Vol 95, No 2 (2017)

Properties of fractal functions which are not differentiable in the classical sense but have continuous Weil-type derivatives of variable order at each point are studied. It is shown that the Weierstrass, Takagi, and Besicovitch classical fractal functions have such derivatives. An example of an oscillatory system controlling which requires constructing a fractal control function having a Weil-type derivative of variable order at each point is considered.

Area formulas for classes of Hölder mappings of Carnot groups and the corresponding graph mappings are obtained. The calculation of a nonintrinsic measure is exemplified.

Conditions are determined under which, for pattern recognition problems with standard information (Ω-regular problems), a correct algorithm and a six-level spatial neural network reproducing the calculations performed by the correct algorithm can be constructed. The proposed approach to constructing the neural network is not related to the traditional approach based on minimizing a functional.

A new family of multioperator approximations to derivatives of even and odd orders with inversion of two-point operators is considered. Existence and uniqueness theorems are stated for multioperators of formally arbitrary orders, and their spectral properties are examined. A scheme for a test hyperbolic equation with a multioperator approximation of 36th order is analyzed as an example. The accuracy and convergence of numerical solutions to the test problem are estimated.

This paper continues the authors’ long-time studies concerning biomathematics (see [1]). One direction of research was related to the development of a new area of modern biomathematics, namely, a mathematical theory of genetic codes. The foundations of this theory are described below. To the author’s knowledge, the results presented have not been found in other researchers’ publications.

The asymptotic behavior, as ε → 0, of the solution uε to a variational inequality with nonlinear constraints for the p-Laplacian in an ε-periodically perforated domain when p ∈ (1, 2) is studied.

Mathematical analysis of large genomes is a problem of current interest motivated by the development of DNA sequencing methods. To date, human and some other genomes have been sequenced. Certain characteristics of a genetic code shared by all these genomes are numerically analyzed in this paper. Relying on the results, a new property of the genetic code for overlaps of six and three genes in a single DNA strand is formulated. The choice of three termination codons out of 64 possible genetic code triplets does not influence the cardinality of the sets of nucleotide chains admitting sextuple and triple overlaps of genes.

In order to detect vessel locations in spherical images of retina we consider the problem of minimizing the functional \(\int\limits_0^l {\mathfrak{C}\left( {\gamma \left( s \right)} \right)\sqrt {{\xi ^2} + k_g^2\left( s \right)} ds}\) for a curve γ on a sphere with fixed boundary points and directions. The total length l is free, s denotes the spherical arclength, and k_g denotes the geodesic curvature of γ. Here the smooth external cost C ≥ δ > 0 is obtained from spherical data. We lift this problem to the sub-Riemannian (SR) problem in Lie group SO(3) and propose numerical solution to this problem with consequent comparison to exact solution in the case C = 1. An experiment of vessel tracking in a spherical image of the retina shows a benefit of using SO(3) geodesics.

On the Lie groups PSL₂(ℝ) and SO₃ we consider left-invariant Riemannian metrics with two equal eigenvalues. The global optimality of geodesics is investigated. We find the parametrization of geodesics, the cut locus and the equations for the cut time. When the third eigenvalue of a metric tends to the infinity the cut locus and the cut time converge to the cut locus and the cut time of the sub-Riemannian problem.

An asymptotic solution to the problem of sound pulse propagation in deep sea is derived using the Maslov canonical operator. An example of a waveguide with the Munk sound speed profile and a point source is considered, and an asymptotic expression for the pulse signal time series at the receiver is obtained. The asymptotic solution is compared with the solution computed using the normal mode theory.

An experience of designing integrated hardware and software solutions for high-performance computing in solving modern geophysical problems on the basis of full-wave inversion is described. Problems of designing mass high-performance software systems intended for extensive use in industry are discussed.

An inversion problem for a linear time-invariant MIMO system with possibly unstable zero dynamics is considered. Sufficient conditions for the invertibility of such systems are given, and an algorithm for invertor synthesis is proposed. The results are extended to time-delay systems with commensurable delays.