Let a von Neumann algebra M of operators act on a Hilbert space H, let τ be a faithful normal semifinite trace on M. Let S(M, τ ) be the ∗-algebra of all τ-measurable operators. Assume that X, Y ∈ S(M, τ ). We have (i) if |Y | ≤ |X| then ker(X) ⊂ ker(Y ); (ii) if X is left invertible with X−1 l ∈Mthen ran(X∗) = H. The following generalizes of the Putnam theorem (1951), see also Problem 188 in the book (Halmos P. R. A Hilbert space problem book. D. van Nostrand company, inc., London, 1967): A positive selfcommutator A∗A−AA∗ (A ∈ S(M, τ )) cannot have the inverse in M. Let I be the unit of the algebra M and τ (I) = +∞, let A,B ∈ S(M, τ ) and A = A3. Then the commutator [A,B] cannot have a form λI + K, where λ ∈ C \ {0} and an operator K ∈ S(M, τ ) is τ-compact

With help of a certain family H of separately radial convex functions in Rn two new spaces of rapidly decreasing infinitely differentiable functions in Rn are introduced in the article. One of them, namely, the space G(H), is a subspace of Gelfand-Shilov type space Sα,A, where α = (1, . . . , 1) ∈ Rn,A = (0, . . . , 0) ∈ Rn. Functions of the second space E(H) admit an extension to entire functions in Cn. A description of the space of such extensions is obtained. Under some mild additional restrictions on H an isomorphism between the spaces G(H) and E(H) is established via the Fourier transform

The article is written following the talk by author on the satellite conference “Lobachevsky readings” held in Kazan in July 2022. The talk has presented a short survey of works concerning the history and results of investigations devoted to the realisation of the complete Lobachevsky plane as a two-dimensional surface in a multidimensional Euclidean space. For the present situation the best result for the minimal dimension of ambient space is given by a theorem affirming that Lobachevsky plane can be immersed in R4 as a piecewise analytic surface with C0.1 smoothness in whole

The paper proves that for any c.e. set W, its non-computability is equivalent to the fulfillment of the Recursion theorem (with parameters) in each universal W-computable numbering, as well as its weak precompleteness and nnonsplittability. It is established that the Turing completeness of the c.e. oracle W is equivalent to the existence of a precomplete W-computable numbering for any W-computable family.

An extreme problem related to finding the maximal and minimal areas of the set of circles inscribed into the bounded by two tangent circles

In this paper all Hecke symmetries are given for which the corresponding algebra S(V,R) is Artin-Schelter regular of type E. Also we prove that there exist no Hecke symmetries with regular algebra S(V,R) of type H