# 2019 № 3(44)

**CONTENTS**

**Skopin I. N. Comparison of opportunity modularization in functional and imperative programming styles****Imomnazarov Kh. Kh., Turdiev U.K. Investigation of the cauchy problem for a one-dimensional system of the burgers type equations by the weak approximation method****Samigulina G. A., Samigulina Z. I. Multi-agent scientific research system for predicting dependence ,,structure-properties“ of drug compounds based on modified algorithms of artificial immune systems****Shcherbakova N. G. Models of networks with preferential attachment****Ershov A. P. Сomputability in arbitrary domains and bases**

Institute of Computational Mathematics and Mathematical Geophysics SB HAS,

630090, Novosibirsk, Russia

**COMPARISON OF OPPORTUNITY MODULARIZATION IN FUNCTIONAL AND IMPERATIVE PROGRAMMING STYLES**

**UDK 004**

The article “Comparison of opportunity modularization in functional and imperative programming styles” discusses the issues of expressiveness of the language means proposed for the two programming styles. The solution to these issues is relevant in connection with the preparation for the transition in the near future to the efficient use of ultra-high power computing equipment: the programmer has the opportunity to build calculations using a very large number of processors and cores.

Today, interest in functional computing has grown very much, and a dangerous trend, very typical for the development of computer science, has appeared: one can consider an unconventional functional style as a panacea for all programming problems in the imperative style update. In this regard, *one of the main goals of the article* being proposed is to dispel the myth of the universality of the functional style and find for it, as well as for the imperative style, an adequate place among the methodological approaches to solving programming problems.

This goal is specified as a comparison of means of supporting imperative and functional modularization. The basis of this comparison is the assertion that the gain in the transition from the traditional style to the new is not in what you have to give up, but in a new quality, which the new style gives in comparison with the old one (see D. Hughes article “Why Functional Programming Matters”). With regard to the transition from an imperative to a functional model of computing, this means an answer to the question of language programming tools that provide new qualities of modularity. First of all, these are tools that allow one to implement lazy calculations and handling functions of high orders, as well as a mechanism for eliminating re-counting, called memoization.

The approach to achieving the goals is based on the use of formally defined operations of an abstract computer in the execution of individual constructions of an imperative and functional program. This allows somebody to consider language constructs as templates for connecting parts of a program. As a result, it is possible to uniformly describe the development and use of modules for both the imperative and functional model of computing. One of the main consequences of this approach is the ability to accurately specify the boundaries of the adequate applicability of imperative and functional modularization, as well as the conditions for the correct joint combined use of the means of these two styles. The proposed method is in good agreement with the concept of mixed computing, in that aspect, which back in 1977 A.P. Ershov quite accurately called the essence of translation.

The exposition of imperative and functional modularization completes the list of losses that a programmer should not forget about when switching from using a traditional model of calculations to a functional one: Memory passivity cannot be expressed in a functional style.

- The notion of states of a computational process, which in many cases give its natural decomposition, is lost in a functional language.
- The concept of the context of calculations in a functional language becomes significantly narrower than when working in an imperative style: for and it is important that the contexts are organized hierarchically.
- Managing the ordering of computations over time is contrary to functionality.

All these losses are somehow connected with the concept of common (but not global!) For different data modules for which a functional language cannot offer adequate means of expression. This explains the success of mixed modularity, when the functional parts of the program are invoked for execution within the framework of the operating environment in which they are executed as modules independent of the environment.

**Key words: **modularization, imperative programming, functional programming, computing model, abstract calculator, memoization, style of programming.

