using the Python programming language. The NumPy library is used to represent matrices and perform basic operations. The input is the augmented matrix. An empirical running time estimate has been calculated using the implementation. It has been experimentally shown that the algorithm is more efficient for sparse systems of equations. Obviously, the binary search method allows finding a binary solution to the system when one exists. This observation opens up the possibility of practical use, in particular, for solving problems of mathematical biology.
Programmirovanie. 2025;(2):73-82
73-82
FINITE DECIMAL FRACTIONS AS ENTRIES OF NONSINGULAR MATRICES
Abstract
How can one check, for a given nonsingular real number matrix the entries of which have only a finite number of decimal digits, whether this matrix will remain nonsingular after some decimal digits are arbitrarily added to some (explicitly specified in advance) of its entries? It turns out that this problem is algorithmically solvable. A computer implementation of the proposed algorithmic solution is discussed.
Programmirovanie. 2025;(2):83-90
83-90