Anna A. Lempert

УДК 519.853.6, 517.958 MSC: 05B40, 52C17, 35A18 DOI: 10.21538/0134-4889-2025-31-4-115-131 Исследование А.А. Лемперт выполнено за счет гранта Российского научного фонда, проект № 24-21-00264, https://rscf.ru/project/24-21-00264/. В статье рассматривается задача построения оптимальных по времени маршрутов для одного или нескольких подвижных объектов в динамической среде с

The article discusses the problem of finding the optimal location for a given number of equal geodesic circles without overlapping on a spherical segment. The optimality criterion is to maximize the radius of the circles. This formulation is known as the Tammes problem, and it is a variant of the classical densest packing problem of equal objects in a specified container. We propose a new two-stage numerical method to solve the issue under consideration. The first stage is the construction of an initial approximation using the best-of-known optimal packings of equal circles into a circle of a larger radius, which centers are projected onto a spherical segment in a specific manner. A theorem on the properties of projection is proved, establishing a connection between the radii of elements of plain and spherical packings. The second stage is the improving procedure based on the billiard modeling. The article presents illustrative numerical calculations for spherical segments of different angular sizes. The comparison of results obtained with known ones shows that the proposed method allows finding optimal or close-to-optimal packings in significantly less time than when using the traditional multistart method. Besides, an applied problem of designing a spherical focal surface, which arises in engineering optics, is solved.

The article addresses the problem of increasing the capacity of the Baikal–Amur Mainline, which is part of the Northern Eurasian Corridor and provides the shortest route from Europe to the countries of the Asia-Pacific region via the ports of the Russian Far East. A key focus of the study is the Severobaikalsk section, which currently operates as a single-track system and is located in a specially protected natural area. In this regard, it is a limiting element since its reconstruction projects require environmental expertise. To tackle this issue, we develop a mathematical model based on semi-open queuing networks as a research tool. This modeling apparatus allows us to describe the features of the considered transport system, including single-track traffic, long sections between stations, which leads to the appearance of additional sidings, the presence of through and ring routes, and the significant influence of random factors on train operation. Based on the results of a numerical study of the model, we evaluate the current and maximum permissible section loads and point out bottlenecks in its structure. We compare two alternative ways of increasing the section capacity: creating a double-track connection on the most problematic sections and using a partially batch train schedule. Based on the results of the calculations, we conclude about the possibility and conditions of their implementation.

This article concerns the problem of estimating the throughput and forecasting the operation of a coal transshipment complex that comprises a marine coal terminal and a railway station. Scenario modeling is employed to address this issue. The mathematical model of the transshipment complex has the form of a queuing network, which allows us to take into account the impact of random factors on the arrival of trains and departure of vessels from the system and their handling. In the model, we use the batch marked Markovian arrival process (BMMAP), which allows for the batch arrival of several types of requests, to describe the arrival of different categories of trains. Various queuing systems model particular structural elements of the complex to consider peculiarities of their work. We investigate the coal transshipment complex, which includes one of the largest and most modern coal export terminals in Russia. Based on the results of a numerical study, we estimate its current and maximum throughput, find bottlenecks in the system structure, and forecast its performance after the planned modernization. We also discuss the advantages and limitations of the model presented and its potential extension.

УДК 514.174.2 MSC: 05B40, 52C17, 35A18 DOI: 10.21538/0134-4889-2024-30-1-142-155 Исследования А.А. Лемперт выполнены в рамках госзадания Минобрнауки России по проекту “Теоретические основы, методы и высокопроизводительные алгоритмы непрерывной и дискретной оптимизации для поддержки междисциплинарных научных исследований” (№ гос. регистрации: 121041300065-9). Рассматривается

The paper concerns a nonlinear second-order parabolic evolution equation, one of the well-known objects of mathematical physics, which describes the processes of high-temperature thermal conductivity, nonlinear diffusion, filtration of liquid in a porous medium and some other processes in continuum mechanics. A particular case of it is the well-known porous medium equation. Unlike previous studies, we consider the case of several spatial variables. We construct and study solutions that describe disturbances propagating over a zero background with a finite speed, usually called ‘diffusion-wave-type solutions’. Such effects are atypical for parabolic equations and appear since the equation degenerates on manifolds where the desired function vanishes. The paper pays special attention to exact solutions of the required type, which can be expressed as either explicit or implicit formulas, as well as a reduction of the partial differential equation to an ordinary differential equation that cannot be integrated in quadratures. In this connection, Cauchy problems for second-order ordinary differential equations arise, inheriting the singularities of the original formulation. We prove the existence of continuously differentiable solutions for them. A new example, an analog of the classic example by S.V. Kovalevskaya for the considered case, is constructed. We also proved a new existence and uniqueness theorem of heat-wave-type solutions in the class of piece-wise analytic functions, generalizing previous ones. During the proof, we transit to the hodograph plane, which allows us to overcome the analytical difficulties.

The article concerns the problem of covering the lateral surface of a right circular cylinder or a cone with equal balls. The surface is required to belong to their union, and the balls’ radius is minimal. The centers of the balls must lie on the covered surface. The problem is relevant for mathematics and for applications since it arises in security and communications. We develop heuristic algorithms for covering construction based on a geodesic Voronoi diagram. The construction of a covering is a non-trivial task since the line of intersection of a cylinder or a cone with a sphere is a closed curve of the fourth order. To compare the numerical results with the known ones, we unroll the surface of revolution onto a plane. Another feature is that, we use both Euclidean distance and a special non-Euclidean metric, which can describe the speed of signal propagation in a heterogeneous medium. We also perform a numerical experiment and discuss its results. Meanwhile, it is shown that with a small number of circles covering a planification of the cylindrical surface, their radius is significantly less than for a similar rectangle.

The paper considers a particular variant of the classical optimal packing problem when the container is a sphere, the packed elements are equal spherical caps, and the optimality criterion is to maximize their geodesic radius. At the same time, we deal with a special integral metric to determine the distance between points, which becomes Euclidean in the simplest case. We propose a heuristic numerical algorithm based on the construction of spherical Voronoi diagrams, which makes it possible to obtain a locally optimal solution to the problem under consideration. Numerical calculations show the operability and effectiveness of the proposed method and allow us to draw some conclusions about the properties of packings.