Conditions are studied under which an arbitrary direct sum of essentially (quasi-) injective modules is an essentially (quasi-) injective module. A description of essentially quasi-injective Abelian groups is obtained.

In the L₂ metric, we obtain sharp inequalities between the best joint approximations of 2π-periodic functions f(x, y) differentiable in each of the variables and their successive derivatives f ^(μ,ν) (x, y) (μ = 0, 1, . . . , r; ν = 0, 1, . . . , s) by trigonometric “angles” with double integrals containing mixed moduli of continuity of higher orders of higher derivatives. The sharp values of the upper bound of the best joint approximation of some classes of functions given by the specified moduli of continuity are found.

A classification of h-spaces H_32,3 of non-constant curvature with respect to (non-homothetic) Lie algebras of infinitesimal projective and affine transformations is given.

Let λ₁ and λ₂ be real, λ₁ < λ₂, functions ψ_( λ_i, t) be solutions to the second order quasidifferential equations Lψ_ = λ_{i P}⁰ψ_, i = 1, 2, satisfying a homogeneous boundary condition at point а. We express the number of eigenvalues of operator L, belonging to the interval (λ₁, λ₂) (or the dimension of its spectral projection relative to the interval (λ₁, λ₂), in terms of the number of zeros of the Vronskian composed for the functions ψ_(λ₁,t) and ψ_(λ₂,t).

The paper studies the global moment stability of systems of nonlinear Itô differential equations with delays. The analysis is done by a modified regularization method, known as the W-method, and based on the use of some auxiliary equation with subsequent application of the theory of positively invertible matrices. Sufficient conditions for the global asymptotic moment stability for both sufficiently general and specific systems of Ito equations formulated in terms of parameters of these systems are given. Connections between this stability and the properties of the delay functions are established.

A symmetric variational eigenvalue problem in the Hilbert space with a cone is investigated. New sufficient conditions on the bilinear forms, the Hilbert space, and the cone of the variational problem guaranteeing the existence of a unique normalized positive eigenelement corresponding to a positive simple minimal eigenvalue are proposed and justified. The obtained abstract results are illustrated by the example of the generalized eigenvalue problem for the second order self-adjoint elliptic differential operator.

In this article we considered a difference-differential polynomial of a transcendental entire function and established in the first two theorems criteria for infinitely many solutions for such type of equation. In the 3rd and 4th theorems we introduced k-th order differentiation of a polynomial of a transcendental entire function and its difference operator, and developed some uniqueness results in different aspects using the conception of sharing value. We elaborated our results with some remarks and analysed the results in different aspects and conditions. The results generalize and improve previous results of Zhang-Lin and Benerjee-Majumder. We introduced some notes to draw the gravity of improvement and generalisation of the previous results.