Vol 95, No 3 (2017)
- Year: 2017
- Articles: 26
- URL: https://journals.rcsi.science/1064-5624/issue/view/13847
Mathematics
Nehari method for the generalized Ginzburg–Landau system
Abstract
The Dirichlet problem for the generalized Ginzburg–Landau system is considered. The existence of positive vector solutions is proved in the following three cases: (1) the cross term has weak growth; (2) the interaction constant is large enough; and (3) the cross term has strong growth and the interaction constant is positive and close to zero.
Kantorovich–Wright integral and representation of quasi-Banach lattices
Abstract
The purpose of this paper is two-fold: first, to outline a purely order-based integral of the type of the Kantorovich–Wright integral of scalar functions with respect to a vector measure defined on a δ-ring and taking values in a Kσ-space (that is, a Dedekind σ-complete vector lattice) and, secondly, prove new theorems on the representation of Dedekind complete vector lattices and quasi-Banach lattices in the form of lattices of functions integrable or “weakly” integrable with respect to an appropriate vector measure. In particular, it is shown that, in studying quasi-Banach lattices, when the duality method does not apply, the Kantorovich–Wright integral is more flexible than the Bartle–Dunford–Schwartz integral.
Liouville nonintegrability of sub-Riemannian problems on free Carnot groups of step 4
Abstract
One of the main approaches to the study of the Carnot–Carathéodory metrics is the Mitchell–Gromov nilpotent approximation theorem, which reduces the consideration of a neighborhood of a regular point to the study of the left-invariant sub-Riemannian problem on the corresponding Carnot group. A detailed analysis of sub-Riemannian extremals is usually based on the explicit integration of the Hamiltonian system of Pontryagin’s maximum principle. In this paper, the Liouville nonintegrability of this system for left-invariant sub-Riemannian problems on free Carnot groups of step 4 and higher is proved.
Modular and norm inequalities for operators on the cone of decreasing functions in Orlicz space
Abstract
Modular and norm inequalities are considered for positively homogeneous operators on the cone of all nonnegative functions and on the cone Ω of nonnegative decreasing functions from the weighted Orlicz space with a general weight and a general Young function. A reduction theorem is obtained for the norm of an operator on Ω. This norm is shown to be equivalent to the norm of a modified operator on the cone of all nonnegative functions in the above Orlicz space. A similar theorem is obtained for modular inequalities. The results are based on the application of the principle of duality, which gives a description of the associated Orlicz norm for Ω. We also establish the equivalence of modular inequalities on the cone Ω and modified modular inequalities on the cone of all nonnegative functions in Orlicz space. In the general situation, the forms of these answers are substantially different from the descriptions obtained earlier by P. Drabek, A. Kufner, and H. Heinig under the assumption that the Young function and its complementary function satisfy Δ2-conditions.
S-units and periodicity of square root in hyperelliptic fields
Abstract
For a polynomial f of odd degree, nontrivial S-units can be effectively related to the continued fraction expansion of elements involving the square root of the polynomial f only in the case where S consists of an infinite valuation and a finite valuation determined by a first-degree polynomial h. In the paper, the proof that the quasi-periodicity of the continued fraction expansion of an element of the form \(\frac{{\sqrt f }}{{{h^s}}}\) implies periodicity is completed. In particular, it is proved that the continued fraction expansion of \(\sqrt f \) for f of any degree is quasi-periodic in k((h)) it and only if it is periodic.
On the limit shape of elements of an arithmetic semigroup with an exponentially growing counting function of basis elements
Abstract
We consider an arithmetic semigroup with exponential growth of the counting function of abstract primes. The Bose–Einstein statistics provides the most probable mean occupation numbers in the sense that large deviations of a sum of occupation numbers from the corresponding sum for the Bose–Einstein statistics have small probabilities. The probabilities of large deviations are estimated.
