Showing papers in "Ukrainian Mathematical Journal in 1996"
TL;DR: In this article, a criterion for solvability of a general linear Noether boundary-value problem for systems of integro-differential equations of Fredholm type with degenerate kernel was established.
Abstract: By using methods of the theory of generalized inverse matrices, we establish a criterion of solvability and study the structure of the set of solutions of a general linear Noether boundary-value problem for systems of integro-differential equations of Fredholm type with degenerate kernel.
31 citations
TL;DR: In this article, all closed sesquilinear forms associated with m-sectorial extensions of a densely defined sectorial operator with vertex at the origin are described, and the authors describe all closed m-sectorsial extensions associated with the m-veto operator.
Abstract: We describe all closed sesquilinear forms associated with m-sectorial extensions of a densely defined sectorial operator with vertex at the origin.
30 citations
TL;DR: For the data of sampling from a mixture of several components with varying concentrations, the authors construct nonparametric estimates for the distributions of components and determine the rank correlation coefficient, and prove the consistency of the rank coefficient and the efficiency of the estimates of distributions.
Abstract: For the data of sampling from a mixture of several components with varying concentrations, we construct nonparametric estimates for the distributions of components and determine the rank correlation coefficient. We prove the consistency of the rank coefficient and the efficiency of the estimates of distributions.
26 citations
TL;DR: For a rectangular real matrix, a decomposition in weighted singular numbers and weighted orthogonal matrices was obtained in this paper, and a representation of a weighted pseudo-inverse matrix in terms of weighted Orthogonal Matrices and weighted Singular Numbers was obtained.
Abstract: For a rectangular real matrix, we obtain a decomposition in weighted singular numbers. On this basis, we obtain a representation of a weighted pseudoinverse matrix in terms of weighted orthogonal matrices and weighted singular numbers.
23 citations
TL;DR: In this article, the existence and construction of approximate solutions of nonlinear boundary value problems for ordinary differential equations is investigated and improved estimates necessary for the justification of the numerical-analytic method for the investigation of the existence of the approximate solutions.
Abstract: We establish new improved estimates necessary for the justification of the numerical-analytic method for the investigation of the existence and construction of approximate solutions of nonlinear boundary-value problems for ordinary differential equations
22 citations
TL;DR: There is a very short chain that joins dynamical systems with the simplest phase space (real line) and dynamical system with the most complicated phase space containing random functions, as well as mentioned in this paper.
Abstract: There is a very short chain that joins dynamical systems with the simplest phase space (real line) and dynamical systems with the “most complicated” phase space containing random functions, as well. This statement is justified in this paper. By using “simple” examples of dynamical systems (one-dimensional and two-dimensional boundary-value problems), we consider notions that generally characterize the phenomenon of turbulence—first of all, the emergence of structures (including the cascade process of emergence of coherent structures of decreasing scales) and self-stochasticity.
18 citations
TL;DR: In this article, conditions under which partial differential equations are reducible to equations with a smaller number of independent variables are determined and shown to be necessary and sufficient in the case of a single dependent variable.
Abstract: We determine conditions under which partial differential equations are reducible to equations with a smaller number of independent variables and show that these conditions are necessary and sufficient in the case of a single dependent variable.
15 citations
TL;DR: In this article, the potential and flow functions of three-dimensional potential solenoidal fields with axial symmetry were derived from a Banach algebra of even Fourier series, and the relationship between these functions and the Stokes flow function was established.
Abstract: We obtain new representations of the potential and flow function of three-dimensional potential solenoidal fields with axial symmetry, study principal algebraic analytic properties of monogenic functions of vector variables with values in an infinite-dimensional Banach algebra of even Fourier series, and establish the relationship between these functions and the axially symmetric potential or the Stokes flow function. The developed approach to the description of the indicated fields is an analog of the method of analytic functions in the complex plane used for the description of two-dimensional potential fields.
