![Anagram canonical svg](https://upload.wikimedia.org/wikipedia/commons/a/a3/Anagram_canonical_svg.svg)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
, a canonical, normal, or standard form of a
mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical p ...
is a standard way of presenting that object as a
mathematical expression
In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical symbols can designate numbers ( constants), variables, operations, f ...
. Often, it is one which provides the simplest representation of an object and which allows it to be identified in a unique way. The distinction between "canonical" and "normal" forms varies from subfield to 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.
The canonical form of a
positive integer in
decimal representation
A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator:
r = b_k b_\ldots b_0.a_1a_2\ldots
Here is the decimal separator, i ...
is a finite sequence of digits that does not begin with zero. More generally, for a class of objects on which an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relation ...
is defined, a canonical form consists in the choice of a specific object in each class. For example:
*
Jordan normal form
In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF),
is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to so ...
is a canonical form for
matrix similarity
In linear algebra, two ''n''-by-''n'' matrices and are called similar if there exists an invertible ''n''-by-''n'' matrix such that
B = P^ A P .
Similar matrices represent the same linear map under two (possibly) different bases, with being ...
.
*The
row echelon form
In linear algebra, a matrix is in echelon form if it has the shape resulting from a Gaussian elimination.
A matrix being in row echelon form means that Gaussian elimination has operated on the rows, and
column echelon form means that Gaussian ...
is a canonical form, when one considers as equivalent a matrix and its left product by an
invertible matrix
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplicati ...
.
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 (with
canonicalization
In computer science, canonicalization (sometimes standardization or normalization) is a process for converting data that has more than one possible representation into a "standard", "normal", or canonical form. This can be done to compare diff ...
being the process through which a representation is put into its canonical form). Thus, the equality of two objects can easily be tested by testing the equality of their canonical forms.
Despite this advantage, canonical forms frequently depend on arbitrary choices (like ordering the variables), which introduce 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 testing for equality by putting the difference of two objects in normal form.
Canonical form can also mean a
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
that is defined in a natural (canonical) way.
Definition
Given a set ''S'' of objects with an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relation ...
''R on S'', 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 equality on their canonical forms.
A canonical form thus provides a
classification theorem
In mathematics, a classification theorem answers the classification problem "What are the objects of a given type, up to some equivalence?". It gives a non-redundant enumeration: each object is equivalent to exactly one class.
A few issues relate ...
and more, in that it not only classifies every class, but also gives a distinguished (canonical)
representative
Representative may refer to:
Politics
* Representative democracy, type of democracy in which elected officials represent a group of people
* House of Representatives, legislative body in various countries or sub-national entities
* Legislator, som ...
for each object in the class.
Formally, a canonicalization with respect to an equivalence relation ''R'' on a set ''S'' is a mapping ''c'':''S''→''S'' such that for all ''s'', ''s''
1, ''s''
2 ∈ ''S'':
#''c''(''s'') = ''c''(''c''(''s'')) (
idempotence
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
),
#''s''
1 ''R'' ''s''
2 if and only if ''c''(''s''
1) = ''c''(''s''
2) (decisiveness), and
#''s'' ''R'' ''c''(''s'') (representativeness).
Property 3 is redundant; it follows by applying 2 to 1.
In practical terms, it is often advantageous 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
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ...
, 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.
The uniqueness requirement is sometimes relaxed, allowing the forms to be unique up to some finer equivalence relation, such as allowing for reordering of terms (if there is no natural ordering on terms).
A canonical form may simply be a convention, or a deep theorem. For example, polynomials are conventionally written with the terms in descending powers: it is more usual to write ''x''
2 + ''x'' + 30 than ''x'' + 30 + ''x''
2, although the two forms define the same polynomial. By contrast, the existence of
Jordan canonical form
In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF),
is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to so ...
for a matrix is a deep theorem.
History
According to
OED
The ''Oxford English Dictionary'' (''OED'') is the first and foundational historical dictionary of the English language, published by Oxford University Press (OUP). It traces the historical development of the English language, providing a co ...
and
LSJ, the term ''
canonical
The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical example ...
'' stems from the
Ancient Greek
Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic peri ...
word ''kanonikós'' (''
κανονικός'', "regular, according to rule") from ''kanṓn'' (''
κᾰνών'', "rod, rule"). The sense of
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
,
standard Standard may refer to:
Symbols
* Colours, standards and guidons, kinds of military signs
* Standard (emblem), a type of a large symbol or emblem used for identification
Norms, conventions or requirements
* Standard (metrology), an object th ...