Problem of determining the permittivity in the stationary system of Maxwell equations
Abstract
The stationary system of Maxwell equations for a unmagnetized nonconducting medium is considered. For this system, the problem of determining the permittivity ε from given electric or magnetic fields is studied. It is assumed that the electromagnetic field is induced by a plane wave coming from infinity in the direction ν. It is also assumed that the permittivity is different from a given positive constant ε0 only inside a compact domain Ω ⊂ R3 with a smooth boundary S. To find ε inside Ω, the solution of the corresponding direct problem for the system of electrodynamic equations on the shadow portion of the boundary of Ω is specified for all frequencies starting at some fixed ω0 and for all ν. The high-frequency asymptotics of the solution to the direct problem is studied. It is shown that the information specified makes it possible to reduce the original problem to the well-known inverse kinematic problem of determining the refraction coefficient inside Ω from the traveling times of an electromagnetic wave. This leads to a uniqueness theorem for the solution of the problem under consideration and opens up the opportunity of its constructive solution.
Solutions to systems of linear Fredholm integral equations of the third kind with multipoint singularities
Abstract
A new approach is used to show that the solution for one class of systems of linear Fredholm integral equations of the third kind with multipoint singularities is equivalent to the solution of systems of linear Fredholm integral equations of the second kind with additional conditions. The existence, nonexistence, uniqueness, and nonuniqueness of solutions to systems of linear Fredholm integral equations of the third kind with multipoint singularities are analyzed.
A summation formula for divergent continued fractions
Abstract
Paradoxical formulas which have been proposed by a number of authors for evaluating divergent continued fractions are discussed. The point of the paradoxes is that limit passages (which are to be rigorously defined) result in the convergence real sequences to complex values. These formulas are refined and substantiated on the basis of the theory of uniform distribution supplemented with certain statements of complex analysis.
On the traces of Sobolev functions on Lipschitz surfaces
Abstract
Functions from the Sobolev spaces Wp1(Q) are considered on a unit cube Q ⊂ Rn, and the properties of their traces on Lipschitz surfaces are examined. The relation is found between the Hölder exponent α and the Hausdorff dimension of the family of poor k-dimensional planes Γ on which the traces do not belong to Cα(Γ). For the corresponding families of poor k-dimensional Lipschitz surfaces, estimates in terms of p-modules are obtained.
Pullback attractors for a model of weakly concentrated aqueous polymer solution motion with a rheological relation satisfying the objectivity principle
Abstract
The qualitative dynamics of weak solutions to a nonautonomous model of polymer solution motion (with a rheological relation satisfying the objectivity principle) is studied using the theory of pullback attractors of trajectory spaces. For this purpose, the existence of weak solutions is proved for the model under study, a family of trajectory spaces is defined, trajectory and minimal pullback attractors are introduced, and their existence is proved.
An optimal Berry-Esseen type inequality for expectations of smooth functions
Abstract
We provide an optimal Berry-Esseen type inequality for Zolotarev’s ideal ζ3-metric measuring the difference between expectations of sufficiently smooth functions, like |·|3, of a sum of independent random variables X1,..., Xn with finite third-order moments and a sum of independent symmetric two-point random variables, isoscedastic to the Xi. In the homoscedastic case of equal variances, and in particular, in case of identically distributed X1,..., Xn the approximating law is a standardized symmetric binomial one. As a corollary, we improve an already optimal estimate of the accuracy of the normal approximation due to Tyurin (2009).
On the periodicity of continued fractions in hyperelliptic fields
Abstract
On the basis of a given criterion for the quasi-periodicity of continued fractions for elements of the hyperelliptic field L = K(x)(\(\sqrt f \)), where K is an arbitrary field of characteristic different from 2 and f ∈ K[x] is a square-free polynomial, new polynomials f ∈ Q[x] of odd degree for which the elements of \(\sqrt f \) have periodic continued fraction expansion are found.
Strengthened equilibrium for game problems with side interest of participants
Abstract
A strengthened equilibrium for conflict problems with partially overlapping game sets of participants is proposed, which is of substantial help in determining fair division in cooperative games and makes it possible to refine the hierarchy of all known equilibria.
Fixed-point and coincidence theorems in ordered sets
Abstract
The paper is devoted to the problem of the existence of common fixed points and coincidence points of a family of set-valued maps of ordered sets. Fixed-point and coincidence theorems for families of set-values maps are presented, which generalize some of the known results. The presented theorems, unlike previous ones, do not assume the maps to be isotone or coverable. They require only the existence of special chains having lower bounds with certain properties in the ordered set.
