# Vol 26, No 1 (2018)

**Year:**2018**Articles:**8**URL:**https://journals.rudn.ru/miph/issue/view/1044**DOI:**https://doi.org/10.22363/2312-9735-2018-26-1

## Full Issue

## Mathematics

### Diﬀerential Properties of Generalized Potentialsof the Type Bessel and Riesz Type

#### Abstract

In this paper we study diﬀerential properties of convolutions of functions with kernels thatgeneralize the classical Bessel-Macdonald kernels ... The theory ofclassical Bessel potentials is an important section of the general theory of spaces of diﬀerentiablefunctions of fractional smoothness and its applications in the theory of partial diﬀerentialequations. The properties of the classical Bessel-Macdonald kernels are studied in detail in thebooks of Bennett and Sharpley, S. M. Nikolskii, I. M. Stein, V. G. Mazya. The local behavior ofthe Bessel-Macdonald kernels in the neighborhood of the origin is characterized by the presenceof a power-type singularity ||-. At inﬁnity, they tend to zero at an exponential rate. Therecent work of M. L. Goldman, A. V. Malysheva, and D. Haroske was devoted to the investigationof the diﬀerential properties of generalized Bessel-Riesz potentials.In this paper we study the diﬀerential properties of potentials that generalize the classicalBessel-Riesz potentials. Potential kernels can have nonpower singularities in the neighborhoodof the origin. Their behavior at inﬁnity is related only to the integrability condition, so thatkernels with a compact support are included. In this connection, the spaces of generalized Besselpotentials generated by them belong to the so-called spaces of generalized smoothness. The casewith the satisﬁed criterion for embedding potentials in the space of continuous bounded functionsis considered. In this case, the diﬀerential properties of the potentials are expressed in termsof the behavior of their module of continuity in the uniform metric. Criteria for embedding ofpotentials in Calderon spaces are established and explicit descriptions of the module of continuityof potentials and optimal spaces for such embeddings are obtained in the case when the basespace for potentials is the Lorentz weight space. These results specify the general constructionsestablished in previous works.

**Discrete and Continuous Models and Applied Computational Science**. 2018;26(1):3-12

### A Survey on Queuing Systems with Parallel Servingof Customers. Part II

#### Abstract

This paper is a continuation of the survey of the “fork-join” queuing systems (in the westernclassiﬁcation) or the systems with splitting of queries. Interest in such systems is explainedby a wide range of problems that can be solved with their help, since in fact it is a matter ofparallel processing of data and their applications. For example, this may concern the analysis ofdisk arrays, cloud computing, high-performance services and even the process of picking ordersin a warehouse. In the ﬁrst part of the survey, the main features of the described model (andrelated systems) and its construction were introduced. Also the detailed description of theapproach to obtaining an accurate expression of the average response time in the case of twodevices was presented as well as several methods of approximate analysis of this characteristic(the case when the number of devices is more than two). This part of the survey is devotedto the description of other existing methods for approximating the average response time. Inparticular, the approaches of the approximate analysis of the response time are as follows: thematrix-geometric method, the analysis with the help of order statistics for various types ofdistribution of the service time of subqueries.

**Discrete and Continuous Models and Applied Computational Science**. 2018;26(1):13-27

### A Heterogeneous Fork-Join Queueing System in Which EachJob Occupy All Free Servers

#### Abstract

In this paper, we consider a multiserver queueing system with heterogeneous servers in whicheach job is split to be serviced into a number of tasks, one for each free server. The tasks areserviced independently, but service time depends on weight of the tasks. A job is considered tobe complete only when all the tasks associated with the job have been executed to completion.Applying a matrix-geometric approach, we obtain the exact expression for the stationarydistribution of the number of jobs in the system under exponential assumptions. Using thedistribution, we derive other important performance measures. Special attention is paid to thesojourn time in the queueing system (the time to complete a job). Finally, some numericalexamples and a section of conclusions commenting the main research contributions of thispaper are presented.The results can be used for the performance analysis of multiprocessor systems and othermodern distributed systems.

**Discrete and Continuous Models and Applied Computational Science**. 2018;26(1):28-38

## Mathematical Modeling

### On the Reduction of Maxwell’s Equations in Waveguidesto the System of Coupled Helmholtz Equations

#### Abstract

The investigation of the electromagnetic ﬁeld in a regular homogeneous waveguide reducesto the investigation of two independent boundary value problems for the Helmholtz equation,corresponding to TE- and TM-modes. In the case of an inhomogeneous waveguide TE- andTM-modes are connected to each other, which in numerical experiments can not always be fullytaken into account. In this paper we show how to rewrite the Helmholtz equations in vectorform to express this relationship explicitly.In the article the cylindrical waveguide with perfectly conducting walls is considered, but wedon’t make any assumptions about ﬁlling of waveguide. The introduced approach is based ontwo-dimensional analogue of the theorem known in the theory of elastic bodies as the Helmholtzdecomposition. On its basis, we introduce four potentials, instead of two potentials, usuallyused in the theory of hollow waveguides. It is proved that any solution of Maxwell’s equationsin a waveguide that satisﬁes the boundary conditions of ideal conductivity on the boundariesof a waveguide can be represented with the help of these potentials. The system of Maxwell’sequations is written with respect to these potentials and it is shown that this system has theform of two independent Helmholtz equations in the case of a hollow waveguide.

