''of a function, set of values of a function, image of a function''
...$x \in X$ with $f(x) = y$. Thus, the range of values of a function is the image of its [[domain of definition]], $Y_f = f(X)$.
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...[zero-dimensional space]]. Also, every [[compact space]] is the continuous image of a closed subspace of a Cantor cube.
The terminology is slightly ambiguous: a [[dyadic space]] is a continuous image of a Cantor cube, so that "dyadic discontinuum" could also mean "totally
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...the area $S(E)$ of that part of the convex surface $F$ that has spherical image $E\subset\Omega$. This definition remains meaningful for general convex sur
...lled the spherical image of $x$. The procedure for obtaining the spherical image of a point goes under the name of spherical mapping, cf. [[Spherical map|Sp
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$#C+1 = 33 : ~/encyclopedia/old_files/data/I050/I.0500150 Image of a morphism
A concept similar to that of the image of a mapping of one set into another. However, in category theory there are
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$#C+1 = 12 : ~/encyclopedia/old_files/data/S085/S.0805800 Small image
is closed (cf. [[Closed mapping|Closed mapping]]) if and only if the small image $ f ^ { \sharp } U $
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...ts with $n=1$. After the introduction of a suitable topology, the analytic image is converted to the [[Riemann surface]] of the given function.
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...us mapping|Continuous mapping]]) of topological spaces under which the pre-image of every point is compact. Perfect mappings are akin to continuous mappings
...pre-image, and the Suslin number of the image is equal to that of the pre-image. If a completely-regular $T_1$-space is mapped onto a completely-regular $T
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...ssible to select a finite number of sets so that $y$ is located inside the image of their union. It is particularly important that the product of any collec
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The topological image of a [[Disc|disc]] (cf. also [[Disc, topological|Disc, topological]]).
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A homeomorphic image of a circle (cf. also [[Homeomorphism|Homeomorphism]]). Named after C. Jord
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A continuous image of a [[Borel set]]. Since any Borel set is a continuous
image of the set of irrational numbers, an ${\mathcal A}$-set can be defined as a
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is entirely covered by the image of the disc $ | z | < 1 $
there are points not belonging to the image only if $ f $
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...space]] such that $\pi(x)^*=\pi(x^*)$ for all $x\in A$, where $x^*$ is the image of $x$ under the involution of $A$.
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...w Y$ is a morphism of algebraic varieties, then $f(X)$ (and, moreover, the image under $f$ of any constructible subset in $X$) is a constructible subset in
...ed constructible if $h(X)$ is finite and if for any point $t\in T$ the pre-image $h^{-1}(t)$ is a constructible subset in $X$.
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...onto a topological space $Y$ which is irreducible (that is, $Y$ is not the image of any proper closed subset of $X$, cf. also [[Irreducible mapping|Irreduci
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...xtensions $A \subset \Omega$ and $B \subset \Omega$. It is the same as the image of the homomorphism $ \phi : A \otimes_{k} B \to \Omega$ that maps the tens
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...s such particularly useful for the analysis of geometrical structure in an image. It has been applied in various disciplines, such as mineralogy, medical di
...and their interrelations. In general, this procedure results in non-linear image operators, some of which will be described briefly below.
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A mapping of one topological space to another, under which the image of every closed set is a closed set. The class of continuous closed mapping
...e the above-mentioned properties. The explanation for this is that the pre-image of a point under a continuous closed mapping need not be compact, though in
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...delta$-diagonal are also metrizable. The perfect image and the perfect pre-image of a paracompact $P$-space are also paracompact $P$-spaces.
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...puter vision, aimed at detecting and characterizing discontinuities in the image domain.
...on in terms of edges is also compact in the sense that the two-dimensional image pattern is represented by a set of one-dimensional curves. For these reason
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...all inverse images of points are finite. Mappings under which the inverse image of every compact set is compact are called $ k $-
is open if and only if its complete inverse image $ f ^ { - 1 } ( V) $
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A mapping of one topological space into another under which the image of every open set is itself open.
...by a complete metric. If a [[Paracompact space|paracompact space]] is the image of a [[Complete metric space|complete metric space]] under a continuous ope
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...f the $2$-dimensional sphere $S^2$; also, every cactoid is a monotone open image of $S^2$.