Transmission problem for odd-order differential equations with two time variables and a varying direction of evolution
Abstract
The solvability of a boundary value problem for the differential equation \(h\left( x \right){u_t} + {\left( { - 1} \right)^m}\frac{{{\partial ^{2m + 1}}u}}{{\partial {a^{2m + 1}}}} - {u_{xx}} = f\left( {x,t,a} \right)\) is studied in the case where h(x) has a jump discontinuity and reverses its sign on passing through the discontinuity point. Existence and uniqueness theorems are proved in the case of solutions having all Sobolev generalized derivatives involved in the equation.
Mixed problem for the wave equation with a summable potential and nonzero initial velocity
Abstract
The resolvent approach in the Fourier method, combined with Krylov’s ideas concerning convergence acceleration for Fourier series, is used to obtain a classical solution of a mixed problem for the wave equation with a summable potential, fixed ends, a zero initial position, and an initial velocity ψ(x), where ψ(x) is absolutely continuous, ψ'(x) ∈ L2[0,1], and ψ(0) = ψ(1) = 0. In the case ψ(x) ∈ L[0,1], it is shown that the series of the formal solution converges uniformly and is a weak solution of the mixed problem.
Mathematical Physics
Entropy balance for the one-dimensional hyperbolic quasi-gasdynamic system of equations
Abstract
Entropy balance in the one-dimensional hyperbolic quasi-gasdynamic (HQGD) system of equations is analyzed. In particular, in regular flow regimes, it is shown that the behavior of entropy in the HQGD system is mainly determined by terms involving the natural viscosity and thermal conductivity coefficients. The total entropy production differs from the Navier–Stokes equations for viscous compressible heat-conducting gases by O(τ2) terms, where τ is a relaxation parameter. Additionally, a similar analysis of energy balance is performed for the simpler case of the barotropic HQGD system, which is of interest for some applications.
Quasi-periodic and chaotic relaxation oscillations in a laser model with variable delayed optoelectronic feedback
Abstract
We derive asymptotically discrete mappings which determine the dynamics of relaxation spikes in a model of laser with optoelectronic feedback in the pump circuit. The time delay in the feedback path varies periodically. Bifurcations of mapping’s attractors correspond to the appearance of quasi-periodic and chaotic spiking with special properties.
Numerical modeling of dynamic wave effects in rock masses
Abstract
Spatial dynamic wave effects occurring in rocks with ravines and caverns were studied. The influence exerted by the explosion type and the cavern-to-ravine distance on the formation of spatial dynamic wave patterns and seismograms was analyzed in the case of horizontal and vertical reception lines. The gridcharacteristic method and the full wave joint numerical modeling of elastic and acoustic waves were used.
Inhomogeneous Burgers equation with modular nonlinearity: Excitation and evolution of high-intensity waves
Abstract
Solutions to an inhomogeneous partial differential equation of the second-order like Burgers equation are derived. Instead of the common quadratically nonlinear term, this equation contains the term with modular nonlinearity. This model describes the excitation of elastic waves in dissipative media differently reacting to tensile and compressive stresses. The equation is linear for the functions, preserving the sign. Nonlinear effects are manifested only to alternating functions. The solution for the periodic signal is found. The processes of generation of fundamental and higher harmonics are studied. The stationary wave profile is constructed. For one special kind of right-hand-side of the “modular” equation the solution in the form of S-wave is pointed out which is a bipolar single pulse.
Control Theory
Flight-test-based construction of structurally stable models for the dynamics of large space structures
Abstract
A method is proposed for constructing dynamic models of large space structures (LSS) when their parameter values are uncertain and LSS state measurements in actual operation conditions are incomplete and subject to errors. The method can be used to construct structurally stable dynamic models of LSS when ground-based LSS tests are impossible.
Logical-optimization approach to pursuit problems for a group of targets
Abstract
Problems of automatic action planning and optimal control of unmanned vehicles in an adversarial environment are considered. An approach based on nonmonotonic logic in the language of positively formed formulas for automatically planning a route to a chosen target area is developed. The choice of this area, as well as classification and determination of priority targets, is implemented by means of logical inference with classical semantics. The optimal pursuit problem for the example of a group of three targets is solved.