Computer-free Proof Research Articles

On Monomial Flocks

We study monomial flocks of quadratic cones of PG(3, q), with emphasis on the case where the flock is semifield, providing some nonexistence and some uniqueness results. In addition, we give a computer-free proof of the existence of the sporadic semifield flock of the quadratic cone of PG(3, 35) (and hence of the sporadic translation ovoid of Q(4, 35)), and relate that flock to the sporadic simple group M11.

Feigenbaum-Coullet-Tresser Universality and Milnor's Hairiness Conjecture

We prove the Feigenbaum-Coullet-Tresser conjecture on the hyperbolicity of the renormalization transformation of bounded type. This gives the first computer-free proof of the original Feigenbaum observation of the universal parameter scaling laws. We use the Hyperbolicity Theorem to prove Milnor's conjectures on self-similarity and hairiness of the Mandelbrot set near the corresponding parameter values. We also conclude that the set of real infinitely renormalizable quadratics of type bounded by some N > 1 has Hausdorff dimension strictly between 0 and 1. In the course of getting these results we supply the space of quadratic-like germs with a complex analytic structure and demonstrate that the hybrid classes form a complex codimension-one foliation of the connectedness locus.

On Weiss' geometric characterization of the Rudvalis simple group

We give a computer-free proof for Weiss' theorem of the uniqueness of the Rudvalis simple group acting flag transitively on an extended generalized octagon with empty triangles.

On K�hnel's 9-vertex complex projective plane

We present an elementary combinatorial proof of the existence and uniqueness of the 9-vertex triangulation of C P2. The original proof of existence, due to Kuhnel, as well as the original proof of uniqueness, due to Kuhnel and Lassmann, were based on extensive computer search. Recently Arnoux and Marin have used cohomology theory to present a computer-free proof. Our proof has the advantage of displaying a canonical copy of the affine plane over the three-element field inside this complex in terms of which the entire complex has a very neat and short description. This explicates the full automorphism group of the Kuhnel complex as a subgroup of the automorphism group of this affine plane. Our method also brings out the rich combinatorial structure inside this complex.

A Computer-free proof of a theorem of Weiss and Yoshiara

A Computer-free proof of a theorem of Weiss and Yoshiara

There Exists no Distance-regular Graph with Intersection Array (5,4,3; 1,1,2)

The problem of deciding whether a distance-regular graph with a given set of parameters exists in a very hard one, and far from solution. The set of parameters mentioned in the title is the smallest unsolved case [1]. We will give here a short, computer-free proof that there is no graph with these parameters.

Ovoids ofPG(3, 16) are elliptic quadrics, II

In a previous paper [8] the authors have shown that every ovoid ofPG(3, 16) is an elliptic quadric. The arguments used a computer and also depended on the computer-aided classification of hyperovals ofPG(2, 16) (see [3]). Recently (see [9]) the classification of hyperovals ofPG(2,16) has been obtained without the use of a computer. The present paper completes a computer-free proof that every ovoid ofPG(3,16) is an elliptic quadric.