Alex Ravsky

Alex Ravsky
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics · Department of Functional Analysis

About

88
Publications
7,469
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549
Citations
Introduction
I am from Lviv topological school (http://www.franko.lviv.ua/faculty/mechmat/Departments/Topology/trees.html). Nevertheless, I have publications in theory of (para)topological groups, general topology, combinatorial geometry, combinatorics, graph theory, algebra, and functional analysis. All of them are free, most of them are available online. I do not upload or update my papers at Research Gate. Almost all my recent papers are in arXiv: http://arxiv.org/find/math/1/au%3a+Ravsky/0/1/0/all/0/1#!
Additional affiliations
June 2001 - December 2003
Lviv University
Position
  • Research Assistant
Description
  • I did different mathematics, for instanse, wrote papers, reviews, and developed my skills. Also I was a member of Jury of different mathematics competitions in Lviv.
November 2004 - present
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics
Position
  • Mathematician
Description
  • Now I mainly work in topological algebra and deal with algebraic objects (mainly, groups) endowed with a topology making continuous some of the algebraic operations determined on these objects. That is (para-, semi-, and quasi-) topological groups.
Education
November 1998 - May 2001
Lviv University
Field of study
  • Topological algebra
September 1993 - June 1998
Lviv University
Field of study
  • Mathematics

Publications

Publications (88)
Article
UDC 515.122 We consider the problem of characterization of topological spaces embedded into countably compact Hausdorff topological spaces. We study the separation axioms for subspaces of Hausdorff countably compact topological spaces and construct an example of a regular separable scattered topological space that cannot be embedded into a Urysohn...
Article
Full-text available
It is well known that any graph admits a crossing-free straight-line drawing in $\mathbb{R}^3$ and that any planar graph admits the same even in $\mathbb{R}^2$. For a graph $G$ and $d \in \{2,3\}$, let $\rho^1_d(G)$ denote the smallest number of lines in $\mathbb{R}^d$ whose union contains a crossing-free straight-line drawing of $G$. For $d=2$, th...
Chapter
We study the following combinatorial problem. Given a set of n y-monotone curves, which we call , a determines the order of the wires on a number of horizontal such that any two consecutive layers differ only in swaps of neighboring wires. Given a multiset L of (that is, unordered pairs of wires) and an initial order of the wires, a tangle L if eac...
Preprint
Full-text available
Let $G$ be a paratopological group. Following F. Lin and S. Lin, we say that the group $G$ is pseudobounded, if for any neighborhood $U$ of the identity of $G$, there exists a natural number $n$ such that $U^n=G$. The group $G$ is $\omega$-pseudobounded, if for any neighborhood $U$ of the identity of $G$, the group $G$ is a union of sets $U^n$, whe...
Article
A subset S of a paratopological group G is a suitable set for G, if S is a discrete subspace of G, \(S\cup \{e\}\) is closed, and the subgroup \(\langle S\rangle \) of G generated by S is dense in G. Suitable sets in topological groups were studied by many authors. The aim of the present paper is to provide a start-up for a general investigation of...
Preprint
Full-text available
Given a topological ring $R$, we study semitopological $R$-modules, construct their completions, Bohr and borno modifications. For every topological space $X$, we construct the free (semi)topological $R$-module over $X$ and prove that for a $k$-space $X$ its free semitopological $R$-module is a topological $R$-module. Also we construct a Tychonoff...
Article
We discuss various modifications of separability, precompactness and narrowness in topological groups and test those modifications in the permutation groups S(X) and \(S_{<\omega }(X)\).
Article
A topological group X is called duoseparable if there exists a countable set S⊆X such that SUS=X for any neighborhood U⊆X of the identity. We construct a functor F assigning to each (abelian) topological group X a duoseparable (abelian-by-cyclic) topological group FX, containing an isomorphic copy of X. In fact, the functor F is defined on the cate...
Article
We construct a metrizable Lawson semitopological semilattice $X$ whose partial order $\le_X\,=\{(x,y)\in X\times X:xy=x\}$ is not closed in $X\times X$. This resolves a problem posed earlier by the authors.
Article
Full-text available
p>We investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces (semigroups, resp.). We show that each Hausdorff topological space is a closed subspace of some Hausdorff ω-bounded pracompact topological space and describe open dense subspaces of countably pracompact topological spaces. We construct a pseudocompact topologi...
