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.

Rapid preconditioning of data for accelerating convex hull algorithms

Given a dataset of two-dimensional points in the plane with integer coordinates, the method proposed reduces a set of n points down to a set of s points s ≤ n, such that the convex hull on the set of s points is the same as the convex hull of the original set of n points. The method is O(n). It helps any convex hull algorithm run faster. The empirical analysis of a practical case shows a percentage reduction in points of over 98%, that is reflected as a faster computation with a speedup factor of at least 4.

A method for merging nadir-sounding climate records, with an application to the global-mean stratospheric temperature data sets from SSU and AMSU

A method is proposed for merging different nadir-sounding climate data records using measurements from high-resolution limb sounders to provide a transfer function between the different nadir measurements. The two nadir-sounding records need not be overlapping so long as the limb-sounding record bridges between them. The method is applied to global-mean stratospheric temperatures from the NOAA Climate Data Records based on the Stratospheric Sounding Unit (SSU) and the Advanced Microwave Sounding Unit-A (AMSU), extending the SSU record forward in time to yield a continuous data set from 1979 to present, and providing a simple framework for extending the SSU record into the future using AMSU. SSU and AMSU are bridged using temperature measurements from the Michelson Interferometer for Passive Atmospheric Sounding (MIPAS), which is of high enough vertical resolution to accurately represent the weighting functions of both SSU and AMSU. For this application, a purely statistical approach is not viable since the different nadir channels are not sufficiently linearly independent, statistically speaking. The near-global-mean linear temperature trends for extended SSU for 1980–2012 are −0.63 ± 0.13, −0.71 ± 0.15 and −0.80 ± 0.17 K decade−1 (95 % confidence) for channels 1, 2 and 3, respectively. The extended SSU temperature changes are in good agreement with those from the Microwave Limb Sounder (MLS) on the Aura satellite, with both exhibiting a cooling trend of ~ 0.6 ± 0.3 K decade−1 in the upper stratosphere from 2004 to 2012. The extended SSU record is found to be in agreement with high-top coupled atmosphere–ocean models over the 1980–2012 period, including the continued cooling over the first decade of the 21st century.

Emulation of chiral hydrogenation catalysts by adsorbtion of small molecules onto Ni surfaces

Adsorption of small molecules on the Ni{111} and NiO{111} surfaces is investigated under UHV and elevated pressures (~10-1 mbar) of hydrogen and water. The molecules considered are chosen for their relevance to understanding the mechanism of enantioselective hydrogenation on Raney Nickel modified by chiral molecules. Adsorption of water onto, and its subsequent reaction with, oxygen-covered Ni{111} is dependent on the initial atomic oxygen coverage. An OH species (O1s binding energy 531.5eV), oriented perpendicular to the surface, forms at atomic oxygen coverages <0.25ML. The reaction does not consume all the adsorbed oxygen for coverages ≥0.12ML. The p(2×2) atomic oxygen uperstructure is unreactive, while an OH species is formed on the p(√3×√3) superstructure at binding energy 530.9eV. L-alanine is adsorbed on Ni{111} as a model chiral modifier molecule. At low coverages, alanine forms a presumed tridentate alaninate species for coverages ≥0.11ML at 250K. A minority, bidentate zwitterionic species forms at coverages >0.11ML, but was not observed at 300K. Saturation occurs at 0.25ML. At high alanine coverages (≥0.19ML) and H2 pressure (≥1×10-2 mbar), the tridentate L-alaninate converts to bidentate zwitterionic L-alanine at 300K. Thermal evolution of L-alanine on Ni{111} under varying hydrogen pressures is examined. Adsorption of L-alanine onto hydroxylated NiO{111} at 300K in UHV, mimicking a catalyst surface under aqueous conditions, yields the tridentate alaninate which is immune to the effects of elevated hydrogen pressure. Exposing the L-alanine/Ni{111} adsorption system to water (≤10-1 mbar) oxidises the surface and recreates the L-alanine/hydroxylated NiO{111} system. Pyruvic acid on Ni{111} is examined as a model for hydrogenation substrate adsorption. Behaviour is coverage dependent and several conformations are possible at low coverages (≤0.1ML). Annealing at coverages <0.2ML causes a condensation reaction, releasing water onto the surface. High coverages do not condense and a saturation coverage of ~0.35ML is found.

