**
Number 1, pp. 1-123
Number 2, pp. 125-233
Number 3, pp. 235-342
Number 4, pp. 343-467**

**On the anniversary of Anatoly
Konovalov**

**On the 90th anniversary of Gury
Marchuk**

**Aleksandrov****
**
**Vladimir**** **
**Mikhailovich**

**
A singular solution to the problem of minimizing resource consumption**

**
**

An iterative method of finding a singular solution to the problem of minimizing
resource consumption has been developed. This method is based on the information
about the finite control structure. A condition for existence of a singular
solution is obtained. The limit value for
transferring the time between the normal and the singular solutions is found. A
relation between the variations of the control switching moments and the
variations of the initial conditions of the adjoint system is determined. A
system of linear algebraic equations relating the variations of the initial
conditions of the adjoint system to the variations of the phase coordinates from
a given final state of the system is obtained. The computational algorithm, the
modeling results and

the numerical calculations are presented.

**Keywords:**

**Ambos
Andrey Yurevich**

**
Numerical models of mosaic homogeneous isotropic random fields and
problems of radiative transfer**

**
**

The new algorithms of statistical modeling of radiative transfer through
different types of stochastic homogeneous isotropic media have been created. To
this end a special geometric implementation of «the maximum cross-section
method» has been developed. This implementation allows one to take into account
the radiation absorption by the exponential multiplier factor. The dependence of
a certain class of solution functionals of the radiative transfer equation on
the correlation length and the field type is studied theoretically and by means
of numerical experiments. The theorem about the convergence of these functionals
to the corresponding functionals for an average field with decreasing the
correlation length up to zero has been proved.

**Keywords:**

**
Artemiev Sergey Semenovich, Yakunin Mikhail Aleksandrovich**

**
Analysis of the accuracy of estimates of the first moments of solving SDE
with Wiener and Poisson components by Monte Carlo method
**

In this paper, we investigate the accuracy of estimates of the first moments of
a numerical solution to SDE with the Wiener and the Poisson components by the
generalized Euler explicit method. The exact expressions for the mathematical
expectation and variance of the test SDE solution are obtained. These
expressions allow us to investigate the dependence of the accuracy of estimates
obtained by Monte Carlo method on the values of SDE parameters, the size of an
integration step, and the size of an ensemble of simulated trajectories of the
solution. The results of the numerical experiments are presented.

**Keywords: **

**
Blatov****
**
**Igor**** **
**Anatolevich****,
**
**Kitaeva**** **
**Elena****
**
**Victorovna**

**
Convergence of the adapting grid method of Bakhvalov's type for
singularly perturbed
boundary value problems**

**
**

We consider the Galerkin finite element method for non-self-adjoint boundary
value problems on Bakhvalov's grids. Using the Galerkin projections method the
convergence of a sequence of computational grids with an unknown boundary of the
boundary layer has been proved. Numerical examples are presented.

**Keywords:**

**
Vabishchevich **
**
Petr Nikolaevich****, Grigoriev **
**Alexander Vissarionovich**

**
Numerical modeling of a fluid flow in anisotropic fractured porous media**

**
**

A model of double porosity in the case of an anisotropic fractured porous medium is considered (Dmitriev, Maksimov; 2007). The function of the exchange flow between fractures and porous blocks, which depends on the direction of a flow, is investigated. The flow function is based on the difference between pressure gradients. This feature enables one to take into account anisotropic filtering properties in a more general form. The results of the numerical solution of the model two-dimensional problem are presented. The computational algorithm is based on the finite element spatial approximation and the explicit-implicit temporal approximation.

**Keywords:**

**Krukier Lev Abramovich****,
Krukier Boris L’vovich, Huang Yu-Mei.**

**
The skew-symmetric iterative method for solving the
convection-diffusion-reaction equation with the alternating-sign reaction
coefficient**

**
**

The iterative product, that is, the triangular skew-symmetric method (PTSM) is
used to solve linear algebraic equation systems obtained by approximation of a
central-difference scheme of the first boundary value problem of
convection-diffusion-reaction and standard grid ordering. Sufficient conditions
of a non-negative definiteness of the matrix resulting from this approximation
have been obtained for a non-stationary sign of the reaction coefficient. This
feature ensures the convergence of a sufficiently wide class of iterative
methods, in particular, the PTSM. In the test problems, the compliance of the
theory with computational experiments is verified, and comparison of the PTSM
and the SSOR is made.

