The paradox of soft singularity crossing and its resolution by distributional cosmological quantitities

A cosmological model of a flat Friedmann universe filled with a mixture of anti-Chaplygin gas and dust-like matter exhibits a future soft singularity, where the pressure of the anti-Chaplygin gas diverges (while its energy density is finite). Despite infinite tidal forces the geodesics pass through the singularity. Due to the dust component, the Hubble parameter has a non-zero value at the encounter with the singularity, therefore the dust implies further expansion. With continued expansion however, the energy density and the pressure of the anti-Chaplygin gas would become ill-defined, hence from the point of view of the anti-Chaplygin gas only a contraction is allowed. Paradoxically, the universe in this cosmological model would have to expand and contract simultaneously. This obviosly could not happen. We solve the paradox by redefining the anti-Chaplygin gas in a distributional sense. Then a contraction could follow the expansion phase at the singularity at the price of a jump in the Hubble parameter. Although such an abrupt change is not common in any cosmological evolution, we explicitly show that the set of Friedmann, Raychaudhuri and continuity equations are all obeyed both at the singularity and in its vicinity. We also prove that the Israel junction conditions are obeyed through the singular spatial hypersurface. In particular we enounce and prove a more general form of the Lanczos equation.Comment: 12 pages; to be published in Phys.Rev.

The PREDICTS database: a global database of how local terrestrial biodiversity responds to human impacts

Biodiversity continues to decline in the face of increasing anthropogenic pressures such as habitat destruction, exploitation, pollution and introduction of alien species. Existing global databases of species’ threat status or population time series are dominated by charismatic species. The collation of datasets with broad taxonomic and biogeographic extents, and that support computation of a range of biodiversity indicators, is necessary to enable better understanding of historical declines and to project – and avert – future declines. We describe and assess a new database of more than 1.6 million samples from 78 countries representing over 28,000 species, collated from existing spatial comparisons of local-scale biodiversity exposed to different intensities and types of anthropogenic pressures, from terrestrial sites around the world. The database contains measurements taken in 208 (of 814) ecoregions, 13 (of 14) biomes, 25 (of 35) biodiversity hotspots and 16 (of 17) megadiverse countries. The database contains more than 1% of the total number of all species described, and more than 1% of the described species within many taxonomic groups – including flowering plants, gymnosperms, birds, mammals, reptiles, amphibians, beetles, lepidopterans and hymenopterans. The dataset, which is still being added to, is therefore already considerably larger and more representative than those used by previous quantitative models of biodiversity trends and responses. The database is being assembled as part of the PREDICTS project (Projecting Responses of Ecological Diversity In Changing Terrestrial Systems – www.predicts.org.uk). We make site-level summary data available alongside this article. The full database will be publicly available in 2015

Vectorial Generalized g-Fractional Direct and Iterated Quantitative Approximation by Linear Operators

In this work we consider quantitatively with rates the convergence of sequences of linear operators applied on Banach space valued functions. The results are pointwise estimates with rates. To prove our main results we use an elegant and natural boundedness property of our linear operators by their companion positive linear operators. Our inequalities are generalized g-direct and iterated fractional involving the right and left vector Caputo type generalized g-direct and iterated fractional derivatives, built in vector moduli of continuity. We treat wide and general classes of Banach space valued functions. We give applications to vectorial Bernstein operators. See also[6]

A strong left fractional calculus for banach space valued functions

We develop here a strong left fractional calculus theory for Banach space valued functions of Caputo type. Then we establish many Bochner integral inequalities of various types. This chapter is based on Anastassiou (A strong fractional calculus theory for banach space valued functions, 2017 [5])

Vector abstract fractional korovkin approximation

In this chapter we study quantitatively with rates the convergence of sequences of general Bochner type integral operators, applied on Banach space valued functions, to function values. The results are mainly pointwise, but in the application to vector Bernstein polynomials we end up to obtain a uniform estimate. To prove our main results we have to build a rich background containing many interesting vector fractional results. Our inequalities are fractional involving the right and left vector Caputo type fractional derivatives, built in vector moduli of continuity. We treat very general classes of Banach space valued functions. It follows [7]

Estimating retention benchmarks for salvage logging to protect biodiversity

Forests are increasingly affected by natural disturbances. Subsequent salvage logging, a widespread management practice conducted predominantly to recover economic capital, produces further disturbance and impacts biodiversity worldwide. Hence, naturally disturbed forests are among the most threatened habitats in the world, with consequences for their associated biodiversity. However, there are no evidence-based benchmarks for the proportion of area of naturally disturbed forests to be excluded from salvage logging to conserve biodiversity. We apply a mixed rarefaction/extrapolation approach to a global multi-taxa dataset from disturbed forests, including birds, plants, insects and fungi, to close this gap. We find that 75 ± 7% (mean ± SD) of a naturally disturbed area of a forest needs to be left unlogged to maintain 90% richness of its unique species, whereas retaining 50% of a naturally disturbed forest unlogged maintains 73 ± 12% of its unique species richness. These values do not change with the time elapsed since disturbance but vary considerably among taxonomic groups