Vol 38, No 1 (2017)

Let Q be a smoothmanifold of dimension n ≥ 1. In this paper,we define the vertical lift of multivector fields from Q to T Q and we give some applications in the Poisson geometry. In particular we describe the structure of singular foliation induced by the vertical lift of Poisson structures defined below. On the other hand, given a second order vector field on Q, we defined the horizontal lift of multivector fields from Q to TQ and we study some properties.

We considered the process of density wave propagation in the logistic equation with diffusion, such as Fisher–Kolmogorov equation, and arguments deviation. Firstly, we studied local properties of solutions corresponding to the considered equation with periodic boundary conditions using asymptotic methods. It was shown that increasing of period makes the spatial structure of stable solutions more complicated. Secondly, we performed numerical analysis. In particular, we considered the problem of propagating density waves interaction in infinite interval. Numerical analysis of the propagating waves interaction process, described by this equation, was performed at the computing cluster of YarSU with the usage of the parallel computing technology—OpenMP. Computations showed that a complex spatially inhomogeneous structure occurring in the interaction of waves can be explained by properties of the corresponding periodic boundary value problem solutions by increasing the spatial variable changes interval. Thus, the complication of the wave structure in this problem is associated with its space extension.

Let m, n ≥ 1 are integers and D be a domain in the complex plane ℂ or in the m-dimensional real space ℝ^m. We build positive subharmonic functions on a part of D vanishing on the boundary ∂D of domain D. We use such (test) functions to study the distribution of zero sets of holomorphic functions f on D ⊂ ℂⁿ with restrictions on the growth of f near the boundary ∂D.

In the classical space L₂(−π, π) there exists the unconditional basis {e^kit} (k is integer). In the work we study the existence of unconditional bases in weighted Hilbert spaces L₂(h) of the functions square integrable on an interval (−1, 1) with the weight exp(−h), where h is a convex function. We prove that there exist no unconditional exponential bases in space L₂(h) if for some α < 0 (1 − |t|)^α = O(e^h(t)), t→±1.

The main aim of this paper is to examine the performance of Half-Sweep Geometric Mean (HSGM) iterative method for solving large, dense, third order Newton–Cotes quadrature system associated with numerical solutions of second kind linear Fredholm integral equations. The formulation and implementation of the method are presented. Some numerical analysis are included to verify the efficiency of the method.

In this paper, we give a King-type modification of the Bernstein–Kantorovich operators and study the approximation properties of these operators. We prove that the error estimation of these operators is better than the classical Bernstein–Kantorovich operators. We also give some estimations for the rate of convergence of these operators by using the modulus of continuity. Furthermore, we obtain a Voronovskaya-type asymptotic formula for these operators.

Improved estimators for the kurtosis parameters of two multivariate populations are developed under the assumption that they are equal. Shrinkage and preliminary test estimators are proposed and their asymptotic properties are presented analytically and numerically. Comparisons of the suggested estimators are made on the basis of their asymptotic distributional biases and asymptotic quadratic risks. It is observed that the suggested estimators perform better than the estimator based on the sample data only in a wider range of parametric space.

We define the axiom of indefinite hemi-slant 3-planes and 3-spheres for an indefinite almost Hermitian manifold with lightlike submanifolds. We prove that if an indefinite Kaehler manifold satisfies the axioms of indefinite hemi-slant 3-planes and 3-spheres for some slant angle θ ∈ (0, π/2) then it is an indefinite complex space form.

For non-homogeneous mixed parabolic-hyperbolic type equation in a rectangular area boundary-value problem is solved which was posed by in 1959 I. M. Gelfand. The solution of this problem is constructed as the sum of orthogonal series. Using the spectral analysis method, we establish a uniqueness criterion and prove the existence theorem for the solution of the problem in justifying the convergence of the problem of small denominators. In connection with this set of evaluation of separation from scratch small denominators that are allowed to prove the convergence of the class of regular solutions.

We consider a boundary value problem for a hyperbolic equation in a rectangular domain with non-complete boundary data and integral boundary value conditions of the first kind. The existence and uniqueness of solution of the problem are established bymeans of the spectralmethod. The solution is obtained in the form of the Fourier–Bessel series. Its convergence is proved in the class of regular solutions.

For a double array of random elements {T_m,n, m ≥ 1, n ≥ 1} in a real separable Banach space X, we study the notion of T_m,n converging completely to 0 in mean of order p where p is a positive constant. This notion is stronger than (i) T_m,n converging completely to 0 and (ii) T_m,n converging to 0 in mean of order p as max{m, n} →∞. When X is of Rademacher type p (1 ≤ p ≤ 2), for a double array of independent mean 0 random elements {V_m,n, m ≥ 1, n ≥ 1} in X and a double array of constants {b_m,n, m ≥ 1, n ≥ 1}, conditions are provided under which max_{1≤k≤m,1≤l≤n}||Ʃ_i=1^kƩ_j=1^lV_i,j||/b_m,n converges completely to 0 in mean of order p. Moreover, these conditions are shown to provide an exact characterization of Rademacher type p (1 ≤ p ≤ 2) Banach spaces. Illustrative examples are provided.