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Weakly-contractive
In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance. These maps are the morphisms in the category of metric spaces, Met. Such functions are always continuous functions. They are also called Lipschitz functions with Lipschitz constant 1, nonexpansive maps, nonexpanding maps, weak contractions, or short maps. Specifically, suppose that X and Y are metric spaces and f is a function from X to Y. Thus we have a metric map when, for any points x and y in X, d_(f(x),f(y)) \leq d_(x,y) . \! Here d_X and d_Y denote the metrics on X and Y respectively. Examples Consider the metric space ,1/2/math> with the Euclidean metric. Then the function f(x)=x^2 is a metric map, since for x\ne y, , f(x)-f(y), =, x+y, , x-y, <, x-y, . Category of metric maps The |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. ''Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lipschitz Maps
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the ''Lipschitz constant'' of the function (and is related to the '' modulus of uniform continuity''). For instance, every function that is defined on an interval and has a bounded first derivative is Lipschitz continuous. In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction, is used in the Banach fixed-point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comment
Comment may refer to: Computing * Comment (computer programming), explanatory text or information embedded in the source code of a computer program * Comment programming, a software development technique based on the regular use of comment tags Law * Public comment, a term used by various U.S. government agencies, referring to comments invited regarding a report or proposal * Short scholarly papers written by members of a law review * Comments on proposed rules under the rulemaking process in United States administrative law Media and entertainment * ''Comment'' (TV series), a 1958 Australian television series * ''Comment'' (album), a 1970 album by Les McCann * "Comment", a 1969 song by Charles Wright & the Watts 103rd Street Rhythm Band * ''Comment'', a quarterly journal published by Cardus * ''Comment'', later ''aCOMMENT'', an Australian quarterly literary magazine published 1940-7 * Comment section, a user-generated content feature of Web content allowing readers to publi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contraction Mapping
In mathematics, a contraction mapping, or contraction or contractor, on a metric space (''M'', ''d'') is a function ''f'' from ''M'' to itself, with the property that there is some real number 0 \leq k < 1 such that for all ''x'' and ''y'' in ''M'', : The smallest such value of ''k'' is called the Lipschitz constant of ''f''. Contractive maps are sometimes called Lipschitzian maps. If the above condition is instead satisfied for ''k'' ≤ 1, then the mapping is said to be a non-expansive map. More generally, the idea of a contractive mapping can be defined for maps between metric spaces. Thus, if (''M'', ''d'') and (''N'', ''d) are two metric spaces, then is a contractive mapping if there is a constant such that : |
Hausdorff Distance
In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty set, non-empty compact space, compact subsets of a metric space into a metric space in its own right. It is named after Felix Hausdorff and Dimitrie Pompeiu. Informally, two sets are close in the Hausdorff distance if every point of either set is close to some point of the other set. The Hausdorff distance is the longest distance someone can be forced to travel by an adversary who chooses a point in one of the two sets, from where they then must travel to the other set. In other words, it is the greatest of all the distances from a point in one set to the closest point in the other set. This distance was first introduced by Hausdorff in his book ''Grundzüge der Mengenlehre'', first published in 1914, although a very close relative appeared in the doctoral thesis of Maurice René Fré ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is derived . The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may often be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism. A common example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension one of a vector space are isomorphic and cannot be identified. An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon X\to Y, its inverse f^\colon Y\to X admits an explicit description: it sends each element y\in Y to the unique element x\in X such that . As an example, consider the real-valued function of a real variable given by . One can think of as the function which multiplies its input by 5 then subtracts 7 from the result. To undo this, one adds 7 to the input, then divides the result by 5. Therefore, the inverse of is the function f^\colon \R\to\R defined by f^(y) = \frac . Definitions Let be a function whose domain is the set , and whose codomain is the set . Then is ''invertible'' if there exists a function from to such that g(f(x))=x for all x\in X and f(g(y))=y for all y\in Y. If is invertible, then there is exactly one functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bijective
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set. A function is bijective if it is invertible; that is, a function f:X\to Y is bijective if and only if there is a function g:Y\to X, the ''inverse'' of , such that each of the two ways for composing the two functions produces an identity function: g(f(x)) = x for each x in X and f(g(y)) = y for each y in Y. For example, the ''multiplication by two'' defines a bijection from the integers to the even numbers, which has the ''division by two'' as its inverse function. A function is bijective if and only if it is both injective (or ''one-to-one'')—meaning that each element in the codomain is mappe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' meaning "equal", and μέτρον ''metron'' meaning "measure". If the transformation is from a metric space to itself, it is a kind of geometric transformation known as a motion. Introduction Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space. In a two-dimensional or three-dimensional Euclidean space, two geometric figures are congruent if they are related by an isometry; the isometry that relates them is either a rigid motion (translation or rotati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subcategory
In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, a subcategory of ''C'' is a category obtained from ''C'' by "removing" some of its objects and arrows. Formal definition Let ''C'' be a category. A subcategory ''S'' of ''C'' is given by *a subcollection of objects of ''C'', denoted ob(''S''), *a subcollection of morphisms of ''C'', denoted hom(''S''). such that *for every ''X'' in ob(''S''), the identity morphism id''X'' is in hom(''S''), *for every morphism ''f'' : ''X'' → ''Y'' in hom(''S''), both the source ''X'' and the target ''Y'' are in ob(''S''), *for every pair of morphisms ''f'' and ''g'' in hom(''S'') the composite ''f'' o ''g'' is in hom(''S'') whenever it is defined. These conditions ensure that ''S'' is a category in its own right: its collection of objects is ob(''S'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Metric Spaces
In category theory, Met is a category that has metric spaces as its objects and metric maps ( continuous functions between metric spaces that do not increase any pairwise distance) as its morphisms. This is a category because the composition of two metric maps is again a metric map. It was first considered by . Arrows The monomorphisms in Met are the injective metric maps. The epimorphisms are the metric maps for which the domain of the map has a dense image in the range. The isomorphisms are the isometries, i.e. metric maps which are injective, surjective, and distance-preserving. As an example, the inclusion of the rational numbers into the real numbers is a monomorphism and an epimorphism, but it is clearly not an isomorphism; this example shows that Met is not a balanced category. Objects The empty metric space is the initial object of Met; any singleton metric space is a terminal object. Because the initial object and the terminal objects differ, there are no zero objec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
