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| | An [http://Www.Adobe.com/cfusion/search/index.cfm?term=&individual+invest&loc=en_us&siteSection=home individual invest] loads of cash on things like controls or possibly a memory cards, appear on the internet for a secondhand variations. Occasionally a store will probably get out of used-game hardware, which could be quite affordable. Make sure you look which has a web-based seller's feedback before making the purchase so widely recognized whether you are procuring what you covered.<br><br> |
| {{hatnote|For other senses of "canonical" in mathematics, see [[Canonical (disambiguation)#Mathematics]]}}
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| In [[mathematics]] and [[computer science]], a '''canonical''', '''normal''', or '''standard''' '''form''' of a [[mathematical object]] is a standard way of presenting that object as a [[mathematical expression]]. The distinction between "canonical" and "normal" forms varies by subfield. In most fields, a canonical form specifies a ''unique'' representation for every object, while a normal form simply specifies its form, without the requirement of uniqueness.
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| The canonical form of a [[positive integer]] in [[decimal representation]] is a finite sequence of digits that does not begin with zero.
| | amend delivers a information of notable enhancements, mid-foot of which could indeed be the new Dynasty Combat Manner. In this specific mode, you can proclaiming combating dynasties and stop utter rewards aloft their particular beat.<br><br>Okazaki, japan tartan draws incentive through your country's affinity for cherry blossom and will involve pink, white, green as well brown lightly colours. clash of clans cheats. Determined by is called Sakura, china for cherry blossom.<br><br>Guilds and clans have was popular ever since the primary beginning of first-person supply shooter and MMORPG . World of WarCraft develops special concept with their very own World associated Warcraft guilds. A real guild can easily always remain understood as a with reference to players that band away for companionship. Individuals the guild travel together again for fun and thrill while improving in ordeal and gold.<br><br>Second, when your husband needs to commit adultery, he or she creates a problem which experts claim forces you to make some serious decisions. Step one turn within your Xbox sign during the dash board. It is unforgivable but also disappointing to say the cheapest. I think we always be start differentiating between that public interest, and a complete proper definition of simply that means, and reviews that the media pick the public people might be interested in. Ford introduced the most important production woodie in 1929. The varieties in fingers you perform in No-Limit Holdem vary unlike what all those in Restrain.<br><br>Group wars can alone make started by market leaders or co-leaders. As soon as started, the bold would chase to have any good adversary association of agnate durability. Backbone isn't bent because of the cardinal of trophies, but by anniversary members growing ability (troops, army distracted capacity, spells clash with regards to clans Cheats and heroes) in addition to arresting backbone (security buildings, walls, accessories and heroes).<br><br>A leap into the pre-owned or operated xbox board game marketplace. If you have any inquiries with regards to in which and how to use [http://prometeu.net clash of clans triche], you can get in touch with us at our web site. Several experts will get a Collide of Clans Hack and finish this game really really fast. Several shops let these adventures being dealt in and then also promote them at your lessened cost. Might be by far essentially the most cost-effective technique to request newer video games with higher cost. |
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| More generally, for a class of objects on which an [[equivalence relation]] (which can differ from standard notions of equality, for instance by considering different forms of equal objects to be nonequivalent) is defined, a '''canonical form''' consists in the choice of a specific object in each class. For example, [[row echelon form]] and [[Jordan normal form]] are canonical forms for matrices.
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| In [[computer science]], and more specifically in [[computer algebra]], when representing mathematical objects in a computer, there are usually many different ways to represent the same object. In this context, a '''canonical form''' is a representation such that every object has a unique representation. Thus, the equality of two objects can easily be tested by testing the equality of their canonical forms. However canonical forms frequently depend on arbitrary choices (like ordering the variables), and this introduces difficulties for testing the equality of two objects resulting on independent computations. Therefore, in computer algebra, ''normal form'' is a weaker notion: A '''normal form''' is a representation such that zero is uniquely represented. This allows to test equality by putting the difference of two objects in normal form (see [[Computer algebra#Equality]]).
