''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
4 KB (549 words) - 22:11, 5 June 2020
...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
3 KB (490 words) - 23:53, 10 December 2018
with the same spherical image, in which each $ \Sigma _ {k+ 1} $
In general, the search for the spherical image of an Egorov system $ \Sigma $
4 KB (653 words) - 07:55, 25 April 2022
...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
1 KB (163 words) - 06:23, 28 October 2017
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
1 KB (158 words) - 18:42, 12 August 2023
...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
7 KB (1,119 words) - 10:25, 1 January 2018
..._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
1 KB (197 words) - 19:18, 25 October 2014
...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
1 KB (178 words) - 16:20, 4 October 2014
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.
1 KB (189 words) - 17:00, 23 November 2023
...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
3 KB (538 words) - 20:07, 3 June 2017
...n interval. However, an exponent of uncountable weight is not a continuous image of the Tikhonov cube $ I ^ \tau $.
4 KB (611 words) - 19:38, 5 June 2020
.... 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.
1 KB (205 words) - 12:29, 12 April 2014
...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
1 KB (227 words) - 19:57, 10 December 2017
is the area of the spherical image of $ U $,
is defined as the length of the spherical image of $ C $(
4 KB (529 words) - 19:41, 5 June 2020
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 $,
4 KB (660 words) - 11:51, 1 May 2023
...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
1 KB (212 words) - 21:08, 14 April 2014
...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
1 KB (202 words) - 14:02, 12 November 2023
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
6 KB (949 words) - 07:39, 14 June 2022
...that $X$ be a paracompact feathered space (Arkhangel'skii's theorem). The image of a paracompact feathered space under a perfect mapping is a paracompact f
...<TD valign="top">[3]</TD> <TD valign="top"> V.V. Filippov, "The perfect image of a paracompact feathered space" ''Soviet Math. Dokl.'' , '''8''' (1967)
6 KB (962 words) - 20:32, 23 September 2017
and called image sources.
of the [[Laplace equation|Laplace equation]]. Adding a negative unit charge image at the point $ ( x _ {0} , - y _ {0} ) $
7 KB (1,055 words) - 07:59, 6 June 2020
...f $\mathbf{R}$ (or $\mathbf{C}$), it induces an open mapping, that is, the image of any open set $G \subset X$ is an open set in $\mathbf{R}$ (or $\mathbf{C
1 KB (204 words) - 16:14, 17 September 2017
...ple, there is a Peterson correspondence between a surface and it spherical image (cf. [[Spherical map|Spherical map]]), between a surface and its [[Rotation
..., and the base of the isometry for the new surfaces has the same spherical image as the original ones. For example, a sphere and a surface of constant posit
4 KB (660 words) - 19:40, 3 January 2021
A function (or mapping) is called '''surjective''' if the image of its domain ([[range of values]]) coincides with its range ([[codomain]])
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The image of the Whitehead homomorphism has been completely calculated (cf. [[#Refere
the Whitehead group is a monomorphism and its image is a direct summand in the group $ \pi _ {m} ^ {S} ( S ^ {0} ) $;
9 KB (1,201 words) - 20:31, 16 January 2024
...tes truth, and an empty disjunction denotes falsity) is called the formula image of the sequent.
