Vol 99, No 2 (2019)

The concept of a term and, accordingly, the concept of a formula are extended using new operators. These extensions of the language preserve the expressiveness of \(\Sigma \)-formulas and, at the level of \({{\Delta }_{0}}\)-formulas and terms, they ensure the polynomiality of algorithms for calculating the value of a term and deciding the truth of a \({{\Sigma }_{0}}\)-formula.

A new method is proposed for solving constrained optimization problems with inequality constraints in the case when the system of Kuhn–Tucker necessary optimality conditions is degenerate. This situation arises, for example, when the strict complementarity conditions do not hold for the solution. A reduction of the inequality-constraint problem to an equality-constraint one and the use of a new 2 factor Newton method for efficiently solving the resulting degenerate system of optimality conditions are justified.

Axiom A diffeomorphisms of closed 2-manifold of genus \(p \geqslant 2\) whose nonwandering set contains a perfect spaciously situated one-dimensional attractor are considered. It is shown that such diffeomorphisms are topologically semiconjugate to a pseudo-Anosov homeomorphism with the same induced automorphism of fundamental group. The main result of this paper is as follows. Two diffeomorphisms from the given class are topologically conjugate on perfect spaciously situated attractors if and only if the corresponding homotopic pseudo-Anosov homeomorphisms are topologically conjugate by means of a homeomorphism that maps a certain subset of one pseudo-Anosov homeomorphism onto a subset of the other.

Solutions x(t) of the equation \(\dot {x} = f(x)\), where \(x \in {{{\text{R}}}^{n}}\) and the function f(x) satisfies the Lipschitz condition with an arbitrary vector norm, are considered. It is proved that the lower bound for the distances between successive extrema x_k(t), k = 1, 2, …, n, is \(\frac{\pi }{L}\), where L is the Lipschitz constant. For nonconstant periodic solutions, the lower bound for the periods is \(\frac{{2\pi }}{L}\). These estimates are sharp for norms that are invariant with respect to permutations of indices.

The study is made of the problem of multiple interpolation on an infinite nodes set by the sums of absolutely convergent series of exponentials whose exponents are from a given set. For entire function conditions on nodes and exponents are obtained that give solubility of the problem. A new approach is demonstrated that enable us, for the case of holomorphic function in a domain, to obtain criteria of solubility of the problem for some class of exponents set and for a special class of nodes set. Moreover the necessity of the conditions is proved in great generality namely for arbitrary nodes sets and in the setting of interpolation by functions that are represented as the Laplace transforms of the Radon measures over the exponents set. Solubility is obtained of the global Cauchy problem for convolution operator with data on the nodes set in domain, in the form of the series of exponentials whose exponents belong to a sparse subset of zero set of characteristic function of the operator. The results substantially strengthen the known results on the theme.

For high-order ordinary differential equations containing terms with lower order derivatives, sufficient conditions for the absence of rapidly growing solutions are obtained.

New estimates for the independence numbers of distance graphs with vertices in \({{{\text{\{}}{\kern 1pt} - {\kern 1pt} 1,0,1{\text{\} }}}^{n}}\) are obtained.

A one-parameter family of left-invariant sub-Finsler problems on a four-dimensional nilpotent Lie group of depth 3 with two generators is considered. The indicatrix of sub-Finsler structures is a square rotated by an arbitrary angle in the distribution. Methods of optimal control theory are applied. Abnormal and singular normal trajectories are described, and their optimality is proved. Singular trajectories arriving at the boundary of the reachable set in fixed time are characterized. A bang-bang phase flow is constructed, and estimates for the number of switchings on bang-bang trajectories are obtained. The structure of all normal extremals is described. Mixed trajectories are studied.

Two analogues of Korn’s inequality on Heisenberg groups are constructed. First, the norm of the horizontal differential is estimated in terms of its symmetric part. Second, Korn’s inequality is treated as a coercive estimate for a differential operator whose kernel coincides with the Lie algebra of the isometry group. For this purpose, we construct a differential operator whose kernel coincides with the Lie algebra of the isometry group on Heisenberg groups and prove a coercive estimate for this operator. Additionally, a coercive estimate is proved for a differential operator whose kernel coincides with the Lie algebra of the group of conformal mappings on Heisenberg groups.

The existence of an aperiodic orbit for an outer billiard outside a regular dodecagon is proved. It is shown that almost all orbits of such an outer billiard are periodic, and all possible periods are explicitly listed. The proofs of the theorems make use of computer calculations.

The problem of reconstructing a piecewise Hölder continuous function describing the refractive index of an inhomogeneous obstacle scattering a monochromatic wave is considered. The boundary value scattering problem is reduced to a system of integral equations. The equivalence of the integral and differential formulations of the problem is proved. A two-step method for solving the inverse problem is proposed. A linear integral equation of the first kind is solved at the first step. Sufficient conditions for the uniqueness of its solution in the class of piecewise constant functions are obtained. At the second step, the unknown refractive index is explicitly expressed in terms of the solution obtained at the first step.

The classical Farkas theorem of the alternative is considered, which is widely used in various areas of mathematics and has numerous proofs and formulations. An entirely new elementary proof of this theorem is proposed. It is based on the consideration of a functional that, under Farkas’ condition, is bounded below on the whole space and attains a minimum. The assertion of Farkas’ theorem that a vector belongs to a cone is equivalent to the fact that the gradient of this functional is zero at the minimizer.

Sufficient maximality conditions for graph surfaces on two-step Carnot groups with a sub-Lorentzian structure are deduced.

A completely Liouville integrable Hamiltonian system with two degrees of freedom that describes the dynamics of two vortex filaments in a Bose–Einstein condensate enclosed in a harmonic trap is considered. For a pair of vortices of positive intensity, a bifurcation of three Liouville tori into a single one is detected. Such a bifurcation was previously found in the Goryachev–Chaplygin–Sretensky integrable case in rigid body dynamics. For an integrable perturbation of the intensity ratio parameter, the identified bifurcation proves to be unstable, which leads to bifurcations of the type of two tori into one and vice versa.

A new computer-aided detection (CAD) system for lung nodule detection and selection in computed tomography scans is substantiated and implemented. The method consists of the following stages: preprocessing based on threshold and morphological filtration, the formation of suspicious regions of interest using a priori information, the detection of lung nodules by applying the fractal dimension transformation, the computation of informative texture features for identified lung nodules, and their classification by applying the SVM and AdaBoost algorithms. A physical interpretation of the proposed CAD system is given, and its block diagram is constructed. The simulation results based on the proposed CAD method demonstrate advantages of the new approach in terms of standard criteria, such as sensitivity and the false-positive rate.