15 citations
TL;DR: In this article, the authors introduced the concept of convergence of functionates to integral functions and studied the relationship between this type of convergence and the convergence of solutions of Neumann variational problems and proved a theorem on the selection of a subsequence Γ-convergent to an integral functional.
Abstract: We introduce and study the concept of Γ-convergence of functionateI
s
:W
k,m
(Ω)→ℝ,s=1,2,..., to a functional defined on (W
k,m
(Ω))2 and describe the relationship between this type of convergence and the convergence of solutions of Neumann variational problems. For a sequence of integral functionateI
s
:W
k,m
(Ω)→ℝ, we prove a theorem on the selection of a subsequence Γ-convergent to an integral functional defined on (W
k,m
(Ω))2.
14 citations
TL;DR: In this paper, it was shown that for every sequence (a k) of complex numbers satisfying the conditions Σ(1/|a k |) < ∞ and |a k+1| − |a K | ↗ ∞ (k → ∞), there exists a continuous functionl decreasing to 0 on [0, + ∞] and such thatf(z) = Π(1 −z/| a k | ) is an entire function of finitel-index.
Abstract: We prove that, for every sequence (a k) of complex numbers satisfying the conditions Σ(1/|a k |) < ∞ and |a k+1| − |a k | ↗ ∞ (k → ∞), there exists a continuous functionl decreasing to 0 on [0, + ∞] and such thatf(z) = Π(1 −z/|a k |) is an entire function of finitel-index.
13 citations
TL;DR: In this paper, conditions for the existence of periodic solutions of nonlinear difference equations with continuous argument were established. But these conditions are not applicable to nonlinear systems with continuous arguments, such as the one we consider here.
Abstract: We establish conditions for the existence of periodic solutions of systems of nonlinear difference equations with continuous argument.
TL;DR: In this paper, it was shown that under certain conditions on a positive function continuous on [0, +∞], there exists an entire transcendental functionf of bounded l-index such that lnlnMf(r)lnL(r),r→∞, whereMf (r)=max {|f(z)|: |z|=r} and L(r)=∫ 0rl(t)dt.
Abstract: We prove that, under certain conditions on a positive functionl continuous on [0, +∞], there exists an entire transcendental functionf of boundedl-index such that lnlnMf(r)lnL(r),r→∞, whereMf(r)=max {|f(z)|: |z|=r} andL(r)=∫0rl(t)dt. Ifl(r)=rp-1 forr≥1, 0<ρ<∞, then there exists an entire functionf of boundedl-index such thatMf(r)≈rp.
TL;DR: In this paper, it was shown that linear extensions whose solutions are not almost periodic form a set of the second category inL (representable as an intersection of countably many everywhere dense open subsets).
Abstract: For a discrete dynamical system ω n =ω0+αn, where a is a constant vector with rationally independent coordinates, on thes-dimensional torus Ω we consider the setL of its linear unitary extensionsx n+1=A(ω0+αn)x n , whereA (Ω) is a continuous function on the torus Ω with values in the space ofm-dimensional unitary matrices. It is proved that linear extensions whose solutions are not almost periodic form a set of the second category inL (representable as an intersection of countably many everywhere dense open subsets). A similar assertion is true for systems of linear differential equations with quasiperiodic skew-symmetric matrices.
TL;DR: The boundary value problem with boundary conditions was studied in this article, where the authors introduced a concept of k-regular boundary conditions and deduced the following asymptotic formula for eigenvalues of the boundary-value problem.
Abstract: We study a boundary-value problem x
(n) + Fx = λx, U
h(x) = 0, h = 1,..., n, where functions x are given on the interval [0, 1], a linear continuous operator F acts from a Holder space H
y into a Sobolev space W
1
n+s
, U
h are linear continuous functional defined in the space
$$H^{k_h } $$
, and k
h ≤ n + s - 1 are nonnegative integers. We introduce a concept of k-regular-boundary conditions U
h(x)=0, h = 1, ..., n and deduce the following asymptotic formula for eigenvalues of the boundary-value problem with boundary conditions of the indicated type:
$$\lambda _v = \left( {i2\pi v + c_ \pm + O(|v|^\kappa )} \right)^n $$
, v = ± N, ± N ± 1,..., which is true for upper and lower sets of signs and the constants κ≥0 and c
± depend on boundary conditions.
