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Map algebra is an algebra for manipulating geographic data, primarily fields.Developed by Dr. Dana Tomlin and others in the late 1970s, it is a set of primitive operations in a geographic information system (GIS) which allows one or more raster layers ("maps") of similar dimensions to produce a new raster layer (map) using mathematical or other operations such as addition, subtraction etc.
Maps of certain kinds have been given specific names. These include homomorphisms in algebra, isometries in geometry, operators in analysis and representations in group theory. [2] In the theory of dynamical systems, a map denotes an evolution function used to create discrete dynamical systems. A partial map is a partial function.
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication.
For b ∈ L, let F b be the map / (). Then F b ≠ F c if b ≠ c. Moreover, the K-linear transformations from L to K are exactly the maps of the form F b as b varies over the field L. When K is the prime subfield of L, the trace is called the absolute trace and otherwise it is a relative trace. [4]
In mathematics a positive map is a map between C*-algebras that sends positive elements to positive elements. A completely positive map is one that satisfies a stronger, more robust condition. A completely positive map is one that satisfies a stronger, more robust condition.
A bilinear map is a function: such that for all , the map (,) is a linear map from to , and for all , the map (,) is a linear map from to . In other words, when we hold the first entry of the bilinear map fixed while letting the second entry vary, the result is a linear operator, and similarly for when we hold the second entry fixed.
In general topology, an embedding is a homeomorphism onto its image. [3] More explicitly, an injective continuous map : between topological spaces and is a topological embedding if yields a homeomorphism between and () (where () carries the subspace topology inherited from ).
The first diagram is the dual of the one expressing associativity of algebra multiplication (called the coassociativity of the comultiplication); the second diagram is the dual of the one expressing the existence of a multiplicative identity. Accordingly, the map Δ is called the comultiplication (or coproduct) of C and ε is the counit of C.