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	Cokernel - Wikipedia, the free encyclopedia
		In abstract algebra, the cokernel of a homomorphism f : X → Y is the quotient of Y by the image of f.
		The cokernel of a morphism f : X → Y is defined as the coequalizer of f and the zero morphism 0
		In the category of groups, the cokernel of a group homomorphism f : G → H is the quotient of H by the normal closure of the image of f.
	en.wikipedia.org /wiki/Cokernel (425 words)

	Cokernel - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-06)
		In abstract algebra, the cokernel of a homomorphism f : X → Y is the quotient of Y by the image of f.
		The cokernel of a morphism f : X → Y is defined as the coequalizer of f and the zero morphism 0
		In the category of groups, the cokernel of a group homomorphism f : G → H is the quotient of H by the normal closure of the image of f.
	www.encyclopedia-online.info /Cokernel (433 words)

	Cokernel -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-06)
		In (Click link for more info and facts about abstract algebra) abstract algebra, the cokernel of a (Similarity of form) homomorphism f : X → Y is the (The ratio of two quantities to be divided) quotient of Y by the (An iconic mental representation) image of f.
		As with all universal constructions the cokernel, if it exists, is unique (Click link for more info and facts about up to) up to a unique ((biology) similarity or identity of form or shape or structure) isomorphism.
		This cokernel is (Click link for more info and facts about dual) dual to the (Click link for more info and facts about kernels of category theory) kernels of category theory, hence the name.
	www.absoluteastronomy.com /encyclopedia/c/co/cokernel.htm (665 words)

	Normal morphism - Wikipedia, the free encyclopedia
		In that case, we say that a monomorphism is normal if it is the kernel of some morphism, and an epimorphism is normal (or conormal) if it is the cokernel of some morphism.
		In an abelian category, every monomorphism is the kernel of its cokernel, and every epimorphism is the cokernel of its kernel.
		The category of abelian groups is the fundamental example of an abelian category, and accordingly every subgroup of an abelian group is a normal subgroup.
	en.wikipedia.org /wiki/Normal_morphism (292 words)

	Cokernel
		In the case of abelian groups, since every subgroup is normal, the cokernel is just H modulo the image of f:\n:coker(f) = H / im(f).
		In such a category, the coequalizer of two morphisms f and g (if it exists) is just the cokernel of their difference:\n:coeq(f, g) = coker(g - f) In a pre-abelian category (a special kind of preadditive category) the existence of kernels and cokernels is guaranteed.
		In such categories the image and coimage of a morphism f are given by\n:im(f) = ker(coker f)\n:coim(f) = coker(ker f) Abelian categories are even better behaved with respect to cokernels.
	encyclopedia.codeboy.net /wikipedia/c/co/cokernel.html (456 words)

	Pre-Abelian category - Wikipedia, the free encyclopedia
		Ab is preadditive because it is a closed monoidal category, the biproduct in Ab is the finite direct sum, the kernel is inclusion of the ordinary kernel from group theory and the cokernel is the quotient map onto the ordinary cokernel from group theory.
		Although kernels and cokernels are special kinds of equalisers and coequalisers, a pre-Abelian category actually has all equalisers and coequalisers.
		Note that this notion of image may not correspond to the usual notion of image, or range, of a function, even assuming that the morphisms in the category are functions.
	en.wikipedia.org /wiki/Pre-Abelian_category (882 words)

	Cokernel Definition / Cokernel Research (Site not responding. Last check: 2007-10-06)
		In abstract algebra, the cokernel of a homomorphismIn abstract algebra, a homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant structure.
		cokernel is 0, then f is the cokernel of its kernel.
		cokernel is the space H 1 /T(H 0).
	www.elresearch.com /Cokernel (315 words)

	cokernel -- cokernel of a map (Site not responding. Last check: 2007-10-06)
		-- produces the cokernel of the module homomorphism f
		The result will be a quotient module of the target of f.
		The generators of the cokernel are provided by the generators of the target of
	www.math.rutgers.edu /Macaulay2/0851.html (50 words)

	PlanetMath: supplemental axioms for an Abelian category (Site not responding. Last check: 2007-10-06)
		Every morphism has a kernel and a cokernel.
		Every monic is the kernel of its cokernel.
		This is version 7 of supplemental axioms for an Abelian category, born on 2001-12-12, modified 2004-04-07.
	planetmath.org /encyclopedia/Complete8.html (220 words)

