Vol 57, No 3 (2016)

Let G be the free product of nilpotent groups A and B of finite rank with amalgamated cyclic subgroup H, H ≠ A and H ≠ B. Suppose that, for some set π of primes, the groups A and B are residually F_π, where F_π is the class of all finite p-groups. We prove that G is residually F_π if and only if H is F_π-separable in A and B.

We study the properties of the Lebesgue constants of the Walsh system L_n(W), n ∈ N, and apply the results to the theory of Banach limits. We show that the sequence \(\left\{ {\frac{{{L_n}\left( W \right)}}{{{{\log }_2}n}},n \geqslant 2} \right\}\) does not belong to the space of almost convergent sequences ac, which reveals their extremely irregular behavior. Several results of the opposite nature are obtained for some special means of these constants.

Optimal cubature formulas are constructed for calculations of multidimensional integrals in weighted Sobolev spaces. We consider some classes of functions defined in the cube Ω = [-1, 1]^l, l = 1, 2,..., and having bounded partial derivatives up to the order r in Ω and the derivatives of jth order (r < j ≤ s) whose modulus tends to infinity as power functions of the form (d(x, Г))^-(j-r), where x ∈ Ω Г, x = (x₁,..., x_l), Г = ∂Ω, and d(x, Г) is the distance from x to Г.

It is known that there are normal plane maps (NPMs) with minimum degree δ = 5 such that the minimum degree-sum w(S₅) of 5-stars at 5-vertices is arbitrarily large. The height of a 5-star is the maximum degree of its vertices. Given an NPM with δ = 5, by h(S₅) we denote the minimum height of a 5-stars at 5-vertices in it.

Lebesgue showed in 1940 that if an NPM with δ = 5 has no 4-stars of cyclic type \(\overrightarrow {\left( {5,6,6,5} \right)} \) centered at 5-vertices, then w(S₅) ≤ 68 and h(S₅) = 41. Recently, Borodin, Ivanova, and Jensen lowered these bounds to 55 and 28, respectively, and gave a construction of a \(\overrightarrow {\left( {5,6,6,5} \right)} \)-free NPM with δ = 5 having w(S₅) = 48 and h(S₅) = 20.

In this paper, we prove that w(S₅) ≤ 51 and h(S₅) ≤ 23 for each (\(\overrightarrow {\left( {5,6,6,5} \right)} \)-free NPM with δ = 5.

Considering the character systems of Vilenkin groups, we obtain some Sidon-type inequalities that generalize the results by Móricz and Avdispahić–Pepić. We apply the inequalities to the theory of strong approximation of functions by series in character systems. Our results are analogs of the trigonometric results by Totik, Leindler, Németh, and Siddiqi.

We prove that, given a sequence {ak}_k=1^∞ with a_k ↓ 0 and {ak}_k=1^∞ ∉ l₂, reals 0 < ε < 1 and p ∈ [1, 2], and f ∈ L^p(0, 1), we can find f ∈ L^p(0, 1) with mes{f ≠ f < ε whose nonzero Fourier–Walsh coefficients c_k(f) are such that |c_k(f)| = a_k for k ∈ spec(f).

We prove an analog of the Dodds–Fremlin–Wickstead Theorem on compact domination for homogeneous orthogonally additive polynomials in Banach lattices. The proof is based on linearization of the polynomials which was established earlier by the author.

A classification is achieved of the 2-groups all of whose finite subgroups are generated by two elements.

We tabulate all knots in the oriented thickened Klein bottle having diagrams with three crossings and less. For proving that the knots are distinct, we use a generalization of the Kauffman bracket polynomial in four variables.

We consider the extension of an orthogonally additive operator from a lateral ideal and a lateral band to the whole space. We prove in particular that every orthogonally additive operator, extended from a lateral band of an order complete vector lattice, preserves lateral continuity, narrowness, compactness, and disjointness preservation. These results involve the strengthening of a recent theorem about narrow orthogonally additive operators in vector lattices.