The correlations of finite Desarguesian planes, Part II: The classification (I)

This paper continues the classification of the correlations of planes of odd nonsquare order. Part I (Generalities) – see reference [1]-included introductory definitions and results (Section 1), algebraic preliminaries (Section 2), as well as a discussion of equivalent correlations (Section 3) and of their general properties (Section 4). The classification proper revolves around a special polynomial which can have one, two, or q + 1 zeros, or no zeros at all, and each of these four possibilities leads to different families of correlations. The present article contains Section 5, devoted to the cases in which the correlation is defined by a diagonal matrix (Subsection 5.1) or the polynomial in the preceding paragraph possesses q + 1 zeros (Subsection 5.2), one zero (Subsection 5.3) and two zeros (Subsection 5.4). Subsection 5.5 presents certain results to be used in the subsequent sections.     

The correlations of finite Desarguesian planes, Part III: The classification (II)

This paper continues the classification of the correlations of planes of odd nonsquare order. Part I (Generalities) included introductory definitions and results (Section 1), algebraic preliminaries (Section 2), as well as a discussion of equivalent correlations (Section 3) and of their general properties (Section 4). The classification proper revolves around a special polynomial which can have one, two, or q + 1 zeros, or no zeros at all, and each of these four possibilities leads to different families of correlations. Part II contained Section 5, devoted to the cases in which the correlation is defined by a diagonal matrix (Subsection 5.1) or the polynomial in the preceding paragraph possesses q + 1 zeros (Subsection 5.2), one zero (Subsection 5.3) and two zeros (Subsection 5.4). Subsection 5.5 presented certain results to be used in the subsequent sections. The present article contains Section 6, devoted to the case in which the above-mentioned polynomial has no zeros.     

The correlations of finite Desarguesian planes, Part I: Generalities

The polarities of Desarguesian planes have long been known. This author has undertaken to classify the correlations of finite Desarguesian planes in general. In [6] we have presented all the correlations with identity companion automorphism which are not polarities, of these planes. In this sequence of papers, we classify the correlations of planes of order $ p^{2^{i}(2n+1)}, n \neq 0 $, with companion automorphism ( $p^{2^{i}t}$ ), p an odd prime, $ t \neq 0 $. This represents a complete classification of the correlations of planes of odd nonsquare order (i = 0). Some of the correlations of planes of odd square order ($ t \neq 0 $ ) are also covered by the present analysis.When the companion automorphism is not trivial, the problem, naturally, becomes more involved, and a great deal begins to hinge upon the order of the plane being odd or even, and also a square or a nonsquare.The correlations of planes of order $ 2^{2^{i}(2n+1)}, n \neq 0 $, with companion automorphism $ 2^{2^{i}t}, t \neq 0 $, and especially those of planes of order $ p^{2^{i}(2n+1)}, i \neq 0 $, with companion automorphism $ p^{2^{j}(2r+1)}, j > i $ require a substantially different treatment, and will be the object of separate efforts.     

A proof of the theorem of Pappus in finite Desarguesian affine planes

Without using the representation theorem and a theorem of J.H.M. WEDDERBURN we show that every finite Desarguesian affine plane is Pappian.     

Constructions of unitals in Desarguesian planes

We present a new construction of non-classical unitals from a classical unital U in . The resulting non-classical unitals are B-M unitals. The idea is to find a non-standard model Π of with the following three properties:

(i)

points of Π are those of ;

(ii)

lines of Π are certain lines and conics of ;

(iii)

the points in U form a non-classical B-M unital in Π.

Our construction also works for the B-T unital, provided that conics are replaced by certain algebraic curves of higher degree.     

Conic blocking sets in Desarguesian projective planes

Define a conic blocking set to be a set of lines in a Desarguesian projective plane such that all conics meet these lines. Conic blocking sets can be used in determining if a collection of planes in projective three-space forms a flock of a quadratic cone. We discuss trivial conic blocking sets and conic blocking sets in planes of small order. We provide a construction for conic blocking sets in planes of non-prime order, and we make additional comments about the structure of these conic blocking sets in certain planes of even order.     

Computable dimensions of Pappusian and Desarguesian projective planes

Computable presentations of projective planes are studied. Based on an interpretation of a class of fields (associative skew fields) within a class of Pappusian (Desarguesian) projective planes, it is proved that the question whether there exists a computable presentation for a Pappusian (Desarguesian) projective plane reduces to asking if there exists a computable presentation for a corresponding field (associative skew field). It is stated that the computable dimension of a Pappusian (Desarguesian) projective plane coincides with that of a corresponding field (associative skew field).     

On the algebras of symmetries (groups of collineations) of designs from finite Desarguesian planes with applications in statistics

Decomposition into a direct sum of irreducible representations of the representation of the full collineation group of a finite Desarguesian plane, as a group of matrices permuting the flags of the plane and the simple components of the corresponding commutant algebra, have been worked out here for the projective plane PG(2, 2) and the affine plane EG(2, 3). The dimension and the components of the covariance matrix of the observations from a design derived from such a plane, which commutes with such a permutation representation of the full collineation group of the plane, are thus determined. This paper is in the spirit of earlier works by, James (1957), Mann (1960), 6., 7., McLaren (1963), and Sysoev and Shaikan (1976). A. T. James, Ann. Math. Statist.28 (1957), 993–1002, H. B. Mann, Ann. Math. Statist.31 (1960), 1–15, E. J. Hannan, Research Report (Part. (I)), Summer Research Institute, Australian Math. Soc. and Methuen's Monographs on Applied Probability and Statistics, Supplementary Review Series in Applied Probability, Vol. 3, A. D. McLaren, Proc. Cambridge Philos. Soc.59 (1963), 431–450, and L. P. Sysoev and M. E. Shaikin, Avtomat. i Telemekh.5 (1976), 64–73.     

Multiple blocking sets and multisets in Desarguesian planes

In AG(2, q ²), the minimum size of a minimal (q ? 1)-fold blocking set is known to be q ³ ? 1. Here, we construct minimal (q ? 1)-fold blocking sets of size q ³ in AG(2, q ²). As a byproduct, we also obtain new two-character multisets in PG(2, q ²). The essential idea in this paper is to investigate q ³-sets satisfying the opposite of Ebert’s discriminant condition.     

On resolvable finite Minkowski planes

Among the known finite Minkowski planes we determine an infinite family of examples admitting a partition of the set of blocks into equivalence classes, each of which in turn partitions the point set; in particular non-miquelian finite Minkowski planes with this property exist.work done within the activity of GNSAGA of CNR and supported by the Italian Ministry of Public Education