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We consider effect of Einstein, Podolsky, Rosen in connection with quantum mechanic and relative theory. We sow that may introduce in quantum mechanic the individual state for quantum particle which eliminate the contradiction quantum mechanics with relative theory of type long-range action between correlated particles. In article we develop the apparatus for individual state introducing in quantum mechanic formalism and build the EPR effect model without contradictory. We describe the general mechanism of Bell inequalities infringement and analogical effects.

The second part of paper is devoted to final study of three classic partial differential equations (Laplace, Diffusion and Wave) solution using simple numerical methods in terms of Cellular Automata. Specificity of this solution has been shown by different examples, which are related to the hexagonal grid. Also the next statements that are mentioned in the first part have been proved: the matter conservation law and the offensive effect of excessive hexagonal symmetry.

From the point of CA view diffusion equation is the most important. While solving of diffusion equation at the infinite time interval we can find solution of boundary value problem of Laplace equation and if we introduce vector-variable we will solve wave equation (at least, for scalar). The critical requirement for the sampling of the boundary conditions for CA-cells has been shown during the solving of problem of circular membrane vibrations with Neumann boundary conditions. CA-calculations using the simple scheme and Margolus rotary-block mechanism were compared for the quasione-dimensional problem “diffusion in the half-space”. During the solving of mixed task of circular membrane vibration with the fixed ends in a classical case it has been shown that the simultaneous application of the Crank–Nicholson method and taking into account of the second-order terms is allowed to avoid the effect of excessive hexagonal symmetry that was studied for a simple scheme.

By the example of the centrally symmetric Neumann problem a new method of spatial derivatives introducing into the postfix CA procedure, which is reflecting the time derivatives (on the base of the continuity equation) was demonstrated. The value of the constant that is related to these derivatives has been empirically found in the case of central symmetry. The low rate of convergence and accuracy that limited within the boundaries of the sample, in contrary to the formal precision of the method (4-th order), prevents the using of the CAmethods for such problems. We recommend using multigrid method. During the solving of the quasi-diffusion equations (two-dimensional CA) it was showing that the rotary-block mechanism of CA (Margolus mechanism) is more effective than simple CA.

Computer-aided assembly planning for complex products is an important engineering and scientific problem. The assembly sequence and content of assembly operations largely depend on the mechanical structure and geometric properties of a product. An overview of geometric modeling methods that are used in modern computer-aided design systems is provided. Modeling geometric obstacles in assembly using collision detection, motion planning, and virtual reality is very computationally intensive. Combinatorial methods provide only weak necessary conditions for geometric reasoning. The important problem of minimizing the number of geometric tests during the synthesis of assembly operations and processes is considered. A formalization of this problem is based on a hypergraph model of the mechanical structure of the product. This model provides a correct mathematical description of coherent and sequential assembly operations. The key concept of the geometric situation is introduced. This is a configuration of product parts that requires analysis for freedom from obstacles and this analysis gives interpretable results. A mathematical description of geometric heredity during the assembly of complex products is proposed. Two axioms of heredity allow us to extend the results of testing one geometric situation to many other situations. The problem of minimizing the number of geometric tests is posed as a non-antagonistic game between decision maker and nature, in which it is required to color the vertices of an ordered set in two colors. The vertices represent geometric situations, and the color is a metaphor for the result of a collision-free test. The decision maker’s move is to select an uncolored vertex; nature’s answer is its color. The game requires you to color an ordered set in a minimum number of moves by decision maker. The project situation in which the decision maker makes a decision under risk conditions is discussed. A method for calculating the probabilities of coloring the vertices of an ordered set is proposed. The basic pure strategies of rational behavior in this game are described. An original synthetic criterion for making rational decisions under risk conditions has been developed. Two heuristics are proposed that can be used to color ordered sets of high cardinality and complex structure.

Method which allows to obtain explicit form of the solution as a part of power series of the argument step is developed. Formalization of characteristics of the algorithm analogous to operations of a computer is performed. The operation of transfer from one rank to another leads to a probability scheme of the algorithm that averages unknown intermediate steps in higher ranks of the series. The stochastic characteristics of the method are studied and illustrated. Examples of solving nonlinear equations and systems of nonlinear differential equations are presented.

Semiclassical approximation formalism is developed for the multidimensional Fokker–Planck–Kolmogorov equation with non-local and nonlinear drift vector with respect to a small diffusion coefficient D, D→0, in the class of trajectory concentrated functions. The Einstein−Ehrenfest system of (0, M)-type is obtained. A family of semiclassical solutions localized around a point driven by the Einstein−Ehrenfest system accurate to O(D^(M+1)/2) is found.

In the field of naval architecture the most competent recommendations in verification and validation of the numerical methods were developed within an international workshop on the numerical prediction of ship viscous flow which is held every five years in Gothenburg (Sweden) and Tokyo (Japan) alternately. In the workshop “Gothenburg–2000” three modern hull forms with reliable experimental data were introduced as test cases. The most general case among them is a containership KCS, a ship of moderate specific speed and fullness. The paper focuses on a numerical research of KCS hull flow, which was made according to the formal procedures of the workshop with the help of CEA FlowVision. Findings were compared with experimental data and computational data of other key CEA.

