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A map is a function, as in the association of any of the four colored shapes in X to its color in Y. In mathematics, a map or mapping is a function in its general sense. [1] These terms may have originated as from the process of making a geographical map: mapping the Earth surface to a sheet of paper. [2] The term map may be used to distinguish ...
A map : is called a closed map or a strongly closed map if it satisfies any of the following equivalent conditions: Definition: f : X → Y {\displaystyle f:X\to Y} maps closed subsets of its domain to closed subsets of its codomain; that is, for any closed subset C {\displaystyle C} of X , {\displaystyle X,} f ( C ) {\displaystyle f(C)} is a ...
Some authors call a function F : X → 2 Y a set-valued function only if it satisfies the additional requirement that F(x) is not empty for every x ∈ X; this article does not require this. Definition and notation: If F : X → 2 Y is a set-valued function in a set Y then the graph of F is the set Gr F := { (x, y) ∈ X × Y : y ∈ F(x) }.
Cartographic design or map design is the process of crafting the appearance of a map, applying the principles of design and knowledge of how maps are used to create a map that has both aesthetic appeal and practical function. [1]
Intuitively, P(X,Y) means F(X) = Y. Then whenever F(X) would appear in a statement, you can replace it with a new symbol Y of type U and include another statement P(X,Y). To be able to make the same deductions, you need an additional proposition: For all X of type T, for some unique Y of type U, P(X,Y).
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.
A map is a local diffeomorphism if and only if it is a smooth immersion (smooth local embedding) and an open map. The inverse function theorem implies that a smooth map f : X → Y {\displaystyle f:X\to Y} is a local diffeomorphism if and only if the derivative D f x : T x X → T f ( x ) Y {\displaystyle Df_{x}:T_{x}X\to T_{f(x)}Y} is a linear ...
Because the world is much more complex than can be represented in a computer, all geospatial data are incomplete approximations of the world. [9] Thus, most geospatial data models encode some form of strategy for collecting a finite sample of an often infinite domain, and a structure to organize the sample in such a way as to enable interpolation of the nature of the unsampled portion.