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2017 year, number 4
K.V. Voronin^{1}, Yu.M. Laevsky^{1,2}
^{1}Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Acad. Lavrentieva 6, Novosibirsk, 630090, Russia ^{2}Novosibirsk State University, Pirogova st., 2, Novosibirsk, 630090, Russia
Keywords: смешанный метод конечных элементов, тепловой поток, схема расщепления, схема предикторкорректор, mixed finite element method, heat flux, splitting scheme, predictorcorrector scheme
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In this paper we propose and study the flux predictorcorrector scheme in the threedimensional case. This scheme is depleted of drawbacks of that constructed on the basis of the DouglasGunn prototypescheme. The scheme proposed demonstrates the second order of accuracy.

A. Gasnikov^{1,2}, E. Gasnikova^{1}, P. Dvurechensky^{2,3}, A. Mohammed^{1}, E. Chernousova^{1}
^{1}Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Russia, 141700 ^{2}Institute for Information Transmission Problems RAS, Bolshoy Karetny per. 19, build. 1, Moscow, Russia, 127051 ^{3}Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, Berlin, Germany, 10117
Keywords: марковская цепь, эргодическая теорема, мультиномиальное распределение, концентрация меры, оценка максимального правдоподобия, Google problem, градиентный спуск, автоматическое дифференцирование, степенной закон распределения, Markov chain, ergodic theorem, multinomial distribution, measure concentration, maximum likelihood estimate, Google problem, gradient descent, automatic differentiation, power law distribution
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In Part 1 of this paper, we consider the webpages ranking problem also known as the problem of finding the PageRank vector or Google problem. We discuss the connection of this problem with the ergodic theorem and describe different numerical methods to solve this problem together with their theoretical background, such as Markov Chain Monte Carlo and equilibrium in a macrosystem.

A.V. Kelmanov^{1,2}, S.M. Romanchenko^{1}, S.A. Khamidullin^{1}
^{1}Sobolev Institute of Mathematics SB RAS, 4 Acad. Koptyug avenue, Novosibirsk, Russia, 630090 ^{2}Novosibirsk State University, Pirogova st., 2, Novosibirsk, 630090, Russia
Keywords: последовательность, евклидово пространство, минимум суммы квадратов расстояний, трудность, FPTAS, euclidean space, sequence, minimum sum of squared distances, hardness, FPTAS
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We consider a strongly hard Euclidean problem of finding a subsequence in a finite sequence. The criterion of the solution is a minimum sum of squared distances from the elements of a sought subsequence to its geometric center (centroid). It is assumed that a sought subsequence contains a given number of elements. In addition, a sought subsequence should satisfy the following condition: the difference between the indices of each previous and subsequent points is bounded with given lower and upper constants. We present an approximation algorithm of solving the problem and prove that it is a fully polynomialtime approximation scheme (FPTAS) when the space dimension is bounded by a constant.

Yu. Kurdyaeva^{1}, S. Kshevetskii^{1}, N. Gavrilov^{2}, E. Golikova^{3}
^{1}Immanuel Kant Baltic Federal University IKBFU, Building 14 A, Nevskogo str., Kaliningrad ^{2}SaintPetersburg University, Ulyanovskaya str., Peterhof, SaintPetersburg, Russia ^{3}A.M. Obukhov Institute of Atmospheric Physics RAS, Pyzhevskii per., 3, Moskou, Russia, 119017
Keywords: численное моделирование, модель атмосферы, акустикогравитационные волны, нелинейность, корректность, граничная задача, суперкомпьютерная программа, numerical simulation, atmospheric model, acousticgravity waves, nonlinearity, correctness, boundary problem, supercomputer program
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Currently, there are international microbarograph networks, with high resolution recording the wave pressure variations on the Earth's surface. This increases the interest in the problems of wave propagation in the atmosphere from variations in the atmospheric pressure. A complete system of nonlinear hydrodynamic equations for an atmospheric gas with lower boundary conditions in the form of wavelike variations on the Earth's surface is considered. Since the wave amplitudes near the Earth's surface are small, linearized equations are used in the analysis of the problem correctness. With the help of the wave energy functional method, it is shown that in the nondissipative case, the solution of the boundary value problem is uniquely determined by the variable pressure field on the Earth's surface. The corresponding dissipative problem is correct if, in addition to the pressure field, suitable conditions on the velocity and temperature on the Earth's surface are given. In the case of an isothermal atmosphere, the problem admits analytical solutions that are harmonic in the variables x and t . A good agreement between numerical solutions and analytical ones is shown. The study has shown that in the boundary value problem, the temperature and density can rapidly vary near the lower boundary. An example of the solution of a threedimensional problem with variable pressure on the Earth's surface, taken from experimental observations, is given. The developed algorithms and computer programs can be used to simulate the atmospheric waves from pressure variations on the Earth's surface.

