Tetiana Osipchuk

• Doctor of Philosophy
• Researcher at National Academy of Sciences of Ukraine

The present work concerns generalized convex sets in the real multi-dimensional Euclidean space, known as weakly 1-convex and weakly 1-semiconvex sets. An open set is called weakly 1-convex (weakly 1-semiconvex) if, through every boundary point of the set, there passes a straight line (a closed ray) not intersecting the set. A closed set is called...

The present work concerns generalized convex sets in the real multi-dimensional Euclidean space, known as weakly $1$-convex and weakly $1$-semiconvex sets. An open set is called weakly $1$-convex (weakly $1$-semiconvex) if, through every boundary point of the set, there passes a straight line (a closed ray) not intersecting the set. A closed set is...

The present work concerns generalized convex sets in the Euclidean plane, known as weakly 1-semiconvex. An open set is called weakly 1-semiconvex if every boundary point of the set is the initial point of a closed ray not intersecting the set. A closed set is called weakly 1-semiconvex if it can be approximated from the outside by a family of open...

The notion of lineally convex domains in the finite-dimensional complex space \(\mathbb {C}^n\) and some of their properties are generalized to the finite-dimensional space \(\mathcal {A}^n\), \(n\ge 2\), that is the Cartesian product of n commutative and associative algebras \(\mathcal {A}\). Namely, a domain in \(\mathcal {A}^n\) is said to be (l...

The present work considers the properties of generally convex sets in the plane known as weakly 1-convex. An open set is called weakly 1-convex if for any boundary point of the set there exists a straight line passing through this point and not intersecting the given set. A closed set is called weakly 1-convex if it is approximated from the outside...

We consider a class of generalized convex sets in the real plane known as weakly 1-convex sets. For a set in the real Euclidean space ℝn, n ≥ 2, we say that a point of the complement of this set to the entire space ℝn is an m-nonconvexity point of the set, m=1,n−1¯, if any m-dimensional plane passing through this point crosses the indicated set. An...

In the present work we study properties of generally convex sets in the n-dimensional real Euclidean space Rn, (n>1), known as weakly m-convex, m=1,...,n-1. An open set of Rn is called weakly m-convex if, for any boundary point of the set, there exists an m-dimensional plane passing through this point and not intersecting the given set. A closed se...

The present work considers properties of generally convex sets in the n-dimensional real Euclidean space ℝn, n > 1, known as weakly m-semiconvex, m = 1, 2, … , n − 1. For all that, the subclass of not m-semiconvex sets is distinguished from the class of weakly m-semiconvex sets. A set of the space ℝn is called m-semiconvex if, for any point of the...

The present work considers properties of generally convex sets in the $n$-dimensional real Euclidean space $\mathbb{R}^{n}$, $n>1$, known as weakly $m $-semiconvex, $m=1,2,\ldots ,n-1$. For all that, the subclass of not $m$-semiconvex sets is distinguished from the class of weakly $m$-semiconvex sets. A set of the space $\mathbb{R}^{n}$ is called \...

The present work considers the properties of generally convex sets in the $n$-dimensional real Euclidean space $\mathbb{R}^n$, $n>1$, known as weakly $m$-convex, $m=1,2,\ldots,n-1$. An open set of $\mathbb{R}^n$ is called weakly $m$-convex if for any boundary point of the set there exists an $m$-dimensional plane passing through this point and not...

The topological properties of classes of generally convex sets in multidimensional real Euclidean space $\mathbb{R}^n$, $n\ge 2$, known as $m$-convex and weakly $m$-convex, $1\le m<n$, are studied in the present work. A set of the space $\mathbb{R}^n$ is called \textbf{\emph{$m$-convex}} if for any point of the complement of the set to the whole sp...

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...

The present work considers properties of classes of generally convex sets in the plane known as 1-semiconvex and weakly 1-semiconvex. More specifically, it is proved that open, weakly 1-semiconvex but not 1-semiconvex set with smooth boundary in the plane consists of not less than four connected components.

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 conn...

Problems related to the determination of the minimal number of balls that generate a shadow at a fixed point in the multi-dimensional Euclidean space R^n are considered in the present work. Here, the statement ”a system of balls generate shadow at a point” means that any line passing through the point intersects at least one ball of the system. The...

Problems, related to the determination of the minimal number of balls that generate a shadow at a fixed point in the multi-dimensional Euclidean space $ \mathbb{R}^n $, are considered in present work. Here, the statement "a system of balls generate shadow at a point" means that any line passing through the point intersects at least one ball of the...

Properties of two classes of generally convex sets in the n-dimentional real Euclidean space, called m-semiconvex and weakly m-semiconvex, 1<=m<n, are investigated in the present work. In particular, it is established that an open set with smooth boundary in the plan which is weakly 1-semiconvex but not 1-semiconvex consists minimum of four simply...

In the present work, the problem about shadow, generalized on domains of space $\mathbb{R}^n$, $n\le 3$, is investigated. Here the shadow problem means to find the minimal number of balls satisfying some conditions an such that every line passing through the given point intersects at least one ball of the collection. It is proved that to generate t...

The main goal of the paper is to solve some problems about shadow for the sphere generalized on the case of the ellipsoid. Here, the essence of the problem is to find the the minimal number of non-overlapping balls with centers on the sphere which are not holding the center of the sphere and such that every line passing through the center of the sp...

The necessary and sufficient conditions of local linear convexity of domains with smooth boundary in multidimensional generalized quaternion space ℍ α,β n , n≥2, are found.

Based on the research of rat brains conducted at the Center for Molecular Biology and Neurosciences and the Institute of Basic Medical Sciences, University of Oslo, Norway, the mapping of the model of cortex map to the model of pontine nuclei maps in rat brais is constructed.

We establish a criterion for the local linear convexity of sets in the two-dimensional quaternion space $ {\mathbb{H}^2} $ that are analogs of bounded Hartogs domains with smooth boundary in the two-dimensional complex space $ {\mathbb{C}^2} $ .

In terms of nonnegativity and positivity of some special quadratic differential forms we give, respectively, necessary and sufficient conditions for locally generally convexity of domains with smooth boundary in multidimensional space that is the Cartesian product of the Universal Clifford algebras.