Dictionary of Distances by Michel Marie Deza, Elena Deza

By Michel Marie Deza, Elena Deza

This booklet comes out of want and urgency (expressed specially in components of knowledge Retrieval with admire to picture, Audio, web and Biology) to have a operating device to check data.The e-book will supply strong source for all researchers utilizing arithmetic in addition to for mathematicians themselves. within the time while over-specialization and terminology fences isolate researchers, this Dictionary you need to be "centripedal" and "oikoumeni", delivering a few entry and altitude of imaginative and prescient yet with no taking the course of medical vulgarisation. This tried stability is the most philosophy of this Dictionary which outlined its constitution and style.Key features:- Unicity: it's the first booklet treating the fundamental concept of Distance in entire generality.- Interdisciplinarity: this Dictionary is bigger in scope than majority of thematic dictionaries.- Encyclopedicity: whereas an Encyclopedia of Distances turns out now too tough to provide, this e-book (by its scope, brief introductions and association) offers the most fabric for it and for destiny tutorials on a few elements of this material.- Applicability: the distances, in addition to distance-related notions and paradigms, are supplied in ready-to-use fashion.- Worthiness: the necessity and urgency for such dictionary was once nice in different large components, esp. info Retrieval, photo research, Speech acceptance and Biology.- Accessibility: the definitions are effortless to find via topic or, in Index, via alphabetic order; the introductions and definitions are reader-friendly and maximally autonomous one from one other; nonetheless the textual content is dependent, within the 3D HTML kind, by means of hyperlink-like boldfaced references to comparable definitions. * Covers a wide variety of matters in natural and utilized arithmetic* Designed to be simply applied--the distances and distance-related notions and paradigms are able to use* is helping clients speedy find definitions by means of topic or in alphabetical order; stand-alone entries comprise references to different entries and assets for additional research

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A metric space is called c-uniformly perfect, 0 < c 1, if this diameter is at least 2cr. , 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. The function f is uniformly continuous if, for any ε > 0, there exists δ > 0 such that, for any x, y ∈ X, inequality dX (x, y) < δ implies inequality dY (f (x), f (y)) < ε.

Every UC metric space is complete. • Polish space A Polish space is a complete separable metric space. A metric space is called Souslin space if it is a continuous image of a Polish space. , a non-negative real function μ on the Borel sigma-algebra F of X with the following properties: μ(∅) = 0, μ(X) = 1, and μ( n An ) = n μ(An ) for any finite or countable collection of pairwise disjoint sets An ∈ F. Given a topological space (X, τ ), a sigma-algebra on X is a collection F of subsets of X with the following properties: ∅ ∈ F, X\U ∈ F for U ∈ F, and n An ∈ F for a finite or countable collection {An }n , An ∈ F.

26 [ • Point-set distance] Part I: Mathematics of Distances If d is the symmetric difference metric, then dSt = 2d ∅ , where, for a given metric d on a set X and a given point p ∈ X, the transform metric d p on X is defined by d p (x, y) = d(x, y) . d(x, p) + d(y, p) + d(x, y) B| The biotope distance (or Tanimoto distance) |A |A∪B| is the special case of Steinhaus distance, obtained for the counting measure μ(A) = |A|. • Point-set distance Given a metric space (X, d), the point-set distance d(x, A) between a point x ∈ X and a subset A of X is defined as inf d(x, y).

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