A perfect secret sharing scheme allows a secret K to be shared among a set of participants in such a way that only qualified subsets of participants can recover the secret, and unqualified subsets of participants obtain no information regarding the secret. In this paper, we propose a construction of perfect secret sharing schemes with uniform, generalized access structures of rank 3. Compared w...

Two different constructions are given of a rank 8 arc-transitive graphwith 165 vertices and valency 8,whose automorphismgroup isM11. One involves 3-subsets of an 11-set while the other involves 4-subsets of a 12-set, and the constructions are linkedwith theWitt designs on 11, 12 and 24 points. Four different constructions are given of a rank 9 arc-transitive graph with 55 vertices and valency 6...

The aim of the paper is to give an effective formula for the calculation of the probability that a random subset of an affine geometry AG(r−1, q) has rank r. Tables for the probabilities are given for small ranks. The expected time to the first moment at which a random subset of an affine geometry achieves the rank r is derived.

The relative ranks of the monoid of endomorphisms of a strong independence algebra of infinite rank modulo two distinguished subsets are calculated. These subsets are the group of automorphisms and the endomorphisms α satisfying α2 = α. The extra generators are characterised in each case. 2000 Mathematics Subject Classification. 20M20, 08A35.

Determinantal point processes (DPPs) are an elegant model for encoding probabilities over subsets, such as shopping baskets, of a ground set, such as an item catalog. They are useful for a number of machine learning tasks, including product recommendation. DPPs are parametrized by a positive semi-definite kernel matrix. Recent work has shown that using a low-rank factorization of this kernel pr...

In this paper we investigate stable vector bundles on the complex projective plane P by means of their minimal free resolution. The fundamental background for this study is the work of Bohnhorst and Spindler [BS92] and their idea to use admissible pairs to characterize the stability of rank-n vector bundles on P and to give a stratification of the relative moduli space in constructible subsets....

It is an open problem if an elementary p-group of rank k ≥ 3 does admit full-rank normalized factorization into two of its subsets such that one of the factors has p elements. The paper provides an answer in the p ≤ 7 special case.

The knowledge of the metric entropy of precompact subsets of operators on finite dimensional Euclidean space is important in particular in the probabilistic methods developped by E. D. Gluskin and S. Szarek for constructing certain random Banach spaces. We give a new argument for estimating the metric entropy of some subsets such as the Grassmann manifold equipped with natural metrics. Here, th...

We give several examples of designs and antidesigns in classical finite polar spaces. These types of subsets of maximal totally isotropic subspaces generalize the dualization of the concepts of m-ovoids and tight sets of points in generalized quadrangles. We also consider regularity of partial spreads and spreads. The techniques that we apply were developed by Delsarte. In some polar spaces of ...