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Alex Ravsky

Publications and source records attributed to Alex Ravsky.

At least 19 recordsLinked to original sources
arXiv2026

Time and Supply Fairness in Electricity Distribution using $k$-times bin packing

Given items of different sizes and a fixed bin capacity, the bin-packing problem is to pack these items into the minimum number of bins such that the sum of the item sizes in each bin does not exceed the capacity. We define a new variant, k-times bin-packing (kBP), in which the goal is to pack the items so that each item appears exactly k times in k different bins. We generalize existing approximation algorithms for bin-packing to solve kBP and analyze their performance ratios. The fair electricity division problem motivates the study of kBP. The goal is to allocate the available supply among households using some fairness criteria, such as the egalitarian principle. We prove that every electricity division problem can be solved by k-times bin-packing for some finite k, which depends only on the number of households. We implement generalizations of the First-Fit and First-Fit Decreasing bin-packing algorithms to solve kBP and apply them to real electricity demand data. We show that our generalizations outperform existing heuristic solutions to the same problem in terms of the egalitarian allocation of connection time. We study another variant of the egalitarian allocation problem, in which the goal is to maximize the minimum number of watts allocated to a household. For this variant, we prove an impossibility result: there does not exist such a k that depends only on the number of agents. This impossibility result motivates us to develop four different heuristic algorithms to solve the egalitarian allocation of watts problem. We evaluate the heuristics by summing the minimum watts allocated to any household in each hour, yielding a fairness metric that reflects the lowest watt allocation across all hours. A higher total minimum of watts indicates a more equitable distribution. Thus, we establish new benchmarks for fair allocation of watts.

cs.DS↗
arXiv2025

Linear Geometry: flats, ranks, regularity, parallelity

Linear Geometry describes geometric properties that depend on the fundamental notion of a line. In this paper we survey basic notions and results of Linear Geomery that depend on the flat hulls: flats, exchange, rank, regularity, modularity, and parallelity.

math.HO↗
arXiv2025

Steiner systems $S(2,6,226)$ and $S(2,6,441)$ exist

Via computer search, we found seven non-isomorphic $1$-rotational Steiner systems $S(2,6,226)$ and six point-transitive Steiner systems $S(2,6,441)$, resolving two of $29$ previously undecided cases for $S(2,6,v)$.

math.CO↗
arXiv2025

New Steiner systems $S(2,6,v)$ with block length 6

In this paper various Steiner systems $S(2,k,v)$ for $k = 6$ are collected and enumerated for specific constructions. In particular, two earlier unknown types of $1$-rotational designs are found for the groups $SL(2,5)$ and $((\mathbb Z_3 \times \mathbb Z_3) \rtimes \mathbb Z_3) \times \mathbb Z_5$. Also new Steiner systems $S(2,6,96), S(2,6,106), S(2,6,111)$ are listed.

math.CO↗
arXiv2023

$k$-times bin packing and its application to fair electricity distribution

Given items of different sizes and a fixed bin capacity, the bin-packing problem is to pack these items into a minimum number of bins such that the sum of item sizes in a bin does not exceed the capacity. We define a new variant called \emph{$k$-times bin-packing ($k$BP)}, where the goal is to pack the items such that each item appears exactly $k$ times, in $k$ different bins. We generalize some existing approximation algorithms for bin-packing to solve $k$BP, and analyze their performance ratio. The study of $k$BP is motivated by the problem of \emph{fair electricity distribution}. In many developing countries, the total electricity demand is higher than the supply capacity. We prove that every electricity division problem can be solved by $k$-times bin-packing for some finite $k$. We also show that $k$-times bin-packing can be used to distribute the electricity in a fair and efficient way. Particularly, we implement generalizations of the First-Fit and First-Fit Decreasing bin-packing algorithms to solve $k$BP, and apply the generalizations to real electricity demand data. We show that our generalizations outperform existing heuristic solutions to the same problem in terms of the egalitarian allocation of connection time.

cs.DS↗
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↗
arXiv2021

On pseudobounded and premeager paratopological groups

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 $U^n$, where $n$ is a natural number. The group $G$ is premeager, if $G\ne N^n$ for any nowhere dense subset $N$ of $G$ and any positive integer $n$. In this paper we investigate relations between the above classes of groups and answer some questions posed by F. Lin, S. Lin, and S\'anchez.

math.GN↗
arXiv2021

Semitopological modules

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 space $X$ whose free semitopological $R$-module is not a topological $R$-module.

math.FA↗
arXiv2020

On unconditionally convergent series in topological rings

We define a topological ring $R$ to be \emph{Hirsch}, if for any unconditionally convergent series $\sum_{n\in\omega} x_i$ in $R$ and any neighborhood $U$ of the additive identity $0$ of $R$ there exists a neighborhood $V\subseteq R$ of $0$ such that $\sum_{n\in F} a_n x_n\in U$ for any finite set $F\subset\omega$ and any sequence $(a_n)_{n\in F}\in V^F$. We recognize Hirsch rings in certain known classes of topological rings. For this purpose we introduce and develop the technique of seminorms on actogroups. We prove, in particular, that a topological ring $R$ is Hirsch provided $R$ is locally compact or $R$ has a base at the zero consisting of open ideals or $R$ is a closed subring of the Banach ring $C(K)$, where $K$ is a compact Hausdorff space. This implies that the Banach ring $\ell_\infty$ and its subrings $c_0$ and $c$ are Hirsch. Also we prove that for every $p\in[1,2]$ the Banach ring $\ell_p$ is Hirsch. On the other hand, for any distinct numbers $p,q\in[1,\infty]$ the commutative Banach ring $\ell_p\oplus i\ell_q$ is not Hirsch. Also for any $p\in (1,\infty)$, the (noncommutative) Banach ring $L(\ell_p)$ of continuous endomorphisms of the Banach ring $\ell_p$ is not Hirsch. We do not know whether the Banach rings $\ell_p$ are Hirsch for $p\in(2,\infty)$.

math.GN↗
arXiv2020

Suitable sets for paratopological groups

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 investigation of suitable sets for paratopological groups, looking to what extent we can (by proving propositions) or cannot (by constructing examples) generalize to paratopological groups results which hold for topological groups, and to pose a few challenging questions for possible future research. We shall discuss when paratopological groups of different classes have suitable sets. Namely, we consider paratopological groups (in particular, countable) satisfying different separation axioms, paratopological groups which are compact-like spaces, and saturated (in particular, precompact) paratopological groups. Also we consider the permanence of a property of a group to have a suitable set with respect to (open or dense) subgroups, products and extensions.

math.GN↗
arXiv2020

Each topological group embeds into a duoseparable topological group

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 functor $F$ is defined on the category of unital topologized magmas. Also we prove that each $\sigma$-compact locally compact abelian topological group embeds into a duoseparable locally compact abelian-by-countable topological group.

math.GN↗