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...tremal disconnectedness is not preserved by perfect mappings. However, the image of an extremally-disconnected space under a continuous open mapping is an e
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...et: \(X\) satisfies the first axiom of countability; \(X\) is a continuous image of an ordered compactum; \(X\) is hereditarily normal; \(X\) is hereditaril
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A '''Suslin space''' is a continuous image of a Polish space.
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...rplane section in $\PP^n$. The Veronese mapping is a closed imbedding; its image $v_m(\PP^n)$ is called a Veronese variety, and is defined by the equations
in $\PP^n$ its image with respect to the Veronese mapping $v_m$ is the intersection of the Veron
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a pre-image $ x \in L _ {n} $
will then be the image of an element $ z \in Z _ {n-} 1 ( K _ {\mathbf . } ) $
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..., depending on the orientation. Certain knots can differ from their mirror image owing to the fact that their branched coverings are asymmetric varieties.
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...nding to the deformation over a principal base, as well as to the Clifford image of the asymptotic lines of a surface in an elliptic space (which is the [[R
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It follows from the Tarski–Seidenberg theorem that the image under a polynomial mapping $ \mathbf R ^ {n} \rightarrow \mathbf R ^ {m
...t on a real-analytic manifold is, by definition, a set that is locally the image of a semi-analytic set under an analytic mapping. The points of a subanalyt
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...ts of the kernel $\ker d$ are known as ''cycles''; and the elements of the image $\mathrm{im}\, d$ are called ''boundaries''. The ''homology'' of $C$ is th
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...rithmic capacity|logarithmic capacity]]. As an example, any function whose image does not contain a disc of a fixed radius $\epsilon > 0$ centred at $a +
...proved that an analytic function is in $\operatorname{BMOA}$ provided its image [[Riemann surface]] (viewed as spread out over the complex plane) has the f
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The Lindelöf principle states that under these conditions: 1) the inverse image $ \widetilde{D} _ \rho $
lies inside the inverse image $ D _ \rho $
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...an open connected set. The basic property here is that of openness of the image, which follows from Rouché's theorem or from the principle of the argument
say, then the image $ f ( X) \subset Y $
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..., touching an arc of the circle $L$ (the image of $K$) at that point; this image is represented in Fig. band is the Zhukovskii profile.
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A continuous mapping $h : f(X) \rightarrow X$ of the image $f(X)$ of a continuous mapping $f : X \rightarrow Y$ of topological spaces
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is always the image of some horizontal distribution $ \Delta $
is obtained as the image $ \pi ^ {*} \overline \Delta \; $,
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such that the image of any set $ U $
while the inverse image $ f ^ { - 1 } ( y) $
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...z$ vary monotonically on the image of the trajectory in $|z|<\alpha$. Each image of a trajectory twists around the point $z=0$ and behaves asymptotically li
...ha$ in the other direction. The modulus of $z$ varies monotonically on the image of the trajectory in $|z|<\alpha$. Different images of trajectories have di
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...ontinuum. The topological product of continua is a continuum, a continuous image of a continuum is a continuum, the components of a Hausdorff compactum are
Every locally connected metric continuum is a continuous image of a closed segment (the Hahn–Mazurkiewicz theorem). A non-degenerate con
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...continuous mapping $g : X \to Q^n$ that coincides with $f$ on the inverse image $f^{-1} S^{n-1}$ of the boundary $S^{n-1}$ of $Q^n$ and takes $X$ into $S^{
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...ath.org/legacyimages/c/c025/c025720/c02572018.png" /> the complete inverse
image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
...ediaofmath.org/legacyimages/c/c025/c025720/c02572049.png" />, that is, the
image of any bounded set <img align="absmiddle" border="0" src="https://www.encyc
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...orithms, aimed at detecting and classifying the nature of junctions in the image domain. A main reason why corner detection is important is that junctions p
One way of detecting junctions directly from image intensities consists of finding points at which the gradient magnitude $| \
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...try with respect to a circle). The centre of an inversion does not have an image. An inversion with negative power $k$ is equivalent to the inversion with t
Sometimes an [[ideal point]] $\infty$ is regarded as the image of the centre of an inversion under this inversion, especially when one con
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...mbedding]]. If a curve $X$ is not hyper-elliptic and has genus 2, then its image in the projective space $P^{g-1}$ under a canonical imbedding has degree $2
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...$u$ of a set $X$ (in general endowed with some structure) into itself. The image of an element $x \in X$ under the transformation $u$ is denoted by $u(x)$,
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...encyclopediaofmath.org/legacyimages/c/c025/c025710/c0257105.png" /> of its
image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
a) the inverse
image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
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...eta$ if $\alpha$ is a limit ordinal number; $Z_\alpha$ is the complete pre-image of the centre of the quotient group $G/Z_\beta$ if $\alpha = \beta+1$ is a
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...f $S$ is a normal sub-semi-group if and only if it is the complete inverse image of the [[unit element]] under some homomorphism of $S$ onto a [[Monoid|semi
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...pace of codimension 1. There is always some element $a \in F$ not in the image of $A$, and so the corresponding Artin-Schreier polynomial has no root in
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The geometric image of the modulus $|f(z)|$ of an analytic function $f(z)$, $z=x+iy$. The analy
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The indicatrix of tangents is also called the spherical tangent image. See also [[Spherical indicatrix|Spherical indicatrix]].
