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Metric Projection
In mathematics, a metric projection is a function that maps each element of a metric space to the set of points nearest to that element in some fixed sub-space. Formal definition Formally, let ''X'' be a metric space with distance metric ''d'', and let ''M'' be a fixed subset of ''X''. Then the metric projection associated with ''M'', denoted ''pM'', is the following set-valued function from ''X'' to ''M'':p_M(x) = \arg\min_ d(x,y)Equivalently:p_M(x) = \ = \The elements in the set \arg\min_ d(x,y) are also called elements of best approximation. This term comes from constrained optimization: we want to find an element nearer to ''x'', under the constraint that the solution must be a subset of ''M''. The function ''pM'' is also called an operator of best approximation. Chebyshev sets In general, ''pM'' is set-valued, as for every ''x'', there may be many elements in ''M'' that have the same nearest distance to ''x''. In the special case in which ''pM'' is single-valued, the set ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), function called a metric or distance function. Metric spaces are a general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a Conceptual metaphor , metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different bra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). The term is also frequently used metaphorically to mean a measurement of the amount of difference between two similar objects (such as statistical distance between probability distributions or edit distance between string (computer science), strings of text) or a degree of separation (as exemplified by distance (graph theory), distance between people in a social network). Most such notions of distance, both physical and metaphorical, are formalized in mathematics using the notion of a metric space. In the social sciences, distance can refer to a qualitative measurement of separation, such as social distance or psychological distance. Distances in physics and geometry The distance between physical locations can be defined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set-valued Function
A set-valued function, also called a correspondence or set-valued relation, is a mathematical function that maps elements from one set, the domain of the function, to subsets of another set. Set-valued functions are used in a variety of mathematical fields, including optimization, control theory and game theory. Set-valued functions are also known as multivalued functions in some references, but this article and the article Multivalued function follow the authors who make a distinction. Distinction from multivalued functions Although other authors may distinguish them differently (or not at all), Wriggers and Panatiotopoulos (2014) distinguish multivalued functions from set-valued functions (which they called ''set-valued relations'') by the fact that multivalued functions only take multiple values at finitely (or denumerably) many points, and otherwise behave like a function. Geometrically, this means that the graph of a multivalued function is necessarily a line of z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
