This paper is dedicated to the study of eT -reducibility — the most common in the intuitive sense of algorithmic reducibility, which is both enumeration reducibility and decidable one. The corresponding structure of degrees — upper semilattice of eT -degrees is considered. It is shown that it is possible to correctly define the jump operation on it by using the T-jump or e-jump of sets. The local properties of eT -degrees are considered: totality and computably enumerable. It is proved that all degrees between the smallest element and the first jump in DeT are computably enumerable, moreover, these degrees contain computably enumerable sets and only them. The existence of non-total eT -degrees is established. On the basis of it, some results have been obtained on the relations between, in particular, from the fact that every eT -degree is either completely contained in some T -or e-degrees, or completely coincides with it, it follows that non-total e-degrees contained in the T-degrees, located above the second T -jump, coincide with the corresponding non-total eT -degrees.

In this paper, the existing approaches to the assessment of computer networks performance are considered. The standard structure of a network of the application layer of the OSI model using the example of SBIS3 application (product of Tensor Company) is treated.

Further, two approaches allowing to analyze degradations in a network are considered - on the basis of aggregated data and the operational analysis.

The degradation study of more than 60 000 request types between two versions of application which works on the basis of the computer network is the cornerstone of the first decision. Each type of requests is described by four based metrics, each metrics representing a time series. The input data are aggregated every 10 minutes before an analysis algorithm. Further, the threshold criteria based on mathematical expectation and dispersion within two adjacent versions of the software are used. Such an approach allows to significantly reduce time for the analysis of potential problems in case of updates within the computer network.

The second decision is based on not aggregated input data. It consists of detail information about all requests, there are data section of the computer network. A threshold criterion is based on durations in the selected queue. This analysis type allows to diagnose the errors with problem clients.

The article describes a background and some steps in the implementation of an industrial solution to build manageable mesh overlay network on top of a complete or partially non-manageable underlay network. The overlay network has (or may have) some features from the software defined networks world. We call this solution as SD-WAN Lite to highlight that this solution is not (and will not be in a visible future) a complete SD-WAN solution.

When exploiting software vulnerabilities such as buffer overflows, code reuse techniques are often used today. Such attacks allow you to bypass the protection against the execution of code in the stack, which is implemented at the software and hardware level in modern information systems. At the heart of these attacks lies the detection, in the vulnerable program of suitable areas, of executable code — gadgets — and chaining these gadgets into chains. The article proposes a way to protect applications from attacks that use code reuse. For this purpose, features that distinguish the chains of gadgets from typical chains of legal basic blocks of the program are highlighted. The appearance of an atypical chain of the base block during program execution may indicate the execution of a malicious code. An algorithm for identifying atypical chains has been developed. A feature of the algorithm is that it is focused on identifying all currently known techniques of re-execution of the code. The developed algorithm is based on a modified QEMU virtualization system. One of the hallmarks of the chain of gadgets is the execution at the end of the chain of instructions of the processor used to call the function of the operating system. For the Linux operating system based on the x86/64 architecture, experiments have been conducted showing the importance of this feature in detecting the execution of the malicious code.

The problem of key distribution in a community for providing secure communication between its participants is studied. To solve this problem, key predistribution systems can be used, in which each user receives some key information that can later be used to independently calculate required shared secret keys for conferences they participate in. Such key distribution systems can be based on different structures, such as error-correcting codes and combinatorial designs. The drawback of such systems is the possibility of collusive attacks, when traitors within the system can form a coalition and use their key information to try to calculate shared secret keys of other users. But the secrecy of keys is guaranteed by the system when the number of traitors in the coalition does not exceed a threshold defined by the system structure. In this paper, a key distribution system is based on combinatorial designs and, in particular, on Hadamard 3-design that guarantees the secrecy of communications in the presence of coalitions with less than three users. New notions of combinatorial span and combinatorial rank of a subset of Hadamard code that are required for the study of the resilience of the system to collusive attacks are introduced. The probability of successful collusive attack on an arbitrary conference against the cardinality of coalition is calculated for this system.

This paper is a logical continuation of the paper about possible approaches to solving the “Useful Proof-of-work for blockchains” problem. We suggest some alternative ways for searching useful tasks for Proof-of-work systems. These ways are based on the process of the multiple and independent repetition of a simple experiment. The experiment is to chose an element independently and uniformly from a quite large set and then to check if the chosen element has a specific rare property. In the classic blockchain of Bitcoin this experiment is a so-called hash-puzzle. In these terms the process of solving a hash-puzzle may be replaced by searching rare astronomical objects or Go positions with specific conditions. Moreover, we describe a possible attack on the blockchain systems in which the task instance generation algorithm is replaced by the algorithm of selecting the task instance from the existing database with public access for publication of task instances and discuss the way of protection.

In recent years, discrete approaches have been widely used in mathematical modeling of physicochemical processes. Cellular automata-based methods greatly simplify modeling procedures in many cases. In particular, this is important when using models in the form of partial differential equations systems to analyze the transfer of a substance in inhomogeneous media. In some cases, it is quite difficult to set the boundary conditions correctly if the object of study has boundaries of complex shape. It is also difficult to use mathematical physics classical equations if one cannot neglect the influence of stochastic effects on the process flow. The lattice gas models considered in the article are one of the types of cellular automata. Until now they have not been widely adopted, despite the fact that the first works on their use appeared about forty years ago. It is known, however, that lattice gases successfully describe a number of hydrodynamic phenomena, and the results obtained do not contradict the generally accepted views on the physical nature of continuous media motion processes. When using models of lattice gases, there are often questions about the correctness of the use of discrete models in various flow regimes. The second problem is a large-scale transition from model discrete parameters to generally accepted macroscopic characteristics of flows, such as flow velocity, viscosity and density of the medium, etc. It is also necessary to take into account that the indicated parameters in the lattice model are dimensionless, and the corresponding real macroscopic parameters have dimension. In this paper, an attempt is made to propose a method of large-scale transition, as well as to indicate the areas of practical use of some models of lattice gases.