1 KB (231 words) - 13:27, 14 February 2020
that is, the image of every non-identity element of a group under a regular automorphism must
2 KB (234 words) - 08:10, 6 June 2020
[[Vector space|vector space]] into itself under which the image of the
sum of two vectors is the sum of their images and the image of the
9 KB (1,480 words) - 17:14, 20 January 2022
be a non-singular closed subvariety. The weak pre-image of the ideal $ I $
...a sequence of blowing-ups with non-singular centres for which the weak pre-image $ I $
7 KB (965 words) - 08:11, 6 June 2020
...y non-empty open subset is [[ Everywhere-dense set|everywhere dense]]. The image of an irreducible topological space under a continuous mapping is irreducib
2 KB (241 words) - 08:04, 3 January 2016
...ry filter which is finer than a Cauchy filter is also a Cauchy filter. The image of a Cauchy filterbase under a uniformly-continuous mapping is again a Cauc
2 KB (279 words) - 21:05, 2 May 2016
...ition that $f(z)$ is bounded in $g$ by the more general condition that the image of $g$ under the mapping $w=f(z)$ is not dense in the $w$-plane. The streng
2 KB (264 words) - 21:15, 1 November 2014
...space whereby any two of its points can be joined by a path (a continuous image of a segment). An open connected subset is called a domain. Domains and con
...t. For closed maps the property of being monotone is equivalent to the pre-image of each connected subset being connected.
8 KB (1,209 words) - 06:00, 22 April 2023
...ical space|weight]], and every compactum can be obtained as the continuous image of a closed set of a Cantor cube of the same weight. These facts generalize
2 KB (246 words) - 17:35, 27 February 2018
...space $Y$ such that each point $y\in Y$ has a neighbourhood $U(y)$ the pre-image of which under $p$ is a union of open subsets that are mapped homeomorphica
...lement of $H$. The point $q(1)$ for the paths of one class is taken as the image of this class; this defines $p$. The topology in $X$ is uniquely determined
4 KB (700 words) - 17:04, 13 June 2020
...–Hirzebruch problem for quasi-complex manifolds consists in describing the image of the mapping $ \phi $.
Then the image of the homomorphism
5 KB (680 words) - 08:07, 6 June 2020
onto its image in $ P ^ {N} $
onto its image in $ P ^ { \mathop{\rm dim} | n K | } $.
7 KB (986 words) - 19:41, 5 June 2020
the image of $ \sigma ^ {n} $
...ual, cycles and boundaries are those chains which belong to the kernel and image of $ \partial ^ {n} $,
8 KB (1,130 words) - 08:14, 6 June 2020
and the image of entourages in $ X $
2 KB (257 words) - 17:46, 4 June 2020
A mapping of one space into another under which the pre-image of each point is compact (cf. [[Compact space|Compact space]]). The require
2 KB (261 words) - 16:55, 7 February 2011
...dules remain valid for semi-linear mappings. In particular, the kernel and image of a semi-linear mapping are submodules; semi-linear mappings of free modul
2 KB (264 words) - 22:23, 2 November 2013
...geometric figures [[#References|[7]]]. (In stereology a multi-dimensional image has to be described through its intersections with straight lines or planes
...of fractional dimensions [[#References|[5]]], mathematical morphology and image analysis [[#References|[6]]], random shape theory [[#References|[7]]].
7 KB (1,051 words) - 09:27, 22 August 2014
...ruence is formed by the normals of a surface with the isothermic spherical image of curvature lines.
2 KB (305 words) - 08:11, 6 June 2020
is the image of $ g _ {2} $
2 KB (302 words) - 22:10, 5 June 2020
...r element. When a corresponding topology has been introduced, the analytic image becomes the Riemann surface for the given function.
The construct here called the [[Analytic image|analytic image]] is variously called the analytic configuration, the analytic entity and t
13 KB (1,972 words) - 20:02, 15 January 2024
the inverse image $ f ^ { - 1 } ( U) $
2 KB (233 words) - 08:09, 6 June 2020
...sed in a naive and non-technical sense as follows. View a curve $C$ as the image of a finite or infinite interval in Euclidean space $E^n$. Let $\phi$ be a
2 KB (316 words) - 11:16, 26 March 2023
...ve) can be constructed; this is a Cantor curve that contains a topological image of every Cantor curve.
