arXiv subjects

T. M. Osipchuk

Publications and source records attributed to T. M. Osipchuk.

2 recordsLinked to original sources

arXiv2020

On local linear convexity generalized to commutative algebras

A commutative associative algebra A with an identity over the field of real numbers which has a basis, where all elements are invertible, is considered in the work. Moreover, among matrixes consisting of the structure constants of A, there is at least one that is non-degenerate. The notion of linearly convex domains in the multi-dimensional complex space and some of their properties are generalized to the space that is the Cartesian product of n algebras A. Namely, the separate necessary and sufficient conditions of the local A-linear convexity of domains with smooth boundary in the space are obtained in terms of nonnegativity and positivity of formal quadratic differential form in A, respectively.

math.CV↗

arXiv2020

On semiconvex sets in the plane

The present work considers the properties of classes of generally convex sets in the plane known as $1$-semiconvex and weakly $1$-semiconvex. More specifically, the examples of open and closed weakly $1$-semiconvex but non $1$-semiconvex sets with smooth boundary in the plane are constructed. It is proved that such sets consist of minimum four connected components. In addition, the example of closed, weakly $1$-semiconvex, and non $1$-semiconvex set in the plane consisting of three connected components is constructed. It is proved that such a number of components is minimal for any closed, weakly $1$-semiconvex, and non $1$-semiconvex set in the plane.

math.MG↗