G+++ and Brane Solutions

We demonstrate that the very extended G+++ group element of the form $g_A=\exp(-{\frac{1}{(\beta,\beta)}\ln N}\beta \cdot H)\exp((1-N)E_\beta)$ describes the usual BPS, electric, single brane solutions found in G+++ theories.Comment: One new equation, added references, corrected typos and minor changes, 42 pages, 6 figures, LaTeX2

Operator product expansions in four-dimensional superconformal field theories

The operator product expansion in four-dimensional superconformal field theory is discussed. The OPE takes a particularly simple form for chiral operators, in N=1 and N=2, and for analytic operators, in N=2 and N=4. It is argued that the Green's functions of such operators can be determined up to constants.The operator product expansion in four-dimensional superconformal field theory is discussed. The OPE takes a particularly simple form for chiral operators, in $N=1$ and $N=2$, and for analytic operators, in $N=2$ and $N=4$. It is argued that the Green's functions of such operators can be determined up to constants.The operator product expansion in four-dimensional superconformal field theory is discussed. It is demonstrated that the OPE takes a particularly simple form for certain classes of operators. These are chiral operators, principally of interest in theories with N = 1 or N = 2 supersymmetry, and analytic operators, of interest in N = 2 and N = 4. It is argued that the Green's functions of such operators can be determined up to constants

Climate adaptation by crop migration.

Many studies have estimated the adverse effects of climate change on crop yields, however, this literature almost universally assumes a constant geographic distribution of crops in the future. Movement of growing areas to limit exposure to adverse climate conditions has been discussed as a theoretical adaptive response but has not previously been quantified or demonstrated at a global scale. Here, we assess how changes in rainfed crop area have already mediated growing season temperature trends for rainfed maize, wheat, rice, and soybean using spatially-explicit climate and crop area data from 1973 to 2012. Our results suggest that the most damaging impacts of warming on rainfed maize, wheat, and rice have been substantially moderated by the migration of these crops over time and the expansion of irrigation. However, continued migration may incur substantial environmental costs and will depend on socio-economic and political factors in addition to land suitability and climate