We present a new proof of the convergence of the N −particle Schrödinger dynamics for bosons towards the dynamics generated by the Hartree equation in the mean-field limit. For a restricted class of two-body interactions, we obtain convergence estimates uniform in , up to an exponentially small remainder. For = 0, the classical dynamics in the mean-field limit is given by the Vlasov equation.

Mathematical results on some models describing the motion of a tracer particle through a Bose-Einstein condensate are described. In the limit of a very dense, very weakly interacting Bose gas and for a very large particle mass, the dynamics of the coupled system is determined by classical non-linear Hamiltonian equations of motion. The particle's motion exhibits deceleration corresponding to friction (with memory) caused by the emission of Cerenkov radiation of gapless modes into the gas.Precise results are stated and outlines of proofs are presented. Some technical details are deferred to forthcoming papers.

Simple proofs of uniqueness of the thermodynamic limit of KMS states and of the decay of equilibrium correlations are presented for a large class of quantum lattice systems at high temperatures. New quantum correlation inequalities for general Heisenberg models are described. Finally, a simplified derivation of a general result on power-law decay of correlations in 2D quantum lattice systems with continuous symmetries is given, extending results of Mc Bryan and Spencer for the 2D classical XY model.We dedicate this note to the memory of our friend Oscar E. Lanford III.1991 Mathematics Subject Classification. 82B10, 82B20, 82B26, 82B31.

At sufficiently high Larmor frequencies, traveling electromagnetic waves along a magnet bore can be used for remote magnetic resonance excitation and detection, effectively using the bore as a waveguide. So far, this approach has relied only on the lowest waveguide modes and thus has not supported multiple-channel operation for radiofrequency shimming and parallel imaging. In this work, this limitation is addressed by establishing a larger number of propagating modes and tapping their spatial field diversity with multiple waveguide ports. The number of available modes is increased by loading with dielectric inserts; the ports are implemented by stub and loop couplers at the end of a waveguide extension. The resulting traveling-wave array, operated at 298 MHz in a 7T whole-body magnet, is shown to enable radiofrequency shimming as well as parallel imaging with commonly used acceleration factors. The last part of the study concerns the amount of dielectric loading that is required. For the given Larmor frequency and bore dimensions, it is found that rather few water-filled inserts, occupying~5% of the bore crosssection, are sufficient for effective parallel imaging. Magn Reson Med 66:290-300,

RF shimming and parallel imaging in a watermelon. a: Relative receive sensitivities of ports 1-7 in a central transverse slice. b: Array image obtained with a uniform RF shim and full Fourier encoding. Note that the high image intensity in the center reflects genuine image contrast unrelated to RF nonuniformity. c, d: SENSE reconstructions from data obtained by 2.5-and 4.0-fold aliasing of the full-Fourier data from the article by Brunner et al (pp 288-298).