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Performing high-precision linear measurements is one of the main tasks of modern engineering geodesy. Consequently, the development and creation of high-precision laser rangefinders and refractometers with a relative measurement error of not more than 3.10-7, becomes an urgent scientific and technical problem. Wide theoretical and experimental studies in the problem laboratory of geodetic measurements of NUACA have accumulated a sufficient amount of experimental results for the construction of modern high-precision light meter with the determination of the residual part of the phase cycle with an error of 0.03-0.05 mm. The article discusses issues related to improving the accuracy of linear measurements developed in the NUACA of high-precision light rangefinder. A two-phase modulation measurement method is proposed, when signals shifted by 180° are formed optically using a phase plate at λ/2. This modulation method of linear measurements provided the phase error of linear measurements mφ = 0.03-0.05 mm. The article also discusses the issue of reducing the modulation power. For this purpose, it is proposed to install a high-quality buffer Q-resonator between the high-quality light modem and the low-quality microwave oscillator.

In the paper we study a class of two-dimensional integral equations on a quarter-plane with monotone nonlinearity and substochastic kernel. With specific representations of the kernel and nonlinearity, an equation of this kind arises in various fields of natural science. In particular, such equations occur in the dynamical theory of $p$-adic open-closed strings for the scalar field of tachyons, in the mathematical theory of the geographical spread of a pandemic, in the kinetic theory of gases, and in the theory of radiative transfer in inhomogeneous media. \newline We prove constructive theorems on the existence of a nontrivial nonnegative and bounded solution. For one important particular case, the existence of a one-parameter family of nonnegative and bounded solutions is also established. Moreover, the asymptotic behavior at infinity of each solution from the given family os studied. At the end of the paper, specific particular examples (of an applied nature) of the kernel and nonlinearity that satisfy all the conditions of the proven statements are given.

The scheme of a high-precision laser rangefinder with a crystal light modulator, which has the ability to work both on the compensation and on the two-phase mode of the linear measurements modulating method, is considered. The two-phase method is implemented by introducing another electro-optical crystal EOC in the receiving path, rotated 90° relative to the first one around the optical Z axis. In this case, a second signal is formed in the optical channel of the light rangefinder, shifted relative to the first signal by 180°. The measurement of the residual phase cycle occurs when the periodically received signals are equal. A theoretical analysis of the proposed scheme is carried out and in this case it is shown that, the receiving points fixing sensitivity increases, leading to an increase in the accuracy of linear measurements in 3-4 times, estimated by the phase determination error value of mφ = 0.02-0.03 mm.