Vol 16, No 4 (2012)

The unique solvability of boundary value problem with Saigo operators for the thirdorder equation with multiple characteristics was investigated. The uniqueness theorem with constraints of inequality type on the known functions and different orders of generalized fractional integro-differentiation was proved. The existence of solution is equivalently reduced to the solvability of Fredholm integral equation of the second kind.

The multidimensional inverse problem of determining spatial part of integral member kernel in integro-differential wave equation is considered. Herein, the direct problem is represented by the initial-boundary problem for this with zero initial data and Neyman’s boundary condition as Dirac’s delta-function concentrated on the boundary of the domain $(x, t) \in \mathbb R^{n+1}$, $z > 0$. As information in order to solve the inverse problem on the boundary of the considered domain the traces of direct problem solution are given. The significant moment of the problem setup is such a circumstance that all given functions are real analytical functions of variables $x \in \mathbb R^n$. The main result of the work is concluded in obtaining the local unique solvability of the inverse problem in the class of continuous functions on variable $z$ and analytical on other spatial variables. For this, by means of singularity separation method, the inverse problem is replaced by the initial-boundary problem for the regular part of the solution of this problem. Further, direct and inverse problems are reduced to the solution of equivalent system of Volterra type integro-differential equations. For the solution of the latter, the method of Banach space scale of real analytical functions is used.

Limiting forms of distribution of length of the maximum series of successes in random binary sequences, formed in Bernulli–Markov’s chain and in Polya’s scheme which is an equivalent to local trends of a time series of strictly stationary process are investigated. More simple and added proof of theorems of the law of the big numbers for series of both types is offered. For series of the second type the effect of the cyclic bimorphism of the limiting law with degeneration on one of the phases and the convergence according to the probability on set no more, than four consequent values of the natural series is established.

The method for solving the boundary value problem of the beam’s creep and creep rupture strength under condition of the pure bending based on the rod type structural model is proposed. Energy criterion of local element destruction is introduced. Comparative analysis of structural model calculated data and the quarter beam of D16T alloy at $T = \rm 250~^\circ C$ curvature value found by experiment is performed. Calculated data agree with those found by experiment. Correlation of the calculated data based on the proposed method with those based on phenomenological model of energy type creep is performed.

The tensor models of damage accumulation for isotropic materials are considered. The material functions of aluminum alloy D16T inelastic deformation based on the results of tests of tension, torsion, tension with torsion with different relations of axial and shear deformations are defined. Approximations of experimental data by analytical expressions are proposed. These expressions include the dependence on the first and second invariants of the strain tensor. Comparison of experimental and theoretical dependences attests to adequacy of the proposed mathematical models.

The effective algorithm of static calculation of geometrically nonlinear compound thin structures is offered. Linear differential equations of moment theory are used. Nonlinearity is considered by assigning unknown initial angular displacement of each segment retaining the form of the generating line. Unknown values of algebraic equations resolving system are the arbitrary constants of the general solution and the initial angles of generating lines rotation. Linearization is realized by Newton–Raphson iterative method and provides the high precision of results.

Continuum percolation of the hard prolate ellipsoids of rotation with permeable shell has been investigated. It is the model of phase transition sol–gel. Ellipsoids are located in the cube randomly. For each set of parameters 100 tests are spent. For each test the finding of the percolation cluster is the main task. The fraction of the packing for which the probability of the percolation cluster appearance is equal 0.5, is called a percolation threshold. Value of the percolation threshold corresponds to the gel point. Dependence of value of the percolation threshold on thickness of permeable shell and aspect ratio has been obtained. In addition to the percolation threshold the other characteristics of the model have been obtained, such as: the size distribution of clusters, the average cluster size, the strength and the fractal dimension of the percolation cluster, the average value and the distribution of neighbors of an element, the critical exponents.

We propose a new approach to generating a pair of initial beams for a polarization converter that operates by summing up two opposite-sign circularly polarized beams. The conjugated pairs of vortex beams matched with laser modes are generated using binary diffractive optical elements. The same binary element simultaneously serves two functions: a beam shaper and a beam splitter. Two proposed optical arrangements are compared in terms of alignment complexity and energy efficiency. The diffractive optical elements in question have been designed and fabricated. Natural experiments that demonstrate the generation of vector higher-order cylindrical beams have been conducted.

The influence of the hot stamping temperature on the structural state of brass 59Cu−3,5Mn−2,5Al−0,5Fe−0,4Ni and the character of blocking rings destruction of the synchronizer of the car transfer change box have been investigated. It is shown that the lack of ductile $\alpha$-phase in the investigated alloy in combination with high temperature of heating ($780~^\circ \rm C$) before stamping leads to intergranular brittle fracture under the action of residual stresses. Decrease in temperature stamping to $700~^\circ \rm C$ with the same structural condition of the brass excludes cracking of the brass rings under the effect of residual stresses and changes the character of destruction under bending load with rock-like to pits-like one.

The purpose of the given paper is reviewing of mathematical resources of enciphering of the source text, allowing to ensure the simplicity of appropriate decryption; the source text is a sequence of integer weight coefficients. The composer of the cipher checks how the condition of separability of these coefficients is satisfied. The selection rule provides usage of operations of the lower or upper roundoff. The generated cipher is represented by the values of the finite sums.

For a complete hyperbolic equation of the third order with variable coefficients in the infinite rectangle the problem with two integral conditions and conjugation on the characteristic plane (Problem I) is considered. As auxiliary Darboux problem is solved by Riemann method which is much simplified by the special presentation of one of the boundary conditions. Taking Darboux problem as a basis for the solution, authors reduce the Problem I to the uniquely solvable integral equation, which gives an explicit solution to the Problem I.

This work addresses the problems of the deformed basis layer thickness assigning in the mathematical modeling of the building – foundation – soil systems on large foundation plates. It also contains the comparison of deformed soil layer thickness computation results for two methods: numerical and analytical.