25,340 research outputs found
On the logarithmic probability that a random integral ideal is -free
This extends a theorem of Davenport and Erd\"os on sequences of rational
integers to sequences of integral ideals in arbitrary number fields . More
precisely, we introduce a logarithmic density for sets of integral ideals in
and provide a formula for the logarithmic density of the set of so-called
-free ideals, i.e. integral ideals that are not multiples of any
ideal from a fixed set .Comment: 9 pages, to appear in S. Ferenczi, J. Ku{\l}aga-Przymus and M.
Lema\'nczyk (eds.), Chowla's conjecture: from the Liouville function to the
M\"obius function, Lecture Notes in Math., Springe
Optimal Interleaving on Tori
We study t-interleaving on two-dimensional tori, which is defined by the property that any connected subgraph with t or fewer vertices in the torus is labelled by all distinct integers. It has applications in distributed data storage and burst error correction, and is closely related to Lee metric codes. We say that a torus can be perfectly t-interleaved if its t-interleaving number – the minimum number of distinct integers needed to t-interleave the torus – meets the spherepacking lower bound. We prove the necessary and sufficient conditions for tori that can be perfectly t-interleaved, and present efficient perfect t-interleaving constructions. The most important contribution of this paper is to prove that the t-interleaving numbers of tori large enough in both dimensions, which constitute by far the majority of all existing cases, is at most one more than
the sphere-packing lower bound, and to present an optimal and efficient t-interleaving scheme for them. Then we prove some bounds on the t-interleaving numbers for other cases, completing a general picture for the t-interleaving problem on 2-dimensional tori
A sufficient condition for a number to be the order of a nonsingular derivation of a Lie algebra
A study of the set N_p of positive integers which occur as orders of
nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of
characteristic p>0 was initiated by Shalev and continued by the present author.
The main goal of this paper is to show the abundance of elements of N_p. Our
main result shows that any divisor n of q-1, where q is a power of p, such that
, belongs to N_p. This extends its special
case for p=2 which was proved in a previous paper by a different method.Comment: 10 pages. This version has been revised according to a referee's
suggestions. The additions include a discussion of the (lower) density of the
set N_p, and the results of more extensive machine computations. Note that
the title has also changed. To appear in Israel J. Mat
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