Superrigidity for irreducible lattices and geometric splitting

We prove general superrigidity results for actions of irreducible lattices on CAT(0) spaces; first, in terms of the ideal boundary, and then for the intrinsic geometry (including for infinite-dimensional spaces). In particular, one obtains a new and self-contained proof of Margulis' superrigidity theorem for uniform irreducible lattices in non-simple groups. The proofs rely on simple geometric arguments, including a splitting theorem which can be viewed as an infinite-dimensional (and singular) generalization of the Lawson-Yau/Gromoll-Wolf theorem.Comment: Improved version of earlier preprint. Definitions 3, 5 and proof of Theorem 55 modifie

From microbial communities to cells

The eukraotic cell, the unit of structure of protoctists, plants, fungi, and animals, is not at all homologous to prokaryotic cells. Instead the eukaryotic cell is homologous to communities of microorganisms such as those of the sulfuretum. This research is based on the hypothesis that at least four different interacting community members entered the original associations that, when stabilized, led to the emergence of eukaryotic cells. These are: (1) host nucleocytoplasm (thermoplasma like archaebacteria); (2) mitochrondria (paracoccus or bdellovibryo like respiring bacteria; and (3) plastids (cyanobacteria) and undulipodia. Tubulin like protein was found in the free living spirochete Spirochaeta bajacaliforniensis and in several other spirochetes. The amino acid sequence was to see if the spirochete protein is homologous to the tubulin of undulipodial and mitotic spindle microtubules

Back of the envelope estimates of environmental damage costs in Mexico

For developing countries, budget constraints help set the agenda on mitigating environmental damage, one of the indelible marks of our era. Political considerations often dictate the measures taken. There are no firm analytical formulas to help even environmentally conscious policymakers rank needs and remedies. A developing country such as Mexico - the focus of this paper - cannot afford an in-depth study of every environmental issue. Policymakers need to be provided with rough,"back-of-the envelope"estimates of the economic costs of various environmental problems. This allows them to rank the issues and act. In this paper the author applied existing methods to estimate the costs stemming from different environmental problems in Mexico. Although the examples are from Mexico, the method can be useful in other developing countries as well. The author how creative use of U.S. and other data can help provide simple estimates of the likely costs of soil erosion, air pollution, mining of underground waters, and estimates of the health effects of water and solid waste pollution, lack of sanitation, and the ingestion of food contaminated by polluted irrigation. The assumptions underlying all calculations are conservative. Some environmental damage issues, such as loss of biodiversity, were too complex to permit quantification.Water Conservation,Economic Theory&Research,Health Monitoring&Evaluation,Environmental Economics&Policies,Pollution Management&Control

Flows on homogeneous spaces and Diophantine approximation on manifolds

We present a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces. This approach yields a new proof of a conjecture of Mahler, originally settled by V. Sprindzhuk in 1964. We also prove several related hypotheses of A. Baker and V. Sprindzhuk formulated in 1970s. The core of the proof is a theorem which generalizes and sharpens earlier results on non-divergence of unipotent flows on the space of lattices.Comment: 19 pages. To appear in Annals of Mathematic

Quantitative Version of the Oppenheim Conjecture for Inhomogeneous Quadratic Forms

A quantitative version of the Oppenheim conjecture for inhomogeneous quadratic forms is proved. We also give an application to eigenvalue spacing on flat 2-tori with Aharonov-Bohm flux