References

- ASANOVIC K. et al. The landscape of parallel computing research: a view from Berkeley. Technical Report No. UCB/EECS-2006-183. Berkeley: University of California, EECS Department, December 18, 2006. [Electron res.]: www.eecs.berkeley.edu/Pubs/TechRpts/2006/EECS-2006-183. pdf (access date: 24.05.2019).
- Backus J. Can Programming be Liberated from von Neumann style? A Functional Style and its Algebra of Programs. Comm. ACM, 21, 1978.
- McCarthy. 91 function — Wikipedia. [Electron res.]: https://en.wikipedia.org/wiki/ McCarthy\_91\_function\#Knuth.27s\_generalization (access data: 24.05.2019).
- Gorodnjaja L.V. Fundamentals of functional programming. Lectures. Moscow: Internet University of Information Technology, 2004. ISBN 5-9556-0008-6.
- Hughes J. Why Functional Programming Matters // Computer Journal. N 32 (2). 1989.
- NEPEJIVODA N.N., Skopin I.N. Foundations of Programming. Moscow-Izhevsk: RChD, 2003.
- Dahl O. J., Dijkstra E. W., Hoare C. A. R. Structured Programming. Moscow: Mir, 1975.
- LlSKOV B., Guttag J.V. Abstraction and Specification in Program Development (MIT Electrical Engineering and Computer Science). The MIT Press, 1986, ISBN-10: 0262121123.
- Stroustrup B. What is Object-Oriented Programming? // IEEE Software. 1988. V. 5 (3).
- KOSVICHENKO A. Zachem ge nugna virtvalizacija? [Electron res.]: https://habrahabr.ru/ post/91503/ (access data: 24.05.2019).
- Sylvan S. Why does Haskell matter? [Electron res.]: http://www.dtek.chalmers.se/ ~sylvan/haskell/why_does_haskell_matter .html (access data: 24.05.2019).
- Ershov A.P. О sushchnosti translvacii. Preprint N 6, Novosibirsk: VC SO AN SSSR, 1977.
- N. WlNSTANLEY. What the hell are Monads? 1999. [Electron res.]: http://www. abercrombiegroup.co.uk/~noel/research/monads.html (access data: 24.05.2019).
- Keene S. E. Object-Oriented Programming in Common Lisp: A Programmer’s Guide to CLOS. Addison-Weslev (Reading, Massachusetts, 1989).
- Skopin I. N. An Approach to the Construction of Robust Systems of Interacting Processes // In: Parallel PROGRAMMING: Practical Aspects, Models and Current Limitations. NOVA science publishers. Series: Mathematics Research Developments. Editor: M.S. Tarkov. 2014, ISBN: 978-163321-957-1.
- Cejitin G.S. Na putv к sborochnomu Programmirovaniju // Programmirovanie. 1990. N 1. P. 78-99.

**Bibliographic reference**: Skopin I. N. Comparison of opportunity modularization in functional and imperative programming styles //journal “Problems of informatics”. 2019, № 3. P. 4-19. DOI: 10.24411/2073-0667-2019-00008

**Imomnazarov Kh. Kh., Turdiev* U.K.**

Institute of Computational Mathematics and Mathematical Geophysics of the SB HAS, Novosibirsk, Russia

*Karshi branch of TUIT, Karshi, Uzbekistan

**INVESTIGATION OF THE CAUCHY PROBLEM FOR A ONE-DIMENSIONAL SYSTEM OF THE BURGERS TYPE EQUATIONS BY THE WEAK APPROXIMATION METHOD**

**UDK 532.5**

This paper is concerned with obtaining a system of Burgers-tvpe equations in one limit case from the system of equations of a two-fluid medium. The system in question differs from the system of equations of a two-fluid medium by the absence of pressure and the incompressibility conditions. For this reason, the problems associated with the Burgers-tvpe system are called a system without pressure for a two-fluid medium. In the case when the dissipative function does not depend on the viscosity coefficients of the medium, we will call the Burgers-tvpe non-viscous system or the Hopf-tvpe system. In the one-dimensional case, we call it also the Riemann-tvpe system of equations, which is a simple quasilinear system of equations. The system of equations of a two-fluid medium and the system of equations of the Burgers type have much in common. For example, the quadratic nonlinear terms - due to the phase velocities - respond to advective terms, corresponding to the dependence of sound on the amplitude of sound waves and linear terms caused by viscosities and the friction coefficient, which are responsible for the attenuation of the sound waves, where inproperties of solutions are completely different. With a Burgers-tvpe equation system with disappearing viscosity coefficients and the friction coefficient, both strong (shock waves) and weak discontinuities are formed, while the solutions of a two-fluid system, do not possess such features. However, the scope of applicability of the system proposed is not limited to the examples given, such systems arise in many problems, which is what determines its importance. A study of the system of the Burgers-tvpe equations arising in the nonlinear acoustics is presented. The proposed mathematical model is due to the combination of a conservative nonlinear system with a dissipative term; here, the dissipation is due to both the viscosity of subsystems and the inter-component friction coefficient (analogous to the Darcy coefficient), for which equivalent diffusion representations can be effectively used. The Cauchy problem for a one-dimensional system of the Burgers-tvpe equations arising in a two-fluid medium is considered. The system under study is quasilinear, and analytical research methods do not allow one to obtain solutions to the Cauchy problem. One of the main methods for carrying out theoretical studies of the mathematical models of a two-fluid medium and applying them to solving important practical problems is numerical methods. Consequently, when studying efficient numerical algorithms, one of the main methods for their construction, is the method of weak approximation of differential equations. The weak approximation method, steering a middle course between the differential problem and the corresponding difference model can be used in two versions: as one of the methods for studying the correctness of the problem; as a method for constructing and rigorous mathematical analysis of the corresponding difference splitting schemes. The latter from this point of view are simple difference approximations of differential problems in fractional steps. In this paper, using the weak approximation method, we prove the existence and uniqueness of the solution to the Cauchy problem for a one-dimensional Burgers-tvpe system in the dissipative approximation.

