The natural mapping from the integers to the congruence classes modulo " n " that takes an integer to its congruence class modulo " n " respects products.
12.
The space " Y " above is aimed at subspace " X " if and only if the natural mapping j \ colon Y \ to \ operatorname { Aim } ( X ) is an isometric embedding.
13.
The natural mapping between the multimodal input, which is provided by several interaction modalities ( visual and auditory channel and sense of touch ), and information and tasks imply to manage the typical problems of human-human communication, such as ambiguity.
14.
If " Z " is a closed subspace of a Banach space " Y " then there exists a " canonical " surjection Q _ Z : Y \ to Y / Z defined via the natural mapping y \ mapsto y + Z.
15.
There is a natural mapping between the integers 0, . . ., " n " ! & minus; 1 and permutations of " n " elements in lexicographic order, which uses the factorial representation of the integer, followed by an interpretation as a Lehmer code.
16.
When " G " is a Hausdorff abelian topological group, the group \ widehat { G } with the compact-open topology is a Hausdorff abelian topological group and the natural mapping from " G " to its double-dual " G ^ ^ " makes sense.
17.
There is a natural mapping between the integers 0, . . ., " n " ! " 1 ( or equivalently the numbers with " n " digits in factorial representation ) and permutations of " n " elements in lexicographical order, when the integers are expressed in factoradic form.
18.
Possibly what's confusing you ( it confuses me ) is that there are at least two different kinds of subtyping in programming languages : t d " u can mean t ?" u or it can mean t E " u ?" v for some v . What they have in common is a natural mapping from t to u, but it's an injection in one case and a surjection in the other.
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