arXiv subjects
Publications and source records attributed to Ivan Hetman.
In this note two Steiner systems $S(2,7,505)$, two Steiner systems $S(2,7,589)$, and ten Steiner systems $S(2,8,624)$ are presented. This resolves two of $21$ undecided cases for block designs with block length $7$, and one of $37$ cases for block designs with block length $8$, mentioned in Handbook of Combinatorial Designs.
Whereas Steiner systems $S(2,k,v)$ with block length $k \le 5$ have large amount of examples and the existence is established for all admissible $v$, for $k\ge 6$ only few examples are known even for decided cases. In this paper the existence of $S(2,9,369)$ is established and some new examples for other admissible pairs $(k,v)$ are given. In particular, lots of new unitals of order $6$ (or $S(2,7,217)$) together with $S(2,7,175)$, $S(2,7,259)$, $S(2,8,120)$, $S(2,8,504)$, $S(2,9,513)$ are presented. Found examples suggest two conjectures on infinite series of designs.
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.
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 $S(2,6,v)$.
In this note six Steiner systems $S(2,8,225)$ and four Steiner systems $S(2,9,289)$ are presented. This resolves two of $129$ undecided cases for block designs with block length $8$ and $9$, mentioned in Handbook of Combinatorial Designs.
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.
In this paper new $1$-rotational 2-Steiner systems for different admissible $v,k$ pairs are introduced. In particular, $1$-rotational unitals of order $4$ are enumerated.
In this paper we introduce enumeration of unitals of order $5$, which are also Steiner systems $S(2,6,126)$, where automorphism group acts transitively and effectively on points or fixes one point.
In this paper new Steiner systems $S(2,6,111)$, $S(2,6,121)$, $S(2,6,126)$, $S(2,7,169)$, $S(2,7,175)$ and possibly others with point-transitive (commutative except $S(2,6,111)$ case) automorphism groups are introduced.
In this paper new Steiner systems $S(2,6,121)$, $S(2,6,126)$, $S(2,7,169)$ are introduced. Also some non-existence results for line lengths $7..11$ are presented. There is no solid proof that presented algorithm is exhaustive or correct, but it produces same results on already known difference families for line lengths $3..6$. Due to calculation-based approach this paper probably won't be published, but will be submitted to arxiv as it contains some new results
Let $X$ be a Banach space and $Conv_H(X)$ be the space of non-empty closed convex subsets of $X$, endowed with the Hausdorff metric $d_H$. We prove that each connected component of the space $Conv_H(X)$ is homeomorphic to one of the spaces: a singleton, the real line, a closed half-plane, the Hilbert cube multiplied by the half-line, the separable Hilbert space, or a Hilbert space of density not less than continuum.
For a Banach space $X$ by $Conv_H(X)$ we denote the space of non-empty closed convex subsets of $X$, endowed with the Hausdorff metric. We prove that for any closed convex set $C\subset X$ and its metric component $H_C=\{A\in Conv_H(X):d_H(A,C)<\infty\}$ in $Conv_H(X)$, the following conditions are equivalent: (1) $C$ is approximatively polyhedral, which means that for every $ε>0$ there is a polyhedral convex subset $P\subset X$ on Hausdorff distance $d_H(P,C)<ε$ from $C$; (2) $C$ lies on finite Hausdorff distance $d_H(C,P)$ from some polyhedral convex set $P\subset X$; (3) the metric space $(H_C,d_H)$ is separable; (4) $H_C$ has density $dens(H_C)<\mathfrak c$; (5) $H_C$ does not contain a positively hiding convex set $P\subset X$. If the Banach space $X$ is finite-dimensional, then the conditions (1)--(5) are equivalent to: (6) $C$ is not positively hiding; (7) $C$ is not infinitely hiding. A convex subset $C\subset X$ is called {\em positively hiding} (resp. {\em infinitely hiding}) if there is an infinite set $A\subset X\setminus C$ such that $\inf_{a\in A}dist(a,C)>0$ (resp. $\sup_{a\in A}dist(a,C)=\infty$) and for any distinct points $a,b\in A$ the segment $[a,b]$ meets the set $C$.
We prove that a closed convex subset $C$ of a complete linear metric space $X$ is polyhedral in its closed linear hull if and only if no infinite subset $A\subset X\backslash C$ can be hidden behind $C$ in the sense $[x,y]\cap C\not = \emptyset$ for any distinct points $x,y\in A$.