Article
We obtain many results and solve some problems about feebly compact paratopological groups. We obtain necessary and sufficient conditions for such a group to be topological. One of them is the quasiregularity. We prove that each 2-pseudocompact paratopological group is feebly compact and that each Hausdorff σ-compact feebly compact paratopological...
Article
Let $X$ be a real separable normed space $X$ admitting a separating polynomial. We prove that each continuous function from a subset $A$ of $X$ to a real Banach space can be uniformly approximated by restrictions to $A$ of functions, which are analytic on open subsets of $X$. Also we prove that each continuous function to a complex Banach space fro...
Article
Full-text available
An effective way to reduce clutter in a graph drawing that has (many) crossings is to group edges that travel in parallel into bundles. Each edge can participate in many such bundles. Any crossing in this bundled graph occurs between two bundles, i.e., as a bundled crossing. We consider the problem of bundled crossing minimization: A graph is given...
Preprint
Full-text available
A paratopological group $G$ has a {\it suitable set} $S$. The latter means that $S$ is a discrete subspace of $G$, $S\cup \{e\}$ is closed, and the subgroup $\langle S\rangle$ of $G$ generated by $S$ is dense in $G$. Suitable sets in topological groups were studied by many authors. The aim of the present paper is to provide a start-up for a general...
Article
Under Martin's Axiom we construct a Boolean countably compact topological group whose square is not countably pracompact.
Article
Let κ be an infinite cardinal. A topological space X is κ-bounded if the closure of any subset of cardinality ≤κ in X is compact. We discuss the problem of embeddability of topological spaces into Hausdorff (Urysohn, regular) κ-bounded spaces, and present a canonical construction of such an embedding. Also we construct a (consistent) example of a s...
Article
Full-text available
A topological semigroup is monothetic provided it contains a dense cyclic subsemigroup. The Koch problem asks whether every locally compact monothetic monoid is compact. This problem was opened for more than sixty years, till in 2018 Zelenyuk obtained a negative answer. In this paper we obtain a positive answer for Koch’s problem for some special c...
Article
Full-text available
We construct a metrizable semitopological semilattice X whose partial order P = {(x, y) ∈ X × X : xy = x} is a non-closed dense subset of X × X. As a by-product we find necessary and sufficient conditions for the existence of a (metrizable) Hausdorff topology on a set, act, semigroup or semilattice, having a prescribed countable family of convergen...
Preprint
Let $X$ be a real separable normed space $X$ admitting a separating polynomial. We prove that each continuous function from a subset $A$ of $X$ to a real Banach space can be uniformly approximated by restrictions to $A$ of functions which are analytic on open subsets of $X$. Also we prove that each continuous function to a complex Banach space from...
Preprint
Full-text available
Under Martin's Axiom we construct a Boolean countably compact topological group whose square is not countably pracompact.
Preprint
Full-text available
We study the following combinatorial problem. Given a set of $n$ y-monotone curves, which we call wires, a tangle determines the order of the wires on a number of horizontal layers such that the orders of the wires on any two consecutive layers differ only in swaps of neighboring wires. Given a multiset $L$ of swaps (that is, unordered pairs of wir...
Preprint
Full-text available
We discuss various modifications of separability, precompactnmess and narrowness in topological groups and test those modifications in the permutation groups $S(X)$ and $S_{<\omega}(X)$.
Preprint
Full-text available
A topological group $X$ is called $duoseparable$ if there exists a countable set $S\subseteq X$ such that $SUS=X$ for any neighborhood $U\subseteq X$ of the unit. We construct a functor $F$ assigning to each (abelian) topological group $X$ a duoseparable (abelain-by-cyclic) topological group $FX$, containing an isomorphic copy of $X$. In fact, the...
Article
A subset D of an abelian group is decomposable if ∅≠D⊂D+D. In the paper we give partial answers to an open problem asking whether every finite decomposable subset D of an abelian group contains a non-empty subset Z⊂D with ∑Z=0. For every n∈N we present a decomposable subset D of cardinality |D|=n in the cyclic group of order 2n−1 such that ∑D=0, bu...
Chapter
An effective way to reduce clutter in a graph drawing that has (many) crossings is to group edges that travel in parallel into bundles. Each edge can participate in many such bundles. Any crossing in this bundled graph occurs between two bundles, i.e., as a bundled crossing. We consider the problem of bundled crossing minimization: A graph is given...