TL;DR: In this paper, for the nonlinear system of partial differential equations, which describes the evolution of temperature and density in TOKAMAK plasmas, multiparameter families of exact solutions are constructed by the Lie-method reduction of initial systems of equations to a system of ordinary differential equations.
Abstract: For the nonlinear system of partial differential equations, which describes the evolution of temperature and density in TOKAMAK plasmas, multiparameter families of exact solutions are constructed. The solutions are constructed by the Lie-method reduction of initial systems of equations to a system of ordinary differential equations. Examples of non-Lie ansatze and exact solutions are also presented.
TL;DR: In this article, the problem of averaging Dirichlet problems for nonlinear elliptic second-order equations in domains with fine-grained boundary was studied and a pointwise estimate for solutions of the model nonlinear boundary-value problem was given.
Abstract: We study the problem of averaging Dirichlet problems for nonlinear elliptic second-order equations in domains with fine-grained boundary. We consider a class of equations admitting degeneration with respect to the gradients of solutions. We prove a pointwise estimate for solutions of the model nonlinear boundary-value problem and construct an averaged boundary-value problem under new structural assumptions concerning perforated domains. In particular, it is not assumed that the diameters of cavities are small as compared to the distances between them.
TL;DR: In this article, the authors establish consistency conditions for equations with additional restrictions in a Hilbert space, suggest and justify iterative methods for the construction of approximate solutions, and describe the relationship between these methods and the Sokolov projection-iterative method.
Abstract: We establish consistency conditions for equations with additional restrictions in a Hilbert space, suggest and justify iterative methods for the construction of approximate solutions, and describe the relationship between these methods and the Sokolov projection-iterative method.
TL;DR: For a linear operatorS in a Hilbert space ℋ, the relationship between the singularity of S and the density of the set k is investigated in this paper, where D(S) ∩ℛ(kS) = 0.
Abstract: For a linear operatorS in a Hilbert space ℋ, the relationship between the following properties is investigated: (i)S is singular (= nowhere closable), (ii) the set kerS is dense in ℋ, and (iii)D(S)∩ℛ(S)={0}.
TL;DR: By using decomposable subgroups of the generalized Poincare group P(1,4), the authors performed a symmetry reduction of a nonlinear five-dimensional wave equation to differential equations with a smaller number of independent variables.
Abstract: By using decomposable subgroups of the generalized Poincare group P(1,4), we perform a symmetry reduction of a nonlinear five-dimensional wave equation to differential equations with a smaller number of independent variables. On the basis of solutions of the reduced equations, we construct some classes of exact solutions of the equation under consideration.
TL;DR: In this paper, the authors studied the well-posedness of a problem with multipoint conditions with respect to time in a tube domain for linear hyperbolic equations of order 2n (n ≥ 1) with coefficients depending onx.
Abstract: By using the metric approach, we study the problem of classical well-posedness of a problem with multipoint conditions with respect to time in a tube domain for linear hyperbolic equations of order 2n (n ≥ 1) with coefficients depending onx. We prove metric theorems on lower bounds for small denominators appearing in the course of the solution of the problem.
TL;DR: In this paper, the authors established estimates for the orders of the best approximations of the classes of functions of many variablesB 1,θ r andB p,α r by orthogonal projections of functions from these classes onto the subspaces of trigonometric polynomials.
Abstract: In the spaceL q, 1
TL;DR: In this paper, the structure of the distribution of a random variable for which elements of the corresponding elementary continued fraction are independent random variables is completely studied, and it is shown that the distribution is pure and absolute continuity is impossible.