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===Image enhancement.===
...nd to be generated by sharp edges. Thus, one can reduce the contrast of an image by diminishing the higher-order terms of its Walsh–Fourier series, and in
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...(G_f) $ of the second component does not produce the codomain but only the image of the domain.
$ f_\ast (S) $ is called the ''image'' of $S$ under $f$, usually denoted as $f(S)$, and
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...lign="top"> W. Sierpiński, "Sur une courbe cantorienne qui contient une image binniro que et continue de toute courbe donnée" ''C.R. Acad. Sci. Paris''
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...modulo $J$ (i.e. for every idempotent of $R/J$ there is an idempotent pre-image in $R$), then $R$ is isomorphic to the full matrix ring of a [[local ring]]
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...encountered analogues of the Paley–Wiener theorem are a description of the image of the space $ C _ {0} ^ \infty ( G) $
...itely-differentiable functions of compact support and a description of the image of the space $ S ( G) $
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A continuous image of a segment filling the interior of a square (or triangle). It was discove
...onditions a) $X$ is a locally connected space and b) $X$ is the continuous image of an interval, are equivalent.
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...the image of an intersection is the sum of the images. In particular, the image of a point is a hyperplane and vice versa. A necessary and sufficient condi
its image is the orthogonal complement of $ V $
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is the image $ D ^ {3} $
and 3) the pre-image $ \phi ^ {-} 1 ( D _ {i} ) $
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is contained in the image of the homomorphism
the group which is the image of the natural homomorphism
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...al surface carries a focal net consisting of geodesic lines. The spherical image of the developable surfaces of a Guichard congruence is a [[Chebyshev net|C
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...an additive category there exists an image $\operatorname{Im}(u)$ and a co-image $\operatorname{Coim}(u)$, then there exists a unique morphism $u\colon\oper
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with the same spherical image, in which each $ \Sigma _ {k+ 1} $
In general, the search for the spherical image of an Egorov system $ \Sigma $
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...e $S^n$ into the Euclidean space $E^n$ there exist antipodes with a common image; 2) Any mapping of the sphere $S^n$ into itself in which the images of anti
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An example of a curve containing the topological image of any curve (and, in addition, of every one-dimensional separable metrizab
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...(Y) \to \CH(X)$ is a homomorphism of rings, and for $f$ proper, the direct-image homomorphism $f_*: \CH(X)\to \CH(Y)$ is a homomorphism of $\CH(Y)$-modules.
...g that preserves the degree and commutes with the inverse-image and direct-image homomorphisms.
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that is the image of $ E $
The image of $ E $
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...] real-valued function on it takes all intermediate values. The continuous image of a connected space, the [[topological product]] of connected spaces, and
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...h was later termed quasi-continuity, where it is required that the inverse image of every open set is semi-open, i.e., is contained in the closure of its in
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at this point (i.e. the dimension of the image $ df ( T _ {x _ {0} } M) $
The set of all critical points is called the critical set, the image $ f( x _ {0} ) $
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...hat is a $T_2$-space is a compactum, and every compactum is the continuous image of the Cantor set (Aleksandrov's theorem). The product of a finite or count
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...s) in an image (a picture), by applying a coordinate transformation to the image such that all the points belonging to a line (or a curve) are mapped into a
image plane gives rise to a two-parameter family. When using the normal parametri
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The image (under a translation) of a vector subspace $M$ with one-dimensional quotien
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a pre-image) of the space onto the plane $ \Pi ^ \prime $
the image) is called the projection of $ A $.
