The concept of time reversal (TR) of scalar wave is reexamined
from basic principles. Five different time reversal
mirrors (TRM) are introduced and their relations are analyzed.

The asymptotic analysis of the near-field focusing property is
presented. It is shown that to have a subwavelength focal spot
the TRM should involve dipole fields. The monopole TR is
extremely ineffective to focus below wavelength as the focal
spot size decreases logarithmically with the distance between
the source and TRM.

Contrary to the matched field processing and the phase processor,
both of which resemble TR, TR in a weak- or non-scattering medium
is usually biased in the longitudinal direction. This is true for
all five TR schemes. On the other hand, the TR focal spot has
been shown repeatedly in the literature, both theoretically and
experimentally, to be centered at the source point when the
medium is multiply scattering. A reconciliation of the two
seemingly conflicting results is found in the random fluctuations
in the intensity of the Green function for a multiply scattering
medium and the notion of scattering-enlarged effective aperture.

This is a report on a joint work with Isabelle Catto, Norbert Mauser and Saber Trabelsi. The Multiconfiguration time dependent Hartree Fock Method (MCTDHF) is a nonlinear approximation of a linear system of /N/ quantum particles with binary interaction. It combines the principle of the Hartree Fock and the Galerkin approximation. The main difficulty is the introduction of a global (in space) density matrix $\Gamma(t) $ which may degenerate. By construction this approximation formally preserves the mass and the energy of the system. The conservation of energy can be used to balance the singularities Coulomb potential and to provide sufficient conditions for the global in time invertibility of $\Gamma(t)$.

In numerical computations this matrix is very often regularized (changed into $\Gamma(t) +\epsilon(t)$). In this situation the energy is no more conserved
and the mathematical analysis done in $L^2$ relies on Strichartz type estimates.

We review some aspects of the spectral theory of the critically coupled Almost Mathieu Operator connected with the structure of the famous associated "Hofstadter's Butterfly." We present a new result (joint with Mira Shamis) establishing that for a topologically generic set of irrational frequencies, the Hausdorff dimension of the spectrum of the critical Almost Mathieu Operator is zero. This result is based a new approach which combines certain inductive WKB-type estimates with Green function techniques and provides more detailed information than what has been previously achieved using more elaborate semiclassical approaches.