Related papers: Extensions of Steiner Triple Systems
An algebraic process for the construction of an autotopism for a non-Steiner C-loop is described and this is demonstrated with an example using a known finite C-loop. In every C-loop, two of its parastrophes are not equivalent(equal) it, if…
The focus of this paper is on the study of specific circle formations known as orthogonal Pappus chains and the related incidence results that involve points of tangency between the circles in the construction. These chains give rise to new…
We consider the problem of embedding the Steiner points of a Steiner tree with given topology into the rectilinear plane. Thereby, the length of the path between a distinguished terminal and each other terminal must not exceed given length…
We develop a technique for estimating the inner eccentricity in hierarchical triple systems, with the inner orbit being initially circular, while the outer one is eccentric. We consider coplanar systems with well separated components and…
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…
We prove several structural properties of Steiner triple systems (STS) of order 3w+3 that include one or more transversal subdesigns TD(3,w). Using an exhaustive search, we find that there are 2004720 isomorphism classes of STS(21)…
We present a concise self-contained inversive geometry solution of the three-circle problem of Steiner of constructing a circle that intersects each of the three given circles at one of the three given angles.
Characterization results for equality cases and for rigidity of equality cases in Steiner's perimeter inequality are presented. (By rigidity, we mean the situation when all equality cases are vertical translations of the Steiner's symmetral…
In this paper we study the boundedness of extension operators associated with spheres in vector spaces over finite fields.In even dimensions, we estimate the number of incidences between spheres and points in the translated set from a…
We construct Steiner triple systems without parallel classes for an infinite number of orders congruent to $3 \pmod{6}$. The only previously known examples have order $15$ or $21$.
In this paper, we attempt to develop the Schreier theory for two special types extensions of multiplicative Lie algebras.
A method to construct trihamiltonian extensions of a separable system is presented. The procedure is tested for systems, with a natural Hamiltonian, separable in classical sense in one of the four orthogonal separable coordinate systems of…
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 $S(2,6,v)$.
We extend the theory of ground states of classical Heisenberg spin systems published previously by a closer investigation of the convex Gram set of $N$ spins. This is relevant for the present purpose since the set of ground states of a…
Cycle switching is a particular form of transformation applied to isomorphism classes of a Steiner triple system of a given order $v$ (an $STS(v)$), yielding another $STS(v)$. This relationship may be represented by an undirected graph. An…
A mixed Ising-Heisenberg spin system consisting of triangular XXZ-Heisenberg spin clusters assembled into a chain by alternating with Ising spins interacting to all three spins in the triangle is considered. The exact solution of the model…
We generalize the idea of "extension of Hamiltonian systems" -- developed in a series of previous articles -- which allows the explicit construction of Hamiltonian systems with additional non-trivial polynomial first integrals of…
The existence of large sets of Kirkman triple systems (LKTSs) is one of the best-known open problems in combinatorial design theory. Steiner quadruple systems with resolvable derived designs (RDSQSs) play an important role in the recursive…
The Euclidean Steiner problem is the problem of finding a set $St$, with the shortest length, such that $St \cup A$ is connected, where $A$ is a given set in a Euclidean space. The solutions $St$ to the Steiner problem will be called…
A Kirkman triple system of order $v$, KTS$(v)$, is a resolvable Steiner triple system on $v$ elements. In this paper, we investigate an open problem posed by Doug Stinson, namely the existence of KTS$(v)$ which contain as a subdesign a…