Courant-Dorfman algebras and their cohomology

We introduce a new type of algebra, the Courant-Dorfman algebra. These are to Courant algebroids what Lie-Rinehart algebras are to Lie algebroids, or Poisson algebras to Poisson manifolds. We work with arbitrary rings and modules, without any regularity, finiteness or non-degeneracy assumptions. To each Courant-Dorfman algebra (\R,\E) we associate a differential graded algebra \C(\E,\R) in a functorial way by means of explicit formulas. We describe two canonical filtrations on \C(\E,\R), and derive an analogue of the Cartan relations for derivations of \C(\E,\R); we classify central extensions of \E in terms of H^2(\E,\R) and study the canonical cocycle \Theta\in\C^3(\E,\R) whose class $[\Theta]$ obstructs re-scalings of the Courant-Dorfman structure. In the nondegenerate case, we also explicitly describe the Poisson bracket on \C(\E,\R); for Courant-Dorfman algebras associated to Courant algebroids over finite-dimensional smooth manifolds, we prove that the Poisson dg algebra \C(\E,\R) is isomorphic to the one constructed in \cite{Roy4-GrSymp} using graded manifolds.Comment: Corrected formulas for the brackets in Examples 2.27, 2.28 and 2.29. The corrections do not affect the exposition in any wa

Comparison of quantum mechanical and classical trajectory calculations of cross sections for ion-atom impact ionization of negative - and positive -ions for heavy ion fusion applications

Stripping cross sections in nitrogen have been calculated using the classical trajectory approximation and the Born approximation of quantum mechanics for the outer shell electrons of 3.2GeV I$^{-}$ and Cs$^{+}$ ions. A large difference in cross section, up to a factor of six, calculated in quantum mechanics and classical mechanics, has been obtained. Because at such high velocities the Born approximation is well validated, the classical trajectory approach fails to correctly predict the stripping cross sections at high energies for electron orbitals with low ionization potential.Comment: submitted to Phys. Rev.

Institutional and interpersonal trust as an indicator of the health of modern society

In modern society an important role was played by trust as a necessary condition for social communication in society, for understanding, harmony and cooperation. The article set relatively low levels of social and institutional trust in Russian society. As a result of changes for the last twenty years there has been disappointment with the population of the liberal economy. Distrust of all important social institutions means failure of the government institutions to provide effective positive impact on socio-economic processes in society

Bubble chambers

XI+371hlm.;23c