2 KB (298 words) - 06:29, 30 May 2020
the linear group $\def\Ad{\mathop{\textrm{Ad}}} \Ad G$ which is the image of the Lie group or algebraic group $G$ under the adjoint representation (c
2 KB (278 words) - 17:24, 20 January 2022
of its image. For $ g \leq 3 $
and its image is the Kummer variety $ K( X, \Theta ) $(
11 KB (1,642 words) - 14:55, 7 June 2020
by the Stiefel numbers it is necessary to find its image. This problem is analogous to the Milnor–Hirzebruch problem for Chern cla
can be in the image of the mapping $ \phi $
5 KB (796 words) - 08:23, 6 June 2020
A simple closed curve is a topological image of a circle in the plane.
2 KB (292 words) - 18:20, 16 April 2014
...function necessarily has zeros. It follows from Cauchy's theorem that the image of an interval on the real line under a continuous mapping into the real li
...suming two distinct values, also assumes any value between them; hence the image of $ X $
5 KB (824 words) - 15:35, 4 June 2020
...nt sheaf of a flabby sheaf by a flabby sheaf is itself a flabby sheaf. The image of a flabby sheaf under a continuous mapping is a flabby sheaf. If $X$ is p
2 KB (276 words) - 18:47, 13 October 2014
is well-defined and is a closed imbedding. The image $ \phi ( P ^ {n} \times P ^ {m} ) $
2 KB (264 words) - 08:12, 6 June 2020
...{ * }$. Without loss of generality one may assume that $G ^ { * }$ is the image of $| \zeta | > 1$ under a univalent conformal mapping (cf. also [[Conforma
...], [[#References|[a2]]], [[#References|[a4]]]. If, for example, $C$ is the image of $| \zeta | = 1$ under a univalent conformal mapping of an annulus $r < |
7 KB (989 words) - 20:48, 23 January 2024
denotes the [[small image]] of the set $ O _ {i} $
2 KB (274 words) - 11:41, 18 February 2022
A set $E\subset \mathbb R^n$ is the image of a rectifiable curve if and only if it is compact, connected and it has f
...ctive), the length of the curve is then the $\mathcal{H}^1$ measure of its image (cp. with Lemma 3.2 of {{Cite|Fa}}). More generally, in the presence of sel
4 KB (658 words) - 13:56, 14 December 2012
...'faithful'': $G$ may be regarded as a subgroup of $S_X$. In any case, the image of $\rho$ is a [[permutation group]] on $X$.
2 KB (330 words) - 16:58, 25 November 2023
...nd every $y\in f(X)$ there is a neighbourhood $U_y\subset Y$ whose inverse image $f^{-1}(U_y)$ splits into a discrete system of open sets in $X$ of diameter
2 KB (318 words) - 19:08, 21 November 2018
the image of this set, $ f ( E) $,
...if (respectively, only if — see below for the missing "if" ) the inverse image $ f ^ { - 1 } ( \{ x \} ) $
6 KB (847 words) - 04:11, 6 June 2020
...if every non-trivial finitely generated subgroup has a non-trivial finite image.)
2 KB (294 words) - 22:07, 14 January 2017
...30/c0262303.png" /> is a convex function if and only if the tangent to the
image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.or
...lopediaofmath.org/legacyimages/c/c026/c026230/c02623048.png" /> of the pre-
image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.or
19 KB (2,650 words) - 17:07, 7 February 2011
A subset of a complete separable metric space that is a continuous image of the space of irrational numbers. The concept of an analytic set was intr
is a subset of that space that is the image of a closed subset of the space of irrational numbers under an upper semi-c
12 KB (1,915 words) - 18:47, 5 April 2020
...e. The metric completion of a metric totally-bounded space is compact. The image of a totally-bounded space under a uniformly continuous mapping is a totall
2 KB (349 words) - 13:16, 12 December 2013
is one-to-one and the image $ f ( I ^ {p} ) $
2 KB (333 words) - 16:43, 4 June 2020
...of a module coincides with the set of endomorphisms having an inessential image.