**Keywords:**

**
Singh P., Kadalbajoo M.K., Sharma K.**

**
Probability density function of leaky integrate-and-fire model with Lèvy
noise and its numerical approximation**

**
**

We investigate a numerical analysis of a leaky integrate-and-fire model with
Lèvy noise. We consider a neuron model in which the probability density function
of a neuron in some potential at any time is modeled by a transport equation.
Lèvy noise is included due to jumps by excitatory and inhibitory impulses. Due
to these jumps the resulting equation is a transport equation containing two
integrals in the right-hand side (jumps). We design, implement, and analyze
numerical methods of finite volume type. Some numerical examples are also
included.

**Keywords:**

**
Tanana****
**
**Vitaliy**** **
**Pavlovich****,
**
**Vishnyakov****
**
**Evgeniy****
**
**Yurevich****, **
**Sidikova**** **
**Anna****
**
**Ivanovna**

**
About an approximate solution to the Fredholm integral equation of the first
kind by the residual method**

**
**

The Tikhonov finite-dimensional approximation was applied to an integral equation of the first kind. This allowed us to use the variation regularization method of choosing the regularization parameter residuals from the principle of reducing the problem to a system of linear algebraic equations. The estimate of accuracy of the approximate solution with allowance for the error of the finite-dimensional problem approximation has been obtained. The use of this approach is illustrated on an example of solving an inverse boundary value problem for the heat conductivity equation.

**Keywords:**

**
Shreifel Igor Semyonovich, Eliseev Ivan Nikolaevich**

**
Theoretical basis of the iterative process of the joint assessment of
difficulties in tasks and levels of training students**

**
**

In this paper, we study the iterative process of the joint numerical assessment
of levels of training students and difficulties in tasks of diagnostic tools
using the dichotomous response matrix *A* of *N *x
*M* size, with allowance for the
contribution of tasks of different difficulty to the assessments obtained. It is
shown that not for any matrix *A* there
exist infinite iterative sequences, and in the case of their existence, they do
not always converge. A wide range of sufficient conditions for their convergence
have been obtained, which are based on the following: 1) matrix
*A* contains at least three different
columns; 2) if one places the columns of *A*
in non-decreasing order of column sums, then for any position of the vertical
dividing line between the columns there exists a row, which has, at least, one
unity to the left of the line, and at least one zero to the right of the line.
It is established that the response matrix *
A* obtained as a result of testing reliability satisfies these two
conditions. The properties of such matrices have been studied. In particular,
the equivalence of the above conditions of primitiveness of the square matrix
*B* of order
*M* with the entries *b _{ij}*
=∑

**Keywords:**

**
Babicheva Galina Andreevna, Kargapolova Nina Aleksandrovna, Ogorodnikov Vasilii
Aleksandrovich**

**
Special algorithms for simulation of homogeneous random fields**

**
**

In this paper, two new algorithms for the simulation of homogeneous random
fields are proposed. Both algorithms are based on the widespread algorithm “in
rows and columns” for the simulation of the Gaussian fields with special
correlation functions. Applying the algorithms developed makes possible to
efficiently simulate homogeneous random fields with non-convex correlation
functions.

**Keywords:**
homogeneous random field, stochastic simulation, randomization.

**
Zhitnikov****
**
**Vladimir**** **
**Pavlovich****,
**
**Sherykhalina****
**
**Nataliya**** **
**Mikhailovna****,
**
**Muksimova****
**
**Roza****
**
**Ravilevna**

**
The peculiarities of error accumulation in solving problems for simple
equations of mathematical physics by finite difference methods**

**
**

A
mixed problem for a one-dimensional heat equation with several versions of
initial and boundary conditions is considered. Explicit and implicit schemes are
applied for the solution. The sweep method and the iteration methods are used
for the implicit scheme for solving the implicit system of equations. The
numerical filtration of a finite sequence of results obtained for different
grids with an increasing number of nodal points is used to analyze errors of the
method and rounding. In addition, to investigate the rounding errors, the
results obtained with several lengths of the machine word mantissa are compared.
The numerical solution of the mixed problem for the wave equation is studied by
similar methods.

The occurrence of deterministic dependencies of the error in the numerical
method and the rounding on spatial coordinates, time and the number of nodes is
revealed. The source models to describe the behavior of errors in terms of time
are based on the analysis of the results of numerical experiments for different
versions of conditions of problems. In accord with such models, which were
verified by the experiment, the errors can increase, decrease or stabilize
depending on conditions over time similar to changing the energy or mass.