, or
archetype
The concept of an archetype (; ) appears in areas relating to behavior, historical psychology, and literary analysis.
An archetype can be any of the following:
# a statement, pattern of behavior, prototype, "first" form, or a main model that ot ...
has been used in many disciplines. Mathematical usage is attested in a 1738 letter from
Logan. The German term ''kanonische Form'' is attested in a 1846 paper by
Eisenstein, later the same year
Richelot uses the term ''Normalform'' in a paper, and in 1851
Sylvester writes:
In the same period, usage is attested by
Hesse
Hesse (, , ) or Hessia (, ; german: Hessen ), officially the State of Hessen (german: links=no, Land Hessen), is a States of Germany, state in Germany. Its capital city is Wiesbaden, and the largest urban area is Frankfurt. Two other major histor ...
("Normalform"),
Hermite
Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.
Hermi ...
("forme canonique"),
Borchardt ("forme canonique"), and
Cayley ("canonical form").
In 1865, the
Dictionary of Science, Literature and Art
The ''Dictionary of Science, Literature and Art'' was a single-volume reference work published in the mid-19th century by Longman's in the United Kingdom and by Harper Brothers in the United States. At the time it was considered a highly successf ...
defines canonical form as:
Examples
Note: in this section, "
up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' wi ...
" 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.
Large number notation
Standard form is used by many mathematicians and scientists to write extremely
large numbers
Large numbers are numbers significantly larger than those typically used in everyday life (for instance in simple counting or in monetary transactions), appearing frequently in fields such as mathematics, cosmology, cryptography, and statistical m ...
in a more concise and understandable way, the most prominent of which being the
scientific notation
Scientific notation is a way of expressing numbers that are too large or too small (usually would result in a long string of digits) to be conveniently written in decimal form. It may be referred to as scientific form or standard index form, o ...
.
Number theory
*
Canonical representation of a positive integer
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...
*Canonical form of a
continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
Linear algebra
Algebra
Geometry
In
analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineerin ...
:
*The equation of a line: ''Ax'' + ''By'' = ''C'', with ''A
2'' + ''B''
2 = 1 and ''C'' ≥ 0
*The equation of a circle:
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
In mathematics, a linear equation is an equation that may be put in the form
a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coeffic ...
and
slope-intercept form
In mathematics, a linear equation is an equation that may be put in the form
a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficien ...
.
Convex polyhedra
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
can be put into
canonical form
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. Often, it is one which provides the simplest representation of an ...
such that:
*All faces are flat,
*All edges are tangent to the unit sphere, and
*The centroid of the polyhedron is at the origin.
Integrable systems
Every differentiable
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
has a
cotangent bundle. That bundle can always be endowed with a certain
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
, called the
canonical one-form
In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T^Q of a manifold Q. In physics, it is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus prov ...
. This form gives the cotangent bundle the structure of a
symplectic manifold, and allows vector fields on the manifold to be integrated by means of the
Euler-Lagrange equations, or by means of
Hamiltonian mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
. Such systems of integrable
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s are called
integrable system
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
s.
Dynamical systems
The study of
dynamical systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...
overlaps with that of
integrable systems
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
; there one has the idea of a
normal form (dynamical systems) In mathematics, the normal form of a dynamical system is a simplified form that can be useful in determining the system's behavior.
Normal forms are often used for determining local bifurcations in a system. All systems exhibiting a certain type ...
.
Three dimensional geometry
In the study of manifolds in three dimensions, one has the
first fundamental form
In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of . It permits the calculation of curvature and me ...
, the
second fundamental form
In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by \mathrm (read "two"). Together with the first fundame ...
and the
third fundamental form.
Functional analysis
Classical logic
*
Negation normal form In mathematical logic, a formula is in negation normal form (NNF) if the negation operator (\lnot, ) is only applied to variables and the only other allowed Boolean operators are conjunction (\land, ) and disjunction (\lor, ).
Negation normal for ...
*
Conjunctive normal form
In Boolean logic, a formula is in conjunctive normal form (CNF) or clausal normal form if it is a conjunction of one or more clauses, where a clause is a disjunction of literals; otherwise put, it is a product of sums or an AND of ORs. As a can ...