	Ferndale, California . Marriage . United States Census Bureau (Site not responding. Last check: 2007-10-06)
		In the category of groups, the cokernel of a group homomorphism f : G H is the quotient group quotient of H by the normal closure of the image of f.
		In the case of abelian groups, since every subgroup is normal, the cokernel is just H modulo the image of f : :coker f = H im f.
		Smoking is the process of curing, cooking, or seasoning food by exposing it for long periods of time to the smoke from a wood fire.
	www.uk.fraquisanto.net /Ferndale,_California (259 words)

	On the exponent of the cokernel of the forget-control map on K_0-groups, by Francis X. Connolly and Stratos Prassidis (Site not responding. Last check: 2007-10-06)
		On the exponent of the cokernel of the forget-control map on K_0-groups, by Francis X. Connolly and Stratos Prassidis
		For groups that satisfy the Farrell-Jones conjecture, we show that the cokernel of the "forget-control" is built up of the Nil-groups that correspond to the infinite, virtually infinite cyclic subgroups.
		We use the previous result of the two authors to calculate the exponent of that cokernel.
	www.math.uiuc.edu /K-theory/0417 (84 words)

	Kernel (category theory) : Kernel (categories) (Site not responding. Last check: 2007-10-06)
		The dual concept to that of kernel is that of cokernel[?].
		That is, the kernel of a morphism is its cokernel in the opposite category[?], and vice versa.
		In this situation, the kernel of the cokernel[?] of any morphism (which always exists in an abelian category) turns out to be the image[?] of that morphism; in symbols:
	www.eurofreehost.com /ke/Kernel_(categories)_3.html (418 words)

	Index - Bvio (Site not responding. Last check: 2007-10-06)
		The index of a subgroup is the number of its left cosets (which is equal to the number of its right cosets).
		The index of a Fredholm operator is the dimension of its kernel minus the dimension of its cokernel.
		The index of a real quadratic form Q is defined (but not always consistently) as p − q where Q can be written as a difference of p squared linear terms and qsquared linear terms.
	bvio.ngic.re.kr /Bvio/index.php/Index (291 words)

	Chain Maps
		Returns the cokernel complex of f and the projection of the cokernel onto the codomain of f.
		The connecting homomorphism in degree n of the short exact sequence of chain complexes given by the chain maps f and g.
		The long exact sequence on homology for the exact sequence of complexes given by the chain maps f and g as a chain complex with the homology group in degree i for the Cokernel of the complex C appearing in degree 3i.
	www.umich.edu /~gpcc/scs/magma/text1014.htm (1337 words)

	Talk:Empty set - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-06)
		If you like algebra, then you might want to ask yourself what is the cokernel of the group homomorphism sgn defined on the symmetric group S
		Then the cokernel is trivial, since the image of f is the entire group {1,−1}.
		On the other hand, you could work in a context where you take signs to belong to the group U(1) of unit complex numbers (which is just what you want in some applications to quantum physics), and then the cokernel is an infinite group (in fact isomorphic to U(1) again).
	en.wikipedia.org /wiki/Talk:Empty_set (3390 words)

	Operations on Mappings (Site not responding. Last check: 2007-10-06)
		Given a mapping f with domain A and codomain B, return the image of A in B as a substructure of B. This function is currently available only for certain intrinsic maps.
		Given the homomorphism f with domain A and codomain B, return the kernel of f as a substructure of A. This function is currently available only for certain intrinsic maps.
		Given the homomorphism f with domain A and codomain B, return the cokernel B/I of f as a quotient structure of B, where I is the image of A in B. This function is currently available only for certain intrinsic maps.
	www.math.ufl.edu /help/magma/text181.html (246 words)

	Re: natural transformation
		Sometimes, K and Q are called "kernel" and "cokernel" instead of the morphisms.
		Note that this is not a definition of kernel and cokernel, since exactness is defined in terms of kernel and *cokernel.[] I can give you the general definition of kernel and cokernel** if you want, but I think what you really need is the definition in the case of KVect.
		The kernel of the cokernel is also called "image", and the cokernel of the kernel is also called "coimage".
	www.lns.cornell.edu /spr/1999-12/msg0020049.html (2042 words)

	Cokernel (Site not responding. Last check: 2007-10-06)
		En álgebra abstracta, el cokernel de un homomorfismo f: &rarr X; Y es el cociente de Y por la imagen de f.
		Uno puede también definir el cokernel en el marco general de la teoría de la categoría.
		En la categoría de grupos, el cokernel de un homomorfismo f del grupo: &rarr de G; H es el cociente de H al lado del encierro normal de la imagen de f.
	www.yotor.net /wiki/es/co/Cokernel.htm (443 words)

