Let A^sem = {A ∈ A : Re A = A^∗A} be the set of all semiorthogonal projections of the unital C∗-algebra A, I be the identity of A. The formula U = 2A − I (A ∈ A^sem) defines a bijection between the set Asem and the set of all isometries from A. For any natural number n ≥ 2, there exists a non-commutative polynomial of degree n that yields a semi-orthogonal projection when substituted for an arbitrary set A₁, . . . , A_n ∈ A^sem. Each element A ∈ A^sem is hyponormal and lies in the unit ball of the C^∗-algebra A. If A ∈ Asem, then A² is hyponormal. If A, A² ∈ Asem, then A is a projection. If A ∈ A^sem and A = A_n for some n ∈ N, n ≥ 2, then A is a normal element, and A is a projection for n = 2.

We establishe that the eff                         ely non-degenerate numbering of any field of the finite characteristic is negative.

We prove relations between regularity of two-dimensional and one-dimensional languages. Each two-dimensional language is corresponded to two sequences of one-dimensional languages corresponding to rows and columns of the two-dimensional language. For each of the following conditions the existence of both regular and nonregular two-dimensional languages is proved: all row and all column languages are regular; all row languages are regular, all column languages are nonregular; all column languages are regular, all row languages are nonregular; both all row and all column languages are nonregular.

We prove that for weakly transitive modal logics equipped with the universal modality whose satisfiability problem is already decidable in PSPACE, adding the connectivity axiom does not increase the complexity. Moreover, we present an algorithm that solves the satisfiability task within the same complexity class.

A generalized random process with values in a measurable space is defined as a complex-valued finite additive cylindrical measure on the space of trajectories with values in the measurable space. Using this extension of the concept of a random process, we aim to obtain a representation of solutions to the evolutionary equation by averaging functionals on the space of trajectories of a random process. For this purpose, a bijective mapping of the space of operator valued functions into a set of complex valued finite additive cylindrical measures on the trajectory space is constructed and investigated. Limit theorems for generalized random processes are obtained. In the second part of the survey, the application of the constructed bijective mapping to the obtaining of perturbed semigroups and evolutionary families of operators in the form of Feynman–Kac formulas will be considered.

We prove that the choice of an initial point of rounding of a closed plane curve influences its exterior geometry.