International Mathematics Research Notices - recent issues

Shelling-Type Orderings of Regular CW-Complexes and Acyclic Matchings of the Salvetti Complex

Delucchi, E. — 2008-02-08

Motivated by the work of Salvetti and Settepanella [24, Combinatorial Morse theory and minimality of hyperplane arrangements, Remark 4.5], we give a purely combinatorial description of a class of discrete Morse functions having a minimal number of critical cells for the Salvetti complex of any linear arrangement. We start by studying certain total orderings of the cells of shellable regular CW-complexes, and use them to construct maximum acyclic matchings of the given complex. We apply this technique to the classical zonotope shellings. A new combinatorial stratification of the Salvetti complex allows us to paste such matchings and describe a class of maximum acyclic matchings of the whole complex. The construction can be done, so that the critical cells can be constructed from the chambers via the nbc sets. The results hold for abstract oriented matroids.

Asymptotic Behavior of Infinity Harmonic Functions Near an Isolated Singularity

Savin, O., Wang, C., Yu, Y. — 2008-02-06

In this paper, we prove that if n≥ 2 and x₀ is an isolated singularity of a non-negative infinity harmonic function u, then either x₀ is a removable singularity of u or u(x)=u(x₀)+c|x–x₀|+o(|x–x₀|) near x₀ for some fixed constant c!=0. In particular, if x₀ is nonremovable, then u has a local maximum or a local minimum at x₀. We also prove a Bernstein-type theorem, which asserts that if u is a uniformly Lipschitz continuous, one-side bounded infinity harmonic function in then it must be a cone function with center at 0.

Bounded gaps between products of primes with applications to ideal class groups and elliptic curves

Thorne, F. — 2008-02-06

In their recent papers, Goldston, Graham, Pintz, and Yildirim [12, 13] use a variant of the Selberg sieve to prove the existence of small gaps between E₂ numbers, that is, square-free numbers with exactly two prime factors. We apply their techniques to prove similar bounds for E_r numbers for any r >= 2, where these numbers are required to have all of their prime factors in a set of primes . Our result holds for any of positive density that satisfies a Siegel–Walfisz condition regarding distribution in arithmetic progressions. We also prove a stronger result in the case that satisfies a Bombieri–Vinogradov condition. We were motivated to prove these generalizations because of recent results of Ono [22] and Soundararajan [25]. These generalizations yield applications to divisibility of class numbers, nonvanishing of critical values of L-functions, and triviality of ranks of elliptic curves.