**Keywords:**

**
Kabanikhin Sergey Igorevich, Krivorotko Olga Igorevna**

**
A numerical algorithm for computing tsunami wave amplitude
**

A numerical algorithm for computing tsunami wave front amplitude is proposed.
The first step consists in solving an appropriate eikonal equation. The eikonal
equation is solved by the Godunov approach and the bicharacteristic method. The
qualitative comparison of the two above methods is described. Then a change in
variables associated with the eikonal solution is introduced. At the last step,
using the expansion of the fundamental solution of shallow water equations in
the sum of singular and regular parts, we obtain the Cauchy problem for the wave
amplitude. This approach allows one to reduce computer costs. The numerical
results are presented.

**Keywords:**
shallow
water equations, tsunami amplitude, fundamental solution, eikonal equation,
finite difference approach.

**
Kansal **
**Munish****, Kanwar V., Bhatia **
**Saurabh**

**
Optimized mean based second derivative-free families of Chebyshev-Halley
type methods**

**
**

In this paper, we present new interesting fourth-order optimal families of
Chebyshev-Halley type methods free from second-order derivatives. In terms of
computational cost, each member of the families requires two functions and one
first-order derivative evaluation per iteration, so that their efficiency
indices are 1.587. It is found by way of illustration that the proposed methods
are useful in high precision computing environment. Moreover, it is also
observed that larger basins of attraction belong to our methods, whereas the
other methods are slow and have darker basins, while some of the methods are too
sensitive to the choice of the initial guess.

**Keywords:**
basins of attraction, Newton's method, King's methods, optimal iterative
methods, efficiency index.

**Korneev
Vladimir Dmitrievich, Sveshnikov Viktor Mitrofanovich**

**
Parallel algorithms and domain decomposition technologies for solving
three-dimensional boundary value problems on quasi-structured grids**

**
**

A new approach to the decomposition method of a three-dimensional computational
domain into subdomains, adjoint without overlapping, which is based on a direct
approximation of the Poincare-Steklov equation at the conjugation interface, is
proposed. With the use of this approach, parallel algorithms and technologies
for three-dimensional boundary value problems on quasi-structured grids are
presented. The experimental evaluation of the parallelization efficiency on the
solution of the model problem on quasi-structured parallelepipedal coordinated
and uncoordinated grids is given.

**Keywords:**
boundary value
problems, domain decomposition methods, Poincare-Steklov equation,
quasistructured grids, algorithms and technologies of parallelization.

**Monakhov
Oleg Gennad'evich, Monakhova Èmiliya Anatol'evna, Pant **
**Millie**

**
Application of differential evolution algorithm for optimization of
strategies based on financial time series**

**
**

An approach to optimization of trading strategies (algorithms) based on
indicators of financial markets and evolutionary computation is described. A new
version of the differential evolution algorithm for the search for optimal
parameters of trading strategies for the trading profit maximization is used.
The experimental results show that this approach can considerably improve the
profitability of the trading strategies.

**Keywords:**
trading
strategy, parallel genetic algorithm, technical analysis, financial indicator,
template, evolutionary computation.

**Stepanova
LarisaValentinovna, Yakovleva Ekaterina Michailovna**

**
Asymptotics of eigenvalues of the nonlinear eigenvalue problem arising
from the near mixed-mode crack-tip stress-strain field problems**

**
**

In the present paper, approximate analytical and numerical solutions to
nonlinear eigenvalue problems arising in the nonlinear fracture mechanics in
analysis of stress-strain fields near a crack tip under a mixed mode loading are
presented. Asymptotic solutions are obtained by the perturbation method (the
small artificial parameter method). The artificial small parameter is a
difference between the eigenvalue corresponding to the nonlinear eigenvalue
problem and the eigenvalue related to the linear “undisturbed” problem. It is
shown that the perturbation technique gives an effective method of solving
nonlinear eigenvalue problems in the nonlinear fracture mechanics. Comparison of
numerical and asymptotic results for different values of the mixity parameter
and hardening exponent shows good agreement. Thus, the perturbation theory
technique for studying nonlinear eigenvalue problems is offered and applied to
eigenvalue problems arising from the fracture mechanics analysis in the case of
a mixed mode loading.

