Comment on 'Molecular arrangement in water: random but not quite'

Accurate high energy x-ray diffraction data are presented on liquid water measured at room temperature. Sources of both systematic and statistical errors within the experiment are considered and data consistency checks are discussed. It is found that the resulting x-ray pair distribution function is smoothly varying in real space and shows no evidence of small peaks in the 3-5???region. Our results are in contrast to the recent findings reported in Petkov et al 2012 J. Phys.: Condens. Matter 24 155102.     

The Cauchy problem for wave maps on a curved background

We consider the Cauchy problem for wave maps ${u: {\mathbb R}\times M \to N,}$ for Riemannian manifolds (M, g) and (N, h). We prove global existence and uniqueness for initial data, u[0]?=?(u ₀, u ₁), that is small in the critical norm ${\dot{H}^{\frac{d}{2}}\times \dot{H}^{\frac{d}{2}-1}(M; TN),}$ in the case (M, g) = ${({\mathbb R}^4, g),}$ where g is a small perturbation of the Euclidean metric. The proof follows the method introduced by Statah and Struwe in (Commun Pure Appl Math 47(5):719–754, 1994) for proving global existence and uniqueness of small data wave maps ${u : {\mathbb R}\times {\mathbb R}^d \to N}$ in the critical norm, for d?≥ 4. In our argument we employ the Strichartz estimates for variable coefficient wave equations established by Metcalfe and Tataru in (To appear in Math Ann).     

Asymptotic Stability of Harmonic Maps on the Hyperbolic Plane under the Schrödinger Maps Evolution

We consider the Cauchy problem for the Schrödinger maps evolution when the domain is the hyperbolic plane. An interesting feature of this problem compared to the more widely studied case on the Euclidean plane is the existence of a rich new family of finite energy harmonic maps. These are stationary solutions, and thus play an important role in the dynamics of Schrödinger maps. The main result of this article is the asymptotic stability of (some of) such harmonic maps under the Schrödinger maps evolution. More precisely, we prove the nonlinear asymptotic stability of a finite energy equivariant harmonic map under the Schrödinger maps evolution with respect to non-equivariant perturbations, provided obeys a suitable linearized stability condition. This condition is known to hold for all equivariant harmonic maps with values in the hyperbolic plane and for a subset of those maps taking values in the sphere. One of the main technical ingredients in the paper is a global-in-time local smoothing and Strichartz estimate for the operator obtained by linearization around a harmonic map, proved in the companion paper [36]. © 2021 Wiley Periodicals LLC.