A multifractal zeta function for Gibbs measures supported on cookie-cutter sets

Starting with the work of Lapidus and van Frankenhuysen a number of papers have introduced zeta functions as a way of capturing multifractal information. In this paper we propose a new multifractal zeta function and show that under certain conditions the abscissa of convergence yields the Hausdorff multifractal spectrum for a class of measures.

On the cardinality and complexity of the set of codings for self-similar sets with positive Lebesgue measure

Let λ1,…,λn be real numbers in (0,1) and p1,…,pn be points in Rd. Consider the collection of maps fj:Rd→Rd given by fj(x)=λjx+(1−λj)pj. It is a well known result that there exists a unique nonempty compact set Λ⊂Rd satisfying Λ=∪nj=1fj(Λ). Each x∈Λ has at least one coding, that is a sequence (ϵi)∞i=1 ∈{1,…,n}N that satisfies limN→∞fϵ1…fϵN(0)=x. We study the size and complexity of the set of codings of a generic x∈Λ when Λ has positive Lebesgue measure. In particular, we show that under certain natural conditions almost every x∈Λ has a continuum of codings. We also show that almost every x∈Λ has a universal coding. Our work makes no assumptions on the existence of holes in Λ and improves upon existing results when it is assumed Λ contains no holes.

On univoque points for self-similar sets

Let K⊆R be the unique attractor of an iterated function system. We consider the case where K is an interval and study those elements of K with a unique coding. We prove under mild conditions that the set of points with a unique coding can be identified with a subshift of finite type. As a consequence, we can show that the set of points with a unique coding is a graph-directed self-similar set in the sense of Mauldin and Williams (1988). The theory of Mauldin and Williams then provides a method by which we can explicitly calculate the Hausdorff dimension of this set. Our algorithm can be applied generically, and our result generalises the work of Daróczy, Kátai, Kallós, Komornik and de Vries.

An introductory perspective to horticulture: plants for people and places

Horticulture is “the first of all the arts and sciences”. This definition indicates both the breadth and depth of the discipline and its early inception as mankind changed from being hunter-gatherers to cultivators. Intensive crop production which is a form of horticulture preceded more extensive agricultural practices. From that time onwards the intricate involvement of horticulture in man’s life has become very apparent by its multitude of applications and the interests of those involved. These extend from the provision of foodstuffs and nutritional benefits through pharmaceuticals to aspects of rest and relaxation onto encouraging physical and mental well-being. Horticulture is therefore, a discipline with many components and as such that it can mean different things in the varying context of its use. This chapter introduces the meanings of horticulture as expressed by the authors who have contributed to this Trilogy of Books. They have analysed in considerable depth “Horticulture” as expressed in its facets of production, environment and society. Horticulture has impact and expression in each of these fields of human activity. This chapter also sets Horticulture into the wider context of the world of plants and their intensive cultivation both in their use by mankind and in the natural world. The aim is to demonstrate the depth and breadth of human activity associated with this discipline for it stretches from crop production, through landscape design and maintenance and into aspects of society and its expression in the arts and humanities. Horticulture touches almost every aspect of human activity. Increasingly Horticulture has significant importance in contributing towards the mitigation of the major problems which now face life on Earth such as:- climate change, food security, the loss of natural biodiversity, pollution, resource erosion and over-population. Indeed despite or perhaps because of its antiquity and therefore its strong connection between science, technology and practice horticulture can offer solutions that might allude other disciplines.

Arnold v. Britton: a ratio and two sets of obiter dicta

Arnold v Britton marks the final stage of the longstanding dispute as to the correct interpretation of a number of 99-year leases of chalets on a leisure park at Oxwich, in the Gower peninsula, near Swansea. The aspect of the case which has attracted most discussion has, understandably, been its main ratio: the proper way to construe a provision of a lease which arguably has an absurd result. This will be considered in this case-note. The judgment of the Supreme Court – particularly the judgment of Lord Neuberger PSC – does, however contain some observations on the possible reform of the law on service charges which are of interest to those engaged in this field. It also contains some obiter comments on ‘letting schemes’ which are – in the view of the present author – highly unorthodox. These three rather disparate issues which are raised by this case will be considered in turn. As they have little in common with each other, they will be considered as separate sections.

