Alexander Georgievich Losev

This work is devoted to the development of capacitary techniques in the study of the properties of solutions to partial differential equations on non-compact Riemannian manifolds. One of the central problems in the classification theory of non-compact Riemannian manifolds, namely determining their parabolic (or hyperbolic) type, was uniquely solved in terms of the non-triviality of the capacitary potential of the exterior of an arbitrary compact set. Subsequent research has introduced other capacitary characteristics of non-compact Riemannian manifolds, including those generated by the Schrödinger operator and other elliptic operators. First and foremost, we note the Liouville function generated by the Schrödinger operator. The application of this capacitary characteristic made it possible to prove a statement providing an exact estimate for the dimension of the space of bounded solutions to the stationary Schrödinger equation in terms of massive sets. It is also known that the triviality of the Liouville function generated by a linear or semilinear elliptic operator is equivalent to the triviality of the set of bounded solutions of the corresponding equation. Other characteristics employed in geometric analysis include capacitary potentials and harmonic measures. This paper investigates the asymptotic behavior of the capacitary characteristics of model Riemannian manifolds, which generalize spherically symmetric ones. The geometric structure of such manifolds allows us to derive precise statements and even construct a fairly comprehensive theory of the elliptic operators under consideration and their generalizations. Exact estimates are obtained for the rate of convergence to the limit value at “infinity” for the Liouville function of the exterior of a compact set, for the harmonic measure, for the capacitary potential, and for the Liouville function of the manifold, all associated with the stationary Schrödinger equation. Работа посвящена развитию емкостной техники в изучении свойств решений уравнений в частных производных на некомпактных римановых многообразиях. Одна из основных проблем классификационной теории некомпактных римановых многообразий об определении параболичности (гиперболичности) типа была однозначно решена в терминах нетривиальности емкостного потенциала внешности произвольного компактного множества. В дальнейших исследованиях появились и другие емкостные характеристики некомпактных римановых многообразий, в том числе порожденные оператором Шрёдингера и другими эллиптическими операторами. Прежде всего отметим функцию Лиувилля, порожденную оператором Шрёдингера. Отметим, что именно специфические свойства функций Лиувилля многообразий позволили сформулировать и доказать утверждение о размерности пространств ограниченных решений стационарного уравнения Шрёдингера в терминах массивных множеств. Кроме того известно, что тривиальность функции Лиувилля, порожденной линейным или полулинейным эллиптическим оператором, эквивалентна тривиальности множества ограниченных решений соответствующего уравнения. Также в геометрическом анализе применяются такие характеристики, как емкостные потенциалы и гармонические меры. В данной статье исследуется асимптотическое поведение емкостных характеристик модельных римановых многообразий, обобщающих сферически-симметричные. Геометрическое строение таких многообразий позволяет получать точные утверждения и даже строить достаточно полную теорию рассматриваемых эллиптических операторов и их обобщений. Получены точные оценки скорости стремления к предельному значению на «бесконечности» функции Лиувилля внешности компакта, гармонической меры, емкостного потенциала и функции Лиувилля многообразия, ассоциированных со стационарным уравнением Шрёдингера.

The study examines the specifics of training machine learning algorithms on small datasets and addresses the task of forming a training set with high representativeness. It is known that class imbalance in objects, typical for small datasets, negatively affects the performance of algorithms. To mitigate this issue, various data synthesis methods have been developed in machine learning to supplement existing datasets and equalize the number of objects per class. However, these methods do not solve the problem of insufficient representativeness. This article proposes a method for constructing a representative training dataset by specifying the distribution that best corresponds to reality. The distribution is formed for each feature within the informative areas. Informative areas contain characteristic values of features that are most significant for distinguishing classes of objects. The proposed method of constructing areas is based on the idea of gradual expansion, accompanied by an increase in the informativeness of the areas. At the same time, informativeness is understood as a measure reflecting how well objects of different classes can be separated using the considered area. To form a complementary dataset, a generation method has been developed. As a result of its application, the complementary dataset is combined with the original one and forms the specified distribution in the informative area. This distribution can be determined either based on expert knowledge about the subject area, if the true distribution is known, or obtained as a result of computational experiments aimed at finding the most effective option. The applicability of the method is demonstrated by solving the problem of determining the level of temperature anomalies of the mammary glands. It is shown that the considered temperature features are characterized by a normal distribution. Increasing the representativeness of the training set allowed training a classic classification algorithm – logistic regression – with an accuracy comparable to a multilayer neural network. This approach to the formation of a training dataset opens up the possibility of creating more transparent and interpretable artificial intelligence systems.

Biochemical oxygen demand (BOD) is one of the most important water/wastewater quality parameters. BOD₅ is the amount of oxygen consumed in 5 days by microorganisms that oxidize biodegradable organic materials in an aerobic biochemical manner. The primary objective of this research is to apply microbial fuel cells (MFCs) to reduce the time requirement of BOD₅ measurements. An artificial neural network (ANN) has been created, and the predictions we obtained for BOD₅ measurements were carried out within 6–24 h with an average error of 7%. The outcomes demonstrated the viability of our AI MFC/BES BOD₅ sensor in real-life scenarios.

The paper investigates the asymptotic behavior of solutions of the inhomogeneous stationary Schrödinger equation on non-compact model Riemannian manifolds. Exact conditions for the unique solvability of the Dirichlet problem for the equation with continuous boundary data at <> and some other boundary value problems are found.