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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. [ABSTRACT FROM AUTHOR]

In this paper, we consider a constructive proof of the termination of the normal form (NF) algorithm for multivariate polynomials, as well as the related concept of admissible ordering < on monomials. In classical mathematics, the well-quasiorder property of relation < is derived from Dickson's lemma, and this is sufficient to justify the termination of the NF algorithm. In provable programming based on constructive type theory (Coq and Agda), a somewhat stronger condition (in constructive mathematics) of the well-foundedness of the ordering (in its constructive version) is required. We propose a constructive proof of this theorem (T) for < , which is based on a known method that we refer to here as the "pattern method." This theorem on the well-foundedness of an arbitrary admissible ordering is also important in itself, independently of the NF algorithm. We are not aware of any other works on constructive proof of this theorem. However, it turns out that it follows, not very difficultly, from the results achieved by other researchers in 2003. We program this proof in the Agda language in the form of our library AdmissiblePPO-wellFounded of provable computational algebra programs. This development also uses the theorem to prove termination of the NF algorithm for polynomials. Thus, the library also contains a set of provable programs for polynomial algebra, which is significantly larger than that needed to prove Theorem T. [ABSTRACT FROM AUTHOR]

For describing specialized mathematical structures, it is preferable to use a special formalism rather than a general one. However, tradition often prevails in this case. For example, to describe rotations in the three-dimensional space or to describe motion in the Galilean or Minkowski spaces, vector (or tensor) formalism, rather than more specialized formalisms of Clifford algebra representations, is often used. This approach is historically justified. The application of specialized formalisms, such as spinors or quaternions, has not become a mainstream in science, but it has taken its place in solving practical and engineering problems. It should also be noted that all operations in theoretical problems are carried out precisely with symbolic data, and manipulations with multidimensional geometric objects require a large number of operations with the same objects. And it is in such problems that computer algebra is most powerful. In this paper, we want to draw attention to one of these specialized formalisms—the formalism of geometric algebra. Namely, it is proposed to consider options for the implementation of geometric algebra in the framework of the symbolic computation paradigm. [ABSTRACT FROM AUTHOR]

In this paper, we propose an algorithm for solving the Cauchy problem for a two-dimensional difference equation with constant coefficients at a point, based on its coefficients and the initial data for the Cauchy problem, which are defined in a "strip." For this purpose, computer algebra methods are employed. To automate the process of computing the solution, the algorithm is implemented in MATLAB, with the input data being the matrix of coefficients of the two-dimensional polynomial difference equation, the coordinates of the initial data matrix (which defines the structure of the difference equation), and the coordinates of the point that determines the dimension of the initial data matrix. The output of the algorithm is a solution of the Cauchy problem (with the initial data defined in a "strip") for the two-dimensional difference equation, which is a function value at the desired point. [ABSTRACT FROM AUTHOR]

Geometric algebra is based on the works by Grassmann and Clifford. Its main objects of research are polyvectors (p-vectors) and multivectors. Polyvectors, together with the exterior product, implement the Grassmann algebra, while multivectors with the geometric product implement the Clifford algebra. Multivector algebra generalizes many operations and objects of analytic geometry and differential geometry (e.g., vector and mixed products, normal vectors and binormals, etc.) to the multidimensional case, as well as provides their geometric interpretation. Complex numbers and quaternions are isomorphic to multivectors of a special kind. This paper applies certain ideas of geometric algebra to solve problems that occur in computer geometry. For this purpose, the Grassmann.jl package for Julia is used. [ABSTRACT FROM AUTHOR]

The integrability of a two-dimensional autonomous polynomial system of ordinary differential equations (ODEs) with a degenerate singular point at the origin that depends on six parameters is investigated. The integrability condition for the first quasihomogeneous approximation allows one of these parameters to be fixed on a countable set of values. The further analysis is carried out for this value and five free parameters. Using the power geometry method, the system is reduced to a non-degenerate form through the blowup process. Then, the necessary conditions for its local integrability are calculated using the method of normal forms. In other words, the conditions for the parameters under which the original system is locally integrable near the degenerate stationary point are found. By resolving these conditions, we find seven two-parameter families in the five-dimensional parametric space. For parameter values from these families, the first integrals of the system are found. The cumbersome calculations that occur in the problem under consideration are carried out using computer algebra. [ABSTRACT FROM AUTHOR]

Calculations of optical devices in the geometrized Maxwell's theory use well-known formalisms of general theory of relativity and differential geometry. In particular, for such calculations it is required to know the analytical form of the geodesic equations, which leads to the need to calculate a large number of monotonous mathematical expressions. One of the purposes of computer algebra is to facilitate the researcher's work by automating cumbersome symbolic computations. Thus, the use of computer algebra systems seems to be quite an obvious way. Several free implementations of symbolic computations for the apparatus of general relativity are considered. A practical example of symbolic computations for the geometrized Maxwell's theory is given. [ABSTRACT FROM AUTHOR]

Based on the Bochner–Martinelli integral representation, an algorithm for determining the number of zeros of a system of holomorphic functions is developed. We search for zeros of the system in a polycube. The use of computer algebra methods in this problem is due to the computational complexity of the developed algorithms and final results. The algorithm is implemented in Maple, which makes it possible to significantly facilitate the necessary computations. [ABSTRACT FROM AUTHOR]

An algorithm for solving the Cauchy problem for a three-dimensional difference equation with constant coefficients by given coefficients of the difference equation and initial data in a parallelepiped. The algorithm is based on the Apanovich and Leirnartas theorems characterizing well-posedness of Cauchy problems and uses computer algebra methods to achieve computational efficiency.The algorithm is implemented in Matlab. [ABSTRACT FROM AUTHOR]

The influence of the variability of the masses of celestial bodies on the dynamic evolution of planetary systems is investigated in the case when the masses of the bodies change isotropically at different rates, and the laws of mass change are assumed to be arbitrary given functions of time. The classical problem of n + 1 bodies of variable mass when bodies move around a central star in quasi-elliptic nonintersecting orbits and interact according to the law of universal gravitation is used as a model of multi-planet system. Differential equations of body motion in terms of the osculating elements of aperiodic motion in quasi-conic sections are derived. An algorithm for calculating the perturbing function in the form of power series in small parameters and the derivation of differential equations determining the secular perturbations of the orbital elements are discussed. All symbolic computations are carried out using the computer algebra system Wolfram Mathematica. [ABSTRACT FROM AUTHOR]

A method for calculating the location of all types of level lines of a real polynomial on the real plane is proposed. To this end, the critical points and critical curves of the polynomial and then its critical values (there are a finite number of them) should be calculated. Using this data, all critical level lines and one representative for each noncritical level line corresponding to intervals of values between adjacent critical level lines are found. A scheme for calculating level lines based on algorithms of polynomial computer algebra—Gröbner bases and primary ideal decomposition—is proposed. Software for implementing these calculations is indicated. Nontrivial examples are discussed. [ABSTRACT FROM AUTHOR]