Chapter
The segment number of a planar graph is the smallest number of line segments whose union represents a crossing-free straight-line drawing of the given graph in the plane. The segment number is a measure for the visual complexity of a drawing; it has been studied extensively. In this paper, we study three variants of the segment number: for planar g...
Chapter
We study the following combinatorial problem. Given a set of n y-monotone wires, a tangle determines the order of the wires on a number of horizontal layers such that the orders of the wires on any two consecutive layers differ only in swaps of neighboring wires. Given a multiset L of swaps (that is, unordered pairs of numbers between 1 and n) and...
Preprint
Full-text available
We construct a metrizable Lawson semitopological semilattice $X$ whose partial order $\le_X=\{(x,y)\in X\times X:xy=x\}$ is not closed in $X\times X$. This resolves a problem posed earlier by the authors.
Article
A topologized semilattice X is complete if each non-empty chain C⊂X has inf⁡C∈C¯ and sup⁡C∈C¯. It is proved that for any complete subsemilattice X of a functionally Hausdorff semitopological semilattice Y the partial order ≤X={(x,y)∈X×X:xy=x} of X is closed in Y×Y and hence X is closed in Y. This implies that for any continuous homomorphism h:X→Y f...
Preprint
The \emph{segment number} of a planar graph is the smallest number of line segments whose union represents a crossing-free straight-line drawing of the given graph in the plane. The segment number is a measure for the visual complexity of a drawing; it has been studied extensively. In this paper, we study three variants of the segment number: for p...
Preprint
Full-text available
We investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces (semigroups, resp.). We prove that each Hausdorff topological space can be embedded as a closed subspace into an H-closed topological space. However, the semigroup of $\omega{\times\omega}$-matrix units cannot be embedded into a topological semigroup which is an...
Preprint
The note contains a few results related to separation axioms and automatic continuity of operations in compact-like semitopological groups. In particular, is presented a semiregular semitopological group $G$ which is not $T_3$. We show that each weakly semiregular compact semitopological group is a topological group. On the other hand, constructed...
Preprint
Full-text available
In this paper we consider the problem of characterization of topological spaces that embed into countably compact Hausdorff spaces. We study the separation axioms of subspaces of countably compact Hausdorff spaces and construct an example of a regular separable scattered topological space which cannot be embedded into an Urysohn countably compact t...
Preprint
Full-text available
We discuss the problem of embeddibility of a topological space into a Hausdorff $\omega$-bounded space, and present two canonical constructions of such an embedding.
Preprint
Full-text available
A subset $S$ of an Abelian group is $decomposable$ if $ \emptyset\ne S\subset S+S$. In the paper we give partial answer to an open problem asking whether every finite decomposable subset $S$ of an Abelian group contains a non-empty subset $T\subset S$ with $\sum T=0$. For every $n\in\mathbb N$ we present a decomposable subset $S$ of cardinality $|S...
Preprint
Full-text available
A topological semigroup is monothetic provided it contains a dense cyclic subsemigroup. The Koch problem asks whether every locally compact monothetic monoid is compact. This problem was opened for more than sixty years, till in 2018 Zelenyuk obtained a negative answer. In this paper we obtain a positive answer for Koch's problem for some special c...
Preprint
Full-text available
We construct a metrizable semitopological semilattice $X$ whose partial order $P=\{(x,y)\in X\times X:xy=x\}$ is a non-closed dense subset of $X\times X$. As a by-product we find necessary and sufficient conditions for the existence of a (metrizable) Hausdorff topology on a set, act, semigroup or semilattice, having a prescribed countable family of...
Preprint
Full-text available
We prove that any topological group $G$ containing a subspace $X$ of the Sorgenfrey line has spread $s(G)\ge s(X\times X)$. Under OCA, each topological group containing an uncountable subspace of the Sorgenfrey line has uncountable spread. This implies that under OCA a cometrizable topological group $G$ is cosmic if and only if it has countable spr...
Preprint
Full-text available
A topological space is defined to be banalytic (resp. analytic) if it is the image of a Polish space under a Borel (resp. continuous) map. A regular topological space is analytic if and only if it is banalytic and cosmic. Each (regular) banalytic space has countable spread (and under PFA is hereditarily Lindel\"of). Applying banalytic spaces to top...
Preprint
Full-text available
We study the following combinatorial problem. Given a set of $n$ y-monotone wires, a tangle determines the order of the wires on a number of horizontal layers such that the orders of the wires on any two consecutive layers differ only in swaps of neighboring wires. Given a multiset $L$ of swaps (that is, unordered pairs of numbers between 1 and $n$...
