Orthomorphisms and the construction of projective planes

We discuss a simple computational method for the construction of finite projective planes. The planes so constructed all possess a special group of automorphisms which we call the group of translations, but they are not always translation planes. Of the four planes of order 9, three admit the additive group of the field as a group of translations, and the present construction yields all three. The known planes of order 16 comprise four self-dual planes and eighteen other planes (nine dual pairs); of these, the method gives three of the four self-dual planes and six of the nine dual pairs, including the ``sporadic' (not translation) plane of Mathon.

     

Blocking Sets in André Planes

We construct new examples of blocking sets in some non-Desarguesian translation planes.     

A Geometric Relationship Between Equivalent Spreads

By Andrè theory, it is well known how to algebraically convert a spread in a projective space to an equivalent spread (representing the same translation plane) in a projective space of different dimension and of different order (corresponding to a subfield of the kernel). The goal of this paper is to establish a geometric connection between any two such equivalent spreads by embedding them as subspaces and subgeometries of an ambient projective spaces. The connection can be viewed as a generalization of a construction due to Hirschfeld and Thas.     

Infinite Nests of Reguli

In this article, infinite versions of t-nests for t=q,(q-1), (q+1),2(q-1) and mixed nests of reguli are constructed. Furthermore, a classification of all group replaceable spreads is given.     

Applications of Number Theory to Ovoids and Translation Planes

In this paper we show that for each prime p7 there exists a translation plane of order p ² of Mason–Ostrom type. These planes occur as six-dimensional ovoids being projections of the eight-dimensional binary ovoids of Conway, Kleidman and Wilson. In order to verify the existence of such projections we prove certain properties of two particular quadratic forms using classical methods form number theory.     

Four-Dimensional Compact Projective Planes with a Solvable Collineation Group

We consider a four-dimensional compact projective plane whose collineation group is six-dimensional and solvable with a nilradical N isomorphic to Nil×R, where Nil denotes the three-dimensional, simply connected, non-Abelian, nilpotent Lie group. We assume that fixes a flag p W, acts transitively on and fixes no point in the set W\p. Under these conditions, we will prove that either contains a three-dimensional group of elations or acts doubly transitively on .     

Ovoids of Parabolic Spaces

A new ovoid in the orthogonal space O(5,3⁵) is presented, along with its associated spreads and (semifield) translation planes. Sundry results on ovoids and spreads are given. In particular, we complete the calculation of the stabilisers of the known O(5,q) ovoids.     

Characterisations of Flock Quadrangles

We characterise the Hermitian and Kantor flock generalized quadrangles of order (q ²,q), q even, (associated with the linear and Fisher–Thas–Walker flocks of a quadratic cone, and the Desarguesian and Betten–Walker translation planes) in terms of a self-dual subquadrangle. Equivalently, we show that a herd which contains a translation oval must be associated with the linear or Fisher–Thas–Walker flock. This result is a consequence of the determination of all functions which satisfy a certain absolute trace equation whose form is remarkably similar to that of an equation arising in recent studies of ovoids in three-dimensional projective space of finite order q.     

Characterization of Translation Planes by Orbit Lengths

The Desarguesian, Hall, and Hering translation planes of order q² are characterized as exactly those translation planes of odd order with spreads in PG (3,q) that admit a linear collineation group with infinite orbits one of length q+1 and i of length (q-q) /i for i=1 or 2.     

Local Schur’s Lemma and Commutative Semifields

A set of linear maps , V a finite vector space over a field K, is regular if to each there corresponds a unique element such that R(x)=y. In this context, Schur’s lemma implies that is a field if (and only if) it consists of pairwise commuting elements. We consider when is locally commutative: at some μ ∈V*, AB(μ)=BA(μ) for all , and has been normalized to contain the identity. We show that such locally commutative are equivalent to commutative semifields, generalizing a result of Ganley, and hence characterizing commutative semifield spreads within the class of translation planes. This enables the determination of the orders |V| for which all locally commutative on V are (globally) commutative. Similarly, we determine a sharp upperbound for the maximum size of the Schur kernel associated with strictly locally commutative . We apply our main result to demonstrate the existence of a partial spread of degree 5, with nominated shears axis, that cannot be extend to a commutative semifield spread. Finally, we note that although local commutativity for a regular linear set implies that the set of Lie products consists entirely of singular maps, the converse is false.     

Lower bounds for coverings of pairs by large blocks

Letnkt be positive integers, andX—a set ofn elements. LetC(n, k, t) be the smallest integerm such that there existm k-tuples ofX B ₁ B ₂,...,B _m with the property that everyt-tuple ofX is contained in at least oneB _i . It is shown that in many cases the standard lower bound forC(n, k, 2) can be improved (k sufficiently large,n/k being fixed). Some exact values ofC(n, k, 2) are also obtained.     

