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Delaunay triangulation and Voronoi partitioning are used to construct computational grids in numerical methods such as the finite element method and the finite volume method. A two-grid technique is considered that utilizes both the nodes of a Delaunay triangulation and the vertices of a Voronoi partitioning. This approach makes it possible to construct operator-difference approximations of the vector calculus operators (gradient, divergence, and curl) on a merged MVD (merged Voronoi-Delaunay) grid which consists of orthodiagonal quadrilaterals. The paper investigates an application of MVD grids for finite element analysis of two-dimensional boundary value problems using a Dirichlet problem for an elliptic equation in an anisotropic medium as an example. Two approaches are examined: Delaunay triangulation with additional Voronoi vertices as extra nodes and direct application of MVD grids. The results of computations on a sequence of progressively refined meshes employing different types of finite elements are presented.

In this paper, we consider problems of obtaining upper bounds for the components of root-mean-square errors for computational constructions of approximation of an unknown probability density for a given sample. Examples are the computer functional kernel and projection algorithms as well as their important special case - the multidimensional analog of the frequency polygon. These bounds are then used in choosing such versions of kernel and projection algorithms that provide a given level of error in a density approximation.

This study propounded a numerical approach for solving a mathematical model of pollutant spread through forest resources using shifted orthogonal Bernoulli polynomials (OBPs). The model is based on a system of ordinary differential equations, which is transformed into an algebraic system using the collocation approach based on shifted OBPs. Newton's method is employed to obtain numerical solutions, and the results are compared with those obtained using the Runge-Kutta method of fourth-order (RK4) to demonstrate the efficiency of the proposed method. The results demonstrate good agreement with the RK4 method, indicating the proposed method's acceptability for modeling pollutant spread through forest resources.

The article studies a problem of optimal control of heating for a homogeneous half-line. The heating problem is set for a heat conduction equation defined on a half-line where the temperature tends to zero at infinity. At the origin of the spatial coordinate a heat flux, i. e. a non-homogeneous boundary condition of the second kind, is given. The heat flux is modeled using a heating function which is a continuous broken line. This choice of the function is explained by the properties of the technical device under study. The article proves the existence of a solution to such a problem and the uniqueness of its classical solution with a certain error. The optimality of the heating control in this paper means that at the boundary x=0 the temperature at any time is maximum permissible (or, in the first time interval, maximum possible), and at the same time does not exceed a certain critical value, which is chosen to be equal to 1. For the optimal control, a recurrence formula is found at different times, it is proven that this is exactly the optimal solution. That is, at large values of the heat flux the critical temperature at the boundary will be exceeded at some point in time, and at the smaller values the temperature will be lower than that allowed by the material. It is also proven that the heat flux found is the exact upper bound of all admissible heat fluxes for a given discrete control and that such a flux is unique.

The real life problems related to engineering and physical systems are theoretically studied using mathematical models and are generally formulated using linear and non-linear differential equations. This work proposes a numerical technique to find approximate solutions of differential equations utilizing polynomials as base approximation functions and metaheuristic optimization algorithms like Differential Evolution (DE) and Particle Swarm Optimization (PSO) for obtaining the optimal values of coefficients of the polynomials in order to get the desired approximate solution. The algorithms for the proposed method have been executed using MATLAB for computer programming. The effectiveness of the approach suggested in this paper is found to be better than or at least comparable to other numerical methods suggested earlier for solving differential equations.

Numerous higher-order iterative methods have been proposed in the literature for locating roots of nonlinear equations. Among these, methods with optimal order are of particular interest due to their superior efficiency. However, not all of them exhibit consistent performance across all scenarios. Some offer low accuracy, while others suffer from slow convergence, yet there are methods that fail to maintain the desired convergence order in certain applications. This paper aims to address these shortcomings. Consequently, we introduce a novel three-point iterative scheme, whose formulation is based on the widely used two-point King's fourth-order method. This scheme attains eighth-order convergence at the cost of four function evaluations per step. As such, it is optimal according to the Kung-Traub conjecture and boasts an efficiency index of 1.682, which surpasses that of Newton's method and many other higher-order techniques. To assess the methods' performance and validate its theoretical properties, we present several numerical examples. Furthermore, we provide a detailed analysis of the complex dynamics through graphical representations of the basins of convergence, comparing our method with those of other established techniques. The computational results and convergence visualizations confirm that our scheme outperforms existing methods in the literature.

