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A proof of the two-dimensional Markus-Yamabe Stability Conjecture and a generalization

Robert Feßler (1995)

Annales Polonici Mathematici

The following problem of Markus and Yamabe is answered affirmatively: Let f be a local diffeomorphism of the euclidean plane whose jacobian matrix has negative trace everywhere. If f(0) = 0, is it true that 0 is a global attractor of the ODE dx/dt = f(x)? An old result of Olech states that this is equivalent to the question if such an f is injective. Here the problem is treated in the latter form by means of an investigation of the behaviour of f near infinity.

An Arzela-Ascoli theorem for immersed submanifolds

Graham Smith (2007)

Annales de la faculté des sciences de Toulouse Mathématiques

The classical Arzela-Ascoli theorem is a compactness result for families of functions depending on bounds on the derivatives of the functions, and is of invaluable use in many fields of mathematics. In this paper, inspired by a result of Corlette, we prove an analogous compactness result for families of immersed submanifolds which depends only on bounds on the derivatives of the second fundamental forms of these submanifolds. We then show how the result of Corlette may be obtained as an immediate...

Combinatoire des simplexes sans singularités I. Le cas différentiable

Jean Cerf (1998)

Annales de l'institut Fourier

On définit le bicomplexe C , , extension naturelle du complexe C engendré par un ensemble simplicial Γ . Ceci permet de définir la notion de ruban de base un cycle de C . La somme directe de l’homologie des colonnes de C , contient, outre l’homologie de C , des groupes dans lesquels se trouvent les obstructions à l’existence de rubans. Si Γ est un sous-ensemble simplicial, stable par subdivision, de l’ensemble des simplexes singuliers d’un espace topologique, l’existence de rubans entraîne l’invariance...

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