# Alexander N. ProkopenyaWarsaw University of Life Sciences - SGGW | SGGW · Department of Applied Informatics

Alexander N. Prokopenya

DSc

## About

78

Publications

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342

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Introduction

Additional affiliations

May 2014 - June 2014

September 2011 - present

June 2008 - present

## Publications

Publications (78)

This paper covers automated settlement of receivables in non-governmental organizations. We tackle the problem with entity matching techniques. We consider setup, where base algorithm is used for preliminary ranking of matches, then we apply several novel methods to increase matching quality of base algorithm: score post processing, cascade model a...

The goal of the study is to determine the differentiation of production types of agricultural holdings in Poland based on the dynamics of changes, spatial differentiation and processes of convergence/divergence in comparison to EU countries. Analyzed materials consist of the data of holdings monitored by the Farm Accountancy Data Network (FADN) sys...

The translational-rotational motion of a non-stationary triaxial body with constant dynamic shape in a non-stationary Newtonian central gravitational field is considered. Differential equations determining translational motion of the triaxial body around a spherical body and its rotation about the center of mass are obtained in terms of the osculat...

The classical problem of three bodies of variable masses is considered in the case when two of the bodies are protoplanets and all the masses vary non-isotropically at different rates. The problem is analyzed in the framework of the planetary perturbation theory in terms of the osculating elements of aperiodic motion on quasi-conic sections. An alg...

The translational-rotational motion of a triaxial body with constant dynamic shape and variable size and mass in a non-stationary Newtonian central gravitational field is investigated. Differential equations of motion of the triaxial body in the relative coordinate system with the origin at the center of a non-stationary spherical body are obtained...

KETCindy is a plug-in for Cinderella2, a Dynamic Geometry Software (DGS), originally developed as a kind of pre-processor for LATEX-based graphical code systems. It is particularly useful for teachers wanting to produce printed materials containing technical drawings, which are to be distributed to students in science courses. KETCindy has enhanced...

A generalized model of the Atwood machine when two bodies can swing in a plane is considered. Combining symbolic and numerical calculations, we have obtained equations of motion of the system and analyzed their solutions. We have shown that oscillations can completely modify motion of the system while the simple Atwood machine demonstrates only the...

The classical two-planet problem of three bodies of variable masses is studied in the general case when the body masses vary anisotropically at different rates. Differential equations of motion in terms of osculating elements of aperiodic motion along quasi-conic sections are derived. An algorithm for computing the perturbation function in the form...

This paper addresses the problem of reconstructing the shape of an unknown Lambertian surface given in the 3D space by a continuously differentiable function \(z = u(x,y)\). The surface is reconstructed from its photometric images obtained by its successive illumination with three different remote light sources. Using computer algebra methods, we s...

This work deals with some properties of synthetic measures designed to differentiate objects in a multidimensional analysis. The aggregate synthetic measures are discussed here to rank the objects including those validating the concentration spread. The paper shows that currently used various measures (based either on a single or a multiple model o...

This paper discusses the special case of reconstructing the unknown Lambertian surface from two-image photometric stereo. Both images are assumed here to be formed by a genuine second-order algebraic surface. The corresponding uniqueness issue is discussed for different pairs of image irradiance equations under various illumination settings. Illust...

We discuss here the problem of solving the system of two nonlinear algebraic equations determining the relative equilibrium positions in the planar circular restricted four-body problem formulated on the basis of the Euler collinear solution of the three-body problem. The system contains two parameters $\mu_1$, $\mu_2$ and all its solutions coincid...

This paper addresses the problem of reconstructing the shape of an unknown Lambertian surface from its two photometric images obtained by successive illumination of the surface with two different remote light sources. Using computer algebra methods, we investigate the conditions of existence and uniqueness of a solution to a system of algebraic equ...

In this paper we consider a general case of the three-body problem with variable masses that change anisotropically at different rates. Due to the change of masses reactive forces appear which significantly complicate the problem. Equations of motion of the system have been derived in Jacobi coordinates for the first time. Using these equations of...

A generalized model of the Atwood machine when one body is constrained to move along a vertical axis while the other one can swing in a plane is considered. Combining symbolic and numerical calculations, we have obtained equations of motion of the system and analyzed their solutions. We have shown that oscillation can completely modify a motion of...

An Atwood machine is a well-known device that consists of two bodies of different masses m 1 , m 2 attached to opposite ends of a massless inextensible thread wound round a massless frictionless pulley (see Ref. [1]). It is assumed that each body can move only along a vertical, and the thread doesn't slip on the pulley. Such Atwood's machine is a s...