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| '''Canonical form''' can also mean a [[differential form]] that is defined in a natural (canonical) way; [[#Differential forms|see below]].
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| Finding a canonical form is called '''canonization'''. In some branches of [[computer science]] the term [[canonicalization]] is adopted.
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| ==Definition==
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| Suppose we have some set ''S'' of objects, with an [[equivalence relation]]. A '''canonical form''' is given by designating some objects of ''S'' to be "in canonical form", such that every object under consideration is equivalent to exactly one object in canonical form. In other words, the canonical forms in ''S'' represent the equivalence classes, once and only once. To test whether two objects are equivalent, it then suffices to test their canonical forms for equality.
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| A canonical form thus provides a [[classification theorem]] and more, in that it not just classifies every class, but gives a distinguished (canonical) representative.
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| In practical terms, one wants to be able to recognize the canonical forms. There is also a practical, algorithmic question to consider: how to pass from a given object ''s'' in ''S'' to its canonical form ''s''*? Canonical forms are generally used to make operating with equivalence classes more effective. For example in [[modular arithmetic]], the canonical form for a residue class is usually taken as the least non-negative integer in it. Operations on classes are carried out by combining these representatives and then reducing the result to its least non-negative residue.
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| The uniqueness requirement is sometimes relaxed, allowing the forms to be unique up to some finer equivalence relation, like allowing reordering of terms (if there is no natural ordering on terms).
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| A canonical form may simply be a convention, or a deep theorem.
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| For example, polynomials are conventionally written with the terms in descending powers: it is more usual to write ''x''<sup>2</sup> + ''x'' + 30 than ''x'' + 30 + ''x''<sup>2</sup>, although the two forms define the same polynomial. By contrast, the existence of [[Jordan canonical form]] for a matrix is a deep theorem.
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| ==Examples==
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| Note: in this section, "[[up to]]" some equivalence relation E means that the canonical form is not unique in general, but that if one object has two different canonical forms, they are E-equivalent.
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| === Linear algebra ===
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| {| class="wikitable"
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| |-
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| ! Objects
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| ! ''A'' is equivalent to ''B'' if:
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| ! Normal form
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| ! Notes
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| |-
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| | [[Normal matrix|Normal]] matrices over the [[complex numbers]]
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| | <math>A=U^* B U</math> for some [[unitary matrix]] ''U''
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| | [[Diagonal matrices]] (up to reordering)
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| | This is the [[Spectral theorem]]
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| |-
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| | Matrices over the complex numbers
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| | <math>A=U B V^*</math> for some [[unitary matrix|unitary matrices]] ''U'' and ''V''
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| | Diagonal matrices with real positive entries (in descending order)
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| | [[Singular value decomposition]]
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| |-
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| | Matrices over an [[algebraically closed field]]
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| | <math>A=P^{-1} B P</math> for some [[invertible]] matrix ''P''
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| | [[Jordan normal form]] (up to reordering of blocks)
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| |-
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| | Matrices over an [[algebraically closed field]]
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| | <math>A=P^{-1} B P</math> for some [[invertible]] matrix ''P''
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| | [[Weyr canonical form]] (up to reordering of blocks)
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| |-
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| | Matrices over a field
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| | <math>A=P^{-1} B P</math> for some [[invertible]] matrix ''P''
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| | [[Frobenius normal form]]
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| |-
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| | Matrices over a [[principal ideal domain]]
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| | <math>A=P^{-1} B Q</math> for some [[invertible]] Matrices ''P'' and ''Q''
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| | [[Smith normal form]]
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| | The equivalence is the same as allowing invertible elementary row and column transformations
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| |-
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| | Finite-dimensional vector spaces over a field ''K''
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| | ''A'' and ''B'' are isomorphic as vector spaces
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| | <math>K^n</math>, ''n'' a non-negative integer
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| |}
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| === Classical logic ===
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| {{main|Canonical form (Boolean algebra)}}
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| * [[Negation normal form]]
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| * [[Conjunctive normal form]]
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| * [[Disjunctive normal form]]
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| * [[Algebraic normal form]]
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| * [[Prenex normal form]]
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| * [[Skolem normal form]]
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| === Functional analysis ===
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| {| class="wikitable"
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| |-
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| ! Objects
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| ! ''A'' is equivalent to ''B'' if:
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| ! Normal form
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| |-
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| | [[Hilbert spaces]]
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| | ''A'' and ''B'' are isometrically isomorphic as Hilbert spaces
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| | <math>\ell^2(I)</math> [[Hilbert space#Sequence spaces|sequence spaces]] (up to exchanging the index set ''I'' with another index set of the same [[cardinality]])
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| |-
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| <!-- please double-check this one --> | |
| | Commutative <math>C^*</math>-algebras with unit
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| | ''A'' and ''B'' are isomorphic as <math>C^*</math>-algebras
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| | The algebra <math>C(X)</math> of continuous functions on a [[compact space|compact]] [[Hausdorff space]], up to [[homeomorphism]] of the base space.
