In this paper we study the existence of the maximal and minimal elements of the set of continuously differentiable convex extensions to $[0,1]^n$ of an arbitrary Boolean function $f_{B}(x_1,x_2,\ldots,x_n)$ and the cardinality of the set of continuously differentiable convex extensions to $[0,1]^n$ of the Boolean function $f_{B}(x_1,x_2,\ldots,x_n)$. As a result of the study, it was established that the cardinality of the set of continuously differentiable convex extensions to $[0,1]^n$ of an arbitrary Boolean function $f_{B}(x_1,x_2,\ldots,x_n)$ is equal to the continuum. It is argued that for any Boolean function $f_{B}(x_1,x_2,\ldots,x_n)$, there is no minimal element among its continuously differentiable convex extensions to $[0,1]^n$. It is proved that for any Boolean function $f_{B}(x_1,x_2,\ldots,x_n)$, the set of its continuously differentiable convex extensions to $[0,1]^n$ has a maximal element only if the number of essential variables of the given Boolean function $f_{B}(x_1,x_2,\ldots,x_n)$ is less than 2.

We consider an NP-hard problem of dynamically distributing virtual machines to servers with placement groups. For each virtual machine, parameters such as required number of resources and creation and deletion timestamps are known. Each server is a composition of NUMA nodes and is placed in a rack. Large virtual machines, which are placed on two nodes of a single server, and small virtual machines, which impose additional conditions on their placement, are considered. Placement groups are associations of subsets of virtual machines with conflict conditions between the subsets. The goal is to pack all virtual machines using the minimum number of server racks within the considered time horizon. A heuristic based on the column generation method is proposed to solve this problem. We analyze a set of static problems at different time points necessary to form a common set of patterns used in the construction of upper bounds. The results of computational experiments on real open instances show an insignificant difference between the lower and upper bounds.

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. Divisible graphs form a special class of multiple graphs. The main peculiarity of them is a possibility to divide the graph into $k$ parts, which are adjusted on the linked edges and which have no common edges. Each part is an ordinary graph.

The multiple tree is a multiple graph with no multiple cycles. The number of edges may be different for multiple trees with the same number of vertices. Also we can consider spanning trees of a multiple graph. A spanning tree is complete if a multiple path joining any two selected vertices exists in the tree if and only if such a path exists in the initial graph.
The problem of the minimum complete spanning tree of a multiple graph is NP-hard even in the case of a divisible graph. In this article, we obtain an exact algorithm for the problem of the minimum complete spanning tree of a divisible multiple graph. Also we define a subclass of divisible graphs, for which the algorithm runs in polynomial time.

This article is devoted to the problem of verifying parallel programs that may contain special types of errors associated with the synchronization of parallel executed threads and access to shared memory. Such errors include deadlocks and data races. There is a division of parallel program verification methods into static and dynamic. The second ones require running the code and allow to check only the current implementation of the program for races, which, if there are a large number of branches, can lead to missing races. Among static methods, analytical methods (for example, based on deductive analysis) and model checking methods are most widely used. However, they are difficult to implement, and model checking still require a significant amount of manual work from the programmer to build such a model. In this regard, it is necessary to use models that can be built automatically. Previously, the authors developed a model based on an extension of Petri nets, which allows automatic creation based on sequential code and its conversion into parallel code. Automatic creation of a model of a parallel program introduces new, previously unused requirements related to the interaction of parallel threads. Thus, this article discusses the features of modeling using extended Petri nets with semantic relations of the main synchronization primitives implemented in most languages and parallel programming technologies for shared memory systems. In the future, these models will be used to search for data races and deadlocks in parallel programs.

Traffic safety in rail transport requires continuous monitoring of the rail condition for timely detection and elimination of defects. One of the methods of non-destructive testing of rails is eddy current flaw detection. The data obtained from eddy current flaw detectors (defectograms) are characterized by a significant volume, which makes it necessary to develop effective methods for their automatic processing and analysis. The analysis of defectograms is complicated by various interferences and noises present in the data. One of the most dangerous interferences that significantly distort the shape of useful signals is prolonged impulse interference. They are characterized by a pronounced square wave shape. Unlike instant impulse interference, prolonged noise cannot be eliminated by classical methods. There are no proven effective methods not only for suppressing square wave interference, but even for detecting it. This article attempts to eliminate this drawback and proposes an effective method for detecting square wave impulse interference on eddy current defectograms, which has good explanatory power. Square signals are explored from the point of view of their probability distribution. SW-characteristic was introduced, which allows to estimate the likelihood of data to the distribution of bipolar impulse signals. The smaller the value of SW-characteristic, the more similar the data distribution is to the distribution of bipolar impulse signals (upon condition that the data are normal). Square wave signals are particular example of bipolar impulse signals. The properties of SW-characteristic were examined. SW-characteristic were calculated for the normal distribution and the distribution of a homoscedastic mixture of two Gaussians. It was shown that the value of SW-characteristic for the normal distribution approximately separates the bimodal mixture of two Gaussians from the unimodal case. These and other properties of SW-characteristic allow using it to detect square wave signals in data by comparing with a threshold, which should satisfy a number of conditions. The application of the criterion based on SW-characteristic was demonstrated on the examples of eddy current defectograms; a comparison was made with criteria based on the EM-algorithm and multiscale disperse entropy. The criterion proposed in the article showed the best results. The use of SW-characteristic for detecting square wave noise has proven its effectiveness in the analysis of eddy current defectograms and can be adapted in the future to work with other types of data.

Fully connected networks of oscillators and their limit systems of integro-differential equations with periodic boundary conditions are considered. It is assumed that the connection is weak, i.e. the coefficient at the integral term is small. In the problem of stability of the zero equilibrium state, the simplest critical cases of loss of stability are distinguished. In these situations, quasi-normal forms are constructed, which are integro-differential equations for which several continuous families of piecewise constant two-step solutions are analytically determined, and their stability is studied. The existence of piecewise constant solutions with more than one discontinuity point is shown. A numerical experiment illustrating the analytical constructions is performed.