No 2 (2014)

This paper deals with the nonexistence of global solutions of quasi-linear backward parabolic equations for p-Laplacian operators: ut = −div Dup−2Du + uq−1u, x,t ∈ Ω × (0,∞) with the Dirichlet boundary condition u = 0 on the boundary ∂Ω × (0,∞) and a bounded integrable initial function u(x,0) = u0(x), where Ω is a smoothly bounded domain in ℝN. We also consider this problem in the case of Ω = ℝN. The problem is analyzed using the test function method, developed by E. L. Mitidieri and S. I. Pohozaev [Mitidieri E., Pohozaev S. I. A Priory Estimates and the Absence of Solutions of Non- linear Partial Differential Equations and Inequalities // Proceedings of the Steklov Institute of Mathematics, 2001. - Vol. 234, No 3. - 362 p. - (in russian).] It is based on deriving a priory estimates for solutions by an algebraic analysis of the integral form of inequalities with an optimal choice of test functions. With the help of this method, we obtain the nonexistence conditions based on the weak formulation of the problem with test functions of the form: φ(x,t) = ±u±(x,t) + εδφR(x,t),for ε > 0,δ > 0, where u+ and u− are the positive and negative parts of the solution u of the problem respectively and φR is a standard cut-off function whose support depends on a parameter R > 0.

One of the main development trends of telecommunication networks is the process of transport networks fotonization that should lead to creation of the whole Optical Transport Network (All-Optical Network, AON). This conception let not to increase reserves that are required for meeting growing demand for data transfer. According to OBS technology packets in ingress node are gathered in bursts. Collisions appears in case of two or more bursts at a time transfer to the same output wavelength. Fiber delay lines (FDL), deflection routing and wavelength conversion are used for their correction. Through the use of FDL bursts are hold for some period of time and using deflection routing they can transfer en reroute but not the main route to the receiver. The whole version of wavelengths let to modify any incoming wavelengths to the outgoing one. In this article the switch in OBS network with FDL, the whole wavelength conversion and deflection routing are regarded. Also the system of equations in the global balance for the steady-state blocking probabilities and formulas for calculation of main productivity characteristics of separate optical fiber are derived.

The two-channel finite-capacity queuing system with several Poisson flows of customers of different types is considered. The service time is distributed according to phase-type which depends on the type of customers and the device of which it is served. On leaving the system there is a buffer in which there is a resequence of customers according to order of their receipt. Functioning of the system is described by uniform Markov process. In the assumption that intensity of flows and service of customers are positive and finite the final probabilities of statuses of Markov process exist, are strictly positive, don’t depend on initial distribution and match the stationary probabilities. For search of these probabilities the equilibrium system of equations is removed. Then possibility of convergence of the received equations to the similar equations for queuing system with resequence of customers with one Poisson flow of summary intensity and the subsequent determination of the type of customers just before arrival on service is set. The last circumstance allowed using results of the previous operations of authors for calculation of stationary distribution of queue length. As a result the recurrent matrix algorithm was developed for calculation of probabilities of statuses of considered system in the conditions of a stationary operation mode.

In the paper the algorithm is proposed for the numerical simulation of the thermal conductivity for the design and optimization of cryogenic cells pulsed (in the millisecond range) feeding the working gases into the electron-stringed source of multiply charged ions. Heating process comes when the electric current passed through one of the layer. A model of the cryogenic cell with four layers (materials) is investigated. The heat transfer into the object is described by the system of heat equations with temperature dependent discontinuous thermal coefficients. The discontinuous thermal coefficients are given by experimental data and approximated by the least-squares method using the polynomial analytical functions. Conjugation condition between materials is considered to be ideal. The results are reported for a common configuration of the cell. The parallel algorithm for modeling thermal processes into four layer model was developed and speedup of the algorithm in depending on number of CPUs is shown.

Problems of modeling of active queue management (AQM) systems have been for a long time in the sphere of interests of authors. One of the areas of work was associated with a dynamic model of the control module Random Early Detection (RED) based on Poisson process driven stochastic differential equations. These equations are used in queuing theory quite recently, and not very well understood. As disadvantages of this approach the authors underlined its non-generic character. We describe the interaction between module RED and protocol TCP Reno. But its extension to other variants of the TCP protocol and the control module is not possible. Our group studied common approaches to modeling of such phenomena. As a result a method for randomization of one-step processes, allowing to obtain new models in a universal manner was developed. In this paper, the authors use this technique to model previously investigated RED module and protocol TCP Reno to demonstrate its applicability to this kind of problems. As a result an extended model of control module type RED for traffic type TCP Reno, was created. It contains previously studied model as a special case.

