In this paper, we study undirected multiple graphs of any natural multiplicity $k>1$. There are edges of three types: ordinary edges, multiple edges and multi-edges. Each edge of the last two types is a union of $k$ linked edges, which connect 2 or $(k+1)$ vertices, correspondingly. The linked edges should be used simultaneously. If a vertex is incident to a multiple edge, it can be also incident to other multiple edges and it can be the common end of $k$ linked edges of some multi-edge. If a vertex is the common end of some multi-edge, it cannot be the common end of another multi-edge.

As for an ordinary graph, we can define the integer function of the length of an edge for a multiple graph and set the problem of the shortest path joining two vertices. Any multiple path is a union of $k$ ordinary paths, which are adjusted on the linked edges of all multiple and multi-edges.
In the article, we optimize the algorithm of finding the shortest path in an arbitrary multiple graph, which was obtained earlier. We show that the optimized algorithm is polynomial. Thus, the problem of the shortest path is polynomial for any multiple graph.

The paper considers the problem of modeling an adaptive control system information exchange for a group of unmanned aerial vehicles (UAVs). The movement of the UAV group is carried out in accordance with the adaptive algorithm for optimal control. Optimal controls are constructed to provide a minimum of the total energy expended. The parameters of the mathematical model of the movement of the UAV group are refined during the flight in accordance with changing external conditions. In accordance with this, the control actions are specified. This task requires significant computing resources and imposes special requirements on the information exchange system between the UAV and the control point. A scheme of information exchange between the UAV and the control point is proposed, which makes it possible to calculate the optimal parameters of the transmitting devices.

Numerical study of various processes leads to the need for clarification (extensions) of the limits of applicability of computational constructs and modeling tools. In this article, we study the differentiability in the space of Lebesgue integrable functions and the consistency of this concept with fundamental computational constructions such as Taylor expansion and finite differences is considered. The function $f$ from $L_1 [a; b]$ is called $(k,L)$-differentiable at the point $ x_0$ from $(a; b),$ if there exists an algebraic polynomial $P,$ of degree no higher than $k,$ such that the integral over the segment from ${x_0}$ then ${x_0+h}$ for $f-P$ there is $o(h^{k+1}).$ Formulas are found for calculating coefficients of such $P,$ representing the limit of the ratio of integral modifications of finite differences $ {\bf\Delta}_h^m(f,x) $ to $ h^m\!, \; m=1, \cdots, k. $ It turns out that if $f\!\in\!W_1^{l}[a; b],$ and $f^{(l)}$ is $(k,L)$-differentiable at the point $x_0,$ then $f$ is approximated by a Taylor polynomial up to $ o\big((x{-}x_0)^{l+k}\big),$ and the expansion coefficients can be found in the above way. To study functions from $L_1$ on a set, a discrete "global" construction of a difference expression is used: based on the quotient ${\bf\Delta}_h^m(f, \cdot)$ and $h^m$ the sequence is built $\big{{\bf\Lambda}_n^m[f]\big}$ of piecewise constant functions subordinate to partitions half-interval $[a; b)$ into $n$ equal parts. It is shown that for a $(k,L)$-differentiable at the point $x_0$ function $f$ the sequence $\big{{\bf\Lambda}_n^m[f]\big},\; m=1,\cdots, k, $ converge as $n\to \infty$ at this point to the coefficients of the polynomial approximating the function at it. Using $\big{{\bf\Lambda}_n^k[f]\big}$ the following theorem is established: {\it "$f$ from $L_1[a;b]$ belongs to $C^k[a;b] \Longleftrightarrow $ $f$ is uniformly $(k,L)$-differentiable on $[a;b]$".} A special place is occupied by the study of constructions corresponding to the case $m\!=\!0.$ We consider them in $L_1[Q_0],$ where $Q_0$ is a cube in the space $\mathbb R^d.$ Given a function $ f\!\in\!L_1$ and a partition $\tau_{n}$ of a semi-closed cube $Q_0$ on $\;n^d$ equal semi-closed cubes we construct a piecewise constant function $\Theta_n[f]$, defined as the integral average $f$ on each cube $Q\!\in\!\tau_{n}.$ This computational construction leads to the following theoretical facts: {\it 1)$f$ from $L_1 $ belongs to $L_p, 1 \le p < \infty, \Longleftrightarrow \big{\Theta_n[f] \big}$ converges in $L_p;$ the boundedness of $\big{\Theta_n[f]\big} \; \Longleftrightarrow f\!\in\!L_\infty;$ 2) sequences $\big{\Theta_n[\cdot]\big}$ define on the equivalence classes the operator-projector $\Theta$ in the space $L_1;$ 3) for the function $f\!\in\!L_{\infty}$ we get $ \overline{\Theta [f]}\!\in\!B,$ where $B$ is the space of bounded functions, and $ \overline{\Theta [f]}$ is the function $ \Theta [f](x),$ extended on a set of measure zero and the equality $\;\big\Vert \overline{\Theta [f]}\big\Vert_{B} = \Vert f\Vert_{\infty}.$} Thus, in the family of spaces $L_p$ one can replace $L_{\infty}[Q_0]$ with $B[Q_0].$