Preprint
Full-text available
An effective way to reduce clutter in a graph drawing that has (many) crossings is to group edges into \emph{bundles} when they travel in parallel. Each edge can participate in many such bundles. Any crossing in this bundled graph occurs between two bundles, i.e., as a \emph{bundled crossing}. We minimize the number of bundled crossings. We conside...
Preprint
Full-text available
A topologized semilattice $X$ is complete if each non-empty chain $C\subset X$ has $\inf C\in\bar C$ and $\sup C\in\bar C$. It is proved that for any complete subsemilattice $X$ of a functionally Hausdorff semitopological semilattice $Y$ the partial order $P=\{(x,y)\in X\times X:xy=x\}$ of $X$ is closed in $Y\times Y$ and hence $X$ is closed in $Y$...
Article
Full-text available
We introduce three new classes of pracompact spaces, consider their basic properties and relations with other compact-like spaces.
Preprint
We introduce three new classes of pracompact spaces, consider their basic properties and relations with other compact-like spaces.
Article
Full-text available
Generalizing the famous 14-set closure-complement Theorem of Kuratowski from 1922, we prove that for a set $X$ endowed with $n$ pairwise comparable topologies $\tau_1\subset\dots\subset\tau_n$, by repeated application of the operations of complement and closure in the topologies $\tau_1,\dots,\tau_n$ to a subset $A\subset X$ we can obtains at most...
Article
An $H$-closed quasitopological group is a Hausdorff quasitopological group which is contained in each Hausdorff quasitopological group as a closed subspace. We obtained a sufficient condition for a quasitopological group to be $H$-closed, which allowed us to solve a problem by Arhangel'skii and Choban and to show that a topological group $G$ is $H$...
Article
Full-text available
Storyline visualizations help visualize encounters of the characters in a story over time. Each character is represented by an x-monotone curve that goes from left to right visualizing progression of time. A meeting is represented by having the characters that participate in the meeting run close together for some time. In order to keep the visual...
Article
Full-text available
Given a drawing of a graph, its visual complexity is defined as the number of geometrical entities in the drawing, for example, the number of segments in a straight-line drawing or the number of arcs in a circular-arc drawing (in 2D). Recently, Chaplick et al. introduced a different measure for the visual complexity, the affine cover number, which...
Preprint
Given a drawing of a graph, its \emph{visual complexity} is defined as the number of geometrical entities in the drawing, for example, the number of segments in a straight-line drawing or the number of arcs in a circular-arc drawing (in 2D). Recently, Chaplick et al. [GD 2016] introduced a different measure for the visual complexity, the \emph{affi...
Conference Paper
It is well known that any graph admits a crossing-free straight-line drawing in \(\mathbb {R} ^3\) and that any planar graph admits the same even in \(\mathbb {R} ^2\). For a graph G and \(d \in \{2,3\}\), let \(\rho ^1_d(G)\) denote the minimum number of lines in \(\mathbb {R} ^d\) that together can cover all edges of a drawing of G. For \(d=2\),...
Article
Full-text available
In the paper we study the preservation of pseudocompactness (resp., countable compactness, sequential compactness, $\omega$-boundedness, totally countable compactness, countable pracompactness, sequential pseudocompactness) by Tychonoff products of pseudocompact (and countably compact) to\-pological Brandt $\lambda_i^0$-extensions of semitopologica...
Conference Paper
Storyline visualizations help visualize encounters of the characters in a story over time. Each character is represented by an x-monotone curve that goes from left to right. A meeting is represented by having the characters that participate in the meeting run close together for some time. In order to keep the visual complexity low, rather than just...
Preprint
Storyline visualizations help visualize encounters of the characters in a story over time. Each character is represented by an x-monotone curve that goes from left to right. A meeting is represented by having the characters that participate in the meeting run close together for some time. In order to keep the visual complexity low, rather than just...
Article
Full-text available
In their seminal work, Mustafa and Ray (2009) showed that a wide class of geometric set cover (SC) problems admit a PTAS via local search -- this is one of the most general approaches known for such problems. Their result applies if a naturally defined "exchange graph" for two feasible solutions is planar and is based on subdividing this graph via...
Article
Full-text available
It is well known that any graph admits a crossing-free straight-line drawing in $\mathbb{R}^3$ and that any planar graph admits the same even in $\mathbb{R}^2$. For $d \in \{2,3\}$, let $\rho^1_d(G)$ denote the minimum number of lines in $\mathbb{R}^d$ that together can accommodate all edges of a drawing of $G$, where $\rho^1_2(G)$ is defined for p...
