On order conditions in affine planes

New Flag-Transitive Affine Planes of Even Order

At least [6 n&1 1 (q i+1)] qn 2m1 log2 q of these planes have kernel GF(q). Moreover, we construct 7(mi)6i q mi planes, where the sum is over all sequences (mi) behaving as in the theorem; and then we settle the isomorphism problem for these planes (Theorem 5.2). The proof combines methods in [Wi] and [Ka]: these planes arise from symplectic and orthogonal spreads combined with changing from fi...

Affine and projective planes

Convexity in Topological Affine Planes

We extend to topological affine planes the standard theorems of convexity, among them the separation theorem, the anti-exchange theorem, Radon’s, Helly’s, Carathéodory’s, and Kirchberger’s theorems, and the Minkowski theorem on extreme points. In a few cases the proofs are obtained by adapting proofs of the original results in the Euclidean plane; in others it is necessary to devise new proofs ...

A Note on Tangentially Transitive Affine Planes

Let $1 be a finite affine translation plane. If 21 contains a subplane, collineation group pair (2I0, A) such that: * A leaves each point of 2t0 fixed and acts transitively on the points of / \ { / n 2I0}, for / a line of 21 with / n 2l0 a line of 2l0, then 21 is said to be tangentially transitive with respect to 2l0. Jha in his thesis [3] has proved that if 21 is tangentially transitive with r...

Mubs Inequivalence and Affine Planes

There are fairly large families of unitarily inequivalent complete sets of N+1 mutually unbiased bases (MUBs) in C for various prime powers N . The number of such sets is not bounded above by any polynomial as a function of N . While it is standard that there is a superficial similarity between complete sets of MUBs and finite affine planes, there is an intimate relationship between these large...