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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Canonical form | 2/2 | https://en.wikipedia.org/wiki/Canonical_form | reference | science, encyclopedia | 2026-05-05T07:23:25.219824+00:00 | kb-cron |
The equation of a line: Ax + By = C, with A2 + B2 = 1 and C ≥ 0 The equation of a circle:
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{\displaystyle (x-h)^{2}+(y-k)^{2}=r^{2}}
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. Convex polyhedra can be put into canonical form 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 has a cotangent bundle. That bundle can always be endowed with a certain differential form, called the canonical one-form. 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. Such systems of integrable differential equations are called integrable systems.
=== Dynamical systems === The study of dynamical systems overