Propriété locale

Résumé

On dit d'une certaine propriété mathématique qu'elle est localement vérifiée en un point d'un espace topologique s'il existe un système fondamental de voisinages de ce point sur lequel la propriété est vraie. On dit d'une certaine propriété mathématique qu'elle est localement vérifiée si elle est localement vérifiée en tout point de l'espace topologique considéré. Cette notion se retrouve dans tous les domaines des mathématiques qui utilisent la topologie, en particulier en analyse. Souvent, il suffit que la propriété soit vraie pour un voisinage du point pour qu'elle soit vraie localement en ce point, par exemple :

On dit qu'une fonction \ f : X \to \R définie sur un espace topologique \ X admet en un point \ a de \ X un maximum local s'il existe un voisinage \ V de \ a tel que \ f(a) soit la plus grande valeur de \ f sur \ V ;
On dit qu'un espace top

Source officielle

https://fr.wikipedia.org/wiki/Propri%C3%A9t%C3%A9_locale

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Differentiable manifolds are a certain class of topological spaces which, in a way we will make precise, locally resemble R^n. We introduce the key concepts of this subject, such as vector fields, differential forms, integration of differential forms etc.

Algebraic geometry is the common language for many branches of modern research in mathematics. This course gives an introduction to this field by studying algebraic curves and their intersection theory.

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Local properties of the Hochschild cohomology of C*-algebras

Let A be a C*-algebra, and let X be a Banach A-bimodule. Johnson [B. E. Johnson, 'Local derivations on C*-algebras are derivations', Trans. Amer Math. Soc. 353 (2000), 313-325] showed that local derivations from A into X are derivations. We extend this concept of locality to the higher cohomology of a C*-algebra and show that, for every n is an element of N, bounded local n-cocycles from A((n)) into X are n-cocycles.

Cambridge University Press2008

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Networks provide an informative, yet non-redundant description of complex systems only if links represent truly dyadic relationships that cannot be directly traced back to node-specific properties such as size, importance, or coordinates in some embedding space. In any real-world network, some links may be reducible, and others irreducible, to such local properties. This dichotomy persists despite the steady increase in data availability and resolution, which actually determines an even stronger need for filtering techniques aimed at discerning essential links from non-essential ones. Here we introduce a rigorous method that, for any desired level of statistical significance, outputs the network backbone that is irreducible to the local properties of nodes, i.e. their degrees and strengths. Unlike previous approaches, our method employs an exact maximum-entropy formulation guaranteeing that the filtered network encodes only the links that cannot be inferred from local information. Extensive empirical analysis confirms that this approach uncovers essential backbones that are otherwise hidden amidst many redundant relationships and inaccessible to other methods. For instance, we retrieve the hub-and-spoke skeleton of the US airport network and many specialised patterns of international trade. Being irreducible to local transportation and economic constraints of supply and demand, these backbones single out genuinely higher-order wiring principles.

2017

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Probing the Strength and Fracture Toughness of Hard Second Phases at the Micron Scale

Marco Cantoni, Marta Fornabaio, Lionel Michelet, Andreas Mortensen, Martin Guillermo Mueller, Václav Pejchal, Andreas Rossoll, Goran Zagar

It is known that the mechanical properties of composite materials and alloys are influenced by the intrinsic mechanical properties of their reinforcements or second phases. It is however challenging to determine such local properties, given their small size and irregular shape. Here we present two newly developed methods to probe the local strength and fracture toughness combining focused ion beam (FIB) milling with microtesting techniques and finite element (FE) simulation. To probe local strength Nextel™ 610 nanocrystalline alumina fibers are used as a test bench material. Deep-etching is carried out to expose the fibres over lengths of a few tens of μm from an aluminium matrix. FIB milling is then used on selected fibres to machine a wide notch oriented perpendicular to the axis of the fibre. A nanoindenter with a flat tip is then used to apply a compressive force along the notched fibre's axis, which produces a bending moment in the ligament opposite to the notch leading to failure. The measured response coupled with the FE model of each notched fibre is used to calculate the stress state in the ligament prior to failure. The strength distribution across a series of such tests is statistically analyzed and fractographic analysis with a SEM is performed. To probe local fracture toughness we produce thin chevron-notched cantilever beams by FIB micromachining. These beams are then loaded using a nanoindenter. The method is successfully applied once the procedure to generate sufficiently thin triangular ligaments with the FIB is mastered, which is crucial to ensure that instability occurs after some stable crack growth, leading to meaningful toughness measurements which are minimally influenced by FIB-induced defects. Test data are interpreted using compliance calibration curves determined by FE simulation of each beam. The method is demonstrated on small volumes of fused quartz and alumina fibres and produces reproducible results consistent with expected values.

2015

Chargement

Probing the Strength and Fracture Toughness of Hard Second Phases at the Micron Scale

Marco Cantoni, Marta Fornabaio, Lionel Michelet, Andreas Mortensen, Martin Guillermo Mueller, Václav Pejchal, Andreas Rossoll, Goran Zagar

2015