For research in the field of transformation optics and for the calculation of optically inhomogeneous lenses the method of geometrization of the Maxwell equations seems to be perspective. The basic idea is to transform the coefficients of constitutive equations, namely the dielectric permittivity and magnetic permeability into the effective geometry of space-time (and the vacuum Maxwell equations). This allows us to solve the direct and inverse problems, that is, to find the permittivity and magnetic permeability for a given effective geometry (paths of rays), as well as finding the effective geometry on the base of dielectric permittivity and magnetic permeability. The most popular naive geometrization was proposed by J. Plebanski. Under certain limitations it is quite good for solving relevant problems. It should be noted that in his paper only the resulting formulas and exclusively for Cartesian coordinate systems are given. In our work we conducted a detailed derivation of formulas for the naive geometrization of Maxwell’s equations, and these formulas are written for an arbitrary curvilinear coordinate system. This work is a step toward building a complete covariant geometrization of the macroscopic Maxwell’s equations.

Maxwell’s equations are extremely simple and elegant. However, the specific calculations require much more sophisticated approaches. Thus, in the problems of calculation of nonregular integrated optical waveguides a few basic techniques are used. The authors propose to use the method of adiabatic waveguide modes. This method follows in the footsteps of Luneburg works. Moreover, the method has a clear geometric interpretation. As well as Luneburg equations, the equations obtained by this correspond to the Hamilton equations on the cotangent bundle over the configuration space. Moreover, to calculate ray paths a simple geometrization is used, when the refractive index is represented as a metric of some efficient space. Thus, the phase function is evaluated as an action along the trajectory. Thin-film Luneburg lens is an interesting object in the general theoretical sense as well as in practical one. Its study allows to further describe a class of objects, but it is an essential element for the construction of a purely optical control devices. Thus, the authors consider the method of adiabatic modes most suitable for studies of such object as a thin-film generalized waveguide Luneburg lens.

The task of automatic classification of illness is very important since its solution greatly facilitates the work of a physician in a large amounts of data analysis and diagnosis. There are different approaches to solving this problem. One of them involves the use of production rules. Production rules empower knowledge representation in clinical medicine and effective in building diagnostic systems. Noting the prospects of applying production rules, however, it should be noted that the decision on the basis of their high complexity real problems requires large amounts of computation and restructure a system of rules at change of problem conditions. At the same time, as an effective alternative tool to analyze complex situations and classifications artificial neural networks (ANN) are widely used, which allow to perform the recognition and diagnosis of various phenomena and high complexity objects by training. This paper studies the possibility of using different neural networks to analyze an illness severity on the basis of precedential information. In particular, the problem of determining the severities of acute intervertebral disc protrusion and of acute acute asthma are solved. In order to improve recognition, the training set is expanded by creating additional precedents that that do not contradict the terms of the problem.The artificial neural networks of various configurations and with different activation functions are used: single-layer and multilayer perceptrons.

In this paper we study the static axisymmetric solutions of the vacuum Einstein equations. Among static axisymmetric vacuum solutions of the most interest are the asymptotically flat solutions reducing to the Schwarzschild solution. The purpose of this paper is to obtain a static solution which turned out to be appropriate for describing the gravitational field around an axisymmetric mass distribution. In this paper the method of singular sources os considered and some new applications are presented. By mean of the method of singular sources it is possible to construct gravitational multipoles which generalize the Schwarzschild solution. The linearity of the gravistatic equations makes it possible to solve the problem of superposition of two or several known solutions. The obtained static vacuum axisymmetric generalization of the Schwarzschild solution near two points of horizon has coordinate singularities. In the obtained solution the dipole term is absent, and we have found the corresponding condition. If one considers axially symmetric solutions of gravistatics, then construction of gravitational multipoles becomes ambiguous. It means that different solutions can give asymptotically the same Newtonian limit.

