A Trotter-Kato product formula for a class of non-autonomous evolution equations

In this article dedicated to Professor V. Lakshmikantham on the occasion of the celebration of his 84th birthday, we announce new results concerning the existence and various properties of an evolution system UA+B(t, s)(0 <= s <= t <= T) generated by the sum -(A(t)+B(t)) of two linear, time-dependent and generally unbounded operators defined on time-dependent domains in a complex and separable Banach space B. In particular, writing G(B) for the algebra of all linear bounded operators on B, we can express UA+B(t, s)(0 <= s <= t <= T) as the strong limit in L(B) of a product of the holomorphic contraction semigroups generated by -A(t) and -B(t), thereby getting a product formula of the Trotter-Kato type under very general conditions which allow the domain D(A(t)+B(t)) to evolve with time provided there exists a fixed set D subset of boolean AND D-t epsilon[0,D-T](A(t)+B(t)) everywhere dense in B. We then mention several possible applications of our product formula to various classes of non-autonomous parabolic initial-boundary value problems, as well as to evolution problems of Schrodinger type related to the theory of time-dependent singular perturbations of self-adjoint operators in quantum mechanics. We defer all the proofs and all the details of the applications to a separate publication. (C) 2008 Elsevier Ltd. All rights reserved.

Insights into the dynamics and sources of atmospheric lead and particulate matter in Sao Paulo, Brazil, from high temporal resolution sampling

We present the first results of a study investigating the processes that control concentrations and sources of Pb and particulate matter in the atmosphere of Sao Paulo City Brazil Aerosols were collected with high temporal resolution (3 hours) during a four-day period in July 2005 The highest Pb concentrations measured coincided with large fireworks during celebration events and associated to high traffic occurrence Our high-resolution data highlights the impact that a singular transient event can have on air quality even in a megacity Under meteorological conditions non-conducive to pollutant dispersion Pb and particulate matter concentrations accumulated during the night leading to the highest concentrations in aerosols collected early in the morning of the following day The stable isotopes of Pb suggest that emissions from traffic remain an Important source of Pb in Sao Paulo City due to the large traffic fleet despite low Pb concentrations in fuels (C) 2010 Elsevier BV All rights reserved