Enumeration of generalized Hadamard matrices of order 16 and related designs

We investigate signings of symmetric GDD( , 16, )s over for . Beginning with , at each stage of this process a signing of a GDD( , 16, ) produces a GDD( , 16, ). The initial GDDs ( ) correspond to Hadamard matrices of order 16. For , the GDDs are semibiplanes of order 16, and for the GDDs are semiplanes of order 16 which can be extended to projective planes of order 16. In this article, we completely enumerate such signings which include all generalized Hadamard matrices of order 16. We discuss the generation techniques and properties of the designs obtained during the search. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 119–135, 2009     

A Characterization of Quaternion Planes, Revisited

It is shown that the group PSL₂(H) cannot act effectively on any eight-dimensional stable plane. Together with previous results, this entails that every eight-dimensional stable plane admitting a nontrivial action of SL₂(H) embeds into the projective plane over Hamilton's quaternions H.     

Inherited conics in Hall planes

The existence of ovals and hyperovals is an old question in the theory of non-Desarguesian planes. The aim of this paper is to describe when a conic of $\mathrm{PG}(2,q)$ remains an arc in the Hall plane obtained by derivation. Some combinatorial properties of the inherited conics are obtained also in those cases when it is not an arc. The key ingredient of the proof is an old lemma by Segre–Korchmáros on Desargues configurations with perspective triangles inscribed in a conic.     

On the strength of Gomory mixed-integer cuts as group cuts

Gomory mixed-integer (GMI) cuts generated from optimal simplex tableaus are known to be useful in solving mixed-integer programs. Further, it is well-known that GMI cuts can be derived from facets of Gomory’s master cyclic group polyhedron and its mixed-integer extension studied by Gomory and Johnson. In this paper we examine why cutting planes derived from other facets of master cyclic group polyhedra (group cuts) do not seem to be as useful when used in conjunction with GMI cuts. For many practical problem instances, we numerically show that once GMI cuts from different rows of the optimal simplex tableau are added to the formulation, all other group cuts from the same tableau rows are satisfied.     

d‐homogeneous and d‐ultrahomogeneous linear spaces

A linear space S is d‐homogeneous if, whenever the linear structures induced on two subsets S₁ and S₂ of cardinality at most d are isomorphic, there is at least one automorphism of S mapping S₁ onto S₂. S is called d‐ultrahomogeneous if each isomorphism between the linear structures induced on two subsets of cardinality at most d can be extended into an automorphism of S. We have proved in [11;] (without any finiteness assumption) that every 6‐homogeneous linear space is homogeneous (that is d‐homogeneous for every positive integer d). Here we classify completely the finite nontrivial linear spaces that are d‐homogeneous for d ≥ 4 or d‐ultrahomogeneous for d ≥ 3. We also prove an existence theorem for infinite nontrivial 4‐ultrahomogeneous linear spaces. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 321–329, 2000     

Ternary Operations as Primitive Notions for Constructive Plane Geometry V

In this paper we provide a quantifier-free, constructive axiomatization of metric-Euclidean and of rectangular planes (generalizations of Euclidean planes). The languages in which the axiom systems are expressed contain three individual constants and two ternary operations. We also provide an axiom system in algorithmic logic for finite Euclidean planes, and for several minimal metric-Euclidean planes. The axiom systems proposed will be used in a sequel to this paper to provide ‘the simplest possible’ axiom systems for several fragments of plane Euclidean geometry. Mathematics Subject Classification: 51M05, 51M15, 03F65.     

On the extreme inequalities of infinite group problems

Infinite group relaxations of integer programs (IP) were introduced by Gomory and Johnson (Math Program 3:23–85, 1972) to generate cutting planes for general IPs. These valid inequalities correspond to real-valued functions defined over an appropriate infinite group. Among all the valid inequalities of the infinite group relaxation, extreme inequalities are most important since they are the strongest cutting planes that can be obtained within the group-theoretic framework. However, very few properties of extreme inequalities of infinite group relaxations are known. In particular, it is not known if all extreme inequalities are continuous and what their relations are to extreme inequalities of finite group problems. In this paper, we describe new properties of extreme functions of infinite group problems. In particular, we study the behavior of the pointwise limit of a converging sequence of extreme functions as well as the relations between extreme functions of finite and infinite group problems. Using these results, we prove for the first time that a large class of discontinuous functions is extreme for infinite group problems. This class of extreme functions is the generalization of the functions given by Letchford and Lodi (Oper Res Lett 30(2):74–82, 2002), Dash and Günlük (Proceedings 10th conference on integer programming and combinatorial optimization. Springer, Heidelberg, pp 33–45 (2004), Math Program 106:29–53, 2006) and Richard et al. (Math Program 2008, to appear). We also present several other new classes of discontinuous extreme functions. Surprisingly, we prove that the functions defining extreme inequalities for infinite group relaxations of mixed integer programs are continuous. S.S. Dey and J.-P.P. Richard was supported by NSF Grant DMI-03-48611.