**Keywords:** nonlinear eigenvalue problem, perturbation theory small parameter method,
asymptotics of stress and strain fields in the vicinity of the
mixed-mode crack, mixed-mode loading, power constitutive law,
eigenspectrum.

**Favorskaya Alena Victorovna, Petrov Igor Borisovich **

**
The study of increased order grid-characteristic methods on unstructured
grids
**

**
**

We study the grid-characteristic methods for solving hyperbolic systems
using a high order interpolation on unstructured tetrahedral and triangular
grids for approximation. We consider the interpolation with orders from the
first to the fifth included. Also, one-dimensional finite difference schemes
appropriate for the considered methods are given. We study these schemes in
terms of stability. The grid-characteristic method on unstructured triangular
and tetrahedral grids are successfully used for solving the seismic prospecting
problems, including, seismic prospecting in the conditions of the Arctic shelf
and permafrost, as well as for solving seismic problems, problems of dynamic
deformation and destruction, studying anisotropic composite materials.

**Keywords:**
grid-characteristic method, numerical simulation, unstructured grids, high order
interpolation.

Number 3, pp. 235-342

Averina
Tat'yana Aleksandrovna

**
Using a randomized method of a maximum cross-section for simulating
random structure systems with distributed transitions****
** (in
Russian),
pp.

In this paper we consider the random structure systems with distributed
transitions. A theorem about the form of conditional structure distributions has
been proved. To simulatethese distributions a statistical algorithm using a
randomized method of maximum cross-section is constructed. Also, a modified
version of this algorithm using the simulation of one random number has been
constructed. The algorithms developed were used for the simulation of the
numerical solution of random structure systems with distributed transitions. The
theorem about a weak convergence of the numerical solution, obtained by the
algorithms developed has been proved.

**Keywords:**

**
Zorkal'tsev Valerii Ivanovich**

**
The search for admissible solutions by the interior point algorithms **(in
Russian),
pp.

**
**A family of interior point algorithms for the linear programming
problems is considered. In these algorithms, the entering into the domain of
admissible solution of the original problem is represented as optimization
process of the extended problem. This extension is realized by adding just one
new variable. The main objective of the paper is to give a theoretical
justification of the proposed procedure of entering into the feasible domain of
the original problem, under the assumption of non-degeneracy of the extended
problem. Particularly, we prove that given the constraints of the original
problem being consistent, the procedure leads to a relative interior point of
the feasible solutions domain.

**Keywords:**
interior
point algorithm, linear programming, techniques of arriving at the feasible
solutions region.

Krukier Lev A., Martynova Tatiana S.

**
Preconditioning of GMRES by the skew-Hermitian iterations**** **(in
Russian),
pp.

A class of preconditioners for solving non-Hermitian positive definite
systems of linear algebraic equations is proposed and investigated. It is based
on the Hermitian and skew-Hermitian splitting of the initial matrix. The
generalization for saddle point systems which have

semidefinite or singular (1,1) blocks is given. Our approach is based on an
augmented Lagrangian formulation. It is shown that such preconditioners are
effective for the iterative solution of systems of linear algebraic equations by
the GMRES.

**
Keywords:
**
Hermitian and skew-Hermitian splitting, iterative methods, preconditioning,
Krylov subspace method, saddle point linear system.

Owolabi Kolade M.

**
Mathematical study of two-variable systems with adaptive numerical
methods **(in
Russian),
pp.

In this paper, we consider reaction-diffusion systems arising from
two-component predator-prey models with Smith growth functional response. The
mathematical approach used here is twofold, since the time-dependent partial
differential equations consist of both linear and nonlinear terms. We discretize
the stiff or moderately stiff term with a fourth-order difference operator,
advance the resulting nonlinear system of ordinary differential equations with a
family of two competing exponential time differencing (ETD) schemes, and analyze
them for stability. A numerical comparison of these two methods for solving
various predator-prey population models with functional responses is also
presented. Numerical results show that the techniques require less computational
work. Also in the numerical results, some emerging spatial patterns are
unveiled.

**
Keywords:
**
predator-prey model, ETD methods, nonlinear, pattern formation,
reaction-diffusion, stability, time-dependent PDE, Turing
instability.

