2026.1

A nonlinear two-phase mathematical model of production at a mining plant is proposed, where ore flow is represented as two phases connected to each other through concentration of useful substance contained in the ore. The process of material balancing is studied, which leads to optimization problem with nonlinear constraints. The interior point method is used to solve the problem. Known measured values of useful substance concentrations are taken into account, as well as the results of indirect measurements, ranges of flow values, unknown concentration values, limits on stored resources. Test experiments are performed, and calculation results are demonstrated.

Initial boundary-value problem for the evolutional equation with involution of the space variable and a non-local boundary condition is studied. The uniqueness of the problem’s solution is obtained and classes of initial functions which provide solution’s existence and stability are described.

A certain characterization of the logistic distribution is proposed by the property of identical distribution of two linear forms.

This work addresses the problem of single-variable function approximation within the class of nonlinear approximations with two sets of parameters. The approximating function is linear with respect to the first parameter (which is constrained to be positive), while each term of the approximating function represents a given function with nonlinear dependence on the second parameter. The computational algorithm is based on residual minimization in Hilbert space. The first key aspect of the developed approach involves effectively reducing the problem to a standard linear best approximation problem by discretizing the second parameter over an extended set of points within the admissible segment. The second key feature lies in determining the set of linear approximation parameters at each iteration of the classical non-negative least squares (NNLS) method. Numerical results are presented to demonstrate the capabilities of this computational algorithm for nonlinear function approximation.

This work examines a modification of a spatially distributed Lanchester model, which describes an antagonistic game between two groups in a two-dimensional domain. The model originally emerged in 1916 to describe combat scenarios between two military forces, characterizing events of the First World War. The model accounts for directed strikes, defined by a velocity vector, and the initial displacement of troop concentrations using a step function. The system includes diffusion terms, nonlinear reactions, Gaussian white noise, and a topographic obstacle (a lake of arbitrary shape) that restricts troop movement. The numerical solution is implemented using the finite element method based on domain triangulation to handle irregular domains. The simulation results demonstrate an increase in spatial heterogeneity under the influence of noise and the obstacle, as well as a significant impact of directed movement and initial displacement on conflict dynamics. Stability analysis confirms the system’s stability, including an analysis specific to the finite element method. Sensitivity analysis of the parameters and computational error estimation have been added. Visualization illustrates the evolution of troop densities over time. The advantages of the proposed approach over existing methods include better handling of irregular domains and model scalability.

The paper considers the asymptotic behaviour of the distribution functions of the statistics based on the samples with random sizes in the testing problems conserning univariate parameters. An asymptotic comparison of the tests is carried out in terms of the necessary additional number of such factors (asymptotic deficiency). Two examples illustrating the obtained results are presented. These examples concern truncated Poisson and binomial distributions are considered.