*
Disjunctive normal form
In boolean logic, a disjunctive normal form (DNF) is a canonical normal form of a logical formula consisting of a disjunction of conjunctions; it can also be described as an OR of ANDs, a sum of products, or (in philosophical logic) a ''cluster c ...
*
Algebraic normal form
In Boolean algebra, the algebraic normal form (ANF), ring sum normal form (RSNF or RNF), '' Zhegalkin normal form'', or '' Reed–Muller expansion'' is a way of writing logical formulas in one of three subforms:
* The entire formula is purely tr ...
*
Prenex normal form
A formula of the predicate calculus is in prenex normal form (PNF) if it is written as a string of quantifiers and bound variables, called the prefix, followed by a quantifier-free part, called the matrix. Together with the normal forms in prop ...
*
Skolem normal form
In mathematical logic, a formula of first-order logic is in Skolem normal form if it is in prenex normal form with only universal first-order quantifiers.
Every first-order formula may be converted into Skolem normal form while not changing i ...
*
Blake canonical form
In Boolean logic, a formula for a Boolean function ''f'' is in Blake canonical form (BCF), also called the complete sum of prime implicants, the complete sum, or the disjunctive prime form, when it is a disjunction of all the prime implicants of ...
, also known as the complete sum of prime implicants, the complete sum, or the disjunctive prime form
Set theory
*
Cantor normal form
In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an e ...
of an
ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least n ...
Game theory
*
Normal form game
In game theory, normal form is a description of a ''game''. Unlike extensive form, normal-form representations are not graphical ''per se'', but rather represent the game by way of a matrix. While this approach can be of greater use in identifyi ...
Proof theory
*
Normal form (natural deduction)
An inference of natural deduction is a normal form, according to Dag Prawitz
Dag Prawitz (born 1936, Stockholm) is a Swedish philosopher and logician. He is best known for his work on proof theory and the foundations of natural deduction.
Pr ...
Rewriting systems
The symbolic manipulation of a formula from one form to another is called a "rewriting" of that formula. One can study the abstract properties of rewriting generic formulas, by studying the collection of rules by which formulas can be validly manipulated. These are the "rewriting rules"—an integral part of an
abstract rewriting system
In mathematical logic and theoretical computer science, an abstract rewriting system (also (abstract) reduction system or abstract rewrite system; abbreviated ARS) is a formalism that captures the quintessential notion and properties of rewriting s ...
. A common question is whether it is possible to bring some generic expression to a single, common form, the normal form. If different sequences of rewrites still result in the same form, then that form can be termed a normal form, with the rewrite being called a confluent. It is not always possible to obtain a normal form.
Lambda calculus
*A lambda term is in
beta normal form
In the lambda calculus, a term is in beta normal form if no ''beta reduction'' is possible. A term is in beta-eta normal form if neither a beta reduction nor an ''eta reduction'' is possible. A term is in head normal form if there is no ''beta-red ...
if no beta reduction is possible;
lambda calculus
Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation ...
is a particular case of an abstract rewriting system. In the untyped lambda calculus, for example, the term
doesn't have a normal form. In the typed lambda calculus, every well-formed term can be rewritten to its normal form.
Graph theory
In
graph theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...
, a branch of mathematics, graph canonization is the problem of finding a canonical form of a given graph ''G''. A canonical form is a
labeled graph
In the mathematical discipline of graph theory, a graph labelling is the assignment of labels, traditionally represented by integers, to edges and/or vertices of a graph.
Formally, given a graph , a vertex labelling is a function of to a set ...
Canon(''G'') that is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to ''G'', such that every graph that is isomorphic to ''G'' has the same canonical form as ''G''. Thus, from a solution to the graph canonization problem, one could also solve the problem of
graph isomorphism
In graph theory, an isomorphism of graphs ''G'' and ''H'' is a bijection between the vertex sets of ''G'' and ''H''
: f \colon V(G) \to V(H)
such that any two vertices ''u'' and ''v'' of ''G'' are adjacent in ''G'' if and only if f(u) and f(v) ar ...
: to test whether two graphs ''G'' and ''H'' are isomorphic, compute their canonical forms Canon(''G'') and Canon(''H''), and test whether these two canonical forms are identical.
Computing
In
computing
Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes, and development of both hardware and software. Computing has scientific, e ...
, the reduction of data to any kind of canonical form is commonly called ''data normalization''.