Platov
Gennadii Alekseevich

**
The influence of the shelf zone relief and the coastline geometry on
coastal trapped waves **(in
Russian),
pp.

This paper presents the results of numerical experiments with a model of
the coastal trapped waves, which made it possible to identify two features that
are important in terms of the regional modeling of the shelf zone interaction
with the open ocean. The first feature is the fact that the wave train of this
type may be formed as a result of the wind action at a considerable distance
from the place where their impact may occur. The propagation of waves along the
coastline takes place without significant loss of wave energy, provided that the
coastline and topography of the shelf zone contain no features comparable to the
Rossby radius. However, the wave loses its energy while passing capes, submarine
canyons and in the case when the width of a shelf decreases. For the regional
modeling, the possibility of remote wave generation should be well understood
and taken into account. The second feature is that a propagating wave is able to
spend part of its energy on the formation of density anomalies on a shelf by
raising the intermediate waters of the adjoining offshore areas of the open
ocean. Thus, the coastal trapped waves carry the wind energy from the areas of
the wind impact to other coastal areas, where it can bring about the formation
of density anomalies and other types of motion.

**
Keywords:
**
coastal
trapped waves, shelf zone, marginal seas.

Plotnikov
Mikhail
Yu.,
Shkarupa
Elena
Valer'evna

**
Evaluation of statistical error when calculating velocity and temperature
components by the direct simulation Monte Carlo method ** (in
Russian),
pp.

The direct simulation Monte Carlo method is now widely used to solve the
problems of rarefied gas dynamics. While solving stationary problems a special
feature of the method is using dependent sample values of random variables to
calculate macroparameters of a gas flow. In this paper, the possibility of using
the results of statistical physics to estimate the statistical error of the DSMC
method is theoretically analyzed. A simple approach to approximate evaluating
the statistical error while calculating components of the velocity and
temperature is proposed. The approach is tested on a number of problems.

**
Keywords:**
direct
simulation Monte Carlo method, statistical error, equilibrium statistical
physics.

Rozhenko
Aleksandr Iosifovich, Fedorov Egor Alexandrovich

**
On an algorithm of bilateral restrictions smoothing with spline **(in
Russian),
pp.

In this paper, the problem of constructing a spline
*
σ*
in the Hilbert space satisfying bilateral restrictions
*z ^{-} ≤ A *

**
Keywords:
**
smoothing, spline, Hilbert space, convex programming, reproducing
mapping, radial basis function.

**
****
**

**
Recovering a tsunami source and designing an observational system based
on the
r-solution
method** (in
Russian), pp.

This study deals with the application of the
*r*-solution method to recover the initial tsunami waveform in a
tsunami source area by inverting the remote water-level measurements for a real
event. The inverse problem in question is regarded as the so-called ill-posed
problem and it is regularized by means of the least square inversion using the
truncated Singular Value Decomposition method. The method presented allows one
to control the instability of the numerical solution and to obtain an acceptable
result in spite of the ill-posedness of the problem. Moreover, it is possible to
make a preliminary prediction of the quality of the inversion with a given set
of observational stations and to estimate further changes in the inversion
result after modifying the monitoring system.

**Keywords:** tsunami
numerical modeling, ill-posed inverse problem, regularization, singular value
decomposition and *r*-solution.

**
**
**
Zhukovskaia Tatyana Vladimirovna, Zhukovskiy
Evgeny Semenovich**

**
On iterative methods for solving equations with covering mappings
**

In this paper we propose an
iterative method for solving the equation
ϓ
(*x*,*x*)=*y*, where a mapping
ϓ acts in metric spaces, is covering in the first
argument and Lipschitzian in the second one. Each subsequent element
*x _{i}*

**Keywords:**
iterative methods for solving equations, covering mappings in metric spaces,
approximate solution.

Leonov
Aleksandr Sergeevich

**Regularizing algorithms with optimal and extra-optimal quality
**
(in
Russian), pp.

The notion of a special quality for approximate solutions to ill-posed
inverse problems is introduced. A posteriori estimates of the quality are
studied for different regularizing algorithms (RA). Examples of typical quality
functionals are provided, which arise in solving linear and
nonlinear inverse problems. The techniques and the numerical
algorithm for calculating a posteriori quality estimates for approximate
solutions of general nonlinear inverse problems are developed. The new notions
of optimal and extra-optimal quality of a regularizing algorithm are introduced.
The theory of regularizing algorithms with optimal and extra-optimal quality is
presented, which includes an investigation of optimal properties for estimation
functions of the quality. Examples of regularizing algorithms with extra-optimal
quality of solutions are given, as well as examples of regularizing algorithms
without such property. The results of numerical experiments illustrate a
posteriori quality estimation.