We have investigated the properties of the static cylindrical symmetric configurations of the interacting scalar and spinor fields taking account of ideal fluid with the state equation P = We, where P is pressure, e is energy density, W - arbitrary dimensionless parameter. Parallel with the usual matter with positive W , we have considered some types of ideal fluids with negative pressure (W < 0), that are actively used at present in cosmology (dark matter, cosmic strings, domain walls, quintessential, cosmic vacuum, fantom matter). We have obtained the exact solutions to the equations of the interacting scalar and spinor fields, Einstein equations and ideal fluid motion equation with arbitrary W . We have written the conditions of the regular metrics on the axis of the symmetry of the system and the conditions of the regular (flat or string) metrics. We have considered the influence of the different types of the ideal fluid upon the formation of the soliton-like or string-like configurations in the = 1 3 (ultrarelative system of the interacting fields. We have established that in case of W 1 (space string’s gas), W = − 2 3 matter), W = −(random distribution of the domain walls), 3 W = − 4 3 (phantom matter), regular configurations of the system of the interacting fields and ideal fluid can exist only under some relation among constants in the equations.

We discuss the problem of elaborating the software systems designed for modeling of physical processes on computing systems with hybrid architecture, formulate the basic principles of construction of such complexes, and present an example of their implementation for the GIMM FPEIP complex. Complex GIMM FPEIP is intended for simulation of thermal processes in materials irradiated by heavy ion beams. The construction of GIMM FPEIP complex is done according to the requirements, intergrability of GIMM FPEIP into a complex hierarchical structure GIMM NANO and specification of the problem under solution. The complex includes the computational modules that provide parallel algorithms realized on the basis of MPI and CUDA technologies and meant for performing computations on hybrid computing systems. The GIMM FPEIP complex provides a possibility to include new computational modules and to expand the current database of physical parameters. In the construction of the complex, a module approach to the structure of the complex has been applied. This allowed realizing a number of common modules in the form of separate libraries with the possibility of their use in other software complexes. In particular, with the use of this approach, a 3D modeling complex GIMM FPEIVE was constructed. GIMM FPEIP and GIMM FPEIVE complexes were tested on the multi-core computing complex CICC JINR, on hybrid computing complex K100 (Keldysh Institute of Applied Mathematics) and on ”Lomonosov” supercomputer (Lomonosov Moscow State University).

The equations of parabolic type describe processes of nonlinear heat conductivity, diffusions of the loaded particles in plasma, diffusion and drift of impurity atoms in semiconductor structures, in chemical kinetics. At the numerical solution of practical tasks such there are the difficulties caused by the insufficient capacity and volume of memory of the personal computer. There is a problem of creation of parallel methods and algorithms for the numerical decision the parabolic equations on supercomputers. One of methods of the numerical solution of the multidimensional parabolic equations the locally one dimensional method is. Parallel realization of a locally one method for numerical solutions of the linear and quasi-linear two-dimensional parabolic equations on supercomputers with the distributed memory is offered. The parallel algorithm is constructed taking into account locality of data - the operations and data are distributed between processes in such a way that the considerable part of data is privatized by processes and doesn’t need communication operations. Results of numerical experiments are given.

A second order finite volume scheme on structured meshes is presented for numerical solution of time dependent Maxwell’s equations with discontinuous dielectric permittivity. The scheme is based on approaches of Godunov, Van Leer and Lax Wendroff and employs a special technique for gradient calculation near dielectric permittivity discontinuities. The scheme was tested for problems with linear and curvilinear discontinuities. Test results demonstrate second order of convergence and support second order of approximation in space and time. A parallel implementation of the scheme based on geometric decomposition was developed. Computational region was partitioned into subregions. Computations in each subregion were carried out independently using halo cells. Test results indicate linear scalability. Parallel implementation was applied to modelling photonic crystal devices. Computational results for photonic crystal waveguide with a bend correctly confirm bend configurations and frequencies with zero reflection.

The paper presents a general description of a new method of artificial viscosity (AAV) for solving equations of gas dynamics. In a basis of a method are put research of stability finite difference schemes and classification of discontinuous desicions of gas dynamics equations. The method was adapted for the solution of problems on cartesian and unstructured meshes. With the help of the method a lot of gas dynamics problems were analyzed numerically, for example, the supersonic flow in a channel with a step was calculated.