2 KB (361 words) - 19:20, 2 October 2016
with respect to the image $ \Gamma ^ {*} = f ( \Gamma ) $
with respect to the image $ \Gamma ^ {*} = f ( \Gamma ) $:
6 KB (869 words) - 22:17, 5 June 2020
The area of the complement to the image of a domain under a mapping by a function regular in it is non-negative. Th
of the complement of the image $ F ( \Delta ^ \prime ) $
9 KB (1,232 words) - 07:25, 26 March 2023
...^A$, where $\phi(X)_\alpha = f_\alpha(X)$, is a homeomorphism onto its own image. Then, by definition, $\beta X = [\phi(X)]$ (where $[ \cdot ]$ denotes the
2 KB (335 words) - 10:44, 19 October 2016
set is a one-to-one continuous image of some $ C {\mathcal A} $-
2 KB (333 words) - 06:29, 30 May 2020
...)$ be the irreducible polynomial of $\theta$ over $k$, let $f^*(x)$ be the image of $f(x)$ in the ring $A/\mathfrak{p}[x]$ and let $f^*(x) = f_1^*(x)^{e_1}\
2 KB (331 words) - 11:31, 17 September 2017
is the image of the submodule $ A _ {n} $
2 KB (330 words) - 16:25, 23 November 2023
...sional continuous linear operators on $E$; if $A$ is a Banach algebra, the image of $A$ in this realization is contained in the uniform closure $F(E)$ of th
2 KB (360 words) - 15:29, 15 April 2014
...$X$ is separable (for $Y$ this is not a necessary condition; however, the image of a completely-continuous operator is always separable).
2 KB (372 words) - 20:40, 2 April 2013
..._\sigma$ ($G_\delta$)]]). Each topological space can be represented as the image under a continuous open mapping of a paracompact Hausdorff space with a cou
2 KB (346 words) - 13:09, 27 September 2014
The image of a curve in the three-dimensional Euclidean space $ \mathbf R ^ {3} $
2 KB (378 words) - 08:22, 6 June 2020
...non-bijective) measure-preserving map need not be measurable (indeed, the image of a null set need not be null; try the projection $\R^2\to\R^1$). Neverth
...\in\F$ whenever $A_1\subset\Om_1$ and $A=f^{-1}(A_1)$. In particular, the image $f(\Om)$belongs to $\F_1$. (See {{Cite|Ru|Th. 3-2}} and {{Cite|H|Prop. 9}}
14 KB (2,104 words) - 16:21, 17 April 2012
...ge of every point is at a positive distance from the complement of the pre-image of every neighbourhood of this point.
6 KB (1,004 words) - 19:52, 3 February 2021
...le functions to another that preserves the property that the image and pre-image are the numbers of the same function (the presence of an effective translat
6 KB (919 words) - 08:26, 6 June 2020
...aracterizations of an ellipsoid are as follows: An ellipsoid is the affine image of a [[Sphere|sphere]]; an ellipsoid is a non-degenerate [[Quadric|quadric]
2 KB (309 words) - 22:00, 24 April 2012
..._i$, for all $i = 1,\ldots,n$, and $\phi_a(1)$ is the identity of $S$. The image of a polynomial $f \in k[x_1,\ldots,x_n]$ under this homomorphism is called
is exact, that is, the kernel of one homomorphism is the image of the next. This result is one possible formulation of the [[Hilbert theor
6 KB (1,011 words) - 06:01, 30 September 2023
inverse image); and
direct image).
