Some resolutions of S(5, 8, 24)

There exist 13 mutually disjoint resolutions of the Steiner system S(5, 8, 24). There also exist nine nonisomorphic mutually disjoint resolutions of S(5, 8, 24) where three of the resolutions have the same L₂(23) as an automorphism group and the other six have the same affine group C₂₃¹¹ as an automorphism group. A resolution of S(5, 8, 24) using a group of order 21 is displayed and a 13-dimensional Room-type design is mentioned.     

Cayley graphs on abelian groups

Let A be an abelian group and let ι be the automorphism of A defined by: ι: a ? a^?1. A Cayley graph Γ = Cay(A,S) is said to have an automorphism group as small as possible if Aut(Γ)=A?<ι>. In this paper, we show that almost all Cayley graphs on abelian groups have automorphism group as small as possible, proving a conjecture of Babai and Godsil.     

Parsimonious phylogenetic trees in metric spaces and simulated annealing

Steiner trees for (finite) subsets D of metric spaces S are discussed. For a given (abstract) tree topology over D Steiner interpretations in S are defined and their properties are studied. An algorithm to obtain Steiner interpretations for a given tree topology is given which is efficient if S is the (L₁-) product of small metric spaces, e.g., if S is the sequence space A^l over an alphabet A of small cardinality. A variant of the same algorithm can be used to minimize efficiently and exactly spin glass Hamiltonians of k-meshed graphs. The interpretation algorithm is used as an ingredient for a variant of the stochastic search algorithm called “simulated annealing” which is used to find Steiner trees for various given data sets D in various sequence spaces S = A^l. For all data sets analyzed so far the trees obtained this way are shorter than or at least as short as the best ones derived using other tree construction methods. Two main features can be observed:

• 1.(1) Very often the shape of Steiner trees constructed this way is more or less chain-like. The trees are “long and slim.”
• 2.(2) Generally, the method allows to find many different Steiner trees.

As a consequence, one may conclude that tree reconstruction programs should be executed in an interactive fashion so that additional biological knowledge, not explicitly represented in the data set, can be introduced at various stages of the reconstruction algorithm to reduce the number of possible solutions. Moreover, as the “Simulated Annealing” search procedure is universally applicable, one may also use this algorithm during such an interactive reconstruction program to optimize any other of the known tree reconstruction minimality principles.     

Some Remarks on Steiner Systems

The main purpose of this paper is to introduce Steiner systems obtained from the finite classical generalized hexagons of order q. We show that we can take the blocks of the Steiner systems amongst the lines and the traces of the hexagon, and we prove some facts about the automorphism groups. Also, we make a remark concerning the geometric construction of a known class (KW) of Steiner systems and we deduce some properties of the automorphism group.     

Covering spaces and the Galois theory of commutative Banach algebras

The principal result of this paper is that there is a bijective (functorial) correspondence between the projective separable extensions of a comutative Banach algebra A and the finite covering spaces of its maximal ideal space M(A). As a consequence, a full Galois theory for commutative Banach algebras is developed which is analogous to the (unramified) Galois theory of function fields on compact Riemann surfaces. In case M(A) is a reasonably “nice” space, its profinite fundamental group is identified as the automorphism group of the separable closure of A.     

The smallest non-derived steiner triple system is simple as a loop

The smallest non-derived triple system is simple as a loop. THEOREM.If A, B are Steiner loops, and f:A→B is a homomorphism, then if B and f ^?1 (1) are derivable from Steiner quadruple systems, then so is A.     

Intersections among Steiner systems

A Steiner system S(l, m, n) is a system of subsets of size m (called blocks) from an n-set S, such that each d-subset from S is contained in precisely one block. Two Steiner systems have intersection k if they share exactly k blocks. The possible intersections among S(5, 6, 12)'s, among S(4, 5, 11)'s, among S(3, 4, 10)'s, and among S(2, 3, 9)'s are determined, together with associated orbits under the action of the automorphism group of an initial Steiner system. The following are results: (i) the maximal number of mutually disjoint S(5, 6, 12)'s is two and any two such pairs are isomorphic; (ii) the maximal number of mutually disjoint S(4, 5, 11)'s is two and any two such pairs are isomorphic; (iii) the maximal number of mutually disjoint S(3, 4, 10)'s is five and any two such sets of five are isomorphic; (iv) a result due to Bays in 1917 that there are exactly two non-isomorphic ways to partition all 3-subsets of a 9-set into seven mutually disjoint S(2, 3, 9)'s.     

Controllable Subsets in Graphs

Let X be a graph on ?? vertices with adjacency matrix A, and let S be a subset of its vertices with characteristic vector z. We say that the pair (X, S) is controllable if the vectors A ^r z for r =? 1, . . . , ?? ? 1 span ${\mathbb{R}^{\nu}}$ . Our concern is chiefly with the cases where S =?V(X), or S is a single vertex. In this paper we develop the basic theory of controllable pairs. We will see that if (X, S) is controllable then the only automorphism of X that fixes S as a set is the identity. If (X, S) is controllable for some subset S then the eigenvalues of A are all simple.     

