Solar wind-magnetosphere coupling functions on timescales of 1 day to 1 year

There are no direct observational methods for determining the total rate at which energy is extracted from the solar wind by the magnetosphere. In the absence of such a direct measurement, alternative means of estimating the energy available to drive the magnetospheric system have been developed using different ionospheric and magnetospheric indices as proxies for energy consumption and dissipation and thus the input. The so-called coupling functions are constructed from the parameters of the interplanetary medium, as either theoretical or empirical estimates of energy transfer, and the effectiveness of these coupling functions has been evaluated in terms of their correlation with the chosen index. A number of coupling functions have been studied in the past with various criteria governing event selection and timescale. The present paper contains an exhaustive survey of the correlation between geomagnetic activity and the near-Earth solar wind and two of the planetary indices at a wide variety of timescales. Various combinations of interplanetary parameters are evaluated with careful allowance for the effects of data gaps in the interplanetary data. We show that the theoretical coupling, P�, function first proposed by Vasyliunas et al. is superior at all timescales from 1-day to 1-year.

Reduced relative entropy techniques for a posteriori analysis of multiphase problems in elastodynamics

We give an a posteriori analysis of a semidiscrete discontinuous Galerkin scheme approximating solutions to a model of multiphase elastodynamics, which involves an energy density depending not only on the strain but also the strain gradient. A key component in the analysis is the reduced relative entropy stability framework developed in Giesselmann (2014, SIAM J. Math. Anal., 46, 3518–3539). This framework allows energy-type arguments to be applied to continuous functions. Since we advocate the use of discontinuous Galerkin methods we make use of two families of reconstructions, one set of discrete reconstructions and a set of elliptic reconstructions to apply the reduced relative entropy framework in this setting.

The signature of a rough path: uniqueness

In the context of controlled differential equations, the signature is the exponential function on paths. B. Hambly and T. Lyons proved that the signature of a bounded variation path is trivial if and only if the path is tree-like. We extend Hambly–Lyons' result and their notion of tree-like paths to the setting of weakly geometric rough paths in a Banach space. At the heart of our approach is a new definition for reduced path and a lemma identifying the reduced path group with the space of signatures.