Programmirovanie

Annual report on the work of the research seminar on computer algebra.

An extendable essay is a special format of electronic texts that is more convenient for reading than hypertext. To facilitate the creation and editing of extendable essays, an editor program implemented as a web application is proposed. Using this editor, an extendable essay on the Sage computer algebra system is written. Sage seems to be a good choice for the users who are not familiar with computer algebra systems.

This paper considers the problem of constructing compartmental models of dynamic systems by using a software package for symbolic calculation written in Julia. The software package is aimed at unifying the formalized construction of compartmental models, taking into account the meaningful description of possible interactions among compartments and the influence of various factors on the evolution of systems. An approach to the development of the instrumental and methodological basis for modeling the dynamic systems the behavior of which can be described by one-step processes is developed. The proposed software package enables the symbolic representation of the differential equations of the model in both stochastic and deterministic cases. It is implemented in Julia and uses the Julia Symbolics computer algebra library. A comparison between the Julia Symbolics tools and some other computer algebra systems is carried out. The application of the developed software package to a compartmental model is considered. The results can be used to solve problems of constructing and studying dynamic models in natural sciences that are represented by onestep processes.

In their research, the authors actively exploit different branches of geometry. For geometric constructions, computer algebra approaches and systems are used. Currently, we are interested in computer geometry, more specifically, the implementation of computer graphics. The use of the projective space and homogeneous coordinates has actually become a standard in modern computer graphics. In other words, the problem is reduced to the application of analytic projective geometry. The authors failed to find a computer algebra system that could implement projective geometry in its entirety. Therefore, it was decided to partially implement computer algebra for visualization of algebraic relations. For this purpose, the Asymptote system was employed.

A method to calculate partial sums of some multiple numerical series arising when searching for the resultant of a polynomial and an entire function is proposed. One can apply a symbolic algorithm that uses recurrent Newton formulas to find power sums of roots included in this formula without finding the very roots of the system. The algorithm that implements the proposed approach to calculate partial sums of multiple numerical series is implemented in Maple. Examples of using this algorithm to find partial sums of some classes of multiple numerical series are given.

The representation of elements of free non-associative algebras as a set of multidimensional tables of coefficients is defined. An operation for finding partial derivatives for elements of free non-associative algebras in the same form is considered. Using this representation, a criterion of primitivity for elements of lengths 2 and 3 in terms of matrix ranks, as well as a primitivity test for elements of arbitrary length, is derived. This test makes it possible to estimate the number of primitive elements in free non-associative algebras with two generators over a finite field. The proposed representation allows us to optimize algorithms for symbolic computations with primitive elements. Using these algorithms, we find the number of primitive elements of length 4 in a free non-associative algebra of rank 2 over a finite field.

We have found a lower bound on the rank of a square matrix, where every entry in the leading diagonal is neither zero nor one, but every entry outside the leading diagonal is either zero or one. The rank of such a matrix is at least half the order of the matrix. Under an additional condition, the lower bound is one higher. This condition means that some auxiliary system of linear equations has no binary solution. Examples are given showing the achievability of the lower bound. This lower bound on the rank allows us to reduce the problem of finding a binary solution to a system of linear equations, where the number of linearly independent equations is sufficiently large, to a similar problem in a smaller number of variables. Restrictions on the existence of a large set of solutions are found, each of which differs from binary one by the value of one variable. In addition, we discuss the possibility of certifying the absence of a binary solution to a system of a large set of linear algebraic equations. Estimates of the running time for calculating the rank of a matrix with the SymPy computer algebra system are also given. It is shown that the matrix rank over the field of residues modulo a prime number is calculated in less time than is usually required to calculate the rank of a matrix of the same order over the field of rational numbers.

This paper considers theoretical foundations and mathematical methods of data analysis under the conditions of the Rice statistical distribution. The problem involves joint estimation of the signal and noise parameters. It is shown that this estimation requires the solution of a complex system of essentially nonlinear equations with two unknown variables, which implies significant computational costs. This study is aimed at mathematical optimization of computer algebra methods for numerical solution of the problem of Rician data analysis. As a result of the optimization, the solution of the system of two nonlinear equations is reduced to the solution of one equation with one unknown variable, which significantly simplifies algorithms for the numerical solution of the problem, reduces the amount of necessary computational resources, and enables the use of advanced methods for parameter estimation in information systems with priority of real-time operation. Results of numerical experiments carried out using Wolfram Mathematica confirm the effectiveness of the developed methods for two-parameter analysis of Rician data. The data analysis methods considered in this paper are useful for solving many scientific and applied problems that involve analysis of data described by the Rice statistical model.