Additive twists of Fourier coefficients of symmetric-square lifts

We study the sum of additively twisted Fourier coefficients of a symmetric-square lift of a Maass form invariant under the full modular group. Our bounds are uniform in terms of the spectral parameter of the Maass form, as well as in terms of the additive twist.Comment: 13 pages. v2: fixed the relation between T and t_j on p.2 and added clarification to some reference

Moments of the critical values of families of elliptic curves, with applications

We make conjectures on the moments of the central values of the family of all elliptic curves and on the moments of the first derivative of the central values of a large family of positive rank curves. In both cases the order of magnitude is the same as that of the moments of the central values of an orthogonal family of L-functions. Notably, we predict that the critical values of all rank 1 elliptic curves is logarithmically larger than the rank 1 curves in the positive rank family. Furthermore, as arithmetical applications we make a conjecture on the distribution of a_p's amongst all rank 2 elliptic curves, and also show how the Riemann hypothesis can be deduced from sufficient knowledge of the first moment of the positive rank family (based on an idea of Iwaniec).Comment: 24 page

The Wecken property for random selfmaps on surfaces with boundary

A selfmap is Wecken when the minimal number of fixed points among all maps in its homotopy class is equal to the Nielsen number, a homotopy invariant lower bound on the number of fixed points. All selfmaps are Wecken for manifolds of dimension not equal to 2, but some non-Wecken maps exist on surfaces. We attempt to measure how common the Wecken property is on surfaces with boundary by estimating the proportion of maps which are Wecken, measured by asymptotic density. Intuitively, this is the probability that a randomly chosen homotopy class of maps consists of Wecken maps. We show that this density is nonzero for surfaces with boundary. When the fundamental group of our space is free of rank n, we give nonzero lower bounds for the density of Wecken maps in terms of n, and compute the (nonzero) limit of these bounds as n goes to infinity

The Wecken property for random maps on surfaces with boundary

A selfmap is Wecken when the minimal number of fixed points among all maps in its homotopy class is equal to the Nielsen number, a homotopy invariant lower bound on the number of fixed points. All selfmaps are Wecken for manifolds of dimension not equal to 2, but some non-Wecken maps exist on surfaces. We attempt to measure how common the Wecken property is on surfaces with boundary by estimating the proportion of maps which are Wecken, measured by asymptotic density. Intuitively, this is the probability that a randomly chosen homotopy class of maps consists of Wecken maps. We show that this density is nonzero for surfaces with boundary. When the fundamental group of our space is free of rank n, we give nonzero lower bounds for the density of Wecken maps in terms of n, and compute the (nonzero) limit of these bounds as n goes to infinity.Comment: 20 pages, 1 stunning figur

Improved timber harvest techniques maintain biodiversity in tropical forests

Tropical forests are selectively logged at 20 times the rate at which they are cleared, and at least a fifth have already been disturbed in this way. In a recent pan-tropical assessment, Burivalova et al. demonstrate the importance of logging intensity as a driver of biodiversity decline in timber estates. Their analyses reveal that species richness of some taxa could decline by 50% at harvest intensities of 38 m3 ha-1. However, they did not consider the extraction techniques that lead to these intensities. Here, we conduct a complementary meta-analysis of assemblage responses to differing logging practices: conventional logging and reduced-impact logging. We show that biodiversity impacts are markedly less severe in forests that utilise reduced-impact logging, compared to those using conventional methods. While supporting the initial findings of Burivalova et al., we go on to demonstrate that best practice forestry techniques curtail the effects of timber extraction regardless of intensity. Therefore, harvest intensities are not always indicative of actual disturbance levels resulting from logging. Accordingly, forest managers and conservationists should advocate practices that offer reduced collateral damage through best practice extraction methods, such as those used in reduced-impact logging. Large-scale implementation of this approach would lead to improved conservation values in the 4 million km2 of tropical forests that are earmarked for timber extraction