The classical restricted three-body problem for bodies with variable masses is discussed for the case when the masses of two bodies vary isotropically at different rates and their sum varies according to the joint Meshchersky law. Within the perturbation theory in Hills’s approximation, differential equations determining the secular perturbations o...

We study the stability of relative equilibrium positions in the planar circular restricted four-body problem formulated on the basis of the Euler collinear solution of the three-body problem. The stability problem is solved in a strict nonlinear formulation in the framework of the KAM theory. We obtained algebraic equations determining the equilibr...

The Applications of Computer Algebra (ACA) conference series is devoted to promoting all kinds of computer algebra applications, and encouraging the interaction of developers of computer algebra systems and packages with researchers and users (including scientists, engineers, educators, and mathematicians). Topics include, but are not limited to, c...

This paper discusses the ambiguous shape recovery in two-image photometric stereo for a Lambertian surface. The current uniqueness analysis refers to linearly independent light-source directions p = (0, 0, −1) and q arbitrary. For this case necessary and sufficient condition determining ambiguous reconstruction is governed by a second-order linear...

The problem of reconstructing a Lambertian surface from its two photometric stereo images is discussed. Previously, the solution to this problem was only obtained for a special choice of two light source directions. In this paper, using the computer algebra system Mathematica, the necessary and sufficient conditions for the unique reconstruction of...

In this paper we investigate the case of ambiguous shape reconstruction from two light-source photometric stereo based on illuminating the unknown Lambertian surface. So-far this problem is merely well-understood for two linearly independent light-source directions with one illumination assumed as overhead. As already established, a necessary and s...

We consider Grover's algorithm of quantum search for one or several integers out of N = 2^n, where n is a number of quantum bits in the memory register. There is a black-box or subroutine containing information about hidden integers and it can easily recognize these integers but we do not know which ones out of N they are. To find the hidden items...

A quantum Fourier transform and its application to a quantum algorithm for phase estimation is discussed. It has been shown that the approximate quantum Fourier transform can be successfully used for the phase estimation instead of the full one. The lower bound for the probability to get a correct result in a single run of the algorithm has been ob...

A quantum algorithm for the computation the order of an integer, which uses the quantum Fourier transform, is discussed. The cases of the exact and approximate Fourier transform are considered, and estimates of the probability of the successful solution of the problem that significantly improve the available results are obtained. The quantum algori...

In this work we consider the satellite version of the restricted three-body problem when masses of the primary bodies P_0, P_1 vary isotropically with different rates, and their total mass changes according to the joint Meshcherskii law. Equations of motion of the body P_2 of infinitesimal mass are obtained in terms of the osculating elements of th...

A quantum algorithm for estimating the phase, which determines the eigenvalue of a unitary operator, is discussed. It is assumed that the eigenvector of this operator and the corresponding quantum circuit are given. The memory register where the approximate phase value is stored consists of n qubits, which makes it possible to determine the phase a...

The Mathematica package “QuantumCircuit” for simulation of quantum computation based on the circuit model is described. The package provides a user-friendly interface to specify a quantum circuit, to draw it, and to construct the corresponding unitary matrix for quantum computation defined by the circuit. This matrix enables to compute the final st...

The satellite version of the restricted three-body problem formulated on the basis of classical Gylden–Meshcherskii problem is considered. Motion of the point P
2 of infinitesimal mass about the point P
0 is described in the first approximation in terms of the osculating elements of the aperiodic quasi-conical motion, and an influence of the point...

The classical problem of three bodies with variable masses is considered in the case when the masses of all three bodies vary isotropically. Solutions to the equation of motion in terms of the osculating elements of the aperiodic quasi�conical motion and the secular perturbations of the orbital elements of the system are examined. An algorithm for...

In this paper we consider the problem of quantum error correction and its simulation with the computer algebra system Mathematica. Basic ideas of constructing the quantum error correcting codes are discussed, and some examples of error correction by means of quantum circuits constructed with application of the Mathematica package QuantumCircuit are...

The problem of equilibrium solutions stability in the spatial Newtonian circular restricted four-body problem formulated on the basis of Lagrange's triangular solution is considered. Using the computer algebra system Mathematica, we have constructed Birkhoff's type canonical transformation, reducing the Hamiltonian function to the normal form up to...

