A characterization of Hering's plane of order 27

Hering's translation plane of order 27 has been characterized by its order and the fact that it admits SL(2, 13) in its translation complement (see [1]). We show that, aside from the Desarguian plane and a Generalized, André plane, it is the only plane of order 27 which admits a subgroup of SL(2, 13) of order 13×12.Partially supported by FONDECYT 0343 and ANDES FOUNDATION.     

A translation plane of order 25

In this paper a new translation plane of order 25 is constructed. It has a collineation group acting on the line at infinity as a permutation group Z of order 48 with the properties:

1. Z contains a normal subgroup 1/2M of order 3 such that Z/1/2M is the direct product of an involution with a dihedral group of order 8.
2. The orbits of Z have lengths 2, 12, 12.

     

A translation plane of order 72 with a very small translation complement

A translation plane of order 7² is constructed whose left nucleus is a cyclic group of order 6. Generally the translation complement of a translation plane is quite large when compared to the order of the left nucleus. The plane discussed in this paper has the distinguished feature that its translation complement modulo the scalar collineations group is very small. In fact it is a Dihedral group of order 12 and is the smallest of all the known examples. The translation complement divides the set of ideal points into 9 orbits of lengths 1,1, 6, 6, 6, 6, 6, 6, and 12.     

A translation plane of order 112

A translation plane of order 11² is constructed. Its translation complement is a solvable group of order 1200 and has 9 orbits on the line at infinity. These orbits have lengths 30, 20, 20, 12, 10, 10, 10, 5, 5, respectively.     

The translation planes of order 49

In this paper, we describe an exhaustive computer search for all the translation planes of order 49. The search uses the well-known correspondence between translation planes of order 49 and spreads of lines inPG(3,7). We conclude that there are exactly 1347 translation planes of order 49, and give some of their properties including group sizes and 7-ranks.     

A translation plane of order 192 admitting SL(2,5), obtained by 12-nest replacement

The problem of determining two-dimensional translation planes admitting SL(2,5) in the translation complement has been intensively studied. Most examples arise from multiple-derivation of a Desarguesian plane. In this paper, we construct two translation planes of order 19² admitting SL(2,5), one of which is obtained by 12-nest replacement.      

First order difference system- existence and uniqueness

In this paper, the general solution of the homogeneous matrix difference system is constructed in terms of two fundamental matrix solutions. The general solution of the inhomogeneous matrix difference system is established by the variation of parameters formula. A unique solution of the two-point boundary value problem associated with the matrix difference system is constructed by applying the QR-algorithm and the Bartels-Stewart algorithm.

     

Remarks on the uniqueness of second order ODEs

We are concerned with the uniqueness problem for solutions to the second order ODE of the form x″+f(x, t) = 0, subject to appropriate initial conditions, under the sole assumption that f is non-decreasing with respect to x, for each t fixed. We show that there is non-uniqueness in general; on the other hand, several types of reasonable additional assumptions make the problem uniquely solvable.     

The collineation groups of the translation planes of order 25

Czerwinski and Oakden have classified all the translation planes of order 25. In this paper the author gives the order and orbit lengths on the line at infinity of the collineation groups of these planes.     

A projective plane of order 16

A projective plane of order 16 is constructed. It is a translation plane and appears to be new. The representation of the collineation group on the axis of the plane has a normal subgroup isomorphic to L₃ (2) with factor group isomorphic to S₃. The orbits of this representation have lengths 14 and 3. If two points in the latter orbit are chosen to define a sharply doubly transitive set of permutations, the permutations from the multiplicative loop generate a group isomorphic to A₇. The plane is of Lenz-Barlotti class IVa.1.     

Translation planes of order 27

Translation planes of order 27 are classified. Various invariants play an important role in a computer search.     

The translation planes of order 49 and their automorphism groups

Using isomorphism invariants, we enumerate the translation planes of order 49 and determine their automorphism groups.