Preconditioning 2D integer data for fast convex hull computations

In order to accelerate computing the convex hull on a set of n points, a heuristic procedure is often applied to reduce the number of points to a set of s points, s ≤ n, which also contains the same hull. We present an algorithm to precondition 2D data with integer coordinates bounded by a box of size p × q before building a 2D convex hull, with three distinct advantages. First, we prove that under the condition min(p, q) ≤ n the algorithm executes in time within O(n); second, no explicit sorting of data is required; and third, the reduced set of s points forms a simple polygonal chain and thus can be directly pipelined into an O(n) time convex hull algorithm. This paper empirically evaluates and quantifies the speed up gained by preconditioning a set of points by a method based on the proposed algorithm before using common convex hull algorithms to build the final hull. A speedup factor of at least four is consistently found from experiments on various datasets when the condition min(p, q) ≤ n holds; the smaller the ratio min(p, q)/n is in the dataset, the greater the speedup factor achieved.

Observational evidence for a negative shortwave cloud feedback in middle to high latitudes

Exploiting the observed robust relationships between temperature and optical depth in extratropical clouds, we calculate the shortwave cloud feedback from historical data, by regressing observed and modeled cloud property histograms onto local temperature in middle to high southern latitudes. In this region, all CMIP5 models and observational data sets predict a negative cloud feedback, mainly driven by optical thickening. Between 45° and 60°S, the mean observed shortwave feedback (−0.91 ± 0.82 W m−2 K−1, relative to local rather than global mean warming) is very close to the multimodel mean feedback in RCP8.5 (−0.98 W m−2 K−1), despite differences in the meridional structure. In models, historical temperature-cloud property relationships reliably predict the forced RCP8.5 response. Because simple theory predicts this optical thickening with warming, and cloud amount changes are relatively small, we conclude that the shortwave cloud feedback is very likely negative in the real world at middle to high latitudes.

Moral uncertainty and permissibility: evaluating option sets

In this essay, we explore an issue of moral uncertainty: what we are permitted to do when we are unsure about which moral principles are correct. We develop a novel approach to this issue that incorporates important insights from previous work on moral uncertainty, while avoiding some of the difficulties that beset existing alternative approaches. Our approach is based on evaluating and choosing between option sets rather than particular conduct options. We show how our approach is particularly well-suited to address this issue of moral uncertainty with respect to agents that have credence in moral theories that are not fully consequentialist.

Inter-comparison of stratospheric mean-meridional circulation and eddy mixing among six reanalysis data sets

The stratospheric mean-meridional circulation (MMC) and eddy mixing are compared among six meteorological reanalysis data sets: NCEP-NCAR, NCEP-CFSR, ERA-40, ERA-Interim, JRA-25, and JRA-55 for the period 1979–2012. The reanalysis data sets produced using advanced systems (i.e., NCEP-CFSR, ERA-Interim, and JRA-55) generally reveal a weaker MMC in the Northern Hemisphere (NH) compared with those produced using older systems (i.e., NCEP/NCAR, ERA-40, and JRA-25). The mean mixing strength differs largely among the data products. In the NH lower stratosphere, the contribution of planetary-scale mixing is larger in the new data sets than in the old data sets, whereas that of small-scale mixing is weaker in the new data sets. Conventional data assimilation techniques introduce analysis increments without maintaining physical balance, which may have caused an overly strong MMC and spurious small-scale eddies in the old data sets. At the NH mid-latitudes, only ERA-Interim reveals a weakening MMC trend in the deep branch of the Brewer–Dobson circulation (BDC). The relative importance of the eddy mixing compared with the mean-meridional transport in the subtropical lower stratosphere shows increasing trends in ERA-Interim and JRA-55; this together with the weakened MMC in the deep branch may imply an increasing age-of-air (AoA) in the NH middle stratosphere in ERA-Interim. Overall, discrepancies between the different variables and trends therein as derived from the different reanalyses are still relatively large, suggesting that more investments in these products are needed in order to obtain a consolidated picture of observed changes in the BDC and the mechanisms that drive them.