By Michel Marie Deza
Distance metrics and distances became a vital device in lots of components of natural and utilized arithmetic, and this encyclopedia is the 1st one to regard the topic in complete. The booklet seems simply as study intensifies into metric areas and particularly, distance layout for functions. those distances are relatively the most important, for instance, in computational biology, photograph research, speech acceptance, and knowledge retrieval. the following, an evaluate of the sensible questions coming up in the course of choice of a ''good'' distance functionality has been left apart in desire of a complete directory of the most on hand distances, a great tool for the space layout neighborhood. This reader-friendly reference bargains either self sustaining introductions and definitions, whereas whilst making cross-referencing effortless via hyperlink-like boldfaced references to unique definitions. This top quality ebook is a mixture of reference source and coffee-table e-book
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Additional resources for Encyclopedia of Distances
Xn }, d), its Cayley–Menger matrix is the symmetric (n + 1) × (n + 1) matrix CM (X, d) = 0 eT e D , where D = ((d2 (xi , xj ))) and e is the n-vector all components of which are 1. The determinant of CM (X, d) is called the Cayley–Menger determinant. )2 times squared (n − 1)-dimensional volume of the convex hull of X in Rn−1 . • Gram matrix Given elements v1 , . . , vk of a Euclidean space, their Gram matrix is the symmetric k × k matrix G(v1 , . . , vk ) = (( vi , vj )) of pairwise inner products of v1 , .
The number inf x∈M supy∈M d(x, y). Then diam(M 2 diam(M ), where diam(M ) is the diameter of the set M , with r(M ) = ) in any diam(M ) in any equidistant metric space and r(M ) = diam(M 2 injective metric space. Some authors deﬁne the radius to be the number diam(M ) . , the smallest number R such that the open metric balls of radius R with centers at the elements of M cover X. It is also called the directed Hausdorﬀ distance from X to M . The set M is called an -covering if its covering radius does not exceed .
For example, (R2 , l2 ) is paradoxical for wobbling equivalence but not for isometry equivalence. , a bijective function f : X → Y such that f and f −1 are continuous (the preimage of every open set in Y is open in X). Two metric spaces (X, dX ) and (Y, dY ) are called uniformly isomorphic if there exists a bijective function f : X → Y such that f and f −1 are uniformly continuous functions. ) M¨ obius mapping Given a metric space (X, d) and quadruple (x, y, z, w) of its distinct points, the cross-ratio is the positive number deﬁned by cr((x, y, z, w), d) = d(x, y)d(z, w) .
Encyclopedia of Distances by Michel Marie Deza