Let \(\Omega = A^N\) - be a space of right-sided infinite sequences drawn from a finite alphabet \(A = \{0,1\}\), \(N = {1,2,\dots} \), \[\rho(\boldsymbol{x},\boldsymbol{y}) = \sum_{k=1}^{\infty}|x_{k} - y_{k}|2^{-k} \] - a metric on \(\Omega\), and \(\mu\) - a probability measure on \(\Omega\). Let \(\boldsymbol{\xi_0}, \boldsymbol{\xi_1}, \dots, \boldsymbol{\xi_n}\) - be independent identically distributed points on \(\Omega\). We study the estimator \(\eta_n^{(k)}(\gamma)\) - of the reciprocal of the entropy \(1/h\), that are defined as. \[\eta_n^{(k)}(\gamma) = k \left(r_{n}^{(k)}(\gamma) - r_{n}^{(k+1)}(\gamma)\right),\] where \[r_n^{(k)}(\gamma) =\frac{1}{n+1}\sum_{j=0}^{n} \gamma\left(\min_{i:i \neq j} {^{(k)}} \rho(\boldsymbol{\xi_{i}}, \boldsymbol{\xi_{j}})\right),\] \(\min ^{(k)}\{X_1,\dots,X_N\}= X_k\), если \(X_1\leq X_2\leq \dots\leq X_N\). Number \(k\) and a function \(\gamma(t)\) - are auxiliary parameters. The main result of this paper is

Theorem. Let \(m\) - be the Bernoulli measure with probabilities \(p_0,p_1>0\), \(p_0+p_1=1\), \(p_0=p_1^2\), then \(\forall eps>0\) some continuous function \(\gamma(t)\) such that \[
\left|E\eta_n^{(k)}(\gamma) - \frac1h\right| <eps,\quad DD\eta_n^{(k)}(\gamma)\to 0,n\to \infty. \]

For \(x^{(0)}\in{\mathbb R}^n, R>0\), by \(B=B(x^{(0)};R)\) we denote a Euclidean ball in \({\mathbb R}^n\) given by the inequality \(\|x-x^{(0)}\|\leq R\), \(\|x\|:=\left(\sum_{i=1}^n x_i^2\right)^{1/2}\). Put \(B_n:=B(0,1)\). We mean by \(C(B)\) the space of continuous functions \(f:B\to{\mathbb R}\) with the norm \(\|f\|_{C(B)}:=\max_{x\in B}|f(x)|\) and by \(\Pi_1\left({\mathbb R}^n\right)\) the set of polynomials in \(n\) variables of degree \(\leq 1\), i.e. linear functions on \({\mathbb R}^n\). Let \(x^{(1)}, \ldots, x^{(n+1)}\) be the vertices of \(n\)-dimensional nondegenerate simplex \(S\subset B\). The interpolation projector \(P:C(B)\to \Pi_1({\mathbb R}^n)\) corresponding to \(S\) is defined by the equalities \(Pf\left(x^{(j)}\right)=%f_j:=f\left(x^{(j)}\right).\) Denote by \(\|P\|_B\) the norm of \(P\) as an operator from \(C(B)\) into \(C(B)\). Let us define \(\theta_n(B)\) as minimal value of \(\|P\|\) under the condition \(x^{(j)}\in B\). In the paper, we obtain the formula to compute \(\|P\|_B\) making use of \(x^{(0)}\), \(R\), and coefficients of basic Lagrange polynomials of \(S\). In more details we study the case when \(S\) is a regular simplex inscribed into \(B_n\). In this situation, we prove that \(\|P\|_{B_n}=\max\{\psi(a),\psi(a+1)\},\) where \(\psi(t)=\frac{2\sqrt{n}}{n+1}\bigl(t(n+1-t)\bigr)^{1/2}+\bigl|1-\frac{2t}{n+1}\bigr|\) \((0\leq t\leq n+1)\) and integer \(a\) has the form \(a=\bigl\lfloor\frac{n+1}{2}-\frac{\sqrt{n+1}}{2}\bigr\rfloor.\) For this projector, \(\sqrt{n}\leq\|P\|_{B_n}\leq\sqrt{n+1}\). The equality \(\|P\|_{B_n}=\sqrt{n+1}\) takes place if and only if \(\sqrt{n+1}\) is an integer number. We give the precise values of \(\theta_n(B_n)\) for \(1\leq n\leq 4\). To supplement theoretical results we present computational data. We also discuss some other questions concerning interpolation on a Euclidean ball.

A model of distributed information carriers in the form of stable spatially inhomogeneous structures in optical and fiber-optic communication systems is considered. We study the conditions for the occurrence of such stable spatially inhomogeneous structures of the light wave of the generator of optical radiation. The formation of inhomogeneous structures that occur in a plane orthogonal to the direction of wave propagation is provided by a thin layer of nonlinear medium and a two-dimensional lagging feedback loop with the rotation operator of the spatial coordinates of the light wave in the emission plane of the optical generator. In the space of the main parameters of the generator (a control parameter, the angle of rotation of the spatial coordinates, the magnitude of the delay), the areas of generation of stable spatially inhomogeneous structures are constructed, the mechanisms of their occurrence are analyzed.