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New page: Image:lighterstill.jpg right|frame|<center>Canonical Polyhedra</center> '''''Canonical''''' is an adjective derived from canon. ''Canon'' comes from the Greek w...
[[Image:lighterstill.jpg]]
[[Image:Selfdual.jpg|right|frame|<center>Canonical Polyhedra</center>]]
'''''Canonical''''' is an adjective derived from canon. ''Canon'' comes from the Greek word ''kanon'' "rule" (perhaps originally from ''kanna'' "reed", cognate to ''[[cane]]'') is used in various meanings.
'''''basic, canonic, canonical''''': reduced to the simplest and most significant form possible without loss of generality, e.g. "a basic story line"; "a canonical syllable pattern"
==Religion==
This word is used by theologians and canon lawyers to refer to the [[canon law|canon]]s of the Roman Catholic, Eastern Orthodox and Anglican Churches adopted by ecumenical councils. It also refers to later law developed by local churches and dioceses of these churches. The function of this collection of various "canons" is somewhat analogous to the precedents established in [[common law]] by [[case law]].
In the 20th century, the Roman Catholic Church revised its canon law in 1917 and then again 1981 into the modern [[Canon law (Catholic Church)|Code of Canon Law]]. This code is no longer merely a compilation of papal decrees and conciliar legislation, but a more completely developed body of international church law. It is analogous to the English system of [[statute]] law.
Canonical can also mean "part of the canon", i.e., one of the books comprising a [[biblical canon]], as opposed to [[apocrypha]]l books.
The term is also applied by Westerners to other religions, but in inconsistent ways: for example, in the case of Buddhism one authority, Macmillan ''Encyclopedia of Buddhism'' (Volume One), page 142, refers to "scriptures and other canonical texts", while another, Bechert & Gombrich, ''World of Buddhism'', Thames & Hudson, London, 1984, page 79, says that scriptures can be categorized into canonical, commentarial and pseudo-canonical.
[[Canonization]] is the process by which a person becomes recognized as a [[saint]].
==Literature and art==
The word is also often used when describing bodies of literature or art: those books that all educated people have supposedly read, or are advised to read, make up the "canon", for example the [[Western canon]].
==Mathematics==
Mathematicians have for perhaps a century or more used the word ''canonical'' to refer to concepts that have a kind of uniqueness or naturalness, and are ([[up to]] trivial aspects) "independent of coordinates." Examples include the canonical [[prime number|prime]] [[factorization]] of positive [[integer]]s, the [[Jordan canonical form]] of [[matrix (mathematics)|matrices]] (which is built out of the irreducible factors of the [[characteristic polynomial]] of the matrix), and the canonical decomposition of a [[permutation]] into a product of disjoint cycles. Various functions in mathematics are also canonical, like the canonical [[homomorphism]] of a [[group (mathematics)|group]] onto any of its quotient groups, or the canonical [[isomorphism]] between a finite-dimensional [[vector space]] and its double dual. Although a finite-dimensional vector space and its dual space are isomorphic, there is no canonical isomorphism. This lack of a canonical isomorphism can be made precise in terms of [[category theory]], but one could say at a simpler level that "any isomorphism you can think of here depends on choosing a basis." As stated by Goguen, "To any canonical construction from one species of structure to another corresponds an [[Adjoint functors|adjunction]] between the corresponding categories." (Goguen J. "A categorical manifesto". Math. Struct. Comp. Sci., 1(1):49--67, 1991)
Being canonical in mathematics is stronger than being a conventional choice. For instance, the [[vector space]] '''R'''<sup>''n''</sup> has a [[standard basis]] which is canonical in the sense that it is not just a choice which makes certain calculations easy; in fact most [[linear operator]]s on [[Euclidean space]] take on a simpler form when written as a matrix relative to some basis ''other'' than the standard one (see [[Jordan form]]). In contrast, an abstract ''n''-dimensional real vector space ''V'' would not have a canonical basis; it is isomorphic to '''R'''<sup>''n''</sup> of course, but the choice of isomorphism is not canonical.
The word ''canonical'' is also used for a preferred way of writing something, see the main article [[canonical form]].