Article
Full-text available
We investigate the problem of drawing graphs in 2D and 3D such that their edges (or only their vertices) can be covered by few lines or planes. We insist on straight-line edges and crossing-free drawings. This problem has many connections to other challenging graph-drawing problems such as small-area or small-volume drawings, layered or track drawi...
Article
Full-text available
Given a Tychonoff space $X$, let $F(X)$ and $A(X)$ be respectively the free topological group and the free Abelian topological group over $X$ in the sense of Markov. In this paper, we discuss two topological properties in $F(X)$ or $A(X)$, namely the countable tightness and $\mathfrak G$-base. We provide some characterizations of the countable tigh...
Article
Full-text available
We prove that a topological manifold (possibly with boundary) admitting a continuous cancellative binary operation is orientable. This implies that the M\"obius band admits no cancellative continuous binary operation. This answers a question posed by the second author in 2010.
Article
Full-text available
We prove that a regular topological space $X$ is Tychonoff if and only if its topology is generated by a quasi-uniformity $\mathcal U$ such that for every $U\in\mathcal U$ and $A\subset X$ there is $V\in\mathcal U$ such that $B(\bar A,V)\subset \bar{B(A,U)}$. This characterization implies that each regular paratopological group is Tychonoff and eac...
Article
For any topological space $X$ we study the relation between the universal uniformity $\mathcal U_X$, the universal quasi-uniformity $q\mathcal U_X$ and the universal pre-uniformity $p\mathcal U_X$ on $X$. For a pre-uniformity $\mathcal U$ on a set $X$ and a word $v$ in the two-letter alphabet $\{+,-\}$ we define the verbal power $\mathcal U^v$ of $...
Article
Full-text available
We derive many upper bounds on the submetrizability number and $i$-weight of paratopological groups and topological monoids with open shifts. In particular, we prove that each first countable Hausdorff paratopological group is submetrizable thus answering a problem of Arhangelskii posed in 2002. Also we construct an example of a zero-dimensional (a...
Article
We define a notion of a rotund quasi-uniform space and describe a new direct construction of a (right-continuous) quasi-pseudometric on a (rotund) quasi-uniform space. This new construction allows to give alternative proofs of several classical metrizability theorems for (quasi-)uniform spaces and also obtain some new metrizability results. Applyin...
Article
We introduce so-called cone topologies of paratopological groups, which are a wide way to construct counterexamples, especially of examples of compact-like paratopological groups with discontinuous inversion. We found a simple interplay between the algebraic properties of a basic cone subsemigroup S of a group G and compact-like properties of two b...
Article
We obtain necessary and sufficient conditions when a pseudocompact paratopological group is topological. (2-)pseudocompact and countably compact paratopological groups that are not topological are constructed. It is proved that each 2-pseudocompact paratopological group is pseudocompact and that each Hausdorff \sigma-compact pseudocompact paratopol...
Article
Full-text available
We study structure of inverse primitive pseudocompact semitopological and topological semigroups. We find the conditions when the maximal subgroup of an inverse primitive pseudocompact semitopological semigroup $S$ is a closed subset of $S$ and describe the topological structure of such semiregular semitopological semigroups. Also an analogue of Co...
Article
Full-text available
Given a $G$-space $X$ and a non-trivial $G$-invariant ideal $I$ of subsets of $X$, we prove that for every partition $X=A_1\cup\dots\cup A_n$ of $X$ into $n\ge 2$ pieces there is a piece $A_i$ of the partition and a finite set $F\subset G$ of cardinality $|F|\le \phi(n+1):=\max_{1<x<n+1}\frac{x^{n+1-x}-1}{x-1}$ such that $G=F\cdot \Delta(A_i)$ wher...
Article
Full-text available
We prove that a topological Clifford semigroup $S$ is metrizable if and only if $S$ is an $M$-space and the set $E=\{e\in S:ee=e\}$ of idempotents of $S$ is a metrizable $G_\delta$-set in $S$. The same metrization criterion holds also for any countably compact Clifford topological semigroup $S$.
Article
Full-text available
If K' and K are convex bodies of the plane such that K' is a subset of K then the perimeter of K' is not greater than the perimeter of K. We obtain the following generalization of this fact. Let K be a convex compact body of the plane with the perimeter p and the diameter d and r>1 be an integer. Let s be the smallest number such that for any curve...
