Geometric permutations of balls with bounded size disparity

Geometri Permutations of Balls with Bounded Size Disparity

Sharp Bounds on Geometric Permutations of Pairwise Disjoint Balls in IRd

(i) We prove that the maximum number of geometric permutations, induced by line transversals to a collection of n pairwise disjoint balls in IR d , is (n d?1). This improves substantially the general upper bound of O(n 2d?2) given in 11]. (ii) We show that the maximum number of geometric permutations of a suuciently large collection of pairwise disjoint unit discs in the plane is 2, improving a...

Some Discrete Properties of the Space of Line Transversals to Disjoint Balls

Attempts to generalize Helly’s theorem to sets of lines intersecting convex sets led to a series of results relating the geometry of a family of sets in Rd to the structure of the space of lines intersecting all of its members. We review recent progress in the special case of disjoint Euclidean balls in Rd, more precisely the inter-related notions of cone of directions, geometric permutations a...

A Stream Cipher Based on Chaotic Permutations

In this paper we introduce a word-based stream cipher consisting of a chaotic part operating as a chaotic permutation and a linear part, both of which designed on a finite field. We will show that this system can operate in both synchronized and self-synchronized modes. More specifically, we show that in the self-synchronized mode the stream cipher has a receiver operating as an unknown input o...

Set Systems and Families of Permutations with Small Traces

We study the maximum size of a set system on n elements whose trace on any b elements has size at most k. This question extends to hypergraphs the classical Dirac-type problems from extremal graph theory. We show that if for some b ≥ i ≥ 0 the shatter function fR of a set system ([n], R) satisfies fR(b) < 2 (b − i + 1) then |R| = O(n); this generalizes Sauer’s Lemma on the size of set systems w...