In set theory, the term "canonical" identifies an element as representative of a set. If a set is [[Partition of a set|partitioned]] into [[Equivalence class|equivalence classes]], then one member can be chosen from each equivalence class to represent that class. That representative member is the canonical member. If you have a canonicalizing function, f(x), that maps x to the canonical member of the equivalence class which contains it, then testing whether two items, a and b, are equivalent is the same as testing whether f(a) is identical to f(b).
==Computer science==
Some circles in the field of [[computer science]] have borrowed this usage from [[#Mathematics|mathematicians]]. It has come to mean "the usual or standard state or manner of something"; for example, "the canonical way to organize a [[file system]] is as a [[hierarchy]], with extensions to make it a [[Directed graph#Directed graph|directed graph]]". [[XML Signature]] defines canonicalization as the process of converting [[XML]] content to a canonical form, to take into account changes that can invalidate a signature over that data (from [[JWSDP]] 1.6).
In [[enterprise application integration]], the "canonical data model" is a [[design pattern]] used to communicate between different data formats. It introduces an additional format, called the "canonical format", "canonical document type" or "canonical data model". Instead of writing translators between each and every format (with potential for a [[combinatorial explosion]]), it is sufficient just to write a translator between each format and the canonical format. The Open Applications Group Integration Specification ([[OAGIS]]) is an example of an integration architecture that is based on a canonical data model.
For an illuminating story about the word's use among computer scientists, see the [[Jargon File]]'s entry for the word[http://catb.org/~esr/jargon/html/C/canonical.html].
Some people have been known to use the noun ''canonicality''; others use ''canonicity''. In fields other than computer science, ''canonicity'' is this word's canonical form.
In computer science, a '''canonical name record''' (or CNAME) is a [[Domain_name_system#Types_of_DNS_records | type of DNS record]].
In computer science, a '''canonical number''' is the old designation for a '''MAC code''' on routers and servers.
==Physics==
In [[theoretical physics|theoretical]] [[physics]], the concept of canonical (or conjugate, or canonically conjugate) variables is of major importance. They always occur in complementary pairs, such as [[point (spatial)|spatial location]] '''x''' and [[momentum|linear momentum]] '''p''', [[angle]] ''φ'' and [[angular momentum]] ''L'', and [[energy]] ''E'' and [[time]] ''t''. They can be defined as any coordinates whose [[Poisson bracket]]s give a [[Kronecker delta]] (or a [[Dirac delta]] in the case of [[continuous]] variables). The existence of such coordinates is guaranteed under broad circumstances as a consequence of [[Darboux's theorem]]. Canonical variables are essential in the [[Hamiltonian mechanics|Hamiltonian]] formulation of physics, which is particularly important in [[quantum mechanics]]. For instance, the [[Schrödinger equation]] and the [[uncertainty principle|Heisenberg uncertainty relation]] always incorporate canonical variables. Canonical variables in physics are based on the aforementioned mathematical structure and therefore bear a deeper meaning than being just convenient variables. One facet of this underlying structure is expressed by [[Noether's theorem]], which states that a (continuous) [[symmetry]] in a variable implies an [[invariant (physics)|invariance]] of the conjugate variable, and vice versa; for instance symmetry under spatial displacement leads to
[[conservation of momentum]], and time-independence implies [[conservation of energy|energy conservation]].
In [[statistical mechanics]], the [[canonical ensemble]], the [[grand canonical ensemble]], and the [[microcanonical ensemble]] are archetypal [[probability distributions]] for the (unknown) [[microstate (statistical mechanics)|microscopic state]] of a thermal system, applying respectively in the physical cases of:- a closed system at fixed temperature (able to exchange energy with its environment); an open system at fixed temperature (able to exchange both energy and particles); and a closed thermally isolated system (able to exchange neither). These probability distributions can be applied directly to practical problems in [[thermodynamics]].
==See also==
*[[Western canon|Literary canons]] and [[Canon (fiction)]]
*[[Canonicalization]] is a transformation to get the canonical form.
