A lattice is called well-rounded if its minimal vectors span the corresponding Euclidean space. In this paper we study the similarity classes of well-rounded sublattices of Z2. We relate the set of all such similarity classes to a subset of primitive Pythagorean triples, and prove that it has the structure of a noncommutative infinitely generated monoid. We discuss the structure of a given simi...

The article is devoted to a review of approaches solving the problem identifying paraphrases. This problem's relevance and use in tasks such as plagiarism detection, text simplification, information search are described. Several classes solutions were considered. first approach based on manual rules - it uses manually selected features fundamental properties second lexical similarity various da...

We construct rogue waves (RWs) in a coupled two-mode system with the self-focusing nonlinearity of the Manakov type (equal SPM and XPM coefficients), spatially modulated coefficients, and a specially designed external potential. The system may be realized in nonlinear optics and Bose-Einstein condensates. By means of a similarity transformation, we establish a connection between solutions of th...

We investigate a class of impulsive gravitational waves which propagate either in Minkowski or in the (anti–) de Sitter background. These waves are constructed as impulsive members of the Kundt class P (Λ) of non-twisting, non-expanding type N solutions of vacuum Einstein equations with a cosmological constant Λ. We show that the only non-trivial waves of this type in Minkowski spacetime are im...