When a colloidal droplet with pinned contact line evaporates on a substrate, surface tension produce a fluid flow. Usually the flow directs to the contact line and the solute in the drop is dragged by this flow, where it accumulates. The deposit remains on the edge of the drop and forms a solid ring after complete evaporation of the liquid. The conditions of evaporation and solution properties affect on deposit structure. We propose simple model of sessile thin drop desiccation under nonuniform evaporation. The model is based on mass conservation and has numerical solution. If the droplet is covered by a ”mask” with hole, evaporation primarily occurs under the hole so that surface tension drives a flow of liquid to replenish this loss. In the proposed physical model a radial flow velocity was studed and the redistribution of component in the droplet arising from the nonuniform evaporation under a mask was predicted. The effect of diffusion and hole radius on the component redistribution was studed too.

The methodology of human-computer interaction based on the determination of the eyeball position and its movement direction. For interference and noise elimination from extraneous lighting devices infra-red illumination which allows receiving only one highlight on a cornea from an infra-red light-emitting diode is used. Thus the entrance video-stream in an infra-red range is processed in real time in two parallel processes. One process is used for definition of position of a highlight from a light-emitting diode on corneas of eyes, another process - for definition of position of a pupil and a direction of its moving. Aggregate of co-ordinates of a highlight of a pupil, position of a cornea and a direction of its moving allow remote to control of the computer. For prevention of loss of “object” the original technique of accumulation of the previous positions of an eyeball and forecasting of a direction of its moving is used. The given methodology of contactless control of the computer allows simulating pressing keys of the keyboard and/or mouse movement. The developed hardware and software system for gaze direction tracking be used as an alternative method of input medium, which is closer to the natural way of interaction with the environment, as well as the only way to work with a computer for the users with reduced mobility.

In recent years significantly increased the negative information and psychological impact on the individual and mass consciousness. Therefore, strongly expressed of emergence of aggression, anxiety, despair, hopelessness, depression, a criminal manifestations and mental illness. The research of stability of level of psychical reaction with personal characteristics of a human and with the power of informational influence on him is presented in the article. The adjoin method offered by Kudinov A.N. was applied to research of the loss of stability. The main advantage of the adjoin method is that for use to the problems of dynamic stability studies in various fields of science, technology, biology, medicine and psychology, if their equations can be reduced to the equations of second order, don’t demand the introduction of Lyapunov functions. The adjoin method permits to find the equilibrium positions and to check the stability of nonperturbed state. Also the research of stability by Lyapunov’s method on first approximation is conducted, as a result the stability conditions of psychical reaction with personal characteristics of a human and with the power of informational influence on him are presented.

The well known to specialist in the field of mathematical biology principle of evolutionary optimality, rising to darvinian concept of natural selection, which is based upon mechanisms of survival of the most strong is formulated. Expressed in term of stability of established equilibria in model system, this principle allows to generalize it to the systems, which can be described by mathematical models, including both integro-and partial differential equations. In the presented article the ways to use the author’s evolutionary optimality theory, which initially was constructed for the dynamical systems in Banach spaces, for to find the selection functional, which are to be optimized, for several structured biological systems are indicated. Particularly it is shown that in the case of communities with age and with spatial structure the constructed functionals have a real biological interpretation. As a practical application of the constructed theory the central result of the theory of correlation adaptometry is formulated.

The model of oil spill evolution on the water surface based on quasi-linear convection-diffusion equation is considered. To solve the latter the method of finite differences is used. Two-dimensional problem is reduced to one-dimensional by alternating direction method. For one-dimensional case traditional difference approximations are briefly discussed. And also difference schemes obtained by computer algebra based method of automatic generation of difference schemes are considered. For generated schemes the order of approximation and linear numerical dissipation are estimated. For explicit schemes stability conditions are briefly discussed. In the absence of convection a numerical comparison of traditional implicit difference scheme and one implicit scheme generated automatically with similarity solution of quasi-linear diffusion equation is carried out. This comparison shows that generated implicit scheme allows to obtain the relative error less than for traditional scheme. Using obtained implicit scheme evaluation of oil slick thickness changes in the presence of oil evaporation is done. Two different models were used to estimate oil evaporation rate. The first one is based on hypothesis that oil evaporation is regulated by air boundary layer. The second assumes that oil evaporation is regulated by oil diffusion. Calculations show that the choice of model essentially influences on oil slick thickness and oil total volume changes in time.