For instance,
database normalization
Database normalization or database normalisation (see spelling differences) is the process of structuring a relational database in accordance with a series of so-called normal forms in order to reduce data redundancy and improve data integrity ...
is the process of organizing the
fields
Fields may refer to:
Music
* Fields (band), an indie rock band formed in 2006
* Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song b ...
and
table
Table may refer to:
* Table (furniture), a piece of furniture with a flat surface and one or more legs
* Table (landform), a flat area of land
* Table (information), a data arrangement with rows and columns
* Table (database), how the table data ...
s of a
relational database
A relational database is a (most commonly digital) database based on the relational model of data, as proposed by E. F. Codd in 1970. A system used to maintain relational databases is a relational database management system (RDBMS). Many relatio ...
to minimize
redundancy and dependency.
In the field of
software security
Application security (short AppSec) includes all tasks that introduce a secure software development life cycle to development teams. Its final goal is to improve security practices and, through that, to find, fix and preferably prevent security i ...
, a common
vulnerability
Vulnerability refers to "the quality or state of being exposed to the possibility of being attacked or harmed, either physically or emotionally."
A window of vulnerability (WOV) is a time frame within which defensive measures are diminished, com ...
is unchecked malicious input (see ''
Code injection
Code injection is the exploitation of a computer bug that is caused by processing invalid data. The injection is used by an attacker to introduce (or "inject") code into a vulnerable computer program and change the course of execution. The re ...
''). The mitigation for this problem is proper
input validation. Before input validation is performed, the input is usually normalized by eliminating encoding (e.g.,
HTML encoding) and reducing the input data to a single common
character set
Character encoding is the process of assigning numbers to graphical characters, especially the written characters of human language, allowing them to be stored, transmitted, and transformed using digital computers. The numerical values tha ...
.
Other forms of data, typically associated with
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
(including
audio
Audio most commonly refers to sound, as it is transmitted in signal form. It may also refer to:
Sound
* Audio signal, an electrical representation of sound
*Audio frequency, a frequency in the audio spectrum
* Digital audio, representation of sou ...
and
imaging
Imaging is the representation or reproduction of an object's form; especially a visual representation (i.e., the formation of an image).
Imaging technology is the application of materials and methods to create, preserve, or duplicate images.
...
) or
machine learning
Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence.
Machine ...
, can be normalized in order to provide a limited range of values.
In
content management
Content management (CM) is a set of processes and technologies that supports the collection, managing, and publishing of information in any form or medium. When stored and accessed via computers, this information may be more specifically referre ...
, the concept of a
single source of truth
In information science and information technology, single source of truth (SSOT) architecture, or single point of truth (SPOT) architecture, for information systems is the practice of structuring information models and associated data schemas su ...
(SSOT) is applicable, just as it is in
database normalization
Database normalization or database normalisation (see spelling differences) is the process of structuring a relational database in accordance with a series of so-called normal forms in order to reduce data redundancy and improve data integrity ...
generally and in
software development
Software development is the process of conceiving, specifying, designing, programming, documenting, testing, and bug fixing involved in creating and maintaining applications, frameworks, or other software components. Software development invol ...
. Competent
content management system
A content management system (CMS) is computer software used to manage the creation and modification of digital content (content management).''Managing Enterprise Content: A Unified Content Strategy''. Ann Rockley, Pamela Kostur, Steve Manning. New ...
s provide logical ways of obtaining it, such as
transclusion
In computer science, transclusion is the inclusion of part or all of an electronic document into one or more other documents by reference via hypertext. Transclusion is usually performed when the referencing document is displayed, and is normal ...
.
See also
*
Canonicalization
In computer science, canonicalization (sometimes standardization or normalization) is a process for converting data that has more than one possible representation into a "standard", "normal", or canonical form. This can be done to compare diff ...
*
Canonical basis
In mathematics, a canonical basis is a basis of an algebraic structure that is canonical in a sense that depends on the precise context:
* In a coordinate space, and more generally in a free module, it refers to the standard basis defined by the ...
*
Canonical class In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''.
Over the complex numbers, it ...
*
Normalization (disambiguation)
*
Standardization
Standardization or standardisation is the process of implementing and developing technical standards based on the consensus of different parties that include firms, users, interest groups, standards organizations and governments. Standardization ...
Notes
References
*.
*{{citation , last=Hansen , first=Vagn Lundsgaard , title = Functional Analysis: Entering Hilbert Space , date=2006 , publisher=World Scientific Publishing , isbn=981-256-563-9.
Algebra
Concepts in logic
Mathematical terminology
Formalism (deductive)