Algorithms for searching equilibrium solutions of the circular restricted four�body problem for�mulated on the basis of the triangular Lagrange solutions of the three�body problem are discussed. An algo�rithm is proposed for calculating the bifurcation curve in the plane of system parameters that separates domains of the eight and ten equilibrium s...

We study stability of equilibrium positions in the spatial circular restricted four-body problem formulated on the basis of Lagrange’s triangular solution of the three-body problem. Using the computer algebra system Mathematica, we have constructed Birkhoff’s type canonical transformation, reducing the Hamiltonian function to the normal form up to...

In the paper, the problem of simulation of quantum error correction by means of error correcting codes is discussed. Examples of error correction by means of quantum circuits constructed with the help of the QuantumCircuit package written in the language of the computer algebra system Mathematica are presented.

The problem of equilibrium solutions stability in the spatial Newtonian circular restricted four-body problem formulated on the basis of Lagrange's triangular solution is considered. Using the computer algebra system Mathematica, we have constructed Birkhoff's type canonical transformation, reducing the Hamiltonian function to the normal form up to...

A symbolic algorithm for construction of a real canonical transformation that reduces the Hamiltonian determining motion of an autonomous two-degree-of-freedom system in a neighborhood of an equilibrium state to the normal form is discussed. The application of the algorithm to the restricted planar circular three-body problem is demonstrated. The e...

We consider the stability of equilibrium positions in the planar circular restricted four-body problem formulated on the basis of Lagrange's triangular solution of the three-body problem. The stability problem is solved in a strict nonlinear formulation on the basis of Arnold-Moser and Markeev theorems. Peculiar properties of the Hamiltonian normal...

We consider an application of the Mathematica package QuantumCircuit to simulation of quantum circuits implementing two of the best known quantum algorithms, namely, the Grover search algorithm and the Shor algorithm for order finding. The algorithms are discussed in detail and concrete examples of their application are demonstrated. The main featu...

Algorithms for calculating unitary matrices determined by quantum circuits are discussed. The algorithms are used in the program
QuantumCircuit designed for modeling quantum circuits. Practical implementations of the algorithms as functions written in
the language of the Mathematica system are suggested.

Algorithms for searching equilibrium solutions of the circular restricted four-body problem formulated on the basis of triangular
Lagrange solutions of the three-body problem are discussed. For small values of one of the two system parameters, equilibrium
solutions are found in the form of power series. For large values of this parameter, an algori...

In this paper we briefly describe a Mathematica package for simulation of quantum circuits and illustrate some of its features by simple examples. Unlike other Mathematica-based quantum simulators, our program provides a user-friendly graphical interface for generating quantum circuits and computing the circuit unitary matrices. It can be used for...

In this paper we briefly describe a Mathematica program for simulation of quantum circuits and illustrate some of its facilities by simple examples. Unlike other Mathematica-based quantum simulators, our program provides a user-friendly graphical interface for generating quantum circuits and computing
the circuit unitary matrices. In addition to st...

1) Объединенный институт ядерных исследований 141980 Дубна, Россия gerdt@jinr.ru 2) Университет прикладных наук D-88241 Вайнгартен, Германия kragler@hs-weingarten.de 3) Брестский государственный технический университет Московская 267, 224017 Брест, Белоруссия prokopenya@brest.by Аннотация In this paper we briefly describe a Mathematica program for...

The Newtonian circular restricted four-body problem is considered. We obtain nonlinear algebraic equations determining equilibrium
solutions in the rotating frame and find six possible equilibrium configurations of the system. Studying the stability of
equilibrium solutions, we prove that the radial equilibrium solutions are unstable, while the bis...

The problem of studying the stability of equilibrium solution of the second order non-autonomous Hamiltonian system, containing
a small parameter, is considered. The main steps in solving this problem and application of the computer algebra systems for
doing necessary calculations are discussed. As an example, we analyze stability of some equilibri...

An algorithm for symbolic computation of characteristic exponents of a linear system of differential equations with a periodic matrix represented by a power series in terms of a small parameter is discussed. The algorithm is based on the infinite determinant method. The corresponding procedures implemented in the Mathematica system and computation...

In the given paper we present the first version of our Mathemat-ica package for simulation of quantum circuits. It provides a user-friendly graphical interface to specify a quantum circuit, to draw it and to construct a unitary matrix for quantum computation defined by the circuit. The matrix is computed by means of the linear algebra tools built-i...