**Keywords:**
ill-posed problems, regularizing algorithms, quality of approximate solution, a
posteriori quality estimates, RA with extra-optimal quality.

**Mastryukov Aleksandr Fedorovich**

**
Optimal finite difference schemes for the wave equation
**

This paper considers the solution of the two-dimensional wave equation
with the use of the Laguerre transform. The optimal parameters of finite
difference schemes for this equation have been obtained. Numerical values of
these optimal parameters are specified. The finite difference schemes of second
order with optimal parameters give the accuracy of the solution to the equations
close to the accuracy of the solution by the scheme of fourth order. It is shown
that using the Laguerre decomposition, the number of optimal parameters in
comparison with the Fourier decomposition can be reduced. This reduction leads
to simplification of finite difference schemes and to reduction of the number of
computations, i.e. the efficiency of the algorithm.

**Keywords:****
wave equation, electromagnetic wave, finite-difference, optimal,
accuracy, Laguerre method, linear system of equations.**

**Penenko Aleksey Vladimirovich, Penenko Vladimir Viktorovich,
Tsvetova Elena Aleksandrovna**

**
Sequential data assimilation algorithms in air quality monitoring models
based on weak-constraint variational principle
**

**
**
Data assimilation problem for non-stationary model is considered as a
sequence of the linked inverse problems which reconstruct, taking into account
the different sets of measurement data, the space-time structure of the state
functions. Data assimilation is carried out together with the identification of
additional unknown function, which we interpret as a function of model
uncertainty. The variational principle serves as a basis for constructing
algorithms. Different versions of the algorithms are presented and analyzed.
Based on the discrepancy principle, a computationally efficient algorithm for
data assimilation in a locally one-dimensional case is constructed. The
theoretical estimation of its performance is obtained. This algorithm is one of
the core components of the data assimilation system in the frames of splitting
scheme for the non-stationary three-dimensional transport and transformation
models of atmospheric chemistry.

**Keywords: ** data
assimilation, variational principle, weak-constraint, direct and inverse
problems, model as regularizer, sequential algorithms.

**Plieva Lyubov' Yur'evna**

**
Quadrature interpolation type formulas for hypersingular
integrals in the interval of integration
**

**
**A hypersingular integral on the interval of integration with the weight
function is considered. We prove the spectral ratios for hypersingular integrals
on [-1, 1]. The quadrature formulas for certain integrals with the weight
function are constructed. The estimation error is presented.

**Keywords:** hypersingular integral, quadrature formula, the estimation
error.

**
A difference scheme for a conjugate-operator model of the heat conduction
problem on non-matching grids
**

On non-matching grids discrete analogue conjugate-operator models of heat
conduction, keeping the structure of the original model are constructed.
Numerical experiments show that the difference scheme converges with second
order of accuracy for the case of discontinuous parameters of the medium in the
Fourier law and non-uniform grids.

**Keywords: **problem
of heat conductivity, mathematical model, discrete analog, non-matching grid,
convergence, difference scheme.

**Cherdantsev Sergei Vasil'evich****,
Cherdantsev Nikolai Vasil'evich**

**
Analysis of the pontoon fluctuations with a seasonally changing parameter
of stability on
the astir surface of finite water
depth**

It is shown that due to the periodic changes in metacentric heights of a
pontoon on the astir surface of liquids in the sump of an open coal mine, the
pontoon is capable to produce parametric pitching, both in the longitudinal and
in the transverse directions. The equation, describing parametric pitching, is
transformed to the Mathieu equation, whose factors depend both on the own
frequencies and the pontoon floatability features on «calm water», and on the
frequency of fluctuation of a liquid, which, in turn, is defined by the sump
size. The installed regularities between parameters, characterizing parametric
pitching in the longitudinal and transverse directions, and areas of its
instability are revealed.

**Keywords:** sump of
an open coal mine, pontoon, potential of the velocities, frequency of the waves,
waterline, metacentric heights, added masses of liquid, parametric pitching
pontoon, Mathieu equation, Inc-Strutt stability diagram.

**Chugunov Vadim Nikolaevich, Ikramov Khakim
Dododzhanovic**

**
Classification of real pairs of commuting Toeplitz and Hankel matrices
**

We give a complete description of the set of matrix pairs such that
*T* is a real Toeplitz matrix,
*H* is a real Hankel matrix, and
*TH = HT*.

**Keywords: **
Toeplitz matrix, Hankel matrix, circulant, skew-circulant, commutativity.