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Diffeomorphisms of Rn with oscillatory jacobians.

Waldyr M. Oliva, Nelson M. Kuhl, Luiz T. Magalhâes (1993)

Publicacions Matemàtiques

The paper presents, mainly, two results: a new proof of the spectral properties of oscillatory matrices and a transversality theorem for diffeomorphisms of Rn with oscillatory jacobian at every point and such that NM(f(x) - f(y)) ≤ NM(x - y) for all elements x,y ∈ Rn, where NM(x) - 1 denotes the maximum number of sign changes in the components zi of z ∈ Rn, where all zi are non zero and z varies in a small neighborhood of x. An application to a semiimplicit discretization of the scalar heat equation...

Differential equations in metric spaces

Jacek Tabor (2002)

Mathematica Bohemica

We give a meaning to derivative of a function u X , where X is a complete metric space. This enables us to investigate differential equations in a metric space. One can prove in particular Gronwall’s Lemma, Peano and Picard Existence Theorems, Lyapunov Theorem or Nagumo Theorem in metric spaces. The main idea is to define the tangent space 𝒯 x X of x X . Let u , v [ 0 , 1 ) X , u ( 0 ) = v ( 0 ) be continuous at zero. Then by the definition u and v are in the same equivalence class if they are tangent at zero, that is if lim h 0 + d ( u ( h ) , v ( h ) ) h = 0 . By 𝒯 x X we denote...

Directional properties of sets definable in o-minimal structures

Satoshi Koike, Ta Lê Loi, Laurentiu Paunescu, Masahiro Shiota (2013)

Annales de l’institut Fourier

In a previous paper by Koike and Paunescu, it was introduced the notion of direction set for a subset of a Euclidean space, and it was shown that the dimension of the common direction set of two subanalytic subsets, called the directional dimension, is preserved by a bi-Lipschitz homeomorphism, provided that their images are also subanalytic. In this paper we give a generalisation of the above result to sets definable in an o-minimal structure on an arbitrary real closed field. More precisely, we...

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