In this paper, the presence of critical points and bifurcation points of rotating Newtonian polytropes with an index of 0.9 ≤ n ≤ 1.6 has been shown for the first time. The symbolic-numerical calculation error in metric L2 has reached the size of 10 −5 order. The approximate analytical solution of the problem to the above mentioned accuracy has been set forth. The critical value of polytropic curve index n = nk =1.54665 has been calculated which is the highest one among the critical points and bifurcation points.

Boolean Gr¨obner basis have shown their practical efficiency for different problems. Among them are algebraic cryptanalysis HFE (Hidden Field Equations), modeling of quantum computing and boolean satisfiability problem (SAT). Algorithms for computing Gr¨obner basis have exponential complexity for execution time as well as for memory usage. The more appropriate data structure was introduced, which is based on zero-suppressed binary decision diagrams (ZDD). Also we show the relation between ZDD and special recursive notation of polynomials. The recursive notation is the collection of equalities which have had one-to-one correspondence with graphical presentation of ZDDs. We prove lemma which gives the number of nodes estimation for ZDD which represent boolean polynomial with all monoms up to d degree of n variables. Furthermore, we present C++11 ZDD package providing possibility for addition and multiplication of boolean polynomials, multiplication by variable, presentation in compressed recursive form and graphical presentation. The package includes its own implementation of red-black trees, lists, and memory manager. ZDD package was developed for using as internal data structure of the boolean polynomials for computation of involutive Gr¨obner basis.

In this article, use of volume integral equations method for calculations of magnetic systems is considered. GFUN program package based on integral approach to magnetostatics applies the method of collocations and piecewise constant approximations of unknown variables within the elements to discretization of equations. The limitation of this approach is related to singularity of the integral equations kernel. An alternative to collocation method, replacing observation point by integration over discretization elements, is considered in this article. This approach enables one to use higher order approximations for unknown variables. In the context of the finite element method, the piecewise constant and piecewise linear approximations of unknown variables are considered. The problem of computing matrix elements for discretized systems of equations can be reduced to evaluation of sixth-order integrals, singular ones in the general case, over two different elements of the computational region. Possible methods are proposed for calculating this kind of integrals. Iterative processes for solving the arising nonlinear systems of discretized equation are discussed. The proposed approach enables one to build discretizations with higher precision of approximation for the initial volume integral equations of magnetostatics. The proposed method was used for 3D modeling of a dipole magnet. Comparison of results obtained for simulation of the dipole magnet using different versions of integral magnetostatics problem discretization are given.

The excited states of quantum systems are nonstationary, and they break up. These states are called unstable or quasi-stationary. Such states are already observed in the study of scattering problems, the accumulation of particles in the lens (the particle prefers to live inside the lens) is accompanied by a large delay τ (the lifetime of the quasi-level). Here the lifetime of the quasi-level τ = γ−1, width of the quasi-level Г = ~γ and complex energy level E = E1 − iE2,E2 = Γ/2. Investigation of the quasi-stationary states is carried out for the quasi-potential equation with piecewise-constant potentials at various values of the parameter of the equation ε and the potential parameters. A comparative analysis of solutions of the quasi-potential equation for the different values of ε with the solutions of the Schredinger equation is performed. Found that at ε → 0 the solutions of quasi-potential equation tend to the solutions of the Schredinger equation.

Methods of calculation of high order derivatives are considered on a basis: interpolation formulas; “without difference methods of calculation of derivatives”; applications of convolution with replacement of differentiation by integration operation; differentiation with use of quadratures on C. Lanczos; the method of Numerova. The comparative analysis of methods of calculation of high order derivatives on accuracy of calculations with use as the sample of the derivatives calculated in package Maple with 20 digit decimal accuracy is carried out. It is shown that all methods are almost equivalent on accuracy and are reduced to convolution calculation between differentiated function and some window which coefficient depend on an applied method. For carrying out of experiments the special program complex is developed for calculation of high order derivative (up to 7th) the tabulated functions with various step. Grids with steps from 0.005 to 0.1 have been investigated. Irrespective of a method of calculation of derivatives it has been defined that optimum value of step mesh for 64 digit arithmetic’s the step is from 0.01 till 0.05. Value of smooth functions differs less than their accuracy of representation at smaller value of a step, and at greater step - the differentiation error increases. Results of experiments confirm N.N. Kalitkin’s theoretical conclusions.