Software development is often about expanding functionality. To improve reliability in this case, it is necessary to minimize the change in previously written code. For instrumental support of the evolutionary development of programs, a procedural-parametric programming paradigm was proposed, which made it possible to increase the capabilities of the procedural approach. This allows to extend both data and functions painlessly. The paper considers the inclusion of procedural-parametric programming in the C language. Additional syntactic constructions are proposed to support the proposed approach. These constructions include: parametric generalizations, specializations of generalizations, generalizing functions, specialization handlers. Their semantics, possibilities and features of technical implementation are described. To check the possibilities of using this approach, models of procedural-parametric constructions in the C programming language were built. The example in the article demonstrates the flexible extension of the program and support of multiple polymorphism.

The task of named entity recognition (NER) is to identify and classify words and phrases denoting named entities, such as people, organizations, geographical names, dates, events, terms from subject areas. While searching for the best solution, researchers conduct a wide range of experiments with different technologies and input data. Comparison of the results of these experiments shows a significant discrepancy in the quality of NER and poses the problem of determining the conditions and limitations for the application of the used technologies, as well as finding new solutions. An important part in answering these questions is the systematization and analysis of current research and the publication of relevant reviews. In the field of named entity recognition, the authors of analytical articles primarily consider mathematical methods of identification and classification and do not pay attention to the specifics of the problem itself. In this survey, the field of named entity recognition is considered from the point of view of individual task categories. The authors identified five categories: the classical task of NER, NER subtasks, NER in social media, NER in domain, NER in natural language processing (NLP) tasks. For each category the authors discuss the quality of the solution, features of the methods, problems, and limitations. Information about current scientific works of each category is given in the form of a table for clarity. The review allows us to draw a number of conclusions. Deep learning methods are leading among state-of-the-art technologies. The main problems are the lack of datasets in open access, high requirements for computing resources, the lack of error analysis. A promising area of research in NER is the development of methods based on unsupervised techniques or rule-base learning. Intensively developing language models in existing NLP tools can serve as a possible basis for text preprocessing for NER methods. The article ends with a description and results of experiments with NER tools for Russian-language texts.

The paper is devoted to construction of a sentence corpus annotated by the general sentiment into 4 classes (positive, negative, neutral, and mixed), a corpus of phrasemes annotated by the sentiment into 3 classes (positive, negative, and neutral), and a corpus of sentences annotated by the presence or absence of irony. The annotation was done by volunteers within the project “Prepare texts for algorithms” on the portal “People of science”.

The existing knowledge on the domain regarding each task was the basis to develop guidelines for annotators. A technique of statistical analysis of the annotation result based on the distributions and agreement measures of the annotations performed by various annotators was also developed.
For the annotation of sentences by irony and phrasemes by the sentiment the agreement measures were rather high (the full agreement rate of 0.60--0.99), whereas for the annotation of sentences by the general sentiment the agreement was low (the full agreement rate of 0.40), presumably, due to the higher complexity of the task. It was also shown that the results of automatic algorithms of detecting the sentiment of sentences improved by 12–13 % when using a corpus for which all the annotators (from 3 till 5) had the agreement, in comparison with a corpus annotated by only one volunteer.