Dictionary of Distances by Michel Marie Deza, Elena Deza

By Michel Marie Deza, Elena Deza

This publication comes out of want and urgency (expressed in particular in components of data Retrieval with admire to photo, Audio, web and Biology) to have a operating instrument to match data.The booklet will supply strong source for all researchers utilizing arithmetic in addition to for mathematicians themselves. within the time whilst over-specialization and terminology fences isolate researchers, this Dictionary you should be "centripedal" and "oikoumeni", offering a few entry and altitude of imaginative and prescient yet with out taking the course of clinical vulgarisation. This tried stability is the most philosophy of this Dictionary which outlined its constitution and style.Key features:- Unicity: it's the first ebook treating the fundamental suggestion 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 tricky to supply, this e-book (by its scope, brief introductions and association) presents 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 numerous large components, esp. info Retrieval, photo research, Speech reputation and Biology.- Accessibility: the definitions are effortless to find by means of topic or, in Index, by means of alphabetic order; the introductions and definitions are reader-friendly and maximally self reliant one from one other; nonetheless the textual content is based, within the 3D HTML kind, through hyperlink-like boldfaced references to related definitions. * Covers a wide range 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 quick 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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D(x, y) = 0 if and only if x = y; 2. d(x, y) d(x, z) + d(z, y) for all x, y ∈ X; 3. d(x, y) = d(y, x), where α is a fixed order-preserving involution of G. The pair (X, d) is called generalized metric space. If the condition 2. and “only if” in 1. above are dropped, we obtain a generalized distance d, and a generalized distance space (X, d). • Distance on building A Coxeter group is a group (W, ·, 1) generated by the elements {w1 , . . , wn : (wi wj )mij = 1, 1 i, j n}. , an arbitrary symmetric n × n matrix with mii = 1, and other values are positive integers or ∞.

Chapter 5). , the number of edges) of a shortest path connecting two given vertices x and y from V (cf. Chapter 15). 24 [ • Editing metric] Part I: Mathematics of Distances • Editing metric Given a finite set X and a finite set O of (unary) editing operations on X, the editing metric on X is the path metric of the graph with the vertex-set X and xy being an edge if y can be obtained from x by one of the operations from O. • Gallery metric A chamber system is a set X (whose elements are referred to as chambers) equipped with n equivalence relations ∼i , 1 i n.

Examples of such hit-and-miss topology are the Vietoris topology, and the Fell topology. Examples of such weak hyperspace topology are the Hausdorff metric topology, and the Wijsman topology. • Discrete space A discrete space is a topological space (X, τ ) with the discrete topology. It can be considered as the metric space (X, d) with the discrete metric: d(x, x) = 0, and d(x, y) = 1 for x = y. • Indiscrete space An indiscrete space is a topological space (X, τ ) with the indiscrete topology. It can be considered as the semi-metric space (X, d) with the indiscrete semi-metric: d(x, y) = 0 for any x, y ∈ X.

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