==References==
#Macmillan Encyclopedia of Buddhism (Volume One), page 142
#Bechert & Gombrich, World of Buddhism, Thames & Hudson, London, 1984, page 79
#Goguen J. "A categorical manifesto". Math. Struct. Comp. Sci., 1(1):49--67, 1991
[[Category: General Reference]]
[[Category: Languages and Literature]]
[[Image:Selfdual.jpg|right|frame|<center>Canonical Polyhedra</center>]]
'''''Canonical''''' is an adjective derived from canon. ''Canon'' comes from the Greek word ''kanon'' "rule" (perhaps originally from ''kanna'' "reed", cognate to ''[[cane]]'') is used in various meanings.
'''''basic, canonic, canonical''''': reduced to the simplest and most significant form possible without loss of generality, e.g. "a basic story line"; "a canonical syllable pattern"
==Religion==
This word is used by theologians and canon lawyers to refer to the [[canon law|canon]]s of the Roman Catholic, Eastern Orthodox and Anglican Churches adopted by ecumenical councils. It also refers to later law developed by local churches and dioceses of these churches. The function of this collection of various "canons" is somewhat analogous to the precedents established in [[common law]] by [[case law]].
In the 20th century, the Roman Catholic Church revised its canon law in 1917 and then again 1981 into the modern [[Canon law (Catholic Church)|Code of Canon Law]]. This code is no longer merely a compilation of papal decrees and conciliar legislation, but a more completely developed body of international church law. It is analogous to the English system of [[statute]] law.
Canonical can also mean "part of the canon", i.e., one of the books comprising a [[biblical canon]], as opposed to [[apocrypha]]l books.
The term is also applied by Westerners to other religions, but in inconsistent ways: for example, in the case of Buddhism one authority, Macmillan ''Encyclopedia of Buddhism'' (Volume One), page 142, refers to "scriptures and other canonical texts", while another, Bechert & Gombrich, ''World of Buddhism'', Thames & Hudson, London, 1984, page 79, says that scriptures can be categorized into canonical, commentarial and pseudo-canonical.
[[Canonization]] is the process by which a person becomes recognized as a [[saint]].
==Literature and art==
The word is also often used when describing bodies of literature or art: those books that all educated people have supposedly read, or are advised to read, make up the "canon", for example the [[Western canon]].
==Mathematics==
Mathematicians have for perhaps a century or more used the word ''canonical'' to refer to concepts that have a kind of uniqueness or naturalness, and are ([[up to]] trivial aspects) "independent of coordinates." Examples include the canonical [[prime number|prime]] [[factorization]] of positive [[integer]]s, the [[Jordan canonical form]] of [[matrix (mathematics)|matrices]] (which is built out of the irreducible factors of the [[characteristic polynomial]] of the matrix), and the canonical decomposition of a [[permutation]] into a product of disjoint cycles. Various functions in mathematics are also canonical, like the canonical [[homomorphism]] of a [[group (mathematics)|group]] onto any of its quotient groups, or the canonical [[isomorphism]] between a finite-dimensional [[vector space]] and its double dual. Although a finite-dimensional vector space and its dual space are isomorphic, there is no canonical isomorphism. This lack of a canonical isomorphism can be made precise in terms of [[category theory]], but one could say at a simpler level that "any isomorphism you can think of here depends on choosing a basis." As stated by Goguen, "To any canonical construction from one species of structure to another corresponds an [[Adjoint functors|adjunction]] between the corresponding categories." (Goguen J. "A categorical manifesto". Math. Struct. Comp. Sci., 1(1):49--67, 1991)
Being canonical in mathematics is stronger than being a conventional choice. For instance, the [[vector space]] '''R'''<sup>''n''</sup> has a [[standard basis]] which is canonical in the sense that it is not just a choice which makes certain calculations easy; in fact most [[linear operator]]s on [[Euclidean space]] take on a simpler form when written as a matrix relative to some basis ''other'' than the standard one (see [[Jordan form]]). In contrast, an abstract ''n''-dimensional real vector space ''V'' would not have a canonical basis; it is isomorphic to '''R'''<sup>''n''</sup> of course, but the choice of isomorphism is not canonical.
The word ''canonical'' is also used for a preferred way of writing something, see the main article [[canonical form]].
