Alexander N. Prokopenya

Symbolic-numerical methods for searching equilibrium states in a restricted four-body problem

Algorithms for searching equilibrium solutions of the circular restricted four-body problem formulated on the basis of the triangular Lagrange solutions of the three-body problem are discussed. An algorithm is proposed for calculating the bifurcation curve in the plane of system parameters that separates domains of the eight and ten equilibrium solutions. For the parameter values corresponding to the bifurcation curve, the system has nine equilibrium solutions. In the neighborhood of the bifurcation points, the equilibrium solutions are found in the form of power series in terms of a small parameter. The dependence of these solutions on the system parameters is studied numerically. Codes of algorithms implemented in the computer algebra system Mathematica are presented.

Simulation of quantum error correction by means of QuantumCircuit package

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.

Simulation of a quantum algorithm for phase estimation

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 accurate to 2−n with the probability greater than 8/π2. By way of example, computations for the case of the quantum phase shift operator are performed. The simulation of the quantum algorithm and the computation of the eigenvalue are performed using the QuantumCircuit package written in the Wolfram Mathematica computer algebra system. This system is also used to perform all the computations and visualize the results.

Symbolic calculations in studying the problem of three bodies with variable masses

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 calculating the secular part of the perturbing functions and derivation of the differential equations determining the secular perturbations of the orbital elements are discussed. All the relevant symbolic calculations are carried out using the Mathematica computer algebra system.

Simulation of Quantum Error Correction with Mathematica

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 presented.

Orthogonal Illuminations in Two Light-Source Photometric Stereo

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 sufficient condition to disambiguate the entire shape reconstruction process is controlled by the satisfaction of the corresponding second-order linear PDE with constant coefficients in two independent variables. This work extends the latter to an arbitrary pair of light-source directions transforming the above constraint into a special nonlinear PDE. In addition, a similar ambiguity analysis is also performed for a special configuration of two light-source directions assumed this time as orthogonal and contained in the vertical plane. Finally, this work is supplemented by illustrative examples exploiting symbolic computation used within a framework of continuous reflectance map model (i.e. an image irradiance equation) and applied to a genuine Lambertian surfaces.

Hamiltonian normalization in the restricted many-body problem by computer algebra methods

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 expressions obtained for the coefficients of the Hamiltonian normal form substantiate results derived earlier by A. Deprit. Symbolic computations are performed in the computer algebra system Mathematica.

Application of computer algebra for the reconstruction of surfaces from their photometric stereo images

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 the surface from its two images is analyzed in a more general setting. Photometric images of various surfaces are simulated, and the validity of the theoretical results is demonstrated.

Approximate Quantum Fourier Transform and Quantum Algorithm for Phase Estimation

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 obtained. The validity of the results is demonstrated by simulation of the algorithm in case of the phase shift operator using the QuantumCircuit package written in the Wolfram Mathematica language. All relevant calculations and visualizations are done with the Wolfram Mathematica system.

Application of Computer Algebra to Photometric Stereo with Two Light Sources

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 equations that determine the gradient of a function of two variables given by the equation u(x, y) − z = 0. We also analyze necessary and sufficient conditions for unique determination of a second-order algebraic surface from its two images in the general case. Correctness of the theoretical results obtained is confirmed by simulating photometric images of various surfaces.