	Pre-Abelian category (Site not responding. Last check: 2007-10-06)
		given any morphism f: A → B in C, the equaliser of f and the zero morphism from A to B exists (this is the kernel), as does the coequaliser (this is the cokernel).
		Note that the zero morphism in item 3 can be identified as the identity element of the hom-set Hom(A,B), which is an Abelian group by item 1; or as the unique morphism A → O → B, where O is a zero object, guaranteed to exist by item 2.
		We simply construct the equaliser of two morphisms f and g as the kernel of their difference g − f; similarly, their coequaliser is the cokernel of their difference.
	www.worldhistory.com /wiki/P/Pre-Abelian-category.htm (956 words)

	[No title]
		In this paper we will completely describe the cokernel of the operator $L$, defined by \begin{equation} \label{eq1} Ly(z) = z^ny^{(n)}(z)+z^{n-1}a_1(z)y^{(n-1)}(z)+\cdots+a_n(z)y(z) \end{equation} where $a_1(z),\dots, a_n(z)$ are analytic at the point $z=0$.
		We will show that the dimension of the kernel of $L$ is equal to the dimension of the cokernel of $L$.
		So there exists only one integer, $p_1=2$, such that the cokernel of $L_3$ is spanned by $f_0(z)$, a polynomial of degree $r_4=0$, and $f_1(z)$, a polynomial of degree $r_4+ \eta(p_1)=0+(1+3)$.
	www.maths.tcd.ie /EMIS/journals/EJDE/Monographs/Monographs/Volumes/Volumes/2002/12/haile-tex (1726 words)

	Pre-Abelian category - Freecyclopedia.com :: The World Bank of Knowledge (Site not responding. Last check: 2007-10-06)
		Redirected from Pre-Abelian categories A pre-Abelian category is an additive category that has all kernels and cokernels[?].
		exists (this is the kernel), as does the coequaliser[?] (this is the cokernel).
		; similarly, their coequaliser is the cokernel of their difference.
	www.freecyclopedia.com /econtents/pr/Pre-Abelian_categories.html (888 words)

	[No title] (Site not responding. Last check: 2007-10-06)
		The function func is declared as: int func(kernel, cokernel, state) node_t kernel; node_t cokernel; char state; func is called once for each (kernel, cokernel*) pair of the function.
		Note that in this context, the cokernel is then always just a single cube.
		Note that in this context, the cokernel may consist of many cubes.
	www.cs.wisc.edu /~david/courses/cs755/UCBtools/other/octtools/src/misII/extract/extract.doc (356 words)

	GAP Forum: BUGFIX #12 (DANGEROUS problem in 'GroupHomomorphismByImages')
		of the cokernel), which are also too small.
		cokernel that may be too small, and the other causes the computation to
		The cokernel of a 'GroupHomomorphismByImages' from a permutation group to
	www-gap.dcs.st-and.ac.uk /oldsite/Forum/Theissen.1/Heiko.1/BUGFIX__.2/1.html (659 words)

	Coequalizer - Wikpedia (Site not responding. Last check: 2007-10-06)
		(This is the cokernel of the morphism f - g; see the next section).
		In categories with zero morphisms, one can define a cokernel of a morphism f as the coequalizer of f and the parallel zero morphism.
		In preadditive categories it makes sense to add and subtract morphisms (the hom-sets actually form abelian groups).
	www.bostoncoop.net /~tpryor/wiki/index.php?title=Coequalizer (375 words)

	Institute of Mathematics of National Academy of Sciences of Armenia (Site not responding. Last check: 2007-10-06)
		Mkrtchian, On dimensions of kernel and cokernel of Toeplitz operator, pp.
		In this paper we obtain some sufficient conditions for invertibility of $\T_n$ which do not assume permutability of operators $T_{ij}$.
		We also obtain bounds for dimensions of kernel and cokernel of the operator matrices $\T_n$, as well as of vector-valued Toeplitz operator matrices.
	math.sci.am /Journal/1993_2.html (350 words)

	getting started
		We can produce the cokernel with cokernel; no computation is performed.
		The answer is expressed in terms of the subquotient function, which produces subquotient modules.
		We can use resolution to compute a projective resolution of the cokernel of f.
	www.math.kth.se /math/internal/dator/M2/1017.html (1346 words)

	Re: natural transformation
		> Note that this is not a definition of kernel and cokernel, > since exactness is defined in terms of kernel and *cokernel.[] > I can give you the general definition of kernel and cokernel** if you want, > but I think what you really need is the definition in the case of KVect.
		> Then the inclusion map from K to V is the kernel of f, > and the natural map from V to Q := V/K is the cokernel of f.
		> The kernel of the cokernel is also called "image", > and the cokernel of the kernel is also called "coimage".
	www.lns.cornell.edu /spr/1999-12/msg0020073.html (2801 words)

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