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2016 | Buch

General Definitions Kapitel lesen Erstes Kapitel lesen

Encyclopedia of Distances

verfasst von: Michel Marie Deza, Elena Deza

Verlag: Springer Berlin Heidelberg

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This 4-th edition of the leading reference volume on distance metrics is characterized by updated and rewritten sections on some items suggested by experts and readers, as well a general streamlining of content and the addition of essential new topics. Though the structure remains unchanged, the new edition also explores recent advances in the use of distances and metrics for e.g. generalized distances, probability theory, graph theory, coding theory, data analysis.

New topics in the purely mathematical sections include e.g. the Vitanyi multiset-metric, algebraic point-conic distance, triangular ratio metric, Rossi-Hamming metric, Taneja distance, spectral semimetric between graphs, channel metrization, and Maryland bridge distance. The multidisciplinary sections have also been supplemented with new topics, including: dynamic time wrapping distance, memory distance, allometry, atmospheric depth, elliptic orbit distance, VLBI distance measurements, the astronomical system of units, and walkability distance.

Leaving aside the practical questions that arise during the selection of a ‘good’ distance function, this work focuses on providing the research community with an invaluable comprehensive listing of the main available distances.

As well as providing standalone introductions and definitions, the encyclopedia facilitates swift cross-referencing with easily navigable bold-faced textual links to core entries. In addition to distances themselves, the authors have collated numerous fascinating curiosities in their Who’s Who of metrics, including distance-related notions and paradigms that enable applied mathematicians in other sectors to deploy research tools that non-specialists justly view as arcane. In expanding access to these techniques, and in many cases enriching the context of distances themselves, this peerless volume is certain to stimulate fresh research.

Inhaltsverzeichnis

Frontmatter

Mathematics of Distances

Frontmatter
Chapter 1. General Definitions
Abstract
A distance space (X, d) is a set X (carrier) equipped with a distance d.
Michel Marie Deza, Elena Deza
Chapter 2. Topological Spaces
Abstract
A topological space (X, τ) is a set X with a topology τ, i.e., a collection of subsets of X with the following properties:
1.
X ∈ τ, ∅ ∈ τ;
 
2.
If A, B ∈ τ, then A ∩ B ∈ τ;
 
3.
For any collection {A α } α , if all A α  ∈ τ, then ∪ α A α  ∈ τ.
 
Michel Marie Deza, Elena Deza
Chapter 3. Generalizations of Metric Spaces
Abstract
Some immediate generalizations of the notion of metric, for example, quasi-metric, near-metric, extended metric, were defined in Chap. 1 Here we give some generalizations in the direction of Topology, Probability, Algebra, etc.
Michel Marie Deza, Elena Deza
Chapter 4. Metric Transforms
Abstract
There are many ways to obtain new distances (metrics) from given distances (metrics). Metric transforms give new distances as a functions of given metrics (or given distances) on the same set X. A metric so obtained is called a transform metric. We give some important examples of transform metrics in Sect. 4.1.
Michel Marie Deza, Elena Deza
Chapter 5. Metrics on Normed Structures
Abstract
In this chapter we consider a special class of metrics defined on some normed structures, as the norm of the difference between two given elements. This structure can be a group (with a group norm), a vector space (with a vector norm or, simply, a norm), a vector lattice (with a Riesz norm), a field (with a valuation), etc.
Michel Marie Deza, Elena Deza

Geometry and Distances

Frontmatter
Chapter 6. Distances in Geometry
Abstract
Geometry arose as the field of knowledge dealing with spatial relationships. It was one of the two fields of pre-modern Mathematics, the other being the study of numbers.
Michel Marie Deza, Elena Deza
Chapter 7. Riemannian and Hermitian Metrics
Abstract
Riemannian Geometry is a multidimensional generalization of the intrinsic geometry of 2D surfaces in the Euclidean space \(\mathbb{E}^{3}\). It studies real smooth manifolds equipped with Riemannian metrics, i.e., collections of positive-definite symmetric bilinear forms ((g ij )) on their tangent spaces which vary smoothly from point to point. The geometry of such (Riemannian) manifolds is based on the line element d s 2 = ∑ i, j g ij dx i dx j . This gives, in particular, local notions of angle, length of curve, and volume.
Michel Marie Deza, Elena Deza
Chapter 8. Distances on Surfaces and Knots
Abstract
A surface is a real 2D (two-dimensional) manifold M 2, i.e., a Hausdorff space, each point of which has a neighborhood which is homeomorphic to a plane \(\mathbb{E}^{2}\), or a closed half-plane (cf. Chap. 7).
Michel Marie Deza, Elena Deza
Chapter 9. Distances on Convex Bodies, Cones, and Simplicial Complexes
Abstract
A convex body in the n-dimensional Euclidean space \(\mathbb{E}^{n}\) is a convex compact connected subset of \(\mathbb{E}^{n}\). It is called solid (or proper) if it has nonempty interior. Let K denote the space of all convex bodies in \(\mathbb{E}^{n}\), and let K p be the subspace of all proper convex bodies. Given a set \(X \subset \mathbb{E}^{n}\), its convex hull c o n v(X) is the minimal convex set containing X.
Michel Marie Deza, Elena Deza

