Ω-results for Beurling's zeta function and lower bounds for the generalised Dirichlet divisor problem

In this paper we study generalised prime systems for which the integer counting function NP(x) is asymptotically well behaved, in the sense that NP(x)=ρx+O(xβ), where ρ is a positive constant and . For such systems, the associated zeta function ζP(s) is holomorphic for . We prove that for , for any ε>0, and also for ε=0 for all such σ except possibly one value. The Dirichlet divisor problem for generalised integers concerns the size of the error term in NkP(x)−Ress=1(ζPk(s)xs/s), which is O(xθ) for some θ<1. Letting αk denote the infimum of such θ, we show that .

Generalised prime systems with periodic integer counting function

We study generalised prime systems (both discrete and continuous) for which the `integer counting function' N(x) has the property that N(x) ¡ cx is periodic for some c > 0. We show that this is extremely rare. In particular, we show that the only such system for which N is continuous is the trivial system with N(x) ¡ cx constant, while if N has finitely many discontinuities per bounded interval, then N must be the counting function of the g-prime system containing the usual primes except for finitely many. Keywords and phrases: Generalised prime systems. I

Reduced height alleles (Rht) and Hagberg falling number of wheat

Near isogenic lines varying for alleles for reduced height (Rht) and photoperiod insensitivity (Ppd-D1) in cv. Mercia (2005/6 to 2010/11; rht (tall), Rht-B1b, Rht-D1b, Rht-B1c, Rht8c+Ppd-D1a, Rht-D1c, Rht12) and cvs Maris Huntsman and Maris Widgeon (2007/8 to 2010/11; rht (tall), Rht-B1b, Rht-D1b, Rht-B1c, Rht-B1b+Rht-D1b, Rht-D1b+Rht-B1c) were compared at one field site, but within different systems (‘organic’, O, 2005/6 to 2007/8 v ‘intensive’, I, 2005/6 to 2010/11). Further experiments at the site (2006/7 to 2008/9) compared 64 lines of a doubled haploid (DH) population [Savannah (Rht-D1b) × Renesansa (Rht-8c+Ppd-D1a)]. Gibberellin (GA) insensitive dwarfing alleles (Rht-B1b; Rht-B1c; Rht-D1b; Rht-D1c) could reduce α-amylase activity and/or increase Hagberg falling number (HFN) but effects depended greatly on system, background and season. Only Rht-B1c increased grain dormancy despite producing plants taller than Rht-D1c. The GA-sensitive Rht8c+Ppd-D1a in Mercia was associated with reduced HFN but analysis of the DH population suggested this was more closely linked with Ppd-D1a, rather than Rht8c. The severe GA-sensitive dwarfing allele Rht12 was associated with reduced HFN. Instability in HFN over season tended to increase with degree of dwarfing. There was a negative association between mean grain weight and HFN that was in addition to effects of Rht and Ppd-D1 allele.

On a stochastic Trotter formula with application to spontaneous localization models

We consider the relation between so called continuous localization models—i.e. non-linear stochastic Schrödinger evolutions—and the discrete GRW-model of wave function collapse. The former can be understood as scaling limit of the GRW process. The proof relies on a stochastic Trotter formula, which is of interest in its own right. Our Trotter formula also allows to complement results on existence theory of stochastic Schrödinger evolutions by Holevo and Mora/Rebolledo.

A note on an integration by parts formula for the generators of uniform translations on configuration space

An integration by parts formula is derived for the first order differential operator corresponding to the action of translations on the space of locally finite simple configurations of infinitely many points on Rd. As reference measures, tempered grand canonical Gibbs measures are considered corresponding to a non-constant non-smooth intensity (one-body potential) and translation invariant potentials fulfilling the usual conditions. It is proven that such Gibbs measures fulfill the intuitive integration by parts formula if and only if the action of the translation is not broken for this particular measure. The latter is automatically fulfilled in the high temperature and low intensity regime.

Linking number analysis of a self-assembled lemniscular Möbius-metallamacrocycle

A 26 membered, 24π-electron metallamacrocycle containing two Ag(I) centers is shown to be a double-twist Möbius cycle by a linking number analysis.