An$L$^{1}-existence theorem is proved for the nonlinear stationary Boltzmann equation with hard forces and no small velocity truncation—only the Grad angular cut-off-in a setting between two coaxial rotating cylinders when the indata are given on the cylinders.

In the context of two illustrative examples from supersymmetric quantum mechanics we show that the semiclassical analysis of the path integral requires complexification of the configuration space and action, and the inclusion of complex saddle points, even when the parameters in the action are real. We find new exact complex saddles, and show that without their contribution the semi-classical expansion is in conflict with basic properties such as positive-semidefiniteness of the spectrum, and constraints of supersymmetry. Generic saddles are not only complex, but also possibly multi-valued, and even singular. This is in contrast to instanton solutions, which are real, smooth, and single-valued. The multi-valuedness of the action can be interpreted as a hidden topological angle, quantized in units of π in supersymmetric theories. The general ideas also apply to non-supersymmetric theories.

The Burton-Cabrera-Frank (BCF) model for the flow of line defects (steps) on crystal surfaces has offered useful insights into nanostructure evolution. This model has rested on phenomenological grounds. Our goal is to show via scaling arguments the emergence of the BCF theory for noninteracting steps from a stochastic atomistic scheme of a kinetic restricted solid-on-solid model in one spatial dimension. Our main assumptions are: adsorbed atoms (adatoms) form a dilute system, and elastic effects of the crystal lattice are absent. The step edge is treated as a front that propagates via probabilistic rules for atom attachment and detachment at the step. We formally derive a quasistatic step flow description by averaging out the stochastic scheme when terrace diffusion, adatom desorption, and deposition from above are present.

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In [8], Hitchin proved that a compact K/ihlerian twistor space is in fact a projective algebraic manifold. Moreover, it is the twistor space associated to either the Euclidean 4-sphere S 4 or the complex projective plane CF 2 with Fubini-Study metric. The twistor spaces are 3 and the flag of lines in~ p2 respectively. In [10, 11], the author proved the existence of self-dual metric with positive scalar curvature on the connected-sums of two or three copies of the complex projective planes. Their twistor spaces are Moish6zon spaces. In fact, they are the small resolution of the intersection of two quadrics in~ ps with four nodes and the double covering of CP 3 branched along a quartic with thirteen nodes. Recently, the joint work of Donaldson and Friedman [3] produced a general procedure to construct new twistor spaces and hence self-dual manifolds. In this article, we shall follow the spirit of Hitchin's work and prove the