**Discrete and Continuous Models and Applied Computational Science**. 2018;26(1):39-48

### On a Method of Investigation of the Self-Consistent NonlinearBoundary-Value Problem for Eigen-Valueswith Growing Potentials

#### Abstract

One of the most common methods for investigating multiparticle problems in the frameworkof the variational approach is the transition to a nonlinear one-particle problem by introducinga self-consistent ﬁeld that depends on the states of these particles. The paper considers anonlinear boundary value eigenvalue problem for the Schr¨odinger equation with a growingpotential including a dependence on the wave function and a power dependence on the coordinate = where = 1,2,3.... For n = 2, the boundary value problem for the Schr¨odinger equation(linear problem) has an exact solution. For even powers of , it is shown that solutions of sucha problem can be expressed in terms of solutions corresponding to the linear problem, and for= 2 the solution can be obtained in explicit form. The set of solutions obtained for= 2 ischaracterized by equal distances between neighboring eigenvalues. It is shown that the solutionof the nonlinear problem diﬀers from the solution of the linear problem by the shift of theeigenvalues. In the case of a potential higher than the quadratic one, new growing potentialsof a lesser degree appear. For the case of odd values of, the transition is discussed, from theintegro-diﬀerential formulation of the problem to a system of diﬀerential equations which can besolved numerically on the basis of the method of successive approximations, which has provedits eﬀectiveness in the study of the polaron model.

**Discrete and Continuous Models and Applied Computational Science**. 2018;26(1):49-57

### On a Method of Multivariate Density Estimate Basedon Nearest Neighbours Graphs

#### Abstract

A method of multivariate density estimation based on the reweighted nearest neighbours,mimicking the natural neighbours techniques, is presented. Estimation of multivariate densityis important for machine learning, astronomy, biology, physics and econometrics. A 2-additivefuzzy measure is constructed based on proxies for pairwise interaction indices. The neighboursof a point lying in nearly the same direction are treated as redundant, and the contributionof the farthest neighbour is transferred to the nearer neighbour. The calculation of the localpoint density estimate is performed by the discrete Choquet integral, so that the contributionsof the neighbours all around that point are accounted for. This way an approximation to theSibson’s natural neighbours is computed. The method relieves the computational burden of theDelaunay tessellation-based natural neighbours approach in higher dimensions, whose complexityis exponential in the dimension of the data. This method is suitable for density estimates ofstructured data (possibly lying on lower dimensional manifolds), as the nearest neighbours diﬀersigniﬁcantly from the natural neighbours in this case.

**Discrete and Continuous Models and Applied Computational Science**. 2018;26(1):58-73

### Modeling of Extreme Precipitation Fields on the Territoryof the European Part of Russia

#### Abstract

Present work is devoted to the study and development of space-time statistical structures ofextreme type modeling with the use of the max-stable processes. The theory of one-dimensionalextremal values and its extension to the two-dimensional case are considered and for that max-stable processes are introduced and then the main parametric families of max-stable processes(Schlather, Smith, Brown-Resnick, and Extremal-t) are presented. By modifying the maximumlikelihood method, namely using the paired likelihood function, parameter estimates wereobtained for each of the models whose eﬃciency was compared using the Takeuchi informationcriterion (TIC).Resulting models are coherent with classical extreme value theory and allow consistenttreatment of spatial dependence of rainfall. We illustrate the ideas through data, based ondaily cumulative rainfall totals recorded at 14 stations in central European part of Russia forperiod 1966-2016 years. We compare ﬁts of diﬀerent statistical models appropriate for spatialextremes and select the model that is the best for ﬁtting our data. The method can be used inother situations to produce simulations needed for hydrological models, and in particular forthe generation of spatially heterogeneous extreme rainfall ﬁelds over catchments. It is shownthat the most successful model for the data we studied is the model from the extremal-t familywith the Whittle-Matern correlation function.

**Discrete and Continuous Models and Applied Computational Science**. 2018;26(1):74-83

## Computer Science

### Analysis of the File Distribution Time in Peer-to-Peer Network

#### Abstract

Peer-to-peer (P2P) ﬁle sharing systems are responsible for a signiﬁcant part of the Internettraﬃc today. File sharing is perhaps the most popular application among P2P applications. Incomparison with traditional Client/Server ﬁle distribution, P2P ﬁle sharing has some advantages,namely, scalability, bandwidth and others. In this paper we study the minimum distribution timefor getting the entire ﬁle by all of the users in the system, who need this ﬁle. This parameter isclosely associated with the mentioned bandwidth. The expression for the minimum distributiontime uses ﬂuid-ﬂow arguments and includes such terms as the ﬁle size, the upload rates of theseeds and the upload and download rates of the leechers. Using numerical examples and theexpression for the minimum distribution time, we show the eﬃciency of P2P ﬁle sharing. Weconsider the system behaviour, when there are two types of leechers in the system. These typesdiﬀer from each other by their upload bandwidths.

**Discrete and Continuous Models and Applied Computational Science**. 2018;26(1):84-92