The paper proposes a method for calculating the area of a flat area from a photograph based on the use of mathematical analysis methods. To calculate the area, a curved integral of the second kind is used along a closed contour bounding the area under consideration. Defining the boundary in the form of a Bezier spline reduces the calculation of a curved integral to the calculation of several definite integrals from the Bernstein basis polynomials. An explicit form is obtained for integrals of Bernstein basis polynomials. For a third-order Bezier spline, a formula is derived for calculating the area of the area in terms of the coordinates of the reference points of the Bezier curves.

Keywords: cubic spline, Bernstein basis polynomials. Bezier curve, Bezier spline, Green's formula, beta function, gamma function

Explicit formulas for the spectral characteristic and optimal linear filtration operator with a forecast for stochastic L–Markov processes are obtained using methods of spectral analysis of random processes, the theory of functions of a complex variable, and using stochastic differential-difference equations. An interesting example of an optimal filtration operator with a forecast for an L-Markov process with a quasi-rational spectral density generalizing the rational one is constructed for technical applications. It is shown that the forecast filtering operator is the sum of a linear combination of the values of the received signal at some time points and the integral of an exponentially decaying weight function.

Keywords: random process, L-Markov process, prediction filtering, spectral characteristic, filtration operator

The paper develops an algorithm for constructing an optimal lagged filtration operator for an L-Markov process. The explicit formula of the filtration operator is obtained on the basis of methods for calculating stochastic integrals and the theory of analytical functions of a complex variable using spectral analysis and the theory of L-Markov processes. An interesting example of an optimal lagged filtration operator for an L-Markov process is considered, which can be used for modeling and controlling complex stochastic systems. It is shown that this operator is represented as a linear combination of the values of the received signal and an integral with an exponentially decaying function.

Keywords: random process, L-Markov process, noise, lag filtering, spectral characteristic, filtering operator

In this paper, the problem of extrapolating a video signal with a quasi-rational spectral density, which significantly generalizes the rational density, is explicitly solved. The spectral characteristic of video signal extrapolation is constructed using the original method of A.M. Yaglom, a follower of academician A.N. Kolmogorov, who first posed the problem of extrapolation for random sequences and processes. The essence of the method consists in transferring all studies and calculations of spectral characteristics and densities from the real axis to the complex plane. The paper considers a video signal with a quasi-rational spectral density of a special type, interesting for practical applications, in which, as shown by the author using the Chebotarev and Sturm methods, it has all its roots only in an open upper half-plane.

Keywords: random process, video signal, prediction, filtering, spectral characteristic, prediction time

The calculation of the coefficients of the linear best method for restoring the second derivative at zero of a bounded analytical function given in a unit circle by the values of the function and its derivative at specified points forming a regular polygon centered at zero is pointed in that paper. It also determines the error of the best method and finds the corresponding extreme function. It is proved that the extremal function is unique. At the end of the work, formulas are got that can be used to calculate the coefficients of the linear best method. In finding of these formulas, the method of duality of extreme problems was applied, which was deeply developed by S.Y. Havinson. It is proved that these coefficients are the only one.

Keywords: optimal recovery, error of the best method, linear best method, coefficients of the linear best method

The paper presents the formulation of problems of minimization and maximization of a linear functional with inequality constraints on the vector of admissible program motions and equality constraints specified by a linear manifold. An analytical solution is synthesized that determines the projection operator for solving the specified mathematical programming problems with equality constraints and inequalities. An analytical solution is obtained that determines the boundary values ​​of the Lagrange multiplier for the synthesized projection operator. The correctness of the obtained solution is illustrated.

Keywords: mathematical programming, linear functional, projection operators, admissible program motions, stabilization of program motions, SimInTech