Homogeneous and ultrahomogeneous Steiner systems

A Steiner system (or t — (v, k, 1) design) S is said to be homogeneous if, whenever the substructures induced on two finite subsets S₁ and S₂ of S are isomorphic, there is at least one automorphism of S mapping S₁ onto S₂, and is said to be ultrahomogeneous if each isomorphism between the substructures induced on two finite subsets of S can be extended to an automorphism of S. We give a complete classification of all homogeneous and ultrahomogeneous Steiner systems. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 153–161, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10034     

Concerning multiplier automorphisms of cyclic Steiner triple systems

A cyclic Steiner triple system, presented additively over Z _v as a set B of starter blocks, has a non-trivial multiplier automorphism λ ≠ 1 when λB is a set of starter blocks for the same Steiner triple system. When does a cyclic Steiner triple system of order v having a nontrivial multiplier automorphism exist? Constructions are developed for such systems; of most interest, a novel extension of Netto's classical construction for prime orders congruent to 1 (mod 6) to prime powers is proved. Nonexistence results are then established, particularly in the cases when v = (2^β + 1)^α, when v = 9p with p ≡ 5 (mod 6), and in certain cases when all prime divisors are congruent to 5 (mod 6). Finally, a complete solution is given for all v < 1000, in which the remaining cases are produced by simple computations.     

Edge-colorings of cubic graphs with elements of point-transitive Steiner triple systems

A cubic graph G is S-edge-colorable for a Steiner triple system S if its edges can be colored with points of S in such a way that the points assigned to three edges sharing a vertex form a triple in S. We show that a cubic graph is S-edge-colorable for every non-trivial affine Steiner triple system S unless it contains a well-defined obstacle called a bipartite end. In addition, we show that all cubic graphs are S-edge-colorable for every non-projective non-affine point-transitive Steiner triple system S.     

Automorphism free latin square graphs

In this paper, we show that there exists an automorphism free latin square graph of order n for all n ? 7 and that the number of such graphs goes to infinity with n. These results are then applied to the construction of automorphism free Steiner triple systems.     

Definibility in normal theories

This paper initiates an investigation which seeks to explain elementary definability as the classical results of mathematicallogic (the completeness, compactness and Löwenheim-Skolem theorems) explain elementary logical consequence. The theorems of Beth and Svenonius are basic in this approach and introduce automorphism groups as a means of studying these problems. It is shown that for a complete theoryT, the definability relation of Beth (or Svenonius) yields an upper semi-lattice whose elements (concepts) are interdefinable formulas ofT (formulas having equal automorphism groups in all models ofT). It is shown that there are countable modelsA ofT such that two formulae are distinct (not interdefinable) inT if and only if they are distinct (have different automorphism groups) inA. The notion of a concepth being normal in a theoryT is introduced. Here the upper semi-lattice of all concepts which defineh is proved to be a finite lattice—anti-isomorphic to the lattice of subgroups of the corresponding automorphism group. Connections with the Galois theory of fields are discussed.     

A recursive construction of 1-rotational Steiner 2-designs

A Steiner 2-design S(2,k, v) is said to be 1-rotational if it admits an automorphism whose cycle structure is a (v ? 1)-cycle and a fixed point. In this paper, a recursive construction of 1-rotational Steiner 2-designs is given.     

2-Local automorphisms of operator algebras

A not necessarily continuous, linear or multiplicative function θ from an algebra A into itself is called a 2-local automorphism if θ agrees with an automorphism of A at each pair of points in A. In this paper, we study when a 2-local automorphism of a C^∗-algebra, or a standard operator algebra on a locally convex space, is an automorphism.     

New 5-designs with automorphism group PSL(2, 23)

Blocks of the unique Steiner system S(5, 8, 24) are called octads. The group PSL(2, 23) acts as an automorphism group of this Steiner system, permuting octads transitively. Inspired by the discovery of a 5-(24, 10, 36) design by Gulliver and Harada, we enumerate all 4- and 5-designs whose set of blocks are union of PSL(2, 23)-orbits on 10-subsets containing an octad. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 147–155, 1999     

Multiplicative Invariants and Semigroup Algebras

Let G be a finite group acting by automorphism on a lattice A, and hence on the group algebra S=k[A]. The algebra of G-invariants in S is called an algebra of multiplicative invariants. We present an explicit version of a result of Farkas stating that multiplicative invariants of finite reflection groups are semigroup algebras.     

The nonexistence of a certain Steiner system S(3, 12, 112)

Although the automorphism group of a projective plane of order 10, if one exists, must be very small, such a plane could be the derived design of a Steiner system S(3, 12, 112) with a larger group. There are several reasons why the Frobenius group of order 56 is a promising candidate for the latter group. However, in this paper it is shown that there is no S(3, 12, 112) which is fixed by this Frobenius group.     

Steiner intervals and Steiner geodetic numbers in distance-hereditary graphs

A Steiner tree for a set S of vertices in a connected graph G is a connected subgraph of G with a smallest number of edges that contains S. The Steiner interval I(S) of S is the union of all the vertices of G that belong to some Steiner tree for S. If S={u,v}, then I(S)=I[u,v] is called the interval between u and v and consists of all vertices that lie on some shortest u-v path in G. The smallest cardinality of a set S of vertices such that ?_u,v∈SI[u,v]=V(G) is called the geodetic number and is denoted by g(G). The smallest cardinality of a set S of vertices of G such that I(S)=V(G) is called the Steiner geodetic number of G and is denoted by sg(G). We show that for distance-hereditary graphs g(G)?sg(G) but that g(G)/sg(G) can be arbitrarily large if G is not distance hereditary. An efficient algorithm for finding the Steiner interval for a set of vertices in a distance-hereditary graph is described and it is shown how contour vertices can be used in developing an efficient algorithm for finding the Steiner geodetic number of a distance-hereditary graph.     

Unbounded derivations

In this note we discuss various extensions of a normal 1 derivation of a uniformly hyperfinite C¹-algebra. Various approximation theorems are employed to show when said extensions generate automorphism groups of the C¹-algebra. We characterize the “maximal” extension of Sakai and Powers as a graph limit and show when this extension is the closure of the given derivation. We also discuss an identity obeyed by the resolvent of a derivation.