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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...

Dimension globale et classe fondamentale d'un espace

Youssef Rami (1999)

Annales de l'institut Fourier

L’algèbre de Pontryagin d’un espace K -elliptique vérifie le théorème d’Auslander-Buchsbaum-Serre. Nous donnons ici plusieurs caractérisations des espaces K -elliptiques tels que gldim( H * ( Ω S ; K ) ) < et lorsque ( S , K ) est dans le domaine d’Anick. Nous introduisons aussi une suite spectrale “impaire des xt ” et complétons les résultats obtenus par A. Murillo dans le cas rationnel.

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...

Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices

David Cimasoni (2012)

Journal of the European Mathematical Society

Let be a flat surface of genus g with cone type singularities. Given a bipartite graph Γ isoradially embedded in , we define discrete analogs of the 2 2 g Dirac operators on . These discrete objects are then shown to converge to the continuous ones, in some appropriate sense. Finally, we obtain necessary and sufficient conditions on the pair Γ for these discrete Dirac operators to be Kasteleyn matrices of the graph Γ . As a consequence, if these conditions are met, the partition function of the dimer...

Discrete thickness

Sebastian Scholtes (2014)

Molecular Based Mathematical Biology

We investigate the relationship between a discrete version of thickness and its smooth counterpart. These discrete energies are deffned on equilateral polygons with n vertices. It will turn out that the smooth ropelength, which is the scale invariant quotient of length divided by thickness, is the Γ-limit of the discrete ropelength for n → ∞, regarding the topology induced by the Sobolev norm ‖ · ‖ W1,∞(S1,ℝd). This result directly implies the convergence of almost minimizers of the discrete energies...

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