Vol 62, No 5 (2018)

For a normed algebra A and natural numbers k we introduce and investigate the ∥ · ∥ closed classes P_k(A). We show that P₁(A) is a subset of P_k(A) for all k. If T in P₁(A), then Tⁿ lies in P₁(A) for all natural n. If A is unital, U, V ∈ A are such that ∥U∥ = ∥V∥ = 1, VU = I and T lies in P_k(A), then UTV lies in P_k(A) for all natural k. Let A be unital, then 1) if an element T in P₁(A) is right invertible, then any right inverse element T⁻¹ lies in P₁(A); 2) for ßßIßß = 1 the class P₁(A) consists of normaloid elements; 3) if the spectrum of an element T, T ∈ P₁(A) lies on the unit circle, then ∥TX∥ = ∥X∥ for all X ∈ A. If A = B(H), then the class P₁(A) coincides with the set of all paranormal operators on a Hilbert space H.

We study the lattice of varieties of monoids, i.e., algebras with two operations, namely, an associative binary operation and a 0-ary operation that fixes the neutral element. It was unknown so far, whether this lattice satisfies some non-trivial identity. The objective of this paper is to give the negative answer to this question. Namely, we prove that any finite lattice is a homomorphic image of some sublattice of the lattice of overcommutative varieties of monoids (i.e., varieties that contain the variety of all commutative monoids). This implies that the lattice of overcommutative varieties of monoids, and therefore, the lattice of all varieties of monoids does not satisfy any non-trivial identity.

We obtain an analog of the Plana formula, which is essential in obtaining the functional equation for the classical Riemann zeta-function.

We introduce φ-distributions and prove that their set is ametric space. We also consider a Banach space and a Hilbert space of such distributions. The results are applied to differential equations with Laurent’s type coefficients.

We consider the following inversemixed boundary-value problem of aerohydrodynamics. It is required to find a form of an airfoil circulated by a potential flow of an incompressible non-viscous liquid. A part of the profile is known; it is a broken line, convex upwards. The other part is found by the values of velocity potential, given as a function of parameter which can be either the abscissa, or the ordinate of an airfoil point. On a small part of the airfoil, containing the leading edge, the parameter is the ordinate, and for other points of the unknown part it is the abscissa.