Local property

Summary

In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some sufficiently small or arbitrarily small neighborhoods of points). Properties of a point on a function Perhaps the best-known example of the idea of locality lies in the concept of local minimum (or local maximum), which is a point in a function whose functional value is the smallest (resp., largest) within an immediate neighborhood of points. This is to be contrasted with the idea of global minimum (or global maximum), which corresponds to the minimum (resp., maximum) of the function across its entire domain. Properties of a single space A topological space is sometimes said to exhibit a property locally, if the property is exhibited "near" each point in one of the following ways:

Each point has a neighborhood exhibiting the property;

Each point has a neighborhood base of sets exhibiting th

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https://en.wikipedia.org/wiki/Local_property

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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

Irreducible network backbones: unbiased graph filtering via maximum entropy

Alessio Vincenzo Cardillo

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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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

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2015