let $g$ be a finite $p$-soluble group‎, ‎and $p$ a sylow $p$-subgroup of $g$‎. ‎it is proved‎ ‎that if all elements of $p$ of order $p$ (or of order ${}leq 4$ for $p=2$) are‎ ‎contained in the $k$-th term of the upper central series of $p$‎, ‎then the $p$-length of‎ ‎$g$ is at most $2m+1$‎, ‎where $m$ is the greatest integer such that‎ ‎$p^m-p^{m-1}leq k$‎, ‎and the exponent of the image of $p$...

Top-k query is widely used in the search engine and gains great success, which perform the cooperative query by aggregating the database objects’ degree of match for each different query predicate and returning the best k matching objects only. It’s also applied to p2p file-sharing systems. However, top-k query processing in p2p systems is very challenging because the potentially large number o...

We show that regular homogeneous two-weight Z pk -codes where p is odd and k > 2 with dual Hamming distance at least four do not exist. The proof relies on existence conditions for the strongly regular graph built on the cosets of the dual code.

In this paper, we survey recent work on ternary Diophantine equations of the shape Axn + Byn = Czm for m ∈ {2, 3, n} where n ≥ 5 is prime. Our goal is to provide a simple procedure which, given A, B, C and m, enables us to decide whether techniques based on the theory of Galois representations and modular forms suffice to ensure that corresponding ternary equations lack nontrivial solutions in ...

Recently Harris (1999), using probabilistic arguments alone, has given new proofs of the (known) existence, asymptotics and uniqueness of travelling wave solutions to the K-P-P equation. This paper is a sequel to Kyprianou (2000b) which provides alternative probabilistic arguments for supercritical wave speeds. We complete our probabilis-tic analysis here for the more diicult case of critical w...