A function (or mapping) is called '''surjective''' if the image of its domain ([[range of values]]) coincides with its rang
In other words, a function $ f : A \to B $ from a set $A$ to a set $B$ is
1 KB (230 words) - 21:19, 18 December 2014
''of a differentiable function $f: \mathbb R^m \to \mathbb R^n$ with $n\leq m$''. A value $c\in \mathbb R^
differential of $f$ is surjective at every preimage of $c$. The concept can be readily extended to the case w
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A function (or mapping) is called '''injective''' if distinct arguments have distinct
In other words, a function $ f : A \to B $ from a set $A$ to a set $B$ is
1 KB (234 words) - 12:11, 12 December 2013
A function (or mapping) is called '''bijective'''
if it is both [[Injection|injective]] and [[Surjection|surjective]].
1 KB (239 words) - 12:13, 12 December 2013
...ps any $y$ precisely into the element $x$ such that $f(x) =y$. The inverse function $g$ is then characterized by the following property:
...by $f^{-1}$ and hence it is commonly written $f^{-1} (y) = x$. The inverse function must not be confused with the ''preimage'' of $y$, which is usually denoted
10 KB (1,719 words) - 16:56, 30 November 2014
...whose classes are the pre-images of the elements of $B$ (cf. [[Kernel of a function]]).
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* The set $T(\Sigma)$ is [[Computable function|computational]].
...ture and let $A$ be a $\Sigma$-algebra. As usual, the interpretation of a function symbol $f\in F$ in the algebra $A$ may be designated as $f^A$. Then the <i
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...space $ X $, and if $ \chi $ is a continuous and uni-modular scalar-valued function on $ Y $, then
...uch that $ |f(x)| < \epsilon $ for $ x \in X \setminus K $). A uni-modular function $ \chi $ is one for which $ |\chi(y)| = 1 $ for all $ y \in Y $.
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$#C+1 = 23 : ~/encyclopedia/old_files/data/S091/S.0901660 Symmetric function
A function that does not change under any permutation of its independent variables. Th
5 KB (731 words) - 07:17, 24 March 2023
$#C+1 = 39 : ~/encyclopedia/old_files/data/R080/R.0800260 Recursive function,
''partial recursive function''
5 KB (727 words) - 15:34, 4 March 2022
A generalization of the notion of a [[Meromorphic function|meromorphic function]]. Let $ X $
is surjective and defines a bijective mapping of the set of irreducible components. If $
4 KB (563 words) - 08:00, 6 June 2020
in each of which is defined a meromorphic function $ f _ \alpha $;
compatibility condition). It is required to construct a function $ f $
16 KB (2,209 words) - 11:03, 26 March 2023
is holomorphically convex if and only if it admits a proper surjective holomorphic mapping $ \phi $
...x analytic spaces" , ''Contributions to function theory. Internat. Colloq. Function Theory, Bombay 1960'' , Tata Inst. (1960) pp. 1–15</TD></TR><TR><TD val
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...$ as a $\Sigma$-number-algebra and $\alpha\colon C \longrightarrow A$ as a surjective $\Sigma$-algebra-morphism from $C$ onto $A$. Usually, $\alpha$ is chosen to
Using a coordinatization $(C,\alpha)$, the notions of [[Computable function|computability]] (and other areas of theoretical computer science) can be tr
3 KB (404 words) - 09:05, 21 April 2013
is surjective and has the property $ \chi ( 1) = 1 $.
determines a function $ \widetilde{a} $
3 KB (461 words) - 16:43, 4 June 2020
and is a far-reaching generalization of the ordinary exponential function regarded as a mapping of a straight line into itself.
is a surjective mapping of $ M _ {p} $
4 KB (600 words) - 19:38, 5 June 2020
be a surjective holomorphic mapping. Suppose this is an $ m $-
is the different of the extension of algebraic function fields determined by $ f $.
4 KB (523 words) - 08:11, 6 June 2020
...sion $v(x_1\leftarrow t_1,\ldots,x_n\leftarrow t_n)$ denotes an assignment function $v$ changed at variables $x_i$ to $t_i$.
...e|EM85}}. With help of these properties one can show for example that no (surjective) $\Sigma$-algebra-morphism exist between two given $\Sigma$-algebras {{Cite
8 KB (1,390 words) - 18:36, 11 December 2020
An object $ (X,\pi,B) $, where $ \pi: X \to B $ is a continuous surjective mapping of a topological space $ X $ onto a topological space $ B $ (i.e.,
...ld be called a morphism (respectively, an isomorphism) only if the induced function $ f: B \to B_{1} $ were continuous (respectively, a [[Homeomorphism|homeomo
5 KB (754 words) - 01:34, 10 December 2016
($\langle \, .\, ,\, . \, \rangle$ being the scalar product) is the support function, maps this cone isomorphically into the space $C ( S ^ { n - 1 } )$ of cont
...ex bodies. Common examples are the mean width and the Steiner point $s$. A surjective mapping $\psi : \mathcal{K} ^ { n } \rightarrow \mathcal{K} ^ { n }$ with $
4 KB (596 words) - 15:30, 1 July 2020