Distances in Classical Mathematics

Frontmatter
Chapter 10. Distances in Algebra
Abstract
A group (G, ⋅ , e) is a set G of elements with a binary operation ⋅ , called the group operation, that together satisfy the four fundamental properties of closure (x ⋅ y ∈ G for any x, y ∈ G), associativity (x ⋅ (y ⋅ z) = (x ⋅ y) ⋅ z for any x, y, z ∈ G), the identity property (\(x \cdot e = e \cdot x = x\) for any x ∈ G), and the inverse property (for any x ∈ G, there exists an element x −1 ∈ G such that \(x \cdot x^{-1} = x^{-1} \cdot x = e\)).
Michel Marie Deza, Elena Deza
Chapter 11. Distances on Strings and Permutations
Abstract
An alphabet is a finite set \(\mathcal{A}\), \(\vert \mathcal{A}\vert \geq 2\), elements of which are called characters (or symbols). A string (or word) is a sequence of characters over a given finite alphabet \(\mathcal{A}\). The set of all finite strings over the alphabet \(\mathcal{A}\) is denoted by \(W(\mathcal{A})\). Examples of real world applications, using distances and similarities of string pairs, are Speech Recognition, Bioinformatics, Information Retrieval, Machine Translation, Lexicography, Dialectology.
Michel Marie Deza, Elena Deza
Chapter 12. Distances on Numbers, Polynomials, and Matrices
Abstract
Here we consider the most important metrics on the classical number systems: the semiring \(\mathbb{N}\) of natural numbers, the ring \(\mathbb{Z}\) of integers, and the fields \(\mathbb{Q}\), \(\mathbb{R}\), \(\mathbb{C}\) of rational, real, complex numbers, respectively. We consider also the algebra \(\mathcal{Q}\) of quaternions.
Michel Marie Deza, Elena Deza
Chapter 13. Distances in Functional Analysis
Abstract
Functional Analysis is the branch of Mathematics concerned with the study of spaces of functions. This usage of the word functional goes back to the calculus of variations which studies functions whose argument is a function. In the modern view, Functional Analysis is seen as the study of complete normed vector spaces, i.e., Banach spaces.
Michel Marie Deza, Elena Deza
Chapter 14. Distances in Probability Theory
Abstract
A probability space is a measurable space \((\varOmega,\mathcal{A},P)\), where \(\mathcal{A}\) is the set of all measurable subsets of Ω, and P is a measure on \(\mathcal{A}\) with P(Ω) = 1. The set Ω is called a sample space. An element \(a \in \mathcal{A}\) is called an event. P(a) is called the probability of the event a. The measure P on \(\mathcal{A}\) is called a probability measure, or (probability) distribution law, or simply (probability) distribution.
Michel Marie Deza, Elena Deza

Distances in Applied Mathematics

Frontmatter
Chapter 15. Distances in Graph Theory
Abstract
A graph is a pair G = (V, E), where V is a set, called the set of vertices of the graph G, and E is a set of unordered pairs of vertices, called the edges of the graph G. A directed graph (or digraph) is a pair D = (V, E), where V is a set, called the set of vertices of the digraph D, and E is a set of ordered pairs of vertices, called arcs of the digraph D.
Michel Marie Deza, Elena Deza
Chapter 16. Distances in Coding Theory
Abstract
Coding Theory deals with the design and properties of error-correcting codes for the reliable transmission of information across noisy channels in transmission lines and storage devices. The aim of Coding Theory is to find codes which transmit and decode fast, contain many valid code words, and can correct, or at least detect, many errors. These aims are mutually exclusive, however; so, each application has its own good code.
Michel Marie Deza, Elena Deza
Chapter 17. Distances and Similarities in Data Analysis
Abstract
A data set is a finite set comprising m sequences (x 1 j , …, x n j ), j ∈ { 1, …, m}, of length n. The values x i 1, …, x i m represent an attribute S i .
Michel Marie Deza, Elena Deza
Chapter 18. Distances in Systems and Mathematical Engineering
Abstract
In this chapter we group the main distances used in Systems Theory (such as
Transition Systems, Dynamical Systems, Cellular Automata, Feedback Systems) and other interdisciplinary branches of Mathematics, Engineering and Theoretical Computer Science (such as, say, Robot Motion and Multi-objective Optimization).
Michel Marie Deza, Elena Deza