Abstract: The structure of the distribution of a random variable for which elements of the corresponding elementary continued fraction are independent random variables is completely studied. We prove that the distribution is pure and the absolute continuity is impossible, give a criterion of singularity, and study the properties of the spectrum. For the distribution of a random variable for which elements of the corresponding continued fraction form a uniform Markov chain, we describe the spectrum, obtain formulas for the distribution function and density, give a criterion of the Cantor property, and prove that an absolutely continuous component is absent.
TL;DR: In this article, conditions for the uniqueness of a solution of the problem for a system of equations unresolved with respect to the time derivative without initial conditions in a noncylindrical domain were established.
Abstract: We establish conditions for the uniqueness of a solution of the problem for a system of equations unresolved with respect to the time derivative without initial conditions in a noncylindrical domain. The system considered, in particular, contains pseudoparabolic equations.
TL;DR: In this article, the authors consider a system of linear difference equations and prove that the system is exponentially dichotomous on the semiaxis and the exponential dichotomy on the entire axis.
Abstract: We consider a system of linear difference equationsxn+1 =A (n)xn in anm-dimensional real or complex spaceVsum with detA(n) = 0 for some or alln eZ. We study the exponential dichotomy of this system and prove that if the sequence {A(n)} is Poisson stable or recurrent, then the exponential dichotomy on the semiaxis implies the exponential dichotomy on the entire axis. If the sequence {A (n)} is almost periodic and the system has exponential dichotomy on the finite interval {k, ...,k +T},k eZ, with sufficiently largeT, then the system is exponentially dichotomous onZ.
TL;DR: In this article, the authors describe nonlinear Galilei-invariant higher-order equations of Burgers and Korteweg-de Vries types and construct new nonlinear extensions for the AG(1, 1) algebra.
Abstract: We describe nonlinear Galilei-invariant higher-order equations of Burgers and Korteweg-de Vries types We study symmetry properties of these equations and construct new nonlinear extensions for the Galilei algebra AG(1, 1)
TL;DR: In this article, the authors consider the problem of applying the averaging method to the asymptotic approximation of solutions of differential inclusions of standard form in the case where the average of the right-hand side does not exist.
Abstract: We consider the problem of application of the averaging method to the asymptotic approximation of solutions of differential inclusions of standard form in the case where the average of the right-hand side does not exist.
TL;DR: In this paper, the authors prove Tauberian and Abelian theorems for Hankel-type integral transformations, and prove that Hankel type integral transformations can be expressed as integral transformations.
Abstract: We prove Tauberian and Abelian theorems for Hankel-type integral transformations.
TL;DR: In this article, necessary and sufficient conditions for the solvability of inhomogeneous linear boundary value problems for systems of ordinary differential equations with pulse influence in the case where the number of boundary conditions is not equal to the order of the differential system (Noetherian problems).
Abstract: We establish necessary and sufficient conditions for the solvability of inhomogeneous linear boundaryvalue problems for systems of ordinary differential equations with pulse influence in the case where the number of boundary conditions is not equal to the order of the differential system (Noetherian problems). We construct a generalized Green operator for boundary-value problems not all solutions of which can be extended from the left endpoint to the right endpoint of the interval where these solutions are constructed.
TL;DR: In this paper, the inner regularity of solutions and their derivatives with respect to spatial coordinates for a degenerate quasilinear parabolic equation of the second order was established.
Abstract: We establish the inner regularity of solutions and their derivatives with respect to spatial coordinates for a degenerate quasilinear parabolic equation of the second order.
TL;DR: In this paper, the existence of anm-parameter family of solutions that form the central invariant manifold of a nonlinear parabolic equation was proved under certain assumptions, and an abstract scheme that corresponds to energy methods for strongly parabolic equations of arbitrary order was presented.
Abstract: Under certain assumptions, we prove the existence of anm-parameter family of solutions that form the central invariant manifold of a nonlinear parabolic equation. For this purpose, we use an abstract scheme that corresponds to energy methods for strongly parabolic equations of arbitrary order.