In set theory, the term "canonical" identifies an element as representative of a set. If a set is [[Partition of a set|partitioned]] into [[Equivalence class|equivalence classes]], then one member can be chosen from each equivalence class to represent that class. That representative member is the canonical member. If you have a canonicalizing function, f(x), that maps x to the canonical member of the equivalence class which contains it, then testing whether two items, a and b, are equivalent is the same as testing whether f(a) is identical to f(b).
==Computer science==
Some circles in the field of [[computer science]] have borrowed this usage from [[#Mathematics|mathematicians]]. It has come to mean "the usual or standard state or manner of something"; for example, "the canonical way to organize a [[file system]] is as a [[hierarchy]], with extensions to make it a [[Directed graph#Directed graph|directed graph]]". [[XML Signature]] defines canonicalization as the process of converting [[XML]] content to a canonical form, to take into account changes that can invalidate a signature over that data (from [[JWSDP]] 1.6).
In [[enterprise application integration]], the "canonical data model" is a [[design pattern]] used to communicate between different data formats. It introduces an additional format, called the "canonical format", "canonical document type" or "canonical data model". Instead of writing translators between each and every format (with potential for a [[combinatorial explosion]]), it is sufficient just to write a translator between each format and the canonical format. The Open Applications Group Integration Specification ([[OAGIS]]) is an example of an integration architecture that is based on a canonical data model.
For an illuminating story about the word's use among computer scientists, see the [[Jargon File]]'s entry for the word[http://catb.org/~esr/jargon/html/C/canonical.html].
Some people have been known to use the noun ''canonicality''; others use ''canonicity''. In fields other than computer science, ''canonicity'' is this word's canonical form.
In computer science, a '''canonical name record''' (or CNAME) is a [[Domain_name_system#Types_of_DNS_records | type of DNS record]].
In computer science, a '''canonical number''' is the old designation for a '''MAC code''' on routers and servers.
==Physics==
In [[theoretical physics|theoretical]] [[physics]], the concept of canonical (or conjugate, or canonically conjugate) variables is of major importance. They always occur in complementary pairs, such as [[point (spatial)|spatial location]] '''x''' and [[momentum|linear momentum]] '''p''', [[angle]] ''φ'' and [[angular momentum]] ''L'', and [[energy]] ''E'' and [[time]] ''t''. They can be defined as any coordinates whose [[Poisson bracket]]s give a [[Kronecker delta]] (or a [[Dirac delta]] in the case of [[continuous]] variables). The existence of such coordinates is guaranteed under broad circumstances as a consequence of [[Darboux's theorem]]. Canonical variables are essential in the [[Hamiltonian mechanics|Hamiltonian]] formulation of physics, which is particularly important in [[quantum mechanics]]. For instance, the [[Schrödinger equation]] and the [[uncertainty principle|Heisenberg uncertainty relation]] always incorporate canonical variables. Canonical variables in physics are based on the aforementioned mathematical structure and therefore bear a deeper meaning than being just convenient variables. One facet of this underlying structure is expressed by [[Noether's theorem]], which states that a (continuous) [[symmetry]] in a variable implies an [[invariant (physics)|invariance]] of the conjugate variable, and vice versa; for instance symmetry under spatial displacement leads to
[[conservation of momentum]], and time-independence implies [[conservation of energy|energy conservation]].
In [[statistical mechanics]], the [[canonical ensemble]], the [[grand canonical ensemble]], and the [[microcanonical ensemble]] are archetypal [[probability distributions]] for the (unknown) [[microstate (statistical mechanics)|microscopic state]] of a thermal system, applying respectively in the physical cases of:- a closed system at fixed temperature (able to exchange energy with its environment); an open system at fixed temperature (able to exchange both energy and particles); and a closed thermally isolated system (able to exchange neither). These probability distributions can be applied directly to practical problems in [[thermodynamics]].
==See also==
*[[Western canon|Literary canons]] and [[Canon (fiction)]]
*[[Canonicalization]] is a transformation to get the canonical form.
==References==
#Macmillan Encyclopedia of Buddhism (Volume One), page 142
#Bechert & Gombrich, World of Buddhism, Thames & Hudson, London, 1984, page 79
#Goguen J. "A categorical manifesto". Math. Struct. Comp. Sci., 1(1):49--67, 1991
[[Category: General Reference]]
[[Category: Languages and Literature]]