Computer-Related Distances

Frontmatter
Chapter 19. Distances on Real and Digital Planes
Abstract
Any L p -metric (as well as any norm metric for a given norm | | . | | on \(\mathbb{R}^{2}\)) can be used on the plane \(\mathbb{R}^{2}\), and the most natural is the L 2-metric, i.e., the Euclidean metric \(d_{E}(x,y) = \sqrt{(x_{1 } - y_{1 } )^{2 } + (x_{2 } - y_{2 } )^{2}}\) which gives the length of the straight line segment [x, y], and is the intrinsic metric of the plane.
Michel Marie Deza, Elena Deza
Chapter 20. Voronoi Diagram Distances
Abstract
Given a finite set A of objects A i in a space S, computing the Voronoi diagram of A means partitioning the space S into Voronoi regions V (A i ) in such a way that V (A i ) contains all points of S that are “closer” to A i than to any other object A j in A.
Michel Marie Deza, Elena Deza
Chapter 21. Image and Audio Distances
Abstract
Image Processing treats signals such as photographs, video, or tomographic output. In particular, Computer Graphics consists of image synthesis from some abstract models, while Computer Vision extracts some abstract information: say, the 3D description of a scene from video footage of it. From about 2000, analog image processing (by optical devices) gave way to digital processing, and, in particular, digital image editing (for example, processing of images taken by popular digital cameras).
Michel Marie Deza, Elena Deza
Chapter 22. Distances in Networks
Abstract
A network is a graph, directed or undirected, with a positive number (weight) assigned to each of its arcs or edges. Real-world complex networks usually have a gigantic number N of vertices and are sparse, i.e., with relatively few edges.
Michel Marie Deza, Elena Deza

Distances in Natural Sciences

Frontmatter
Chapter 23. Distances in Biology
Abstract
Distances are mainly used in Biology to pursue basic classification tasks, for instance, for reconstructing the evolutionary history of organisms in the form of phylogenetic trees. In the classical approach those distances were based on comparative morphology, physiology, mating studies, paleaontology and immunodiffusion. The progress of modern Molecular Biology also allowed the use of nuclear- and amino-acid sequences to estimate distances between genes, proteins, genomes, organisms, species, etc.
Michel Marie Deza, Elena Deza
Chapter 24. Distances in Physics and Chemistry
Abstract
Physics studies the behavior and properties of matter in a wide variety of contexts, ranging from the submicroscopic particles from which all ordinary matter is made (Particle Physics) to the behavior of the material Universe as a whole (Cosmology).
Michel Marie Deza, Elena Deza
Chapter 25. Distances in Earth Science and Astronomy
Abstract
In Geography, spatial scales are shorthand terms for distances, sizes and areas. For example, micro, meso, macro, mega may refer to local (0.001–1), regional (1–100), continental (100–10,000), global ( > 10,000) km, respectively.
Michel Marie Deza, Elena Deza
Chapter 26. Distances in Cosmology and Theory of Relativity
Abstract
The Universe is defined as the whole space-time continuum in which we exist, together with all the energy and matter within it.
Michel Marie Deza, Elena Deza

Real-World Distances

Frontmatter
Chapter 27. Length Measures and Scales
Abstract
The term length has many meanings: distance, extent, linear measure, span, reach, end, limit, etc.; for example, the length of a train, a meeting, a book, a trip, a shirt, a vowel, a proof. The length of an object is its linear extent, while the height is the vertical extent, and width (or breadth) is the side-to-side distance at 90∘ to the length, wideness. The depth is the distance downward, distance inward, deepness, profundity, drop.
Michel Marie Deza, Elena Deza
Chapter 28. Distances in Applied Social Sciences
Abstract
In this chapter we present selected distances used in real-world applications of Human Sciences. In this and the next chapter, the expression of distances ranges from numeric (say, in m) to ordinal (as a degree assigned according to some rule) and nominal.
Michel Marie Deza, Elena Deza
Chapter 29. Other Distances
Abstract
In this chapter we group together distances and distance paradigms which do not fit in the previous chapters, being either too practical (as in equipment), or too general, or simply hard to classify.
Michel Marie Deza, Elena Deza
Backmatter
Metadaten
Titel
Encyclopedia of Distances
verfasst von
Michel Marie Deza
Elena Deza
Copyright-Jahr
2016
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-662-52844-0
Print ISBN
978-3-662-52843-3
DOI
https://doi.org/10.1007/978-3-662-52844-0

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