We consider the high frequency Helmholtz equation with a variable refraction index *n*^{2}(*x*) (*x* ∈ ℝ^{d}), supplemented with a given high frequency source term supported near the origin *x* = 0. A small absorption parameter α_{ε} > 0 is added, which prescribes a radiation condition at infinity for the considered Helmholtz equation. The semi-classical parameter is ε > 0. We let ε and α_{ε} go to zero *simultaneously*. We study the question whether the prescribed radiation condition at infinity is satisfied *uniformly* along the asymptotic process ε Ⅺ 0.

This question has been previously studied by the first autor in [4], where it is proved that the radiation condition is indeed satisfied uniformly in ε, provided the refraction index satisfies a specific *non-refocusing condition*. The non-refocusing condition requires, in essence, that the rays of geometric optics naturally associated with the high-frequency Helmholtz operator, and that are sent from the origin *x* = 0 at time *t* = 0, should not refocus at some later time *t* > 0 near the origin again.

In the present text we show the *optimality* of the above mentioned non-refocusing condition. We exhibit a refraction index which *does* refocus the rays of geometric optics sent from the origin near the origin again, and we show that the limiting solution *does not* satisfy the natural radiation condition at infinity in that case.

In this paper, we study *fold maps* from *C*^{∞} closed manifolds into Euclidean spaces whose singular value sets are disjoint unions of spheres embedded concentrically. We mainly study homology and homotopy groups of manifolds admitting such maps.

In this paper we study Ulam's cellular automaton, a nonlinear almost equicontinuous two-dimensional cell-model of crystalline growths. We prove that Ulam's automaton contains a linear chaotic elementary cellular automaton (Rule 150) as a subsystem. We also study the application of the inverse ultradiscretization, a method for deriving partial differential equations from a given cellular automaton, to Ulam's automaton. It is shown that the partial differential equation obtained by the inverse ultradiscretization preserves the self-organizing pattern of Ulam's automaton.

We study, using Hepp's method, the propagation of coherent states for a general class of self interacting bosonic quantum field theories with spatial cutoffs. This includes models with non-polynomial interactions in the field variables. We show indeed that the time evolution of coherent states, in the classical limit, is well approximated by time-dependent affine Bogoliubov unitary transformations. Our analysis relies on a non-polynomial Wick quantization and a specific hypercontractive estimate.