Homomorphism (4101 bytes)
3: ...ther of the same type that preserves all the relevant structure.
5: ...re general usage see the [[morphism]] article.</small>
7: ... a map <math>\phi: X \rightarrow Y</math> such that
8: :<math>\phi(u \cdot v) = \phi(u) \circ \phi(v)</math>
9: ... <math>\circ</math> is the operation on <math>Y</math>.

Ring homomorphism (3195 bytes)
1: ...he operations of addition and multiplication. See Also [[Ring theory]].
3: ... are rings and ''f'' : ''R'' <tt>-></tt> ''S'' is a function, we require
4: ...b'') = ''f''(''a'') + ''f''(''b'') for all ''a'' and ''b'' in ''R''
5: ...'ab'') = ''f''(''a'') ''f''(''b'') for all ''a'' and ''b'' in ''R''
10: Directly from these definitions, one can deduce:

Graph homomorphism (2617 bytes)
1: ...hat respects their structure. More concretely it maps
4: ...es to undirected edges or collapses the edge onto a vertex.
5: ...ut changing direction) or collapses the edge onto a vertex
9: ...,E)</math> to a graph <math>G':=(V',E')</math> is a function
10: :<math>f:V \cup E \to V' \cup E'</math>

Algebra homomorphism (279 bytes)
1: ...rrow B</math> such that for all k in K and x,y in A,
6: {{math-stub}}
7: [[Category:Algebra]]

Group homomorphism (6229 bytes)
1: ... that for all ''u'' and ''v'' in ''G'' it holds that
3: ...up>-1</sup>. Hence one can say that ''h'' "is compatible with the group structure".
5: ...though this may be confused as an index or a general subscript.
6: ...rend is to write group homomorphisms on the right of their
7: arguments, omitting brackets, so that ''h''(''x'') becomes simply ''x h''.

Boolean homomorphism (108 bytes)
1: ...[homomorphism]] from one [[Boolean algebra]] into another.
3: {{math-stub}}

Chern-Weil homomorphism (2645 bytes)
1: ...lass]]es. It is a generalization of the [[Chern-Gauss-Bonnet theorem]].
3: ...]] with [[Lie algebra]] <math>\mathfrak g</math>; and let
5: :<math>\mathbb K(\mathfrak g^*)</math>
7: ...h> under the adjoint action of G, so that for instance
8: ...th>f(t_1,\dots,t_k)=f(Ad_g t_1,\dots, Ad_g t_n)</math>

Kernel (1172 bytes)
1: {{disambig}}
3: *'''Botany'''
5: ...n a single seed or [[cereal|grain]], as "a kernel of [[corn]]".
9: ...rounds the ''kernel'', presenting an interface to a user.
11: *'''Mathematics'''

VA Kernel (847 bytes)
1: ...which are expected to not be written in ANSI Standard MUMPS.
3: ...rt to those sytems as the VA Kernel does to [[VistA]].
5: The VA Kernel provides abstractions for:
7: * Menu Management (MenuMan)
8: * Login and Access Security

Dirichlet kernel (3551 bytes)
1: [[fr:Noyau de Dirichlet]]
2: ...s]], the '''Dirichlet kernel''' is the collection of functions
4: :<math>D_n(x)=\sum_{k=-n}^n
5: ...left(n +\frac{1}{2}\right)x\right)}{\sin(x/2)}.</math>
7: It is named after [[Johann Peter Gustav Lejeune Dirichlet]].

Quark (kernel) (303 bytes)
1: ...'ABox''', that let users run old [[AmigaOS]] software compiled for [[MC680x0]] [[processor|processors...

Kernel (function) (308 bytes)
1: ...)''''' is potentially very confusing, because it may refer either to
3: ...l operator'''; for that topic see [[kernel (mathematics)]], or
4: * the [[kernel of a function]].
6: {{msg:disambig}}

Fejér kernel (519 bytes)
1: ...is a positive kernel, giving rise to an [[approximate identity]].
3: The '''Fejer kernel''' is defined as
5: ...th>F_n(x) = \frac{1}{n} \sum_{k=0}^{n-1}D_k(x),</math>
9: :<math>D_k(x)</math>
11: ...Hungarian]] mathematician [[Lipót Fejér]] (1880&amp;ndash;1959).

Kernel of a function (3763 bytes)
1: ...] [[Category:Topology]] [[Category:Abstract algebra]]
2: ...unction (mathematics)|function]] <i>f</i> may be taken to be either
4: ...a of "equivalent as far as the function <i>f</i> can tell", or
5: ...onding [[partition of a set|partition]] of the domain.
7: ...word "kernel" in mathematics; see [[kernel (mathematics)]] for these.

Kernel (mathematics) (3015 bytes)
1: ...gs in [[mathematics]], some related to each other and some not.
3: ==Kernels in algebra==
5: ...ailure of a [[homomorphism]] or [[function (mathematics)|function]] to be [[injective]].
7: ...''X'' which is defined in terms of ''f''. Specifically,
9: ...'f'' is injective [[iff]] the kernel is the diagonal in ''X'' &amp;times; ''X''.

Kernel (algebra) (14647 bytes)
1: [[fr:Noyau (algèbre)]] [[ja:&amp;#26680; (&amp;#25968;&amp;#23398;)]]
2: [[Category:Algebra]]
3: ...]] measures the degree to which the homomorphism fails to be [[injective]].
5: The definition of kernel takes various forms in various contexts.
6: ... relevant to that context) [[if and only if]] the homomorphism is injective.

Security kernel (887 bytes)
1: ...he term '''security kernel''' has the following meanings:
3: ...s the basic security procedures for controlling [[access]] to system resources.
5: ...that must be met before specified programs can be accessed.
7: ...ftware elements of a [[trusted computing base]] that implement the [[reference monitor]] concept.
9: ... the [[National Information Systems Security Glossary]]

Mach kernel (6431 bytes)
1: ...ted and parallel computation. The project at CMU ran from 1985 to 1994.
3: ...atures in common, or wanted to if they didn't already. These included:
5: ...pport for multiple [[application]]s using [[multitasking]]
6: ...pplication, or mini-applications which then multitask
7: ...ti-processor support, in a single machine or over a network

Linux kernel (17176 bytes)
1: ...x kernel|Linux]] [[mascot]] [[Tux]] created by [[Larry Ewing]].]]
3: ...uently improved with the assistance of developers around the world.
5: ... language extensions and [[AT&amp;T]] [[assembly language]].
7: ...e]], the [[source code]] for Linux is [[free software]].
9: ...butions of [[software]] based on this kernel are called [[Linux distribution]]s.

Kernel panic (871 bytes)
1: ... that is useful only to the developers of the operating system.
3: ...A panic may also occur as a result of a hardware failure.
5: ...er to Nifty Doorways (a parody of the Windows operating system.).
7: ==Related articles==
8: *[[Black screen of death]]

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