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Iryna Banakh

Publications and source records attributed to Iryna Banakh.

12 recordsLinked to original sources
arXiv2023

Midconvex sets in Abelian groups

A subset $X$ of an Abelian group $G$ is called $midconvex$ if for every $x,y\in X$ the set $\frac{x+y}2=\{z\in G:2z=x+y\}$ is a subset of $X$. We prove that a subset $X$ of an Abelian group $G$ is midconvex if and only if for every $g\in G$ and $x\in X$, the set $\{n\in\mathbb Z:x+ng\in X\}$ is equal to $C\cap H$ for some order-convex set $C\subseteq \mathbb Z$ and some subgroup $H\subseteq \mathbb Z$ such that the quotient group $\mathbb Z/H$ has no elements of even order. This characterization implies that a subset $X$ of a periodic Abelian group $G$ is midconvex if and only if for every $x\in X$ the set $X-x$ is a subgroup of $G$ such that every element of the quotient group $G/(X-x)$ has odd order. Also we prove that a nonempty set $X$ in a subgroup $G\subseteq\mathbb Q$ is midconvex if and only if $X=C\cap(H+x)$ for some order-convex set $C\subseteq\mathbb Q$, some $x\in X$ and some subgroup $H$ of $G$ such that the quotient group $G/H$ contains no elements of even order.

math.GR↗
arXiv2023

Semiaffine sets in Abelian groups

A subset $X$ of an Abelian group $G$ is called $semiaf\!fine$ if for every $x,y,z\in X$ the set $\{x+y-z,x-y+z\}$ intersects $X$. We prove that a subset $X$ of an Abelian group $G$ is semiaffine if and only if one of the following conditions holds: (1) $X=(H+a)\cup (H+b)$ for some subgroup $H$ of $G$ and some elements $a,b\in X$; (2) $X=(H\setminus C)+g$ for some $g\in G$, some subgroup $H$ of $G$ and some midconvex subset $C$ of the group $H$. A subset $C$ of a group $H$ is $midconvex$ if for every $x,y\in C$, the set $\frac{x+y}2:=\{z\in H:2z=x+y\}$ is a subset of $C$.

math.GR↗
arXiv2023

Metric characterizations of some subsets of the real line

A metric space $(X,d)$ is called a $subline$ if every 3-element subset $T$ of $X$ can be written as $T=\{x,y,z\}$ for some points $x,y,z$ such that $d(x,z)=d(x,y)+d(y,z)$. By a classical result of Menger, every subline of cardinality $\ne 4$ is isometric to a subspace of the real line. A subline $(X,d)$ is called an $n$-$subline$ for a natural number $n$ if for every $c\in X$ and positive real number $r\in d[X^2]$, the sphere $S(c;r):=\{x\in X:d(x,c)=r\}$ contains at least $n$ points. We prove that every $2$-subline is isometric to some additive subgroup of the real line. Moreover, for every subgroup $G\subseteq\mathbb R$, a metric space $(X,d)$ is isometric to $G$ if and only if $X$ is a $2$-subline with $d[X^2]=G_+:= G\cap[0,\infty)$. A metric space $(X,d)$ is called a $ray$ if $X$ is a $1$-subline and $X$ contains a point $o\in X$ such that for every $r\in d[X^2]$ the sphere $S(o;r)$ is a singleton. We prove that for a subgroup $G\subseteq\mathbb Q$, a metric space $(X,d)$ is isometric to the ray $G_+$ if and only if $X$ is a ray with $d[X^2]=G_+$. A metric space $X$ is isometric to the ray $\mathbb R_+$ if and only if $X$ is a complete ray such that $\mathbb Q_+\subseteq d[X^2]$. On the other hand, the real line contains a dense ray $X\subseteq\mathbb R$ such that $d[X^2]=\mathbb R_+$.

math.MG↗
arXiv2022

On symmetrizability and perfectness of second-countable spaces

A symmetrizability criterion of Arhangelskii implies that a second-countable Hausdorff space is symmetrizable if and only if it is perfect. We present an example of a non-symmetrizable second-countable submetrizable space of cardinality $\mathfrak q_0$ and study the smallest possible cardinality $\mathfrak q_i$ of a non-symmetrizable second-countable $T_i$-space for $i\in\{1,2\}$.

math.GN↗
arXiv2020

A semigroup is finite if and only if it is chain-finite and antichain-finite

A subset $A$ of a semigroup $S$ is called a $chain$ ($antichain$) if $xy\in\{x,y\}$ ($xy\notin\{x,y\}$) for any (distinct) elements $x,y\in S$. A semigroup $S$ is called ($anti$)$chain$-$finite$ if $S$ contains no infinite (anti)chains. We prove that each antichain-finite semigroup $S$ is periodic and for every idempotent $e$ of $S$ the set $\sqrt[\infty]{e}=\{x\in S:\exists n\in\mathbb N\;\;(x^n=e)\}$ is finite. This property of antichain-finite semigroups is used to prove that a semigroup is finite if and only if it is chain-finite and antichain-finite. Also we present an example of an antichain-finite semilattice that is not a union of finitely many chains.