**Key words: **two-velocitv hydrodynamics, Burgers type system, weak approximation method.

This work was supported in part by the Russian Foundation of Fundamental Research under grant No 02-05-64939.

References

- Kulikovskii A.G., Sveshnikov E.I., & Chugaynova A.P. Matematicheskie metodi izucheniva razrivnikh resheniv nelinevnikh geperbolicheskikh system uravneniv [Mathematical methods for studying discontinuous solutions of nonlinear hyperbolic systems of equations]. (Leksionnie kursi NOS. Vip. 16), Moskva, 2010 (in Russian).
- Dorovsky V. N. Continual theory of filtration // Sov. Geology and Geophysics, 1989. N 7. P. 34-39.
- DOROVSKY V.N., Perepechko Yu. V. Phenomenological description of two-velocitv media with relaxing shear stresses // J. Appl. Meeh, and Tech. Phvs., 1992. N 3. P. 403-409.
- Dorovsky V.N., Perepechko Yu. V. Theory of the partial melting // Sov. Geology and Geophysics. 1989. N 9. P. 56-64.
- Perepechko Yu. V., Sorokin K.E., Imomnazarov Kh. Kh. The influence of acoustic vibrations on convection in a compressible two-fluid medium. In
*Sbornik materialov Sovremennie problem mekhaniki sploshnoy sredi [Proceedings XVII International Conference "Contemporary Problems of Continuum Mechanics 2014J,*Rostov-on-Don, 2014, P. 166-169 (in Russian). - Demidov G. V., Novikov V. A. On the convergence of the method of weak approximation in a reflexive Banach space // Functional analysis and its applications. 1975. V. 9. N 1. P. 25-30.
- Gegechkori Z. G., Demidov G. V. On the Convergence of the Weak Approximation Method // DAN USSR. 1973. V. 213. N 2. P. 264-266.
- Demidov G. V., Marchuk G. I. A theorem on the existence of a solution to the problem of short-term weather forecast // DAN USSR. 1966. V. 170. N 5. P. 1006-1009.
- Belov Yu. Ya., Demidov G. V. Reshenie zadachi Koshi diva sistemi uravneniv nipa Hopfa metodom slabov approksimasii // Chislennie metodi mexaniki sploshnov sredi. Novosibirsk, VS SO AN SSSR. 1970. T. 1. N 2. S. 3-16.
- Demidov G. V. Nekotorie prilogeniva obobshennov neoremi Kovalevskov // Chislennie metodi mexaniki sploshnov sredi. Novosibirsk, VS SO AN SSSR. 1972. T. 2. N 2. S. 10-32.
- Raputa V. F. Metod slabov approksimasii diva zadachi Koshi v shkale banakhovikh prostranstv // Chislennie metodi mexaniki sploshnov sredi. Novosibirsk, VS SO AN SSSR. 1975. T. 6. N 1. S. 93-96.
- Boyarinsev Yu. E. Regulvarnie I singulvarnie sistemi linevnikh obiknovennikh differensialnikh uravneniv. Novosibirsk: Nauka, 1980.
- Belov Yu. Ya., Kantor S.A. Metod slabov approksimasii. Krasnoyarsk: Krosnovarskiv Gos. Universitet, 1999.
- Belov Yu. Ya. On Estimates of Solutions of the Split Problems for Some Multi-Dimensional Partial Differential Equations // J. of Siberian Federal University. Mathematics and Physics. 2009. V. 3. P. 258-270.
- Yanenko N. N. Metod drobnikh shagov resheniva mnogomernikh zadach matematicheskov fiziki. Novosibirsk: Nauka, Sibirskoe otdelenie, 1967.
- Demidov G.V., Yanenko N.N. Metod slabov approksimasii // Trudi Vsesovuznov konferensii po uravnenivam s chastnimi proizvodnimi. M.: Izd. MGU, 1978. T. 1, S. 100-102.
- Sobolev S.L. Nekotorie primeneniva funksionalnogo analiza v matematicheskov fizike. Novosibirsk: SO AN SSSR, 1962.

