An optimization problem concerning multiplicative functions

In this paper we study the problem of maximizing a quadratic form 〈Ax,x〉 subject to ‖x‖q=1, where A has matrix entries View the MathML source with i,j|k and q≥1. We investigate when the optimum is achieved at a ‘multiplicative’ point; i.e. where x1xmn=xmxn. This turns out to depend on both f and q, with a marked difference appearing as q varies between 1 and 2. We prove some partial results and conjecture that for f multiplicative such that 0

Declining resilience of ecosystem functions under biodiversity loss

The composition of species communities is changing rapidly through drivers such as habitat loss and climate change, with potentially serious consequences for the resilience of ecosystem functions on which humans depend. To assess such changes in resilience, we analyse trends in the frequency of species in Great Britain that provide key ecosystem functions-specifically decomposition, carbon sequestration, pollination, pest control and cultural values. For 4,424 species over four decades, there have been significant net declines among animal species that provide pollination, pest control and cultural values. Groups providing decomposition and carbon sequestration remain relatively stable, as fewer species are in decline and these are offset by large numbers of new arrivals into Great Britain. While there is general concern about degradation of a wide range of ecosystem functions, our results suggest actions should focus on particular functions for which there is evidence of substantial erosion of their resilience.

Climatic risks and impacts in South Asia: extremes of water scarcity and excess

This paper reviews the current knowledge of climatic risks and impacts in South Asia associated with anthropogenic warming levels of 1.5°C to 4°C above pre-industrial values in the 21st century. It is based on the World Bank Report “Turn Down the Heat, Climate Extremes, Regional Impacts and the Case for Resilience” (2013). Many of the climate change impacts in the region, which appear quite severe even with relatively modest warming of 1.5–2°C, pose significant hazards to development. For example, increased monsoon variability and loss or glacial meltwater will likely confront populations with ongoing and multiple challenges. The result is a significant risk to stable and reliable water resources for the region, with increases in peak flows potentially causing floods and dry season flow reductions threatening agriculture. Irrespective of the anticipated economic development and growth, climate projections indicate that large parts of South Asia’s growing population and especially the poor are likely to remain highly vulnerable to climate change.