In the paper the possibility of describing the flights of ultra-fast Alcubierre warp spaceships that change the topology and geometry of 3-dimensional physical space using Ricci flows is demonstrated.

We consider the Project Scheduling Problem with accumulative resources of timedependent cost and the case of limited storage. We prove its NP-hardness in a strong sense. Also a dynamic programming algorithm has been developed and pseudopolynomially solvable cases have been identified.

The article is devoted to the problem of improving the accuracy of classification of phenotypes of diseases of the gastrointestinal tract. In the course of preliminary research, the CatBoost model was identified, which demonstrates the best results in this task, achieving an accuracy of 92.85 % in the training sample and 79.31 % in the test sample. The main goal of this work is to improve the accuracy of predictions of the model solving the problem of classification of diseases by studying the characteristics of the original data set. Data processing methods aimed at increasing the quality of model predictions, including the search and removal of abnormal data in the sample, are proposed. The issue of choosing the criterion of "abnormality" for specific data is also considered. In addition, the article discusses methods of dealing with the problem of class imbalance, which is an important aspect for improving the overall efficiency of classification.

This article deals with a graph invariant called the Kirchhoff index. An explicit analytical formula for the Kirchhoff index of circulant graphs with non-fixed jumps is defined.

In the article an overview of Gauss-Bonnet-Zhen type formulas, some of which were included in the 2009 thr author's review, some of which were missing from the mentioned review, and some were obtained later is given. Special attention is paid to the application of these formulas in the general theory of relativity. The positive energy required to create a 4-dimensional wormhole through which a warp ship flies has been established. The conclusions of the formula in supersymmetric Euclidean field theory and from the Atiyah- Singer index theorem are given.

The article discusses the issues of forecasting sales volumes of automotive batteries of a trade organization by a method based on an arti cial neural network. Signi cant parameters have been determined for forecasting sales volumes of automotive batteries with the required accuracy.

The article provides an overview of methods for constructing motion trajectories for unmanned vehicles, describes approaches, advantages, disadvantages and limitations of the methods.

The Fuzzy Cognitive Map (FCM) method was developed by B. Kosko to extend cognitive maps, which are widely used to represent knowledge in the social sciences. Several algorithms have been proposed for training FCMs. The main objective of the training procedure is to nd a setting of FCM weights that will bring the FCM to a desired stable state. This is achieved by minimizing a well-de ned objective function. The established algorithms heavily depend on the initial approximation of the weight matrix, which is provided by experts. One of the methods for training FCMs is based on particle swarm optimization (PSO). PSO is used to determine the appropriate weight matrices for the system by minimizing a given objective function.

The work is devoted to the task of diagnosing the phenotypes of diseases of the gastrointestinal tract using machine learning models. The main purpose of this work is to find a model that solves the problem of classifying patients in a medical institution. In the course of the work, simple models such as logistic regression and cluster analysis were first considered. Due to the inefficiency of these methods, a search was made for more complex models and methods of data analysis: decision trees, ensembles of trees, artificial neural networks. Work was also carried out on the preliminary processing of the data set and the selection of significant features for classification. As a result of the work, the best model for the data set under consideration was found - CatBoost, which diagnoses phenotypes with an accuracy of 92.85 % in the training sample and 79.31 % in the test sample. In addition, the paper evaluates the applicability of machine learning methods in clinical practice and identifies areas for future research for more accurate diagnosis.

We consider a special three-dimensional Lie algebra 3 of Bianchi type V. A matrix representation of this Lie algebra and the corresponding connected simply connected Lie group 𝑆3 is found and natural coordinates are introduced, which are determined by the matrix representation. Formulas for exponential mapping with respect to natural coordinates are found. Complete groups of autoisometries of the special Lie algebra with respect to the Euclidean or Lorentz scalar product are written out. The leftinvariant Riemannian metric of the Lie group 𝑆3 and the complete group of isometries of the resulting homogeneous manifold are found. This manifold turned out to be a space of constant Ricci curvature.