12 KB (1,730 words) - 22:13, 5 June 2020
...nd $f _ { L } ^ { \leftarrow } : L ^ { Y } \rightarrow L ^ { X }$ (the pre-image operator) by
...\phi ) ^ { \leftarrow } ( b ) = \phi ^ { \text{op} } \circ b \circ f$. An image operator $( f , \phi ) ^ { \rightarrow }$ is also available which, if $\phi
13 KB (1,886 words) - 17:28, 25 November 2023
its image $ T _ {X} ( x) $
2 KB (351 words) - 17:33, 5 June 2020
is the direct image of the sheaf of linear functions on $ k ^ {N+1} $
as its quotient. The direct image of the structure sheaf under the mapping $ k ^ {N+1} \setminus \{ 0 \}
6 KB (950 words) - 12:37, 29 December 2021
...ce into a locally convex space. If a separable locally convex space is the image of a Fréchet space under an open mapping, then it is itself a Fréchet spa
2 KB (359 words) - 22:13, 26 July 2012
In [[#References|[a1]]], J.F. Adams was studying the order of the image of the $J$-homomorphism and related topics. He was led to the following con
2 KB (351 words) - 14:26, 19 April 2014
if among the components of the inverse image of the set $ M \setminus \alpha $
2 KB (382 words) - 08:12, 6 June 2020
is said to be horizontal if the image of its tangential mapping lies in a horizontal subbundle.
is a nilpotent element (see [[#References|[4]]]). The image of the monodromy group $$
12 KB (1,699 words) - 09:49, 20 December 2019
$\Omega^1$. The image of the mapping $H_1(X,{\mathbb Z}) \to (\Omega^1)^*$ thus obtained is a lat
2 KB (366 words) - 18:09, 12 September 2011
is a monomorphism. Below a description of the image of the homomorphism $ \phi $
lies in the image of the homomorphism $ \phi : \Omega _ {2n} ^ {u} \rightarrow A $
8 KB (1,150 words) - 18:42, 13 January 2024
matrices with complex entries; the image of this homomorphism is called the monodromy group $ M ( t _ {0} , G ) $
3 KB (379 words) - 08:01, 6 June 2020
...f Dupin (cf. [[Dupin cyclide|Dupin cyclide]]). They can be obtained as the image of the torus under a conformal transformation of the ambient space.
3 KB (414 words) - 09:10, 26 March 2023
...g in numerical analysis, multi-resolution analysis and pyramidal coding in image processing.
...align="top"> M. Antonini, M. Barlaud, I. Daubechies, P. Mathieu, "Image coding using vector quantization in the wavelet transform domain" , ''IEEE
6 KB (909 words) - 08:28, 6 June 2020
...]; a neighbourhood of the identity of a Banach Lie group is covered by the image of the exponential mapping; there is a correspondence between connected clo
3 KB (427 words) - 22:16, 5 June 2020
...ace is defined by means of the concept of a surface patch — a homeomorphic image of a square in $ E ^ {3} $.
3 KB (385 words) - 14:55, 7 June 2020
A homomorphic image of a subgroup is a subgroup. If a group $G_1$ is isomorphic to a subgroup $
3 KB (467 words) - 14:22, 30 August 2014
...value theorem for a continuous function on a closed interval: A continuous image of a connected topological space in a topological space is also connected.
26 KB (3,622 words) - 17:12, 7 February 2011
$ \mathop{\rm Im} f $ (the image of $ f $),
and $ \mathop{\rm Coim} f $ (the co-image of $ f $),
11 KB (1,695 words) - 20:37, 23 December 2023
...almost uniformly convergent random variables with given weakly convergent image laws" ''Ann. Math. Stat.'' , '''41''' (1970) pp. 284–291</TD></TR></ta
3 KB (375 words) - 08:14, 6 June 2020
(here the bar denotes the image under the reduction $ A \rightarrow A/ \mathfrak m $).
3 KB (386 words) - 22:10, 5 June 2020
The image of this homomorphism, which is a subgroup of $ G $,
3 KB (397 words) - 08:04, 6 June 2020
...tre of a group]]), and coincides with $Z _ { G }$ if $G$ is connected. The image $\text{Ad}( G )$ is called the [[Adjoint group|adjoint group]]; it is a Lie
...ext { ad } )$, coincides with the centre of the Lie algebra $\frak g$. The image, $\operatorname{ad}({\frak g} )$, forms a subalgebra of $\operatorname { De
8 KB (1,184 words) - 16:59, 1 July 2020
...tors $\ad x$ are derivations of $\g$ and are called inner derivations. The image $\ad \g$ is called the adjoint linear Lie algebra and is an ideal in the Li
3 KB (407 words) - 16:57, 27 March 2012
...e a mapping having the property above, with the further condition that the image $ f ( X ) $
3 KB (411 words) - 08:09, 6 June 2020
is by definition the image of a representative of $ P $
3 KB (439 words) - 08:09, 6 June 2020
that annihilates the image of a homomorphism $ \alpha : A \rightarrow B $.