**Bibliographic reference**: Imomnazarov Kh. Kh., Turdiev U.K. Investigation of the cauchy problem for a one-dimensional system of the burgers type equations by the weak approximation method

//journal “Problems of informatics”. 2019, № 3. P. 4-19. DOI: 10.24411/2073-0667-2019-00009

**Samigulina G. A., Samigulina* Z. I.**

Institute of Information and Computational Technologies, 050010, Almaty, Kazakhstan

**MULTI-AGENT SCIENTIFIC RESEARCH SYSTEM FOR PREDICTING DEPENDENCE ,,STRUCTURE-PROPERTIES“ OF DRUG COMPOUNDS BASED ON MODIFIED ALGORITHMS OF ARTIFICIAL IMMUNE SYSTEMS**

**UDK 004.89**

The article deals with the issues of creating multi-agent Smart-system for conducting scientific research for computer molecular design of new drugs with specified properties and prediction of the „structure-property" relationship (QSAR) based on modified algorithms of artificial immune systems and other bio-inspected approaches of artificial intelligence. The main advantages and disadvantages of using various intelligent algorithms when building a Smart — system are given.

One of the problems with computer-aided molecular design of drugs is the „paradox of similarity", when the compounds differ structurally quite insignificantly, but have completely different properties. For example, an optically active substance and its mirror isomer can vary significantly in biological activity. Therefore, it is particularly relevant to develop new non-traditional approaches of artificial intelligence and algorithms that provide the ability to recognize chemical compounds with almost the same structure, but with completely different properties.

The structure of a multi-agent Smart-system for conducting scientific research has been developed based on modified algorithms of artificial immune systems and the functioning of agents has been described.

The work was carried out according to the grant of the Committee Science of the Ministry of Education and Science of the Republic of Kazakhstan (2018-2020), on the topic „Development and analysis of databases for the information system of predicting dependence „structure-property" of drug compounds based on artificial intelligence algorithms".

**Key words: **multiagent Smart-system, drugs, molecular design, prediction of the „structureproperty" relationship (QSAR), artificial immune systems, modified artificial intelligence algorithms.

