Fourier series and interpolation processes possess an important property: their rate of convergence can be upper-bounded if the estimates of the best approximations and Lebesgue constants are known. Lebesgue constants in the polynomial case have been extensively studied. For rational Fourier series, they were estimated by G. S. Kocharyan (1958), and for rational interpolation processes, they have been studied since the works of V. N. Rusak (1962). However, in each case, the estimates were obtained under rather strict restrictions on the poles. In this work, the estimates of Lebesgue constants of interpolating rational trigonometric processes are obtained under conditions of completeness of the corresponding system of rational functions.

The main result of the paper is the continuous embedding of Besov spaces of analytic functions in bounded domains with a Dini-smooth boundary. The proved embedding is consistent with and generalizes the known embeddings of Besov spaces for the circle and the half-plane. The independence of the definition of Besov spaces analytic in the considered domains of functions from the order of the derivative included in its de nition is shown.

More than 50 years ago, Chvátal introduced a new graph invariant that measures how tightly different parts of the graph are connected to each other, which he called graph toughness. From then on a lot of research has been obtained, mainly related to the relationship between toughness conditions and the existence of cyclic structures, in particular, determining whether the graph is Hamiltonian and pancyclic. Many important results have been obtained using spectral graph theory. Bondy in 1976, has suggested the metaconjecture that almost any nontrivial condition on a graph which implies that the graph is Hamiltonian also implies that the graph is pancyclic. We confirm the Bondy’s metaconjecture for 1-tough graphs in terms of spectral radius.

This paper explores the concept of quark and gluon bleaching by diagonalizing the matrix of vector potentials of a non-Abelian gauge field. The proposed approach is implemented using a model problem of quark motion in a constant, uniform chromomagnetic field. Potential applications of this method for solving the general confinement problem are outlined.

This article presents the results of a comprehensive investigation focused on developing and validating a methodology for creating two-group cross-section libraries for the DYN3D reactor code in the formula format (IWQS = 2), utilizing the high-precision Monte Carlo code Serpent. The study specifically targets the VVER-1200 nuclear reactor at the Belarusian Nuclear Power Plant. The methodology involves calculating two-group cross-sections and other constants for various reactor states. It is demonstrated that the largest deviation for k_∞ for all types of fuel assemblies when calculated using the DYN3D code with the obtained library differs from precise Monte Carlo Serpent calculations by approximately 1000 ppm. This level of precision is deemed sufficient for safety analysis calculations using this library for the VVER-1200 reactor. Calculations of boric acid concentrations during the reactor’s first and stationary fuel cycles were performed, along with the distribution of energy release at the beginning of the campaign. The obtained results underscore the reliability and effectiveness of the developed methodology.

The necessity of correcting the perturbations related to light propagation in the turbulent atmosphere and obtaining the diffraction-limited images requires computer processing of a large information flow in real time. Here the straightforward approaches, which ignore the structural properties of the atmospheric turbulence and the active optical system itself, lead to algorithms with complexity proportional to the fourth power of the aperture diameter. In the paper it is shown that taking into account the structural properties of atmospheric turbulence as well as the active optical system itself allows to significantly reduce the complexity coming close to the second power of the diameter. The approach is proposed, which is based on the efficient construction of deformable mirror commands in a multi-conjugate adaptive system in the case of already known turbulence volume distribution taking into account structural properties of the active optical system itself. The efficiency of the proposed approach is demonstrated by the example of the system of adaptive optics of a 30-meter telescope pupil with a 6-layer atmospheric turbulence model.

We considered the problem of estimation of the parameters of the trapezoidal distribution. We built a confidence interval. This confidence interval with high probability contains the value of the random variable with a trapezoidal distribution. We further developed two approximation algorithms for the problem that are based on the maximum likelihood principle. Algorithm uses the enumeration principle of the values of the parameters. The trapezoidal distribution may be used instead of normal distribution when there are upper and lower bounds on the value of the random variable.

The algorithm of the multilayer feed-forward neural network learning is developed. The algorithm realizes an error back-propagation idea. The particularity of the algorithm is its multidimensional-matrix form, which provides its theoretical and algorithmic generality. The program realization of the algorithm is performed as the function of Matlab programming language. In spite of the multidimensional-matrix form of the algorithm, this function is defined fully by the usual matrices. The validity of the algorithm is confirmed by the computer simulation of different approximation problems including the problem of pattern recognition.

This paper examines the problem of land cover classification in densely populated urban environments using ultra-high-resolution Earth observation images. The aim of the study is to develop and validate a neural network algorithm for a high-precision semantic segmentation of urban green spaces using multimodal data. An improved U-Net convolutional neural network architecture, modified for the use with a 7-channel input tensor (RGB, NIR, RedEdge, DSM, and NDVI), is proposed. The approach is based on the Early Fusion strategy, which combines spectral measurements with promising structural characteristics (digital surface model, DSM). Focal Loss is used to overcome the class imbalance. The proposed model reliably separated spectrally identical layers (grass and trees) and eliminated false positives on green anthropogenic objects. The final Mean IoU was 0.725. The recall for detecting forested areas reached 0.93, and for the complex minority class “Shrubs” it reached 0.77. The experiment on an independent test site confirmed the model’s high generalizability (F1-score 0.93). The integration of seven data channels and the use of a modified U-Net are fully justified for the tasks of accurate calculating forest areas and environmental monitoring in the Smart City concept.