3 KB (417 words) - 16:16, 5 June 2020
Then the image of the homomorphism
3 KB (434 words) - 22:14, 5 June 2020
...ed have a pre-image in $\Pi_0$ and not every point of $\Pi_0$ need have an image in $\Pi_1$. This leads to the necessity of completing affine space by so-ca
7 KB (1,119 words) - 13:21, 12 December 2013
one can take the image of the unit under the composition of the canonical mappings $ A \rightarr
3 KB (430 words) - 08:07, 6 June 2020
...ra-morphism $m\colon A \longrightarrow B$, $A/\sim_m$ is isomorphic to the image $m(A)$ of $A$ under $m$ {{Cite|ST99}}.
...with $\sim=\sim^A$ {{Cite|W90}}. This is the case, in turn, iff it is the image of $T(\Sigma)$ under a $\Sigma$-algebra-morphism.
8 KB (1,264 words) - 16:16, 18 February 2013
into the unit disc. Closest to the origin of coordinates is the image of $ z ^ \prime $.
the mapping is lattice conformal and the image of $ Q _ {T} $
8 KB (1,168 words) - 20:13, 12 January 2024
that are in the image of $ X _ {H _ {2} } $
that are in the image of $ X _ {H _ {2} } $.
10 KB (1,362 words) - 08:22, 26 March 2023
...ny magma with cancellation is imbeddable into a quasi-group. A homomorphic image of a quasi-group is a magma with division, that is, a magma in which the eq
3 KB (454 words) - 19:12, 7 January 2016
as well as the image of $ G $
4 KB (534 words) - 11:05, 17 December 2019
...ee [[Tikhonov product|Tikhonov product]]). A topological space that is the image of a compact space under a continuous mapping is a compact space. These pro
compactum; the image of a $ T _ {2} $-
26 KB (3,890 words) - 16:31, 5 June 2020
...ient module of a finitely-generated free module by a submodule which is an image of a finitely-generated free module. Such a ring is called a left coherent
3 KB (479 words) - 17:45, 4 June 2020
...$A$. This property is invariant under continuous mappings. Any continuous image of a Luzin space lying in $Y$ has Lebesgue measure zero and dimension zero.
3 KB (449 words) - 09:03, 2 January 2021
...ble, and when a reasonable topology is introduced in the image and inverse image, then the operator in question is an open mapping, i.e., $ L $ is locally i
7 KB (1,113 words) - 16:28, 16 January 2017
...same Kauffman polynomial. In particular, the knot $9_{42}$ and its mirror image $\overline { 9 } _ { 42 }$ are different but have the same Kauffman polynom
...om the Alexander polynomial, it often distinguishes a knot from its mirror image but, for example, it does not distinguish the knots $11_{255}$ and $11_{257
7 KB (1,046 words) - 17:02, 1 July 2020
...$L$ be a linear system without fixed components on $V$ and let $W$ be the image of the variety $V$ under the mapping given by $L$. The following two theore
3 KB (451 words) - 11:57, 23 April 2017
denotes the image of a set $ E \subset D _ {0} $
3 KB (406 words) - 13:07, 6 January 2024
If the area of the Riemann surface that is the image of $ V $
then the direct image $ f ( B) $
7 KB (1,075 words) - 19:29, 1 January 2021
...of the language under consideration) into the natural number sequence. The image of a word is called its number. Relations between and operations defined on
3 KB (481 words) - 12:48, 14 February 2020
such that the pre-image of any open affine subscheme in $ S $
3 KB (464 words) - 05:59, 19 March 2022
whose image coincides with $ A \cap [ F, F ] $.
