A Fast Algorithm for Computing the p-Curvature

We design an algorithm for computing the $p$-curvature of a differential system in positive characteristic $p$. For a system of dimension $r$ with coefficients of degree at most $d$, its complexity is \softO (p d r^\omega) operations in the ground field (where $\omega$ denotes the exponent of matrix multiplication), whereas the size of the output is about $p d r^2$. Our algorithm is then quasi-optimal assuming that matrix multiplication is (\emph{i.e.} $\omega = 2$). The main theoretical input we are using is the existence of a well-suited ring of series with divided powers for which an analogue of the Cauchy--Lipschitz Theorem holds.Comment: ISSAC 2015, Jul 2015, Bath, United Kingdo

Supergeometry and Arithmetic Geometry

We define a superspace over a ring $R$ as a functor on a subcategory of the category of supercommutative $R$-algebras. As an application the notion of a $p$-adic superspace is introduced and used to give a transparent construction of the Frobenius map on $p$-adic cohomology of a smooth projective variety over the ring of $p$-adic integers.Comment: 14 pages, expanded introduction, more detail

Sound propagation over uneven ground and irregular topography

The goal of this research is to develop theoretical, computational, and experimental techniques for predicting the effects of irregular topography on long range sound propagation in the atmosphere. Irregular topography here is understood to imply a ground surface that is not idealizable as being perfectly flat or that is not idealizable as having a constant specific acoustic impedance. The interest of this study focuses on circumstances where the propagation is similar to what might be expected for noise from low-attitude air vehicles flying over suburban or rural terrain, such that rays from the source arrive at angles close to grazing incidence. The activities and developments that have resulted during the period, August 1986 through February 1987, are discussed

Sound propagation over uneven ground and irregular topography

The acoustic impedance of the surface coverings used in the laboratory experiments on sound diffraction by topographical ridges was determined. The model, which was developed, takes into account full wave effects and the possibility of surface waves and predicts the sound pressure level at the receiver location relative to what would be expected if the flat surface were not present. The sound pressure level can be regarded as a function of frequency, sound speed in air, heights of source and receiver, and horizontal distance from source to receiver, as well as the real and imaginary parts of the surface impedance

Sound propagation over uneven ground and irregular topography

Theoretical, computational, and experimental techniques were developed for predicting the effects of irregular topography on long range sound propagation in the atmosphere. Irregular topography is understood to imply a ground surface that: (1) is not idealizable as being perfectly flat, or (2) that is not idealizable as having a constant specific acoustic impedance. The focus is on circumstances where the propagation is similar to what might be expected for noise from low altitude air vehicles flying over suburban or rural terrain, such that rays from the source arrive at angles close to grazing incidence

Ice Formation on Kaolinite: Insights from Molecular Dynamics Simulations

The formation of ice affects many aspects of our everyday life as well as technologies such as cryotherapy and cryopreservation. Foreign substances almost always aid water freezing through heterogeneous ice nucleation, but the molecular details of this process remain largely unknown. In fact, insight into the microscopic mechanism of ice formation on different substrates is difficult to obtain even via state-of-the-art experimental techniques. At the same time, atomistic simulations of heterogeneous ice nucleation frequently face extraordinary challenges due to the complexity of the water-substrate interaction and the long timescales that characterize nucleation events. Here, we have investigated several aspects of molecular dynamics simulations of heterogeneous ice nucleation considering as a prototypical ice nucleating material the clay mineral kaolinite, which is of relevance in atmospheric science. We show via seeded molecular dynamics simulations that ice nucleation on the hydroxylated (001) face of kaolinite proceeds exclusively via the formation of the hexagonal ice polytype. The critical nucleus size is two times smaller than that obtained for homogeneous nucleation at the same supercooling. Previous findings suggested that the flexibility of the kaolinite surface can alter the time scale for ice nucleation within molecular dynamics simulations. However, we here demonstrate that equally flexible (or non flexible) kaolinite surfaces can lead to very different outcomes in terms of ice formation, according to whether or not the surface relaxation of the clay is taken into account. We show that very small structural changes upon relaxation dramatically alter the ability of kaolinite to provide a template for the formation of a hexagonal overlayer of water molecules at the water-kaolinite interface, and that this relaxation therefore determines the nucleation ability of this mineral