Higher dimensional geometries related to fuzzy odd-dimensional spheres

We study $SO(m)$ covariant Matrix realizations of $\sum_{i=1}^{m} X_i^2 = 1$ for even $m$ as candidate fuzzy odd spheres following hep-th/0101001. As for the fuzzy four sphere, these Matrix algebras contain more degrees of freedom than the sphere itself and the full set of variables has a geometrical description in terms of a higher dimensional coset. The fuzzy $S^{2k-1}$ is related to a higher dimensional coset ${SO(2k) \over U(1) \times U(k-1)}$. These cosets are bundles where base and fibre are hermitian symmetric spaces. The detailed form of the generators and relations for the Matrix algebras related to the fuzzy three-spheres suggests Matrix actions which admit the fuzzy spheres as solutions. These Matrix actions are compared with the BFSS, IKKT and BMN Matrix models as well as some others. The geometry and combinatorics of fuzzy odd spheres lead to some remarks on the transverse five-brane problem of Matrix theories and the exotic scaling of the entropy of 5-branes with the brane number.Comment: 32 pages, v2 : ref and acknowledgment adde

The Dirty Footprint of the Broken Grid: The Impacts of Fossil Fuel Back-up Generators in Developing Countries

Around the world, nearly 1 billion people live without access to electricity, and about 840 million more live with unreliable and intermittent service from electric grids. For many of them, fossil fuel backup generators are the only source of power. But these machines offer a problematic, intermediate solution: their cost of operation is high, they fill neighborhoods and cities with noise pollution, and the exhaust is hazardous to health and the environment. Researched in partnership with the Schatz Energy Research Center at Humboldt State University and the International Institute for Applied Systems Analysis (IIASA) with support from the IKEA Foundation, the report, The Dirty Footprint of the Broken Grid, documents for the first time the economic, environmental, and health effects of fossil fuel generators, and calls for the rapid adoption of clean alternatives. This study explores fundamental questions about the scale and impacts of backup generators that have been largely unanswered beyond anecdote and local or regionally focused studies