Bulletin of the Brazilian Mathematical Society, New Series

Hörmander, Lars

doi: 10.1007/BF02585465pmid: N/A

This paper is devoted to the Cauchy problem for a fully non-linear perturbation $$\left( {\partial _t^2 - \Delta } \right)u + G\left( {u',u''} \right) = 0$$ of the wave equation with three space dimensions and small datau=εu 0 ∂ t u=εu 1;u j ∈C 0 ∞ (ℝ3). HereG∈C ∞ vanishes of second order at the origin. We give an explicit positive lower bound for $$\underline {\lim } _{\varepsilon \to 0} \varepsilon \log T_c $$ whereT c is the lifespan of the solution; it is equal to the lifespan of the limit as ε→0 of a rescaled solution. The main point is an exact determination of the lifespan for the solution of a non-linear first order differential equation in ℝ2 of the form $${{\partial u} \mathord{\left/ {\vphantom {{\partial u} {\partial t}}} \right. \kern-\nulldelimiterspace} {\partial t}} = a\left( {{{\partial u} \mathord{\left/ {\vphantom {{\partial u} {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}}} \right)^2 + {{2bu\partial u} \mathord{\left/ {\vphantom {{2bu\partial u} {\partial x + cu^2 }}} \right. \kern-\nulldelimiterspace} {\partial x + cu^2 }}$$ withu(0,x)=u 0(x)∈C 0 ∞ (ℝ>.

Moser, Jürgen

doi: 10.1007/BF02585466pmid: N/A

In a recent paper [9] the KAM theory has been extended to non-linear partial differential equations, to construct quasi-periodic solutions. In this article this theory is illustrated with three typical examples: an elliptic partial differential equation, an ordinary differential equation and a difference equation related to monotone twist mappings.

Faddeev, L.

doi: 10.1007/BF02585467pmid: N/A

An elementary introduction to the notions of the quantum Lie Groups and quantum Lie algebras is given. The approach is based on the fundamental commutation relations which appeared first in the quantum inverse scattering method.

Feit, Walter

doi: 10.1007/BF02585468pmid: N/A

Some conditions are stated which imply that certain finite groups are Galois groups over some number fields and related fields.

Mata-Lorenzo, Luis

doi: 10.1007/BF02585469pmid: N/A

For a stable map from a closed 3-manifold into the plane, its Stein Factorization gives useful information. Our main result presents the list of changes in the Stein Factorization of the maps in a generic family of homotopies (the Pi-stable ones).

Lovász, László

doi: 10.1007/BF02585470pmid: N/A

We study linear spaces of n×n matrices in which every matrix is singular. Examples are given to illustrate that a characterization of such subspaces would solve various open problems in combinatorics and in computational algebra. Several important special cases of the problem are solved, although often in disguise.

Coates, John

doi: 10.1007/BF02585471pmid: N/A

We propose a definition of thep-adic L-function of a motiveM over ℚ assumingM admits at least one critical point, andp is ordinary forM. This corrects by a power of $$i = \sqrt { - 1} $$ an earlier definition of B. Perrin-Riou and the author.

Lewowicz, Jorge

doi: 10.1007/BF02585472pmid: N/A

Letf be an expansive homeomorphism of a compact oriented surfaceM. We show thatS 2 does not support such anf, and thatf is conjugate to an Anosov diffeomorphism ifM=T 2, and to a pseudo-Anosov map ifM has genus ≥2. These results are consequences of our description of local stable (unstable) sets: everyx∈M has a local stable (unstable) set that consists of the union ofr arcs that meet only atx. For eachx∈M r=2, except for a finite number of points, wherer≥3.