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We give a meaning to derivative of a function $u ℝ \to X$ , 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 $x \in X$ . Let $u, v [0, 1) \to X$ , $u (0) = v (0)$ 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 $lim_{h \to 0^{+}} \frac{d (u (h), v (h))}{h} = 0 .$ By $𝒯_{x} X$ we denote...

Differential equations, Spencer cohomology, and computing resolutions.

Lambe, Larry A., Seiler, Werner M. (2002)

Georgian Mathematical Journal

Differential Invariance of Multiplicity of Analytic Varieties.

Y.-N. Gau, J. Lipman (1983)

Inventiones mathematicae

Differential operators on homogeneous spaces. III. Characteristic varieties of Harish Chandra modules and of primitive ideals.

W. Borho, J.-L. Brylinski (1985)

Inventiones mathematicae

Differential Topology and the Computation of Total Absolute Curvature.

Eberhard Teufel (1981)

Mathematische Annalen

Dimension and decompositions

Craig Zemke (1977)

Fundamenta Mathematicae

Dimension des orbites d'une action de ... sur une variété compacte.

P. Molina, F.J. Turiel (1988)

Commentarii mathematici Helvetici

Dimension Functions of Homotopy Representations for Compact Lie Groups.

Stefan Bauer (1988)

Mathematische Annalen

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( $H_{*} (Ω S; K)) < \infty$ et lorsque $(S, K)$ est dans le domaine d’Anick. Nous introduisons aussi une suite spectrale “impaire des $ℰ xt$ ” 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...

Discontinuous groups in some metric and nonmetric spaces

B. Klotzek (1986)

Beiträge zur Algebra und Geometrie = Contributions to algebra and geometry

Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices

David Cimasoni (2012)

Journal of the European Mathematical Society

Let $\sum$ be a flat surface of genus $g$ with cone type singularities. Given a bipartite graph $Γ$ isoradially embedded in $\sum$ , we define discrete analogs of the $2^{2 g}$ Dirac operators on $\sum$ . 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 $Γ \subset \sum$ 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...

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Differentiable S1 Actions on Homotopy Spheres.

Differentiable structure in a conjugate vector bundle of infinite dimension [Book]

Differentiable structures of elliptic surfaces with cyclic fundamental group

Differentiable structures on a generalized product of spheres.

Differential equations in metric spaces