For prime power q, we determine the minimum number of parallel classes in a resolvable 2-(kq, L 1) covering for almost all values of k. @ 1999 Elsevier Science B.V. All rights reserved In September 1995 Jan van Haastrecht gave a dinner on the occasion of his retirement. He had invited 20 colleagues. The host and the guests were to be seated at three tables each of which had seven places. The di...

G −→ Aut(G/N) ≈ Sq. Let K denote the kernel of the map. Clearly K ⊂ N since each g ∈ K must in particular left translate N back to itself. Thus, since q is the smallest prime divisor of |G|, q = |G/N | | |G/K| = q · (product of primes p ≥ q). On the other hand, the first isomorphism theorem says that G/K is isomorphic to the image of G in Sq, a subgroup of Sq. Thus |G/K| | q!, so that |G/K| = q...

We extend the recently introduced framework of metric distortion to multiwinner voting. In this framework, n agents and m alternatives are located in an underlying space. The exact distances between unknown. Instead, each agent provides a ranking alternatives, ordered from closest farthest. Typically, goal is select single alternative that approximately minimizes total distance agents, worst-ca...

In this paper, we study the p-ary linear code Ck(n, q), q = p , p prime, h ≥ 1, generated by the incidence matrix of points and k-dimensional spaces in PG(n, q). We note that the condition k ≥ n/2 arises in results on the codewords of Ck(n, q). For k ≥ n/2, we link codewords of Ck(n, q)\ Ck(n, q) ⊥ of weight smaller than 2q to k-blocking sets. We first prove that such a k-blocking set is unique...

We study a new problem for cubic graphs: bipartization of a cubic graph Q by deleting sufficiently large independent set I. It can be expressed as follows: Given a connected n-vertex tripartite cubic graph Q = (V,E) with independence number α(Q), does Q contain an independent set I of size k such that Q− I is bipartite? We are interested for which value of k the answer to this question is affir...

For a permutation f of an n-dimensional vector space V over a finite field of order q we let k-affinity(f) denote the number of k-flats X of V such that f(X) is also a k-flat. By k-spectrum(n, q) we mean the set of integers k-affinity(f) where f runs through all permutations of V . The problem of the complete determination of k-spectrum(n, q) seems very difficult except for small or special val...

For a quadric Q over a real field k, we investigate whether finiteness of the Pythagoras number of the function field k(Q) implies the existence of a uniform bound on the Pythagoras numbers of all finite extensions of k. We give a positive answer if the quadratic form that defines Q is weakly isotropic. In the case where Q is a conic, we show that the Pythagoras number of k(Q) is 2 only if k is...

Let b ≥ 2 be a fixed integer. Let sb(n) denote the sum of digits of the nonnegative integer n in the base-b representation. Further let q be a positive integer. In this paper we study the length k of arithmetic progressions n, n+ q, . . . , n+ q(k − 1) such that sb(n), sb(n+ q), . . . , sb(n+ q(k − 1)) are (pairwise) distinct. More specifically, let Lb,q denote the supremum of k as n varies in ...