math.GR↗
arXiv2020

On the asymptotic dimension of products of coarse spaces

We prove that for any coarse spaces $X_1,\dots,X_n$ of asymptotic dimension $\ge 1$, the product $X=X_1\times\dots\times X_n$ has asymptotic dimension $\ge n$. Another result states that a finitary coare space $Z$ has $\mathrm{asdim}(Z)\ge n$ if $Z$ admits an almost free action of the group $\mathbb Z^n$. We deduce these results from the following combinatorial result (that generalized the the Hex Theorem of Gale): for any cover $\mathcal F$ of a discrete box $K=k_1\times \dots \times k_n$, either some set $F\in\mathcal F$ contains a chain connecting two opposite faces of $K$ or there exists a set $B\subset K$ of diameter $\le 1$ such that $|\{F\in\mathcal F:F\cap B\ne\emptyset\}|>n$.

math.GN↗
arXiv2020

The completion of the hyperspace of finite subsets, endowed with the $\ell^1$-metric

For a metric space $X$, let $\mathsf FX$ be the space of all nonempty finite subsets of $X$ endowed with the largest metric $d^1_{\mathsf FX}$ such that for every $n\in\mathbb N$ the map $X^n\to\mathsf FX$, $(x_1,\dots,x_n)\mapsto \{x_1,\dots,x_n\}$, is non-expanding with respect to the $\ell^1$-metric on $X^n$. We study the completion of the metric space $\mathsf F^1\!X=(\mathsf FX,d^1_{\mathsf FX})$ and prove that it coincides with the space $\mathsf Z^1\!X$ of nonempty compact subsets of $X$ that have zero length (defined with the help of graphs). We prove that each subset of zero length in a metric space has 1-dimensional Hausdorff measure zero. A subset $A$ of the real line has zero length if and only if its closure is compact and has Lebesgue measure zero. On the other hand, for every $n\ge 2$ the Euclidean space $\mathbb R^n$ contains a compact subset of 1-dimensional Hausdorff measure zero that fails to have zero length.

math.GN↗
arXiv2020

The continuity of Darboux injections between manifolds

We prove that an injective map $f:X\to Y$ between connected metrizable spaces $X,Y$ is continuous if for every connected subset $C\subset X$ the image $f(C)$ is connected and one of the following conditions is satisfied: (1) $Y$ is a 1-manifold and $X$ is compact; (2) $Y$ is a 2-manifold and $X$ is a closed $n$-manifold of dimension $n\ge 2$; (3) $Y$ is a 3-manifold and $X$ is a simply-connected closed $n$-manifold of dimension $n\ge 3$. This gives a partial answer to a problem of Willie Wong, posed on Mathoverflow.

math.GN↗
arXiv2014

Toehold Purchase Problem: A comparative analysis of two strategies

Toehold purchase, defined here as purchase of one share in a firm by an investor preparing a tender offer to acquire majority of shares in it, reduces by one the number of shares this investor needs for majority. In the paper we construct mathematical models for the toehold and no-toehold strategies and compare the expected profits of the investor and the probabilities of takeover the firm in both strategies. It turns out that the expected profits of the investor in both strategies coincide. On the other hand, the probability of takeover the firm using the toehold strategy is considerably higher comparing to the no-toehold strategy. In the analysis of the models we apply the apparatus of incomplete Beta functions and some refined bounds for central binomial coefficients.

q-fin.GN↗
arXiv2012

On Local Convexity Of Nonlinear Mappings Between Banach Spaces

We find conditions for a smooth nonlinear map $f:U\rightarrow V$ between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some $c$ and each positive $\varepsilon<c$ the image $% f(B_\varepsilon(x))$ of each $\varepsilon$-ball $B_\varepsilon(x)\subset U$ is convex. We give a lower bound on $c$ via the second order Lipschitz constant $\mathrm{Lip}_2(f)$, the Lipschitz-open constant $\mathrm{Lip}_o(f)$ of $f$, and the 2-convexity number $\mathrm{conv}_2(X)$ of the Banach space $% X$.

math.FA↗
arXiv2010

Constructing non-compact operators into $c_0$

We prove that for each dense non-compact linear operator $S:X\to Y$ between Banach spaces there is a linear operator $T:Y\to c_0$ such that the operator $TS:X\to c_0$ is not compact. This generalizes the Josefson-Nissenzweig Theorem.

math.FA↗