References

- Roy K., Kar S., Das R.N. Understanding the Basics of QSAR in for Applications in Pharmaceutical Sciences and Risk Assessment. 2015. NY: Akademic Press, [Electron. Res.]: https://www.elsevier.com/books/ understanding-the-basics-of-qsar-for-applications-in-pharmaceutical-sciences-and-risk- assessment/roy/978-0-12-801505-6 (access date 06.02.2019).
- Roy K. Advances in QSAR Modeling. Applications in Pharmaceutical, Chemical, Food, Agricultural and Environmental Sciences. 2017. Springer, [Electron. Res.]: http://www.springer. com/in/book/9783319568492 (access date 06.02.2019).
- Gal‘bershtam N.M., Baskin 1.1., Palyutin V.A., Zemfirov N.S. Nejronnv‘e seti как metod poiska zavisimostej struktura-svojstvo organicheskikli soedinenij // Uspekhi khimii. 2003. N 72 (7). S. 706-727.
- IVANCIUC О. Structure-activity relationships in aquatic toxicology with artificial immune systems: Mechanism of toxic action classification of polar and nonpolar narcotic pollutants with CLONALG (clonal selection algorithm) //J. Molecular Design. 2007. N 6. P. 106-114.
- ChunLin Liang, Lingxi Peng. An Automated Diagnosis System of Liver Disease using Artificial Immune and Genetic Algorithms // Journal of Medical Systems, 2013. D01:10.1007/sl0916- 013-9932-9.
- Umit Yilmaz, Helmi Md Rais, Said Jadid Abdulkadir. Hybrid Swarm Intelligence Algorithms with Ensemble Machine Learning for Medical Diagnosis // Proceedings of 4th International Conference on Computer and Information Sciences (ICCOINS), 2018.
- Zhu H., Wu J., Gu J. Studies on Immune Clonal Selection Algorithm and Application of Bioinformatics // International Journal of Intelligent Engineering and Systems. 2015. V. 8. N 1. P. 1016.
- Sosnin S.B., Radchenko E.V., Palyulin V.A., Zefirov N.S. Obobshhennv‘j fragmentny‘i podkhod v issledovaniyakh QSAR/QSPR // Doklady
^{1}Akademii nauk. Khimiya. 2015, T. 463. N3. S. 297-300. - Ghasemi, Perez-Sanchez; Mehri, Perez-Garrido. Neural network and deep-learning algorithms used in QSAR studies: merits and drawbacks // Drug Discovery Today. 2018. N 23 (10). P. 1784-1790. D01:10.1016/j.drudis.2018.06.016. PMID29936244.
- Yavuz B.3., Sertkaya C., Yurtay N. Prediction of secondary structures of hemoglobin using clonal selection algorithm // Proceedings of the 7 th Int. Work. Comput. Sci. Eng. 2017. P. 1387-1391.
- SAMIGULINA G. A., SAMIGULINA Z.I. Immune network technology on the basis of Random Forest algorithm for computer aided drag design //J. Lecture Notes in Computer Science. Proceedings of the Conference on Bioinformatics and Biomedical Engineering (IWBBIO 2017). Granada (Spain): Springer, 26-28 April 2017. Part 1. P. 50-61.
- Lucija Brezocnik, Iztok Fister Jr., VlLI Podgorelec. Swarm Intelligence Algorithms for Feature Selection: A Review //J- Applied Sciences. 2018. N 8 (9). P. 1521. [Electron. Res.]: https://doi.org/10.3390/app8091521.
- Ramirez M.R., Ramirez Moreno H.B., Rojas E.M., Hurtado C., Vazquez Nunez S. O. Multi-Agent System Model for Diagnosis of Personality Types // Proceedings of the 12th International Conference Agents and Multi-agent Systems: Technologies and Applications. Smart Innovation, Systems and Technologies book series (SIST). Springer, 2018. Vol. 96. P. 209-214. DOI: 10.1007/978-3-319-92031-3_19.
- IVANOVIC M., NlNKOVIC S. Personalized Healthcare and Agent Technologies // Proceedings of the 11th International Conference Agents and Multi-agent Systems: Technologies and Applications. Smart Innovation, Systems and Technologies book series (SIST). Springer: KES-AMSTA, 2017. V. 74. P. 3-11.
- German S., Shin S., Tsourdos A. // Proceeding of the IEEE 24th Mediterranean Conference on Control and Automation (MED). 2016. P. 1020-1025.
- Samigulina G. A., Massimkanova Zh. A. Multiagent system of recognize on the basis of modified algorithms of swarm intelligence and immune network modeling // Proceedings of the 12th International Conference Agents and Multi-agent Systems: Technologies and Applications (AMSTA- 18). Australia: Springer, 20-22 June 2018. P. 199-208.
- Samigulina G.A., Samigulina Z.I. Modified immune network algorithm based on the Random Forest approach for the complex objects control // Artificial Intelligence Review. 2018. P. 1-17.
- Samigulina G. A., Samigulina Z. I. Development of multi-agent technology for prediction of the „structure-property" dependence of drugs on the basis of modified algorithms of artificial immune systems // Proceedings of International work-conference on bioinformatics and biomedical engineering (IWBBIO2018). Spain, Granada, 25-27 April 2018. P. 1-2.
- Samigulina G.A., Samigulina Z.I. Razrabotka tekhnologii immunnosetevogo modelirovaniva diva komp‘vuternogo molekulvarnogo dizajna lekarstvennv‘kh preparatov (programma diva E‘VM). Svidetebstvo о gosudarstvennoj registraczii prav na ob‘ekt avtorskogo prava v Komitete po pravam intellektuaTnoj sobstvennosti MYu RK. Astana, 28 marta 2011. N 473. 23 s.

**Bibliographic reference**: Samigulina G. A., Samigulina* Z. I. Multi-agent scientific research system for predicting dependence ,,structure-properties“ of drug compounds based on modified algorithms of artificial immune systems //journal “Problems of informatics”. 2019, № 3. P. 31-45. DOI: 10.24411/2073-0667-2019-00010

**MODELS OF NETWORKS WITH PREFERENTIAL**** ATTACHMENT**

UDK 519.177

Modeling is one of the methods of analyzing organizational principles of complex networks that define their topology and behavior. Traditionally networks with no apparent design principles were described as random graph that were first studied by Paul Erdos and Alfred Renvi [1, 2]. But it is known that the topology of real networks deviates from a random graph and many of them self-organize into scale-free (SF) state. The basic motivation of main theoretical models that come under review in this article is to explain the origin of this scale invariance.