This publication provides a current assessment of the synurbic population of Magpie in Belarus. The current distribution of Magpie in urban territories is given based on quantitative surveys of this species in 2024-2025, which covered all the most favourable habitats in 19 of the largest cities and 12 smaller cities in various regions of the country. Studies of the habitat preferences and breeding densities of Magpies were conducted in 2015-2025 in the largest synurbic population of this species, which is found in Minsk. As a result, it was established that the current distribution of synurbic populations of Magpie across the territory of the republic is highly uneven. The total estimated synurbiс population of Magpie in Belarus is 6-6.3 thousand breeding pairs. More than 80 % (5-5.4 thousand breeding pairs) of the entire synurbiс population of this species is concentrated in Minsk. In Baranovichi, the city with the second highest number of breeding pairs of Magpie, there are no more than 130 breeding pairs of this species. In the 13 largest cities in Belarus, Magpie nests in single pairs (up to 10 pairs) or is absent. Our results suggest that synurbiс population groups of Magpie in different cities of Belarus formed independently. The current average breeding density of Magpie in Minsk is 27 pairs/km² and the number of this species has increased compared to 2015-2016. The breeding density of Magpie decreases from individual and low-rise residential buildings (0,26±0,26 pairs/ha), as well as small-area tree and shrub plantations (0,36 ± 0,61 pairs/ha), to multi-storey modern residential buildings (0,2 ± 0,12 pairs/ha) and large-area tree and shrub plantations (0,2 ± 0,1 pairs/ha). Although the synurbic population of Magpie in Minsk is generally increasing, there are noticeable fluctuations in numbers from year to year and a constant redistribution of breeding pairs within the city.

This study presents the results of a model experiment investigating the effects of atmospheric pollutants - sulfuric and nitric acids, ammonium sulfate, and ammonium nitrate - on the epiphytic organisms Parmelia sulcata and Orthotrichum speciosum. The experiment simulated the deposition of polluted atmospheric precipitation. The assessment included spectrophotometric analysis of photosynthetic pigment content (chlorophylls a and b) and evaluation of morphological changes. All tested pollutants induced stress responses manifested by reduced pigment content and structural tissue damage. Ammonium sulfate ((NH₄)₂SO₄) exerted the most pronounced inhibitory effect, leading to sharp pigment decline and morphological degradation. In contrast, ammonium nitrate (NH₄NO₃) had the least damaging effect, likely due to its high bioavailability and balanced nitrogen composition. The lichen P. sulcata exhibited more pronounced changes and higher sensitivity, while the moss O. speciosum showed signs of resistance and compensatory reactions. Cluster analysis confirmed differences between the taxa and revealed two stable response types: sensitive (P. sulcata) and compensatory (O. speciosum). The findings highlight the diagnostic value of a comprehensive approach combining biochemical and morphological indicators for assessing the state of the atmospheric environment within ecological monitoring frameworks.

The article presents the results of the accumulation of metals by the epigeal lichen Flavocetraria nivalis, which is widespread on the island of Vaigach. Lead-zinc ore deposits are one of the sources of environmental pollution on the island. However, the influence of lead-zinc ore mines on metal concentrations in F. nivalis has not been established, since the metal content in lichen only increased with distance from the mines. Concentrations of chemical elements in lichen differ in different biotopes, and they are higher on the cliff tops than in the tundra on the slopes of the native shores. A comparison of the results of metal accumulation on Vaigach Island with other territories has shown that the concentrations of most metals in F. nivalis are lower or close to the Marmorilik area of western Greenland, where lead-zinc ore mines are also located. The excess of zinc and lead at sampling points near the mines in the Maarmorilik area over the concentrations obtained in lichen thallomas on Vaigach Island may be due to the recent closure of mines in Maarmorilik. So, if the mines on Vaigach Island were closed back in the late 30s of the last century, then in the area of Maarmorilik only 20 years ago. The concentration of metals in lichen on Vaigach Island is much lower than in the industrialized areas of the Kola Peninsula, which is especially typical for nickel.

This study evaluates the relationship between soil chemical contamination and the state of the microbiome in urban soils from transportation, transport-industrial, and recreational zones of Bishkek city, as well as the Ala - Archa Nature Park. Total concentrations of Pb, As, Zn, Cu, Co, Mn, Cr, Cd, Hg, and Sb were determined in soils using a DELTA X - ray fluorescence spectrometer. It was shown that the degree of heavy metal contamination depended on traffic intensity and emissions from thermal power plants. Functional diversity of soil microbiomes was assessed via multisubstrate testing, and functional activity was measured by carbon dioxide emission using chromatographic methods. Microbial community health and stability were evaluated by the shape coefficient (d) of the rank distribution of substrate utilization spectra, serving as a measure of microbial system destabilization. A distinct microbial community state was observed in soils from the western part of the city, characterized by reduced substrate utilization, decreased metabolic activity, and community destabilization. The studied heavy metals were grouped according to their influence on biotic indicators such as microbial respiration and substrate consumption. The As - Pb - Cu - Co group was more strongly associated with the integrated heavy metal contamination index (Z_st) and most strongly inhibited substrate-induced respiration and consumption of pentose monosaccharides. The Hg - Cr - Mn group mainly suppressed amino acid and organic acid consumption. The Cd - Zn - Sb group reduced basal respiration intensity and slightly increased oligosaccharide consumption and entropy parameters (H_k and χ²) of substrate utilization spectra. It was found that contamination with the ten studied heavy metals did not exert a direct specific effect on the microbiome, as the overall heavy metal contamination index (Z_st) showed no significant correlations with any biotic indicators. The conclusion was made that only the combined consideration of abiotic (chemical contamination) and biotic (microbiome responses) criteria should be used to assess ecological risks of soil degradation.