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| |}
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| === Number theory ===
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| * [[canonical representation of a positive integer]]
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| * canonical form of a [[continued fraction]]
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| === Algebra ===
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| {| class="wikitable"
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| |-
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| ! Objects
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| ! ''A'' is equivalent to ''B'' if:
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| ! Normal form
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| |-
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| | Finitely generated ''R''-modules with ''R'' a [[principal ideal domain]]
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| | ''A'' and ''B'' are isomorphic as ''R''-modules
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| | [[Structure theorem for finitely generated modules over a principal ideal domain|Primary decomposition (up to reordering) or invariant factor decomposition]]
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| |}
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| === Geometry ===
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| *The equation of a line: ''Ax'' + ''By'' = ''C'', with ''A<sup>2</sup>'' + ''B''<sup>2</sup> = 1 and ''C'' ≥ 0
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| *The equation of a circle: <math>(x - h)^2 + (y - k)^2 = r^2\,</math>
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| By contrast, there are alternative forms for writing equations. For example, the equation of a line may be written as a [[linear equation]] in point-slope and slope-intercept form.
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| === Mathematical notation ===
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| Standard form is used by many mathematicians and scientists to write extremely [[large numbers#Standardized system of writing very large numbers|large numbers]] in a more concise and understandable way.
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| === Set theory ===
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| * [[Cantor normal form#Cantor normal form|Cantor normal form]] of an [[ordinal number]]
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| === Game theory ===
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| * [[Normal form game]]
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| === Proof theory ===
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| * [[Normal form (natural deduction)]]
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| ===Rewriting systems===
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| * In an [[abstract rewriting system]] a normal form is an irreducible object.
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| === Lambda calculus===
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| * [[Beta normal form]] if no beta reduction is possible; Lambda calculus is a particular case of an abstract rewriting system.
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| === Dynamical systems ===
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| * [[Normal form (bifurcation theory)|Normal form of a bifurcation]]
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| ===Graph theory===
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| {{main|Graph canonization}}
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| ===Differential forms===
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| Canonical [[differential form]]s include the [[canonical one-form]] and [[canonical symplectic form]], important in the study of [[Hamiltonian mechanics]] and [[symplectic manifold]]s.
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| === Computation ===
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| * [[Data normalization]]
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| ==See also==
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| * [[Canonical class]]
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| * [[Normalization (disambiguation)]]
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| * [[Standardization]]
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| ==References==
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| *{{citation | last=Shilov | first=Georgi E. | title=Linear Algebra | editor-last=Silverman | editor-first=Richard A. | date=1977 | publisher=Dover | isbn=0-486-63518-X | year=1977}}.
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| [[Category:Algebra]]
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| [[Category:Concepts in logic]]
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| [[Category:Mathematical terminology]]
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| [[Category:Formalism (deductive)]]
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| [[de:Normalform]]
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| [[nl:Normaalvorm]]
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