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By M. Aizenman (Chief Editor)

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Q n )− 2 φn , ψ) n . 75), 1 1 + ∇ Hn ((Q 0 )− 2 φ0 , . . , (Q n )− 2 φn , ψ) · ∇ f {− f 1 1 +H ess(Hn ((Q 0 )− 2 φ0 , . . , (Q n )− 2 φn , ψ))}∇u x = ∇ Hψx . 77) Thus, 1 1 + ∇ Hn ((Q 0 )− 2 φ0 , . . , (Q n )− 2 φn , ψ) · ∇ f {− f +M2 H ess(Hn ((Q 0 )− 2 φ0 , . . , (Q n )− 2 φn , ψ))M2−1 1 f 1 f 1 1 1 −H ess(Hn ((Q 0 )− 2 φ0 , . . , (Q n )− 2 φn , ψ)) + H ess(Hn ((Q 0 )− 2 φ0 , . . 78) W = M2 ∇u x , F = M2 ∇ Hψx . 79) f Let us estimate M2 H ess(Hn ((Q 0 )− 2 φ0 , . . , (Q n )− 2 φn , ψ))M2−1 − H ess(Hn ((Q 0 )− 2 φ0 , .

On extensions of the Brunn-Minkowski and Prkopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. : Grad φ perturbations of massless gaussian fields. Commun. Math. Phys. : Construction and borel summability of infrared 4 by a phase space expansion. Commun. Math. Phys. : Ann. Phys. : Conformal field theory and strings, In: Quantum fields and strings: a course for mathematicians, Volume 2, Providence, RI: Amer.

12c, 75 (1975) Communicated by A. Kupiainen Commun. Math. Phys. 1007/s00220-008-0532-3 Communications in Mathematical Physics Openness of the Set of Non-characteristic Points and Regularity of the Blow-up Curve for the 1 D Semilinear Wave Equation Frank Merle1 , Hatem Zaag2 1 Université de Cergy Pontoise, IHES and CNRS, Département de mathématiques, 2 avenue Adolphe Chauvin, BP 222, 95302 Cergy Pontoise cedex, France. B. Clément, 93430 Villetaneuse, France. fr Received: 8 March 2007 / Accepted: 21 February 2008 Published online: 24 June 2008 – © Springer-Verlag 2008 Abstract: We consider here the 1 D semilinear wave equation with a power nonlinearity and with no restriction on initial data.

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