Volume 10, Issue 4

Let 𝑉 𝑖 , 1 ≤ 𝑖 ≤ 𝑠, be complex topological vector spaces with 𝑉 1 infinite-dimensional and 𝑌 a closed analytic subset of finite codimension of 𝐏(𝑉 1 ) × . . . × 𝐏(𝑉 𝑠 ). Here we show that 𝑌 is algebraic (at least if each 𝑉 𝑖 is a Banach space) and that any two points of 𝑌 may be connected by a chain of 𝑠 + 3 lines contained in 𝑌.

In the recent years, evolution problems with an integral term in the boundary conditions have received a great deal of attention. Such problems, in general, are nonself-adjoint, and this poses the basic source of difficulty, which can considerably complicate the application of standard functional and numerical techniques. To avoid these complications, we have introduced a nonclassical function space to establish a priori estimates without any additional complication as compared to the classical evolution problems. As an example of the applicability of this way of solving problems of this type, we investigate an initial-boundary value problem for a pseudoparabolic equation which combines Neumann and integral conditions.

Given a bisimplicial group 𝐺∗∗ such that 𝑁(𝐺)∗ 𝑞 = {1} for 𝑞 ⩾ 2, a simplicial group is obtained whose Moore complex is a mapping cone of the chain morphism 𝑁(𝐺)∗ 1 → 𝑁(𝐺)∗ 0 . This simplicial group is homotopy equivalent to the diagonal of 𝐺∗∗. In the last section a special case is considered.

Criteria for almost sure boundedness and convergence to zero almost surely of Banach space valued independent Gaussian random elements are found. The obtained statements can be viewed as vector-valued versions of the corresponding results due to N. Vakhania. Moreover, from the obtained statements a strong law of large numbers is derived in the form of Yu. V. Prokhorov.

Chanturia, Wang and Akhobadze proved sufficient conditions for the inclusion of Chanturia, Waterman and Generalized Wiener classes, respectively, in the classes . In this paper we prove that these conditions are necessary for this inclusion to hold.

We have selected a class of hyperbolic quasilinear equations of second order, admitting parabolic degeneracy by the following criterion: they have a general solution represented by superposition of two arbitrary functions. For equations of this class we consider the initial Cauchy problem and nonlocal characteristic problems for which sufficient conditions are established for the solution solvability and uniquness; the domains of solution definition are described.

For any convex polyhedron 𝑃 ⊂ 𝐑 3 and for any natural number 𝑘, let 𝐹 𝑘 (𝑃) denote the number of all faces of 𝑃 with exactly 𝑘 sides. It is well known that 𝐹 𝑘 (𝑃) ≥ 2 for at least one 𝑘. We consider the question whether 𝐹 𝑘 (𝑃) ≥ 3, for at least one 𝑘, and present a solution to it. Some related questions are also discussed.

In this paper, the continuity of multilinear Marcinkiewicz operators on certain Hardy and Herz–Hardy spaces is obtained.

The two-dimensional interface problem of statics of elastic mixtures is investigated when differences between displacement vectors and stress vectors are given on the contact curve. For this problem special potentials with complex densities are constructed and by means of them the contact problem is reduced to a system of integral equations of Fredholm type. Analysis of the equations obtained is given.

The structure of roots is investigated for polynomials in one variable with quaternionic and octonionic coefficients. It is proved that their zero sets consist of a finite number of points and Euclidean spheres of corresponding dimension. The general statements are illustrated by typical examples.

This paper considers superfluous subsemimodules and some applications of them to projective covers of semimodules. The following notions are introduced: a superfluous ∧-subsemimodule, a 𝑘-quasiregular homomorphism, a radical of a ∧-semimodule, a projective cover, and a semiring satisfying the condition 0–1. We present some characteristic properties of superfluous subsemimodules and give some applications of them to projective covers of semimodules, analogous to applications of superfluous submodules to projective covers of modules.

By employing a generalized Riccati technique and an integral averaging technique, new oscillation criteria are established for a class of second-order nonlinear differential equations with damping term. These criteria extend, improve and unify a number of existing results and handle the cases which are not covered by the known criteria. In particular, several interesting examples that illustrate the importance of our results are included.

This paper is concerned with a class of second order half-linear damped differential equations. Using the generalized Riccati transformation and the averaging technique, new oscillation criteria are obtained which are either extensions of or complementary to a number of the existing results.

A result by Móricz, Su, and Taylor from Acta Math. Hungar. 65 (1994), 1–16, was misstated in the authors' paper in Georgian Math. J. 8 (2001), No. 2, 377–388, where due to this misstatement the invalid formulation and proof of a corollary is given. In this correction note, the needed result is correctly stated and a corrected version of the invalid corollary is proved.