The example of a network with in-degree and out-degree power-law distribution is the citation network observed by Derek de Sofia Price in [4, 5]. He discovered that in order the network to have these properties the rate at which a paper gets new citations should be proportional to the number that it already has. He called the process *cumulative advantage.* Now it is known as *preferential attachment* due to [6]. Barabasi-Albert model (BA) incorporates two ingredients: growth and preferential attachment. The algorithm of network evolving starts with m_{0} nodes and at each step one new node with *m < m _{0}* is added. When choosing nodes to which the new node connects it is assumed that the probability П that the new node will be connected to node i depends on the degree

k_{i} of node *i* such that .. Analytical studying predicts the emergence of power law distribution *P (k) ~ k ^{3}*.

*BA* model lays the foundation of many extensions and modifications motivated by the reason that it can’t explain a behavior of all real-world *SF* networks. Thus in [15] the model with edges rewire have been introduced. Krapviskv, Redner in [13] have examined the initial attractiveness П((k_{i}) = C + ki) and the case where probability of attachment to a node is not linear. In [12, 17, 18] the effect of ageing have been investigated and the extended models have been suggested. In the BB-model [20, 21] parameter fitness of the node accounts for the competitive aspect of link obtaining. These added parameters have an effect on a network behavior and can vary the law of degree distribution.

**Key words: **complex network analysis, scale-free networks, networks models, preferential attachment.

References

- Erdos P., RENYI A. On random graphs // Publicationes Mathematicae Debrecen. V. 6. 1959. P. 290-297.
- Erdos P., RENYI A. On the evolution of random graphs // Bull. Inst. Internat. Statist. V. 38, № 4. 1961. P. 343-347.
- Watts D. J., Strogatz S.H. Collective dynamics of “small-world” networks // Nature. 1998. V.393. P. 440-442.
- Price D. J. de Solla. Networks of Scientific Papers // Science. 1965. V. 149. P. 510-515.
- Price D. J. de Solla. A general theory of bibliometric and other cumulative advantage processes // J. of the American Society for Information Science. 1976. V. 27(5-5). P. 292-306.arabasi A.-L., Albert R. Emergence of scaling in random networks // Science. 1999. V. 286. P. 509-512.
- Barabasi A.-L., Albert R., Jeong H. Mean-field theory for scale-free random networks // Phvsica A 272. 1999. P. 173-187.
- Albert R., Barabasi A.-L. Statistical mechanics of complex networks // Rev. Mod. Phys. 2001.V. 47, N. 74. P. 47-97.
- DOROGOVTSEV S. N., Mendes J. F. F. Evolution of networks // Advances in Phys. 2002. V. 51, 1079.
- Dorogovtsev S. N., Mendes J. F. F., Samukhin A. N. Structure of growing network with preferential attachment // Phys. Rev. Lett. 2000. V. 85, 4633.
- Kullmann L., KertESZ J. Preferential growth: exact solution of the time dependent distributions // Phys. Rev. E. 63.051112. 2001.
- KRAPVISKY P.L., Redner S., Leyvraz F. Connectivity of growing random networks // Phys. Rev. Lett. V. 85, 4629. 2000.
- KRAPVISKY P. L., Redner S. Organization of growing random networks // Phys. Rev. E 63, 066123. 2001.
- Albert R., Jeong H., Barabasi A.-L. Error and attack tolerance of complex networks // Nature. 2000. V.406. P. 378-482.
- Albert R., Barabasi A.-L. Topology of evolving networks: local events and universality // Phys. Review Letters. 2000. V. 85. P. 5234-5237.
- Dorogovtsev S. N., Mendes J. F. F. Scaling behavior of developing and decaying networks // Europhes. Lettr. 2000. V. 52, N. 3. P. 33-39.
- Dorogovtsev S.N., Mendes J.F.F. Evolution networks with aging of sites // Phvs. Rev. E62, 1842. 2000.
- Zhu H., Wang X., Zhu J-Y. The effect of aging on network structure // Phys. Rev. E 68, 056121. 2003.
- Hajra К. B., Sen P. Phase transitions in an aging network // Phys. Rev. E 70, 056103. 2004.
- BlANCONl G., Barabasi A.-L. Bose-Einstein condensation in complex networks // Phys. Rev. Lett. 86, 5632. 2001.
- Bianconi G., Barabasi A.-L. Competition and multiscaling in evolving networks // Europhvsics Letters, 54: 436-442, 2001.
- Ergun G, Rodgers G. J. Growing random network with fitness // arXiv: cond-mat/0103423.
- Wang D., Song C., Barabasi A.-L. Quantifying long-term scientific impact // Science. 2013. V.342. P. 127-131.
- VAZQUEZ A. Knowing a network by walking on it: emergence of scaling // arXiv:cond- mat/0006132.
- McGLOHON M., Akoglu L., Faloutsos C. Weighted graphs and disconnected components // PAKDD’10 Proc, of the 14th Pacific-Asia conference on advances in knowledge discovery and data mining. 2010. Vol. P. 2. P. 410-421.
- Kumar R., Raghavan P., Rajagopalan S., Sivakumar D., Tomkins A., Upfal E. Stochastic models for the Web graph // Proc, of the 41th IEEE svmp. on foundations of computer science. 2000. P. 57-65.
- Leskovec J., Kleinberg J. M., Faloutsos C. Graph evolution: Densification and shrinking diameters // ACM Transactions on Knowledge Discovery from Data (TKDD). 2007. V. 1, iss. 1. Art. 2.