3 KB (473 words) - 08:12, 6 June 2020
The image of a class $ x \in h ^ {n} ( X) $
3 KB (443 words) - 08:24, 6 June 2020
onto the set of semi-simple elements of its image.
3 KB (489 words) - 16:35, 17 December 2019
...clopediaofmath.org/legacyimages/b/b015/b015310/b01531052.png" />-th direct
image of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopedi
11 KB (1,513 words) - 17:08, 7 February 2011
is the inverse image of the nil radical (cf. [[Nil ideal|Nil ideal]]) of $ R/A $.
3 KB (505 words) - 08:09, 6 June 2020
...jective object of a category]]) implies that every object is an epimorphic image of a projective object (namely a suitable coproduct of copies of the genera
3 KB (545 words) - 19:41, 5 June 2020
Passing to the Fourier image one sets $ { {J _ \alpha } hat } ( k ) = J ( e ^ {ik \cdot x } \otimes
3 KB (424 words) - 19:41, 5 June 2020
consists of the zero vector only and the image of $ A - \lambda I $
4 KB (546 words) - 17:06, 19 January 2024
denotes the image of $ [ 0 , t ] $
that consists of the image of the points $ x _ {t} $
9 KB (1,365 words) - 18:18, 19 January 2021
is the image of the equilibrium situation of some quasi-informational extension $ \wid
4 KB (563 words) - 08:09, 6 June 2020
...ction of the branching point itself is an interior point with a unique pre-image. In a neighbourhood of a branching point $(u=0,v=0)$ the coordinates $(x,y,
3 KB (495 words) - 13:00, 3 October 2014
there exists points not belonging to the image of the disc $ | z | < 1 $
which is the one-to-one image of a subdomain of $ D $.
7 KB (959 words) - 22:14, 5 June 2020
...um under a [[Perfect mapping|perfect mapping]] is a paracompactum, and the image of a paracompactum under a continuous [[Closed mapping|closed mapping]] is
9 KB (1,401 words) - 14:04, 30 July 2014
...)$ splits, giving in particular that each indecomposable $B$-module is the image of an indecomposable $A$-module either by $\text{Hom}_A( T , - )$ or by $\o
3 KB (507 words) - 16:45, 1 July 2020
is the image of a vector bundle $ E $
3 KB (474 words) - 13:07, 24 December 2020
...ch that $n+1$ meet at the vertices (primitive parallelohedra) is an affine image of the partitioning $\{D\}$ for a lattice. Thus he reduced the study of suc
...iling". The Voronoi problem is whether each parallellohedron is the affine image of a Dirichlet–Voronoi region for a lattice. This is true for $n\leq4$, [
8 KB (1,201 words) - 19:28, 2 June 2016
is called the image of the element $ x \in T ^ {s} ( F, A) $
3 KB (536 words) - 09:05, 8 April 2023
...hi : G \rightarrow G_1$ is a $k$-epimorphism of algebraic groups, then the image $\phi(H)$ of any arithmetic group $H \subset G$ is an arithmetic group in $
4 KB (527 words) - 20:14, 14 October 2014
...up|Rank of an algebraic group]]; [[Reductive group|Reductive group]]). The image of a $ k $ -
4 KB (559 words) - 16:39, 17 December 2019
...chain-complete ordered set $P$ the existence of a point comparable to its image is equivalent to the existence of a fixed point [[#References|[a1]]], [[#Re
...P$ is a comparative retraction (i.e., each point $p$ is comparable to its image $r(p)$), then $p$ has the fixed-point property if and only if $r[P]$ has th
9 KB (1,338 words) - 12:00, 9 January 2016
is called logarithmically convex if the image $ \lambda ( D ^ {*} ) $
4 KB (586 words) - 08:10, 6 June 2020
...N.J.A. Sloane, M.D. Trott, "The Nordstrom Robinson code is the binary image of the octacode" R. Calderbank (ed.) G.D. Forney Jr. (ed.) N. Moayeri (e
3 KB (459 words) - 17:32, 17 April 2016
st coordinate (that is, the image with respect to the mapping $ ( a _ {1} \dots a _ {n} , a _ {n+} 1 ) \map
4 KB (529 words) - 22:11, 5 June 2020
is proper if and only if the inverse image of each compact subset of $ Y $
the inverse image of $ Y ^ \prime $.