Article
We found a solution of the star puzzle (a path on a chessboard from c5 to d4 in 14 straight strokes) in 14 queen moves, which has been claimed by the author as impossible.
Article
Full-text available
A convex subset X of a linear topological space is called compactly convex if there is a continuous compact-valued map $\Phi:X\to exp(X)$ such that $[x,y]\subset\Phi(x)\cup \Phi(y)$ for all $x,y\in X$. We prove that each convex subset of the plane is compactly convex. On the other hand, the space $R^3$ contains a convex set that is not compactly co...
Article
Given a planar graph G, we consider drawings of G in the plane where edges are represented by straight line segments (which possibly intersect). Such a drawing is specified by an injective embedding π of the vertex set of G into the plane. Let be the maximum integer k such that there exists a crossing-free redrawing π′ of G which keeps k vertices f...
Article
We consider approximations of a continuous function on a countable normed Fr\'{e}chet space by analytic and $*$-analytic. Also we found a criterium of the existence of an extension of a continuous function from a dense subspace of a topological space onto the space.
Article
We prove a precise formula for the minimal number K(n) such that every binary word of length $n$ can be divided into K(n) palindromes. Also we estimate the average number $\ol K(n)$ of palindromes composing a random binary word of the length n.
Article
A binary word is symmetric if it is a palindrome or an antipalindrome. We define a new measure of asymmetry of a binary word equal to the minimal number of letters of the word whose deleting from the word yields a symmetric word and obtain upper and lower estimations of this measure. Comment: 4 pages
Article
We consider the Steinhaus geometrical game on cake dividing. Hugo Steinhaus in his popular book [One hundred problems in elementary mathematics (Pergamon Press, Oxford) (1963; Zbl 0116.24102)] (see also B. Grünbaum [Studies in combinatorial geometry and the theory of convex bodies (Nauka Moskau) (1971; Zbl 0229.52001)]) considered the following gam...
Article
Full-text available
A paratopological group $G$ is saturated if the inverse $U^{-1}$ of each non-empty set $U\subset G$ has non-empty interior. It is shown that a [first-countable] paratopological group $H$ is a closed subgroup of a saturated (totally bounded) [abelian] paratopological group if and only if $H$ admits a continuous bijective homomorphism onto a (totally...
Article
A Hausdorff paratopological group G is H-closed if G is closed in each Hausdorff paratopological group containing G. We obtain criteria of H-closedness for some classes of abelian paratopological groups. In particular, for topological groups. Comment: 7 pages
Article
Full-text available
Let $H$ be a closed subgroup of a regular abelian paratopological group $G$. The group reflexion $G^\flat$ of $G$ is the group $G$ endowed with the strongest group topology, weaker that the original topology of $G$. We show that the quotient $G/H$ is Hausdorff (and regular) if $H$ is closed (and locally compact) in $G^\flat$. On the other hand, we...
Article
Full-text available
We prove that a Hausdorff paratopological group G is meager if and only if there are a nowhere dense subset A of G and a countable subset C in G such that CA=G=AC.
Article
Full-text available
We introduce and study oscillator topologies on paratopological groups and define certain related number invariants. As an application we prove that a Hausdorff paratopological group $G$ admits a weaker Hausdorff group topology provided $G$ is 3-oscillating. A paratopological group $G$ is 3-oscillating (resp. 2-oscillating) provided for any neighbo...
Conference Paper
We consider straight-line drawings of a planar graph G with possible edge crossings. The untangling problem is to eliminate all edge crossings by moving as few vertices as possible to new positions. Let fix G denote the maximum number of vertices that can be left fixed in the worst case among all drawings of G. In the allocation problem, we are giv...
Article
Given a planar graph G, we consider drawings of G in the plane where edges are represented by straight line segments (which possibly intersect). Such a drawing is specified by an injective embedding π of the vertex set of G into the plane. Let fix(G, π) be the maximum integer k such that there exists a crossing-free redrawing π ′ of G which keeps k...
Article
A binary word is symmetric if it is a palindrome or an antipalindrome. We define a new measure of asymmetry of a binary word equal to the minimal number of letters of the word whose deleting from the word yields a symmetric word and obtain upper and lower estimations of this measure.
Article
We prove a precise formula for the minimal number K(n) such that every binary word of length n can be divided into K(n) palindromes. Also we estimate the average number K(n) of palindromes composing a random binary word of the length n.

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