Vol 30, No 1 (2019)

We consider the notion of universal function such that a subset of the function’s values defines any function from some set. For the set of linear functions, we consider all the combinations of the number of variables and the number of values, except four-valued functions of two variables.

We consider one of the central intractable problems of logical data analysis – finding maximal independent elements of partial orders (dualization of a product of partial orders). An important particular case is considered with each order a chain. If each chain is of cardinality 2, the problem involves the construction of a reduced disjunctive normal form of a monotone Boolean function defined by a conjunctive normal form. An asymptotic expression is obtained for a typical number of maximal independent elements of products for a large number of finite chains. The derivation of such asymptotic bounds is a technically complex problem, but it is necessary for the proof of existence of asymptotically optimal algorithms for the monotone dualization problem and the generalization of this problem to chains of higher cardinality. An asymptotically optimal algorithm is described for the problem of dualization of a product of finite chains.

We consider the realization of Boolean functions by formulas with restrictions on superpositions of basis functions such that superposition is allowed only by iterative variables. For a number of special symmetrical bases, we establish new high-accuracy bounds of the Shannon function L(n) for the complexity of realization of Boolean functions dependent on n direct variables.

We consider symmetries of tensor decompositions related to an algorithm for computing the commutator of 2 × 2 matrices using 5 multiplications.

An iterative algorithm is described for three-dimensional deconvolution in the Fourier plane using parallel computations on CPU and GPU. The algorithm demonstrates easy scalability and can process any number of input images of any size. It is only limited by the local storage volume.

The article considers mathematical modeling of the electromagnetic field in a nonhomogeneous medium by the integral equation method. The case of high-contrast conducting media is studied in detail, with the conducting nonhomogeneity embedded in a poorly conducting medium. The analysis of the integral equation, in this case, has shown that the solution deteriorates when the conducting nonhomogeneity is inside a low-conductivity layer. It is shown that this effect can be overcome by Dmitriev’s method of elevated background conductivity. The contrast effect is most pronounced for the H -polarized two-dimensional electromagnetic field in a nonhomogeneous medium. The numerical experiment has accordingly been conducted for this particular case. The solution computed by the integral equation method with elevated background conductivity is compared with the solution computed by the finite-difference method. The results of the two methods show excellent fit.