9 KB (1,267 words) - 08:08, 6 June 2020
A topological image of the closed part of the plane comprised between two non-identical concent
4 KB (606 words) - 08:26, 6 June 2020
...9 note, Listing showed that the figure-eight knot, Fig.a1b, and its mirror image are equivalent [[#References|[a2]]]. Because of this, this knot is often ca
3 KB (496 words) - 07:37, 18 March 2023
then the image of $ D $
4 KB (590 words) - 11:46, 30 December 2020
In other words, the image of the frame at a point $ x _ {t} $
uniquely: The image, under the mapping $ ( P _ {n} ) _ {x _ {t} } \rightarrow ( P _ {n} ) _
10 KB (1,455 words) - 06:59, 6 March 2024
has codimension 1 at each point, the image of $ g $
4 KB (550 words) - 22:13, 5 June 2020
of points in the pre-image $ f ^ { - 1 } y $
3 KB (535 words) - 19:39, 5 June 2020
...of the Cantor set; 2) secondly, every metrizable compactum is a continuous image of the Cantor set (Aleksandrov's theorem). This theorem marks the beginning
4 KB (527 words) - 09:11, 26 March 2023
...alent functions. Koebe's $ 1 / 4 $-theorem holds: The domain that is the image of the disc $ | z | < 1 $
the length of the image of $ A $
11 KB (1,524 words) - 02:42, 14 September 2022
is the image of the variety $ X $(
4 KB (558 words) - 14:54, 7 June 2020
from the image of the circle $ | z- a | = \rho $
4 KB (554 words) - 20:40, 16 January 2024
for the image of $ { \mathop{\rm Tr} } _ {H,K } $.
6 KB (984 words) - 08:44, 26 March 2023
and the image of $ K $
4 KB (549 words) - 08:54, 8 April 2023
...logical space: A set of points of a topological space that is a continuous image of an interval is called a curve in the sense of Jordan.
...t of which has an arbitrary small connected neighbourhood) is a continuous image of an interval (a theorem of Mazurkiewicz). Thus, not only a square, but al
19 KB (2,971 words) - 19:30, 26 March 2023
...ks), the fibre of $A\times_C B$ over an element $c\in C$ (i.e. the inverse image of $c$ under the mapping $\a\phi$) is the Cartesian product of the fibres $
3 KB (575 words) - 10:30, 23 November 2013
right $ f ^ { - 1 } \mathcal D _ {Y} $-bimodule. The inverse image functor $ L f ^ { * } $
The direct image functor $ f _ {+} $
24 KB (3,511 words) - 07:03, 10 May 2022
...reducible representation. The mapping thus obtained is surjective, the pre-image of any point of $ M _ {L} $
is finite and for the points of an open everywhere-dense subset this pre-image consists of one element [[#References|[7]]]. A complete description of all
10 KB (1,344 words) - 08:11, 6 June 2020
and is such that any perfect irreducible inverse image of the space $ X $
is the maximal inverse image of the space $ (X, \delta ) $
10 KB (1,505 words) - 04:04, 9 May 2022
...ed to a circle on the plane, which is also considered as the stereographic image of the point outside the sphere onto the plane. The coordinates of a point
4 KB (675 words) - 14:01, 17 March 2023
be the image of $ 1 \in \widetilde{E} {} ^ {0} ( S ^ {0} ) $
4 KB (602 words) - 10:13, 30 January 2022