In the present paper which is an extended version of paper [1] we consider a Mathematica-based package for simulation of quantum circuits. It provides a user-friendly graphical interface to specify a quantum circuit, to draw it, and to construct the unitary matrix for quantum computation defined by the circuit. The matrix is computed by means of th...

In the present paper we briefly describe a Mathematica package which allows users to specify a quantum circuit, to draw it, and to construct the unitary matrix for quantum computation defined by the circuit. For circuits composed from the Toffoli and Hadamard gates the package can also output the corresponding multivariate system over F_2 whose num...

An algorithm is proposed for analytical computing the stability boundaries of the Lagrange triangular solutions in the elliptic restricted three‐body problem. It is based on the infinite determinant method. The algorithm has been implemented by using the computer algebra system Mathematica and the stability boundaries have been determined in the fo...

An algorithm for computing fundamental solutions to a linear system of differential equations with a periodic matrix represented
by a power series in terms of a small parameter is discussed. An algorithm based on the infinite determinant method for determining
boundaries between regions of stability and instability for such a system in the paramete...

An algorithm is proposed for analytical computing the stability boundaries of the Lagrange triangular solutions in the elliptic restricted three-body problem. It is based on the infinite determinant method. The algorithm has been implemented by using the computer algebra system Mathematica and the stability boundaries have been determined in the fo...

Stability of equilibrium solutions in the elliptic restricted many-body problem of Sitnikov's kind is studied. Equations of the disturbed motion are obtained in the form of the Hamiltonian system of differential equations with periodic coefficients. We have found the domains of instability of the system in the parameter space and shown that it is s...

The stability of cylindrical precession of the dynamically symmetric satellite in the Newtonian gravitational field is studied.
We consider the case when a center of mass of the satellite moves in an elliptic orbit, while the satellite rotates uniformly
about the axis of its dynamical symmetry that is perpendicular to the orbit plane. In the case o...

We consider the hamiltonian system of linear differential equations with periodic coefficients. Using the infinite determinant method based on the existence of periodic solutions on the boundaries between the domains of stability and instability in the parameter space we have developed the algorithm for analytical computation of the stability bound...

We consider the hamiltonian system of linear differential equations with periodic coefficients. Using the infinite determinant method based on the existence of periodic solutions on the boundaries between the domains of stability and instability in the parameter space we have developed the algorithm for analytical computation of the stability bound...

We consider the Hill equation with damping describing the parametric oscillations of a torsional pendulum excited by varying the moment of inertia of the rotating body. Using the method of a small parameter, we analytically calculate a fundamental system of solutions of this equation in the form of power series in the excitation amplitude with accu...

In this paper the stability of a new class of exact symmetrical solutions in the Newtonian gravitational (n + 1) -body problem is studied. This class of solution follows from a suitable geometric distribution of the (n+1) -bodies, and initial conditions, so that the solution is represented geometrically by an oscillating regular polygon with n side...

The stability problem for the Hill equation containing two parameters is analyzed using the Mathematica computer algebra system. The characteristic constant is found as a series expansion in powers of a small parameter e. It is shown that the domains of instability are located only between the curves a = a(e) on the a-e plane crossing the axis e =...

In the one-loop approximation we calculate the electronic contribution to the elastic scattering amplitude of a Z-boson in a constant homogeneous magnetic field. Using this amplitude we obtain the probability of the decay of a Z-boson into an e+e- pair, and we investigate its dependence on the boson energy and the external field strength.

The planar central configurations in the newtonian problem of four bodies are studied with the computer algebra system Mathematica. We have shown that in the case of two equal masses there can exist central configurations in the form of isosceles triangle
with three bodies being in its vertices and the fourth body being situated in the axis of symm...

Newton's restricted problem of four bodies is investigated. It has been shown that there are six equilibrium solutions of the equations of motion. Stability of these solutions is analyzed in linear approximation with computer algebra system Mathematica. It has been proved that four radial solutions are unstable while two bisector solutions are stab...

In this note we briefly present the first version of our Mathematica package QuantumCircuit [1] for simulation of quantum circuits [2] and illustrate some of its features by simple examples. To provide a user with a tool for designing and testing quantum algorithms, a simulator program must be user-friendly, allow to input arbitrary quantum circuit...

In the present paper we study parametric oscillations of a torsional pendulum excited by means of varying a moment of inertia of the rotating body. Motion of the system is determined by the second order differential equation with periodic coefficients. We have studied stability of this equation and proved that parametric resonance in the system can...

## Projects

Projects (2)

Investigation of the many-body problem with the aid of computer algebra methods.