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...e $Y$ for which a set $v\subseteq Y$ is open in $Y$ if and only if its pre-image $f^{-1}v$ is open in $X$. If one is given a mapping $f$ of a topological sp
...t mapping be such. The restriction of a quotient mapping to a complete pre-image does not have to be a quotient mapping. More precisely, if $f:X\to Y$ is a
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..._n=0\}$ can conveniently be represented by a Hartogs diagram, viz., by the image of the Hartogs domain under the mapping $('z,z_n)\to('z,|z_n|)$.
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is called the range, or image, of $ R $.
is called the image of $ a $
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...|topological space]] in which any two points can be joined by a continuous image of a simple arc; that is, a space $ X $
...space is connected (cf. [[Connected space|Connected space]]). A continuous image of a path-connected space is path-connected.
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...pace, then a linear representation of $X$ on $E$ is a representation whose image contains only continuous linear operators on $E$. The space $E$ is called t
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...ng]] of a topological space $X$ onto a topological space $Y$ such that the image of every closed set in $X$, other than $X$ itself, is different from $Y$. I
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is the tangent bundle) contains in its image all vectors normal to $ X $
in the pre-image $ V $
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...3$ an edge of regression represents a smooth curve $L$ in $M$, with smooth image $f(L)\subset E^3$, such that for any $p\in f(L)$ the intersection of $M$ wi
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the image of an element $ (a, b) \in A \times A $
...the orthogonal group is called the subgroup of Euclidean motions. The pre-image of the special linear group SGL is called the equi-affine subgroup (cf. [[A
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...pact space. In particular, any finite space is quasi-compact. A continuous image of a quasi-compact space is quasi-compact. A topological product of any num
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...l and if every idempotent of the quotient ring $R/J$ has an idempotent pre-image in $R$. The first condition can be replaced by the requirement of classical
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When domain and image are fixed to certain shapes, pictures become equivalent to many other combi
...also be computed by the jeu de taquin at the image side, to transform the image $ \psi $
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...a function expressing the property that equivalent elements have the same image under the function.
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...m $f$. The functor $f^*$ adjoint to $f*$ on the left is called the inverse-image functor. In particular, the stalk of $\mathcal{F}$ at a geometric point $\e
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...n interval. However, an exponent of uncountable weight is not a continuous image of the Tikhonov cube $ I ^ \tau $.
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.... Gauss in 1814. The image under the spherical map is called the spherical image of $ M ^ {k} $.
is the inverse image of the metric form of $ S ^ {k} $,
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A theory of multi-scale representation of sensory data developed by the image processing and computer vision communities. The purpose is to represent sig
...e a compact way to characterize the local image structure around a certain image point at any scale. Specifically, the output from scale-space derivatives c
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...ke continuum is a pseudo-arc. Every snake-like continuum is the continuous image of a pseudo-arc and the limit of the inverse spectrum of arcs.
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...oup of the [[fundamental group]] $\pi_1(Y,y_0)$, where $y_0 \in Y$, by the image of the group $\pi_1(X,x_0)$, where $p(x_0)=y_0$, under the homomorphism ind
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is the area of the spherical image of $ U $,
is defined as the length of the spherical image of $ C $(
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the image of the hyperbolic disc with centre at the point $ z _ {0} \in D $
...erbolic metric yields a more precise estimate of the domain containing the image of a hyperbolic disc in $ D $,
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...e corresponding points. If $V^n$ is complete, then the equi-distant is the image under the [[Exponential mapping|exponential mapping]] of the vectors of con
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...a cyclic group, that for every $\phi : H \rightarrow G$, if $g$ is in the image of $\phi$ then $\phi$ is an [[epimorphism]] ([[surjection|surjective]] as a
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A [[Topological space|topological space]] containing a homeomorphic image of every topological space of a certain class. Examples are: 1) $ C [ 0 ,
...notion of universal space: Every space in a certain class is a continuous image of the space in question. E.g., the [[Cantor set|Cantor set]] is universal
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