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New bounds are given for the L 2-norm of the solution of the Kuramoto-Sivashinsky equation d t U(x, t) =-(d 2 x + d A x)U{x, t)-U(x, t)d x U(x, t) , for initial data which are periodic with period L. There is no requirement on the antisymmetry of the initial data. The result is limsup||£/(,ί)|| 2 < const. L 8/5 .

We propose a general framework for quantum field theory on the de Sitter space-time (i.e. for local field theories whose truncated Wightman functions are not required to vanish). By requiring that the fields satisfy a weak spectral condition, formulated in terms of the analytic continuation properties of their Wightman functions, we show that a geodesical observer will detect in the corresponding "vacuum" a blackbody radiation at temperature T = 1/2πR. We also prove the analogues of the PCT, Reeh-Schlieder and Bisognano-Wichmann theorems.

In the framework of the &.6f.2£. formalism, the crossing property is proved on the mass shell for amplitudes involving two incoming and two outgoing stable particles with arbitrary masses. Any couple of physical regions in the (s 9 1, u) plane corresponding to crossed processes are shown to be connected by a certain domain of analyticity. For every negative value of ί, the amplitude is analytic in the cut s-plane outside of a large circle.

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