Brief description of a FORTRAN 77 program for calculating energy values, refection and transmission matrices, and corresponding wave functions in a coupled-channel approximation of the adiabatic approach is presented. In this approach, a multidimensional Schr¨odinger equation is reduced to a system of the coupled second-order ordinary differential equations on a finite interval with the homogeneous boundary conditions of the third type at the leftand right-boundary points for continuous spectrum problem, or a set of first, second and third type boundary conditions for discrete spectrum problem. The resulting system of these equations containing the potential matrix elements and first-derivative coupling terms is solved using high-order accuracy approximations of the finite element method.

Mel-Frequency Cepstral Coefficients are in so far the most popular speech features. However, depending on the length of a vocal tract (it is worth mentioning that length of a vocal tract is dependent on sex and other physiologic parameters of a speaker, such as height, and can vary from 13 cm to 18 cm) frequencies of central formants are shifted. The value of the shift can be as large as 25%. This huge difference can lead to a wrong recognition of a new utterance by a previously well-trained model when the utterance was said by a new speaker, thus the system becomes speaker-dependent. Alternative way is to use speaker independent features such as that obtained using Auditory Image Model (AIM) to describe input utterance. In our work we propose AIM based features which are calculated using simulated annealing algorithm. Using Monte-Carlo schemes we investigate statistical properties of maximum likelihood estimates of Gram-Charlier extension of normal density obtained via simulated annealing algorithm, also we compare different methods to solve aforementioned optimization problem.

Complexes of localized states are numerically analyzed in two dynamical systems: directly driven nonlinear Schr¨odinger equation (NLS) and double sine-Gordon equation (2SG). Both systems have a wide range of physical applications. Our numerical approach is based on the numerical continuation with respect to the control parameters of the quiescent (stationary) solutions and stability and bifurcation analysis of the linearized eigenvalue problem. Multisoliton complexes of the NLS equation are studied in the undamped and the weak damping regimes. We show that in the weak damping case the directly driven NLS equation holds stable and unstable multi-soliton complexes. The results are confirmed by means of direct numerical simulations of the time-dependent NLS equation. Properties of the multi-fluxon solutions of 2SG equation are studied depending on the parameter of the second harmonic. We show that the second harmonic changes properties and increases the complexity of coexisting static fluxons of 2SG equation. Results are discussed within the frame of the long Josephson junction model.

An one-dimensional mathematical model described evolution of drop shape and viscous fluid dynamics (vertically averaged radial flow of the liquid) as the result of evaporation from impermeable horizontal surface is presented in this work. The model considers the influence of volume and capillary forces. Non-steady-state approach has been used to describe the process mathematically. The continuity equation and the motion equation for the case of a system with a variable mass are solved numerically with using standard tools of mathematical package Maple. We carry out calculations for cases of different sizes of water drop volumes. Results of modeling have shown that profile of drop which is bigger in size of capillary length (Bond number is greater than unity) differs from shape of spherical segment. The surface of such drop is almost flat. We explain it by domination of the gravity over a surface tension force. The flow of compensation nature is present in evaporated drops of different volumes what is coordinated with experimental data of other authors. Thus radial fluid flow is result of work of both capillary and gravitational forces. The results which we have got will help to describe the coffee ring effect in macrodrops in future.

Actual problem of the theory of stability is creation of strict and effective methods of research of the loss of stability of movement of systems with distributed parameters, in particular for continuous environments. This problem has a huge impact on both theoretical and applied uses. In the article is presented the research of the loss of stability of the equations which describe mathematical models of soil massifs and the bases. These models reflect the character of the soil under load, are based on the laws of structural mechanics and the theory of elasticity. The adjoin method offered by Kudinov A.N. was applied to research of the loss of stability. The main advantage of the adjoin method is that for use to the problems of dynamic stability studies in various fields of science and technology, if their equations can be reduced to the equations of second order, don’t demand the introduction of Lyapunov functions. The developed algorithm allows to find positions of balance of the decision and to check up stability of system by adjoin method. The research of stability of the solution of linear systems by using Lyapunov’s method of first approximation is presented in the article.

