Jan-Christoph Schlage-Puchta

Covering and radius-covering arrays: Constructions and classification

The minimum number of rows in covering arrays (equivalently, surjective codes) and radius-covering arrays (equivalently, surjective codes with a radius) has been determined precisely only in special cases. In this paper, explicit constructions for numerous best known covering arrays (upper bounds) are found by a combination of combinatorial and computational methods. For radius-covering arrays, explicit constructions from covering codes are developed. Lower bounds are improved upon using connections to orthogonal arrays, partition matrices, and covering codes, and in specific cases by computation. Consequently for some parameter sets the minimum size of a covering array is determined precisely. For some of these, a complete classification of all inequivalent covering arrays is determined, again using computational techniques. Existence tables for up to 10 columns, up to 8 symbols, and all possible strengths are presented to report the best current lower and upper bounds, and classifications of inequivalent arrays.

A -group with positive rank gradient

We construct, for d >= 2 and epsilon > 0, a d-generated p-group Gamma which, in an asymptotic sense, behaves almost like a d-generated free pro-p-group. We show that a subgroup of index p(n) needs (d - epsilon)p(n) generators, and that the subgroup growth of Gamma satisfies s(p)(n)(Gamma) > s(p)(n)(F-d(p))(1-epsilon), where F-d(p) is the d-generated free pro-p-group. To do this we introduce a new invariant for finitely-generated groups and study some of its basic properties.

Streamlined subrecursive degree theory

Abstract This paper is divided into two parts. In Part I, we investigate the structure of honest elementary degrees, that is, the degree structure induced on the honest functions by the reducibility relation “being (Kalmar) elementary in”. In Part II, we generalise the degree theory found in Part I. We introduce the reducibility relation “being α -elementary in”, where α is an ordinal ≤ ϵ 0 , and investigate the structure of honest α -elementary degrees. Towards the end of the paper, we discuss relations between our degree theory and provability in Peano Arithmetic.

The structure of maximal zero-sum free sequences

Let n be an integer, and consider finite sequences of elements of the group Z/nZ x Z/nZ. Such a sequence is called zero-sum free, if no subsequence has sum zero. It is known that the maximal length of such a zero-sum free sequence is 2n-2, and Gao and Geroldinger conjectured that every zero-sum free sequence of this length contains an element with multiplicity at least n-2. By recent results of Gao, Geroldinger and Grynkiewicz, it essentially suffices to verify the conjecture for n prime. Now fix a sequence (a_i) of length 2n-2 with maximal multiplicity of elements at most n-3. There are different approeaches to show that (a_i) contains a zero-sum; some work well when (a_i) does contain elements with high multiplicity, others work well when all multiplicities are small. The aim of this article is to initiate a systematic approach to property B via the highest occurring multiplicities. Our main results are the following: denote by m_1 >= m_2 the two maximal multiplicities of (a_i), and suppose that n is sufficiently big and prime. Then (a_i) contains a zero-sum in any of the following cases: when m_2 >= 2/3n, when m_1 > (1-c)n, and when m_2 0 not depending on anything.

Modular arithmetic of free subgroups

Abstract Denote by ƒλ (G ) the number of free subgroups of index λmG , where mG is the least common multiple of the orders of the finite subgroups in G. The present paper develops a general theory for the p-divisibility of ƒλ (G ), where p is a prime dividing mG . Among other things, we obtain an explicit combinatorial description of ƒλ (G ) modulo p, leading to an optimal generalisation of Stothers’ explicit formula for the parity of ƒλ (PSL2 (ℤ)).

The equation ω(n)=ω(n+1)

We prove that there are infinitely many integers

Applications of character estimates to statistical problems for the symmetric group

n

Mean representation number of integers as the sum of primes

such that

Inductive Methods and zero-sum free sequences

n

The irrationality of some number theoretical series

and