A.S. Popov
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Acad. Lavrentieva 6, Novosibirsk, 630090, Russia
Keywords: численное интегрирование, инвариантные кубатурные формулы, инвариантные многочлены, группа вращений икосаэдра, numerical integration, invariant cubature formulas, invariant polynomials, icosahedral group of rotations
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An algorithm of the search for the best (in a sense) cubature formulas on a sphere that are invariant with respect to the transformations of the icosahedral group of rotations with inversion is described. This algorithm is applied to finding the parameters of all the best cubature formulas of this symmetry type up to the 79th order of accuracy. The parameters of the new cubature formulas of the 21st, 25th and 29th orders of accuracy to 16 significant digits are given.

M.V. Urev^{1,2,3}, Kh.Kh. Imomnazarov^{1}, JianGang Tang^{4}
^{1}Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Acad. Lavrentieva 6, Novosibirsk, 630090, Russia ^{2}Novosibirsk State University, Pirogova st., 2, Novosibirsk, 630090, Russia ^{3}Siberian Institute of Management, ul. Nizhegorodskya, 6, Novosibirsk, 630102 ^{4}YiLi Normal University, 448, Jiefang Road, Yinning Xinjiang, P.R. of China
Keywords: переопределенная стационарная система двухскоростной гидродинамики, множитель Лагранжа, метод конечных элементов, overdetermined twovelocity stationary hydrodynamics system, Lagrange multiplier, finite element method
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In this paper we investigate the twovelocity stationary hydrodynamics system with a single pressure and inhomogeneous divergent and boundary conditions for the two velocities. This system is overdetermined. By replacing the unknown functions, the problem is reduced to a homogeneous one. The solution of the resulting system is reduced to the consecutive solutions of the two boundary value problems: the Stokes problem for a single velocity and pressure, and overdetermined system for the other velocity. We present the generalized statements of these problems and their discrete approximation using the finite element method. To solve the overdetermined problem we apply a version of the regularization methods.

V.N. Chugunov^{1}, Kh.D. Ikramov^{2}
^{1}Institute of Numerical Mathematics, Russian Academy of Sciences, Gubkina st., 8, Moscow, Russia, 119333 ^{2}Lomonosov Moscow State University, Moscow, Leninskie Gory, Russia, 119991
Keywords: теплицева матрица, ганкелева матрица, П•циркулянт, квазикоммутирующие матрицы, Toeplitz matrix, Hankel matrix, П•circulant, quasicommuting matrices
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We say that the square matrices A and B are of the same order quasicommute if AB = σ BA for some scalar σ. Classical relations of commutation and anticommutation are particular cases of this definition. We give a complete description of pairs of the quasicommuting Toeplitz and Hankel matrices for σ ≠ ± 1.

V.T. Shevaldin^{1}, O.Ya. Shevaldina^{2}
^{1}Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, S. Kovalevskaja st., 16, Ekaterinburg, Russia, 620990 ^{2}Ural Federal University, 19 Mira street, Ekaterinburg, Russia, 620002
Keywords: константа Лебега, локальные кубические сплайны, равноотстоящие узлы, Lebesgue constants, local cubic splines, equallyspaced knots
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It is proved that the uniform Lebesgue constant (the norm of a linear operator from C to C) of local cubic splines with equallyspaced knots, which preserve cubic polynomials, is equal to 11/9.