Nonlinear waves in liquid with gas bubbles are investigated taken into account liquid viscosity and compressibility and inter phase heat transfer. The nonlinear differential equation for long weakly nonlinear waves is obtained with the help of the reductive perturbation method. At the derivation of the equation higher order corrections in the asymptotic expansion are taken into account. This equation is the generalization of the Burgers equation and describes nonlinear waves in a liquid with gas bubbles in the case of dissipation main influence. The normal form is constructed for the equation with the help of the near-identity transformations. It is shown that the normal form equation is integrable under certain condition on parameters. In this case the equation for nonlinear waves is the second member of the Burgers hierarchy. Exact solution in the form of kink is obtained in the general case. Dependence of this solution on physical parameters is investigated. It is shown that the amplitude of this exact solution decreases when the bubbles radius in the unperturbed state and the liquid viscosity increase.

In the process of mathematical modeling a necessity often arises to smoothly approximate various dependencies which are defined discretely or graphically. Especially if the value of such dependencies are obtained as a result of complex experiments or cumbersome calculations. The inverse transform of continuous simulated objects in discrete digital format that is used for storage and computer processing also requires a certain ordering. It is assumed beforehand that the smooth approximation of discrete set of points on the plane is performed by linear analytical model. Interpolation conditions lead to a system of linear equations with a square matrix. When interpolating polynomials by breaking and rearranging the terms of a power series one can get such basic functions as Lagrange polynomials or Bernstein polynomials. Other methods of interpolation are Newton polynomials, Aitken iterative process, etc. However, these methods realize only some particular cases of all possible approximations of discrete data by arbitrary basis functions and are mainly focused on manual calculations. In computer calculations, it is desirable to find a general algorithm for solutions in order to avoid programming many particular cases. The problem of generalization of existing methods for approximation of discrete data sets (generalized algorithm) and bringing these discrete data to a common form (a discrete unified structure) is considered.

Evolutionary algorithms are in active development for last two decades, due to numerous studies in the field of mathematical biology, and the wide spread of massively parallel computing systems since numerical modeling of biological systems (with significant degree of parallelism) requires significant computations. Swarm optimization algorithms discussed in this article are based on modeling of collective behavior in large colonies of animals, such as ants, bacteria, bees. Such algorithms are universal and applicable to a wide range of computational problems. The present paper is devoted to the new approach to the construction of self adaptive swarm optimization algorithms, which automatically adjusts parameters of the algorithm in the process of its evolution. The idea of building self adaptive evolutionary algorithms is based on the using in the background to the main algorithm (e.g., bacterial foraging algorithm) auxiliary genetic algorithm, the purpose of which is to adjust the parameters of the basic algorithm, providing the highest possible rate of its convergence. The application of the proposed scheme of self-adaptation on the examples of bacterial foraging algorithm and bees algorithms is considered. The results of the numerical study of such algorithms on the standard test problems of continuous optimization, demonstrating the efficiency of the proposed scheme of self-adaptation, are described.

The problem of constructing various types of object detectors in images is still an urgent task, despite the relatively strong set of methods described in the literature. One of the methods that have become standard for the construction of efficient and fast classification, is a Viola-Jones cascade, which is still fundamental to search for objects in the image in real time and which implementation has been included in the open-source computer vision library OpenCV. For the experiments in this study we used the database of images CMU Face Database. In practice, when we use of the algorithms in computer vision the computational complexity becomes a significant factor. Preferably, one should use threshold decision rules or Haarfeatures as classifiers, which gives small the computational complexity. In this paper, the approach to the construction of classifiers of comparable performance for the problem of detecting faces. For the construction of the detector were studied approach involves separating detection process into two stages: construction the descriptor of image, and classification stage. For the phase, which responsible for the classification, were considered two possibilities: a two-layer neural network, i.e. using multilayer perceptron as a “strong” classifier, and a cascade of several such networks of different size. For the phase, which responsible for forming the descriptor, we also have investigated two possibilities. First one - fixed Haar-basis, which gives us a feature-vector of the descriptor of input image. This basis was constructed using the ADABOOST principle. The second possibility, investigated in this paper, was the construction of the basis of fewer required Haarfeatures, every of which more accurately reflects the object characteristics, which was obtained by using Karhunen-Loeve transform. In order to get Haar-features from eigenvectors, they have been quantized. As a result, the classifier built with efficiency which comparable to the Haar cascade.