**Bibliographic reference**: Shcherbakova N. G. Models of networks with preferential attachment //journal “Problems of informatics”. 2019, № 3. P. 46-61. DOI: 10.24411/2073-0667-2019-00011

**FOREWORD ****ТО**** ****THE ARTICLE BY A. P. ERSHOV „COMPUTABILITY IN ARBITRARY DOMAINS AND BASES“**

In this issue, in the section “Turning over the old pages”, the editorial board offers readers the article by A. P. Ershov “Computability in arbitrary fields and bases”.

Reading this article many years ago, I was struck at that time, and now I am still amazed at how A. P. Ershov was able to specifically, not abstractly, foresee that due to the development of programming, the mathematical logic will be an applied discipline.

For programmers-encoders, this article may seem to be uninteresting and unnecessary. Those, who want to understand, even if not completely, may wish to better understand the basis of the modern programming, to understand and realize the connections of set theory, theory of algorithms and mathematical logic and their role in the development of programming in the coming decades.

As an epigraph to this publication, an extract from the same article begs:

The spread of computers and programming make mathematical logic an applied science. The caste of mathematical logicians, the priest-keepers of the sacred fire of the foundations of mathematics, to their amazement finds themselves face to face with the invasion of hordes of programmers, whose mathematical culture is close to barbarism, but who, nevertheless, inspired by their prophets of structural programming, seek to light a torch from the life-giving fire and drag it to them in order to illuminate the corners of their units clogged with lamps, OS / 360 instructions, Fortran forms and other symbols of idolatry. Nevertheless, in order to establish a union of such different cultures, each side - logicians and programmers - needs to overcome their inferiority complexes, to develop a capacity for mutual understanding and, what is more important, to become convinced in the interest to the other side.

The text is technically somewhat out of date, for example, the symbols of idolatry among programmers are now different, but the thoughts expressed are true today. Indeed, the era of programming, based largely on the mathematical logic is beginning. Logic is slowly but surely becoming an applied science, and programming systems in the mass-scale programming are gradually beginning to be replaced by systems for the automatic construction of algorithms and programs on the bases of the axiomatic description of a subject domain. There appear systems, in which an algorithm and a program are taken from the axiomatic description of the subject area, see, for example, Charm and LuNA systems (Victor Malyshkin. Active Knowledge, LuNA and Literacy for Oncoming Centuries // Springer, LNCS, Vol. 9465, P. 292 -303). Nowadays the students, who are trying to minimize their knowledge of mathematics, may have problems in finding a job in the future: they simply will not be able to understand the formulation of problems and how to solve them.

*V**. **E**. **Malyshkin*

**Bibliographic reference: **Ershov A. P. Сomputability in arbitrary domains and bases //journal “Problems of informatics”. 2019, № 3. P. 62-92. DOI: 10.24411/2073-0667-2019-00012