A formal model mechanism of oscillon formation is proposed. These structures were found in a variety of physical systems and a chemical Belousov–Jabotinsky reaction proceeding in an aerosol OT water-inoil microemulsion. Via the proposed mechanism oscillons occur as a result of interaction of two subsystems. In the first subsystem for a proper set of parameters solitary stationary structures may arise as a result of hard local excitation. These structures influence spatial distribution of the second subsystem parameter that leads to local oscillations in the subsystem.

Selection and correct applying of the geosensor — the instrument of mineral geothermobarometry is challenging because of the wide variety of existing geosensors on the one hand and the availability of specific requirements for their use on the other. In this paper, organization of the geosensors within the computer system called interactive registry was proposed for reducing the labor intensity of the geosensors usage and providing information support for them. The article provides a formal description of the thermodynamic geosensor, as a function of the minerals composition and independent parameters, as well as the basic steps of pressure and temperature estimation which are common for all geosensors: conversion to the formula units, calculation of the additional parameters and the calculation of the required values. Existing collections of geosensors made as standalone applications, or as spreadsheets was examined for advantages and disadvantages of these approaches. Additional information necessary to use the geosensor was described: paragenesis, accuracy and range of parameter values, reference and others. Implementation of the geosensors registry as the webbased application which uses wiki technology was proposed. Usage of the wiki technology allows to effectively organize not so well formalized additional information about the geosensor and it’s algorithm which had written in a programming language into a single information system. For information organization links, namespaces and wiki markup was used. The article discusses the implementation of the applications on the top of DokuWiki system with specially designed RESTful server, allowing users to apply the geosensors from the registry to their own data. Programming language R uses as a geosensors description language. RServe server uses for calculations. The unittest for each geosensor allows to check the correctness of it’s implementation. The user interface of the application was developed as DokuWiki plug-in. The example of usage was given. In the article conclusion, the questions of the application security, performance and scaling was discussed.

The article describes an improved paired comparison method, which systematizes in tables the rules of logical conclusions and formulas of checking indices for comparison of technical systems. To achieve this goal, the authors formulate rational rules of logical conclusions in making a paired comparison of the systems. In addition, for the purpose of consistency check of the results of the assessment, the authors introduce parameters such as «the number of scores gained by one system» and «systems’ quality index»; moreover, they design corresponding calculation formulas. For the purposes of practical application of this method to design computer programs, the authors propose to use formalized variants of interconnected tables: a table for processing and systematization of expert information, a table of possible logical conclusions based on the results of comparison of a set number of technical systems and a table of check values in the paired comparison method used in quality assessment of a definite number of technical systems. These tables allow one to organize procedures of the information processing in a more rational way and to predominantly exclude the influence of mistakes on the results of quality assessment of technical systems at the stage of data input. The main positive effect from the implementation of the paired comparison method is observed in a considerable reduction of time and resources needed to organize experts work, process expert information, and to prepare and conduct distant interviews with experts (on the Internet or a local computer network of an organization). This effect is achieved by a rational use of input data of the quality of the systems to be assessed. The proposed method is applied to computer programs used in assessing the effectiveness and stability of large technical systems.

Optical systems with two-dimensional feedback demonstrate wide possibilities for studying the nucleation and development processes of dissipative structures. Feedback allows to influence the dynamics of the optical system by controlling the transformation of spatial variables performed by prisms, lenses, dynamic holograms and other devices. A nonlinear interferometer with a mirror image of a field in two-dimensional feedback is one of the simplest optical systems in which is realized the nonlocal nature of light fields.

A mathematical model of optical systems with two-dimensional feedback is a nonlinear parabolic equation with rotation transformation of a spatial variable and periodicity conditions on a circle. Such problems are investigated: bifurcation of the traveling wave type stationary structures, how the form of the solution changes as the diffusion coefficient decreases, dynamics of the solution’s stability when the bifurcation parameter leaves the critical value. For the first time as a parameter bifurcation was taken of diffusion coefficient.

The method of central manifolds and the Galerkin’s method are used in this paper. The method of central manifolds and the Galerkin’s method are used in this paper. The method of central manifolds allows to prove a theorem on the existence and form of the traveling wave type solution neighborhood of the bifurcation value. The first traveling wave born as a result of the Andronov –Hopf bifurcation in the transition of the bifurcation parameter through the сritical value. According to the central manifold theorem, the first traveling wave is born orbitally stable.

Since the above theorem gives the opportunity to explore solutions are born only in the vicinity of the critical values of the bifurcation parameter, the decision to study the dynamics of traveling waves of change during the withdrawal of the bifurcation parameter in the supercritical region, the formalism of the Galerkin method was used. In accordance with the method of the central manifold is made Galerkin’s approximation of the problem solution. As the bifurcation parameter decreases and its transition through the critical value, the zero solution of the problem loses stability in an oscillatory manner. As a result, a periodic solution of the traveling wave type branches off from the zero solution. This wave is born orbitally stable. With further reduction of the parameter and its passage through the next critical value from the zero solution, the second solution of the traveling wave type is produced as a result of the Andronov –Hopf bifurcation. This wave is born unstable with an instability index of two.

Numerical calculations have shown that the application of the Galerkin’s method leads to correct results. The results obtained are in good agreement with the results obtained by other authors and can be used to establish experiments on the study of phenomena in optical systems with feedback.