Modern network systems (unmanned aerial vehicles groups, social networks, network production chains, transport and logistics networks, communication networks, cryptocurrency networks) are distinguished by their multi-element nature and the dynamics of connections between its elements. A number of discrete problems on the construction of optimal substructures of network systems described in the form of various classes of graphs are NP-complete problems. In this case, the variability and dynamism of the structures of network systems leads to an "additional" complication of the search for solutions to discrete optimization problems. At the same time, for some subclasses of dynamical graphs, which are used to model the structures of network systems, conditions for the solvability of a number of NP-complete problems can be distinguished. This subclass of dynamic graphs includes pre-fractal graphs. The article investigates NP-complete problems on pre-fractal graphs: a Hamiltonian cycle, a skeleton with the maximum number of pendant vertices, a monochromatic triangle, a clique, an independent set. The conditions under which for some problems it is possible to obtain an answer about the existence and to construct polynomial (when fixing the number of seed vertices) algorithms for finding solutions are identified.

Let $G_{k}$ be defined as $G_{k} = \langle a, b;\ a^{-1}ba = b^{k} \rangle$, where $k \ne 0$. It is known that, if $p$ is some prime number, then $G_{k}$ is residually a finite $p$-group if and only if $p \mid k - 1$. It is also known that, if $p$ and $q$ are primes not dividing $k - 1$, $p < q$, and $\pi = \{p,\,q\}$, then $G_{k}$ is residually a finite $\pi$-group if and only if $(k, q) = 1$, $p \mid q - 1$, and the order of $k$ in the multiplicative group of the field $\mathbb{Z}_{q}$ is a $p$-number. This paper examines the question of the number of two-element sets of prime numbers that satisfy the conditions of the last criterion. More precisely, let $f_{k}(x)$ be the number of sets $\{p,\,q\}$ such that $p < q$, $p \nmid k - 1$, $q \nmid k - 1$, $(k, q) = 1$, $p \mid q - 1$, the order of $k$ modulo $q$ is a $p$\-number, and $p$, $q$ are chosen among the first $x$ primes. We state that, if $2 \leq |k| \leq 10000$ and $1 \leq x \leq 50000$, then, for almost all considered $k$, the function $f_{k}(x)$ can be approximated quite accurately by the function $\alpha_{k}x^{0.85}$, where the coefficient $\alpha_{k}$ is different for each $k$ and $\{\alpha_{k} \mid 2 \leq |k| \leq 10000\} \subseteq (0.28;\,0.31]$. We also investigate the dependence of the value $f_{k}(50000)$ on $k$ and propose an effective algorithm for checking a two-element set of prime numbers for compliance with the conditions of the last criterion. The results obtained may have applications in the theory of computational complexity and algebraic cryptography.

Conformance checking methods diagnose to which extent a real system, whose behavior is recorded in an event log, complies with its specification model, e.g., a Petri net. Nonetheless, the majority of these methods focus on checking isolated process instances, neglecting interaction between instances in a system. Addressing this limitation, a series of object-centric approaches have been proposed in the field of process mining. These approaches are based on the holistic analysis of the multiple process instances interacting in a system, where each instance is centered on the handling of an object. Inspired by the object-centric paradigm, this paper presents a replay-based conformance checking method which uses a class of colored Petri nets (CPNs) -- a Petri net extension where tokens in the model carry values of some types (colors). Particularly, we consider conservative workflow CPNs which allow to describe the expected behavior of a system whose components are centered on the end-to-end processing of distinguishable objects. For describing a system’s real behavior, we consider event logs whose events have sets of objects involved in the execution of activities. For replay, we consider a jump strategy where tokens absent from input places of a transition to fire move from their current place of the model to the requested places. Token jumps allow to identify desire lines, i.e., object paths unforeseen in the specification. Also, we introduce local diagnostics based on the proportion of jumps in specific model components. The metrics allow to inform the severity of deviations in precise system parts. Finally, we report experiments supported by a prototype of our method. To show the practical value of our method, we employ a case study on trading systems, where orders from users are matched to trade.

