The full infinite dimensional moment problem on semi-algebraic sets of generalized functions

We consider a generic basic semi-algebraic subset S of the space of generalized functions, that is a set given by (not necessarily countably many) polynomial constraints. We derive necessary and sufficient conditions for an infinite sequence of generalized functions to be realizable on S, namely to be the moment sequence of a finite measure concentrated on S. Our approach combines the classical results about the moment problem on nuclear spaces with the techniques recently developed to treat the moment problem on basic semi-algebraic sets of Rd. In this way, we determine realizability conditions that can be more easily verified than the well-known Haviland type conditions. Our result completely characterizes the support of the realizing measure in terms of its moments. As concrete examples of semi-algebraic sets of generalized functions, we consider the set of all Radon measures and the set of all the measures having bounded Radon–Nikodym density w.r.t. the Lebesgue measure.

A priori error analysis of space–time Trefftz discontinuous Galerkin methods for wave problems

We present and analyse a space–time discontinuous Galerkin method for wave propagation problems. The special feature of the scheme is that it is a Trefftz method, namely that trial and test functions are solution of the partial differential equation to be discretised in each element of the (space–time) mesh. The method considered is a modification of the discontinuous Galerkin schemes of Kretzschmar et al. (2014) and of Monk & Richter (2005). For Maxwell’s equations in one space dimension, we prove stability of the method, quasi-optimality, best approximation estimates for polynomial Trefftz spaces and (fully explicit) error bounds with high order in the meshwidth and in the polynomial degree. The analysis framework also applies to scalar wave problems and Maxwell’s equations in higher space dimensions. Some numerical experiments demonstrate the theoretical results proved and the faster convergence compared to the non-Trefftz version of the scheme.

Advanced source apportionment of size-resolved trace elements at multiple sites in London during winter

Trace element measurements in PM10–2.5, PM2.5–1.0 and PM1.0–0.3 aerosol were performed with 2 h time resolution at kerbside, urban background and rural sites during the ClearfLo winter 2012 campaign in London. The environment-dependent variability of emissions was characterized using the Multilinear Engine implementation of the positive matrix factorization model, conducted on data sets comprising all three sites but segregated by size. Combining the sites enabled separation of sources with high temporal covariance but significant spatial variability. Separation of sizes improved source resolution by preventing sources occurring in only a single size fraction from having too small a contribution for the model to resolve. Anchor profiles were retrieved internally by analysing data subsets, and these profiles were used in the analyses of the complete data sets of all sites for enhanced source apportionment. A total of nine different factors were resolved (notable elements in brackets): in PM10–2.5, brake wear (Cu, Zr, Sb, Ba), other traffic-related (Fe), resuspended dust (Si, Ca), sea/road salt (Cl), aged sea salt (Na, Mg) and industrial (Cr, Ni); in PM2.5–1.0, brake wear, other traffic-related, resuspended dust, sea/road salt, aged sea salt and S-rich (S); and in PM1.0–0.3, traffic-related (Fe, Cu, Zr, Sb, Ba), resuspended dust, sea/road salt, aged sea salt, reacted Cl (Cl), S-rich and solid fuel (K, Pb). Human activities enhance the kerb-to-rural concentration gradients of coarse aged sea salt, typically considered to have a natural source, by 1.7–2.2. These site-dependent concentration differences reflect the effect of local resuspension processes in London. The anthropogenically influenced factors traffic (brake wear and other traffic-related processes), dust and sea/road salt provide further kerb-to-rural concentration enhancements by direct source emissions by a factor of 3.5–12.7. The traffic and dust factors are mainly emitted in PM10–2.5 and show strong diurnal variations with concentrations up to 4 times higher during rush hour than during night-time. Regionally influenced S-rich and solid fuel factors, occurring primarily in PM1.0–0.3, have negligible resuspension influences, and concentrations are similar throughout the day and across the regions.

Tests of sunspot number sequences: 1. Using ionosonde data

More than 70 years ago it was recognised that ionospheric F2-layer critical frequencies [foF2] had a strong relationship to sunspot number. Using historic datasets from the Slough and Washington ionosondes, we evaluate the best statistical fits of foF2 to sunspot numbers (at each Universal Time [UT] separately) in order to search for drifts and abrupt changes in the fit residuals over Solar Cycles 17-21. This test is carried out for the original composite of the Wolf/Zürich/International sunspot number [R], the new “backbone” group sunspot number [RBB] and the proposed “corrected sunspot number” [RC]. Polynomial fits are made both with and without allowance for the white-light facular area, which has been reported as being associated with cycle-to-cycle changes in the sunspot number - foF2 relationship. Over the interval studied here, R, RBB, and RC largely differ in their allowance for the “Waldmeier discontinuity” around 1945 (the correction factor for which for R, RBB and RC is, respectively, zero, effectively over 20 %, and explicitly 11.6 %). It is shown that for Solar Cycles 18-21, all three sunspot data sequences perform well, but that the fit residuals are lowest and most uniform for RBB. We here use foF2 for those UTs for which R, RBB, and RC all give correlations exceeding 0.99 for intervals both before and after the Waldmeier discontinuity. The error introduced by the Waldmeier discontinuity causes R to underestimate the fitted values based on the foF2 data for 1932-1945 but RBB overestimates them by almost the same factor, implying that the correction for the Waldmeier discontinuity inherent in RBB is too large by a factor of two. Fit residuals are smallest and most uniform for RC and the ionospheric data support the optimum discontinuity multiplicative correction factor derived from the independent Royal Greenwich Observatory (RGO) sunspot group data for the same interval.