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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]
Graph of the sine map ( 4-1 ) Orbit diagram of the sine map ( 4-1 ) The bifurcation pattern shown above for the logistic map is not limited to the logistic map . It appears in a number of maps that satisfy certain conditions . The following dynamical system using sine functions is one example :
A function may also be called a map or a mapping, but some authors make a distinction between the term "map" and "function". For example, the term "map" is often reserved for a "function" with some sort of special structure (e.g. maps of manifolds). In particular map may be used in place of homomorphism for the sake of succinctness (e.g ...
However, the exponential function is a holomorphic function with a nonzero derivative, but is not one-to-one since it is periodic. [ 2 ] The Riemann mapping theorem , one of the profound results of complex analysis , states that any non-empty open simply connected proper subset of C {\displaystyle \mathbb {C} } admits a bijective conformal map ...
A function is bijective if and only if it is both surjective and injective. If (as is often done) a function is identified with its graph, then surjectivity is not a property of the function itself, but rather a property of the mapping. [7] This is, the function together with its codomain.
An injective function need not be surjective (not all elements of the codomain may be associated with arguments), and a surjective function need not be injective (some images may be associated with more than one argument). The four possible combinations of injective and surjective features are illustrated in the adjacent diagrams.
Self-concordant function; Semi-differentiability; Semilinear map; Set function; List of set identities and relations; Shear mapping; Shekel function; Signomial; Similarity invariance; Soboleva modified hyperbolic tangent; Softmax function; Softplus; Splitting lemma (functions) Squeeze theorem; Steiner's calculus problem; Strongly unimodal ...
Given a map :, the mapping cylinder is a space , together with a cofibration ~: and a surjective homotopy equivalence (indeed, Y is a deformation retract of ), such that the composition equals f. Thus the space Y gets replaced with a homotopy equivalent space M f {\displaystyle M_{f}} , and the map f with a lifted map f ~ {\displaystyle {\tilde ...