Vol 61, No 7 (2017)

We consider a perturbation of a periodic second order differential operator, defined on the real axis, which is a special case of the Hill operator. The perturbation is realized by a sum of two complex-valued potentials with compact supports. The potentials depend on two small parameters. One of them describes the lengths of the supports of the potentials and the reciprocal to the second one corresponds to the maximum values of the potentials. We obtain a sufficient condition, under fulfillment of which, the eigenvalues arise from the edges of non-degenerate lacunas of continuous spectrum, and construct their asymptotics. We also give a sufficient condition under which the eigenvalues do not arise.

The most important problem in the theory of phenomenologically symmetric geometries of two sets is that of classification of these geometries. In this paper, complexifying the metric functions of some known phenomenologically symmetric geometries of two sets (PSGTS) with the use of associative hypercomplex numbers, we find metric functions of new geometries in question. For these geometries, we find equations of the groups of motions and establish phenomenological symmetry, i.e., find functional relations between metric functions for certain finite number of arbitrary points. In particular, for one-component metric functions of PSGTS’s of ranks (2, 2), (3, 2), (3, 3), we find (n + 1)-component metric functions of the same ranks. For these metric functions, we find finite equations of the groups of motions and equations that express their phenomenological symmetry.

We study properties and ways of classification of threshold functions as well as known estimates of complexity of implementing in the functional elements type of circuits. We determine a dependence of the maximum values of variables weights on their number. Using the intermediate conversion method we obtain a precise upper bound of complexity of implementing arbitrary threshold functions in the functional elements type of circuits.

We study solutions of a polycaloric equation and an equation of mixed parabolichyperbolic type of the second order. We prove the sign-definiteness of the solution in dependence of the right-hand side of the equation. Based on these results we study the sign-definiteness of a solution to a higher-order inhomogeneous equation of mixed parabolic-hyperbolic type in dependence on the right-hand side of the equation.

We propose a formula for the conformalmapping of the upper half-plane onto a polygonal domain, which generalizes the Schwarz–Christoffel equation. It is obtained by terms of partial solution to the Hilbert boundary-value problem with a countable set of singularity points of the coefficients including a turbulence of logarithmic type at the infinity point. We also prove the existence of closed and univalent mappings.

We deal with analogs of multicolor urn schemes such that the number of particles is not more than a given number. We introduce conditions which provide the convergence of random variables which is the maximal number of taken particles of the same color to a random variable that has values zero and one. We prove this convergence in the case when a number of taken particles is not more than a fixed number and number of colors converges to infinity. We also consider the case when the number of taken particles converges to infinity.