To ensure traffic safety of railway transport, non-destructive tests of rails are regularly carried out by using various approaches and methods, including eddy-current flaw detection methods. An automatic analysis of large data sets (defectograms) that come from the corresponding equipment is an actual problem. The analysis means a process of determining the presence of defective sections along with identifying structural elements of railway tracks in defectograms. At the same time, severity estimation of defined defects is also of great interest. This article continues the cycle of works devoted to the problem of automatic recognition of images of defects and rail structural elements in eddy-current defectograms. In the process of forming these images, only useful signals are taken into account, the threshold levels of amplitudes of which are determined automatically from eddy-current data. The article is devoted to the issue of constructing severity estimation of found defects with various lengths. The construction of the severity estimation is based on a concept of the generalized relative amplitude of useful signals. A relative amplitude is a ratio of an actual signal amplitude to a corresponding threshold level of useful signals. The generalized relative amplitude is calculated by using the entropy of the half-normal distribution, which is assumed to be a model for a probability distribution of an appearance of certain relative amplitudes in an evaluated defect. Tuning up the formula for calculating severity estimation of a defect is carried out on the basis of eddy-current records of structural elements. As a reference of the most dangerous defect, the bolted rail joint is considered. It models a fracture of a rail. A reference weak defect is a flash butt weld, a defectogram of which contains signals with low amplitude values. The proposed approach to severity estimation of defects is shown by examples.

Let  $B$ be a Euclidean ball in ${\mathbb R}^n$ and let $C(B)$ be a space of continuos functions $f:B\to{\mathbb R}$ with the uniform norm $\|f\|_{C(B)}:=\max_{x\in B}|f(x)|.$ By $\Pi_1\left({\mathbb R}^n\right)$ we mean a set of polynomials of degree $\leq 1$, i.e., a set of linear functions upon ${\mathbb R}^n$. The interpolation projector  $P:C(B)\to \Pi_1({\mathbb R}^n)$ with the nodes $x^{(j)}\in B$ is defined by the equalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right)$,  $j=1,\ldots, n+1$.The norm of $P$ as an operator from $C(B)$ to $C(B)$ can be calculated by the formula $\|P\|_B=\max_{x\in B}\sum |\lambda_j(x)|.$ Here $\lambda_j$ are the basic Lagrange polynomials corresponding to the $n$-dimensional nondegenerate simplex $S$ with the vertices $x^{(j)}$. Let $P^\prime$ be a projector having the nodes in the vertices of a regular simplex inscribed into the ball. We describe the points $y\in B$ with the property $\|P^\prime\|_B=\sum |\lambda_j(y)|$. Also we formulate some geometric conjecture which implies that $\|P^\prime\|_B$ is equal to the minimal norm of an interpolation projector with nodes in $B$.  We prove that this conjecture holds true at least for $n=1,2,3,4$. 

Functional dataflow programming languages are designed to create parallel portable programs. The source code of such programs is translated into a set of graphs that reflect information and control dependencies. The main way of their execution is interpretation, which does not allow to perform calculations efficiently on real parallel computing systems and leads to poor performance. To run programs directly on existing computing systems, you need to use specific optimization and transformation methods that take into account the features of both the programming language and the architecture of the system. Currently, the most common is the Von Neumann architecture, however, parallel programming for it in most cases is carried out using imperative languages with a static type system. For different architectures of parallel computing systems, there are various approaches to writing parallel programs. The transformation of dataflow parallel programs into imperative programs allows to form a framework of imperative code fragments that directly display sequential calculations. In the future, this framework can be adapted to a specific parallel architecture. The paper considers an approach to performing this type of transformation, which consists in allocating fragments of dataflow parallel programs as templates, which are subsequently replaced by equivalent fragments of imperative languages. The proposed transformation methods allow generating program code, to which various optimizing transformations can be applied in the future, including parallelization taking into account the target architecture.