PHYSICAL AND MATHEMATICAL SCIENCES
Let H G be a space of analytic functions of one variable in simply - connected domain G of the complex plane. It is known that a linear complex convolution operator is generated by a one - variable analytic function, a multivalued one in general. A known problem when all such functions are single - valued is solved. As it turned out, the solution to the problem is connected with the geometry of G domain. Set s G with property s G G G is termed re sidue of G domain. A class of simply connected regions whose residue is a connected set is described. Let the linear operator be continuous in function space, analytical in simply - connected domain G, and let it commute with diffe rentiation. Then it can be reduced to a complex convoluti on operator. It is proved that the function generating such an operator will always be single - valued for regions with a connected residue. When the residue of region G is not connected, there is always a complex convolution operator with a multivalued func tion generating a kernel.
An axial ly symmetric quasistatic thermoelasticity problem on the indentation of a flat - ended cylindrical punch with a constant temperature at its base into the functionally - graded half - space which elasticity modulus, Poisson ratio, heat conductivity and expansion coefficients are independently continuously varying in the boundary layer, is considered. Out of the contact area, the surface is perfectly thermally - insulated and stress - free. The earlier solution, obtained through the combined numerical and analytical ap proach (using Hankel integral transform and the modulating fun ction method) to the unmixed pro b lem on the arbitrary thermomechanical effect upon the inhomogeneous in depth thermoelastic half - space, is applied to solve the problem. The original problem is r educed to the system of dual integral equations. The properties of the dual integral equations kernel transforms allow applying a well - grounded bilateral asymptotic technique which is being actively developed at present. The approximate express ions for d et ermining the thermal flux, the half - space surface displacement, and the contact stresses under the heated stamp base, are obtained with the aid of this method. The numerical values of contact stresses for various cases of the thermomechanical properties va riation in the boundary layer of the half - space are provided. The cases either when values of the thermomechanical coating prope r ties are the same as those of the substrate, or when the property value differs twice (upward or dow n ward) on the surface, and linearly decreases (or goes up) in depth to the value in the substrate, are con sidered.
The applicability of the bionic techniques of artificial bee colonies for the implementation of the classical transpos ition cipher cryptanalysis is considered. The problem is a classical optimization problem to the solution of which the known techniques of artificial bee colonies fallen within a relatively new class of bioinspired opti mization methods are applied. It is shown that this is a subproblem of allocation, and it can b e solved with an artificial bee colony algorithm, as the bee behavior principle is a self - organization delivering a collective swarm goal. А t the first stage, a set of promising areas - sources is formed with the aid of scout - bees, at the second stage, the n eighborhood of these areas is explored with the aid of foraging bees. At this, the main goal of the bee colony is to find a source with a maximum amount of nectar. Solution representation methods (positions in search space) are considered , a formula for de termining an object function value (amount of nectar) is given. It is shown that the target search i s the d etermination of an optimal symbol combination with the highest value of the objective function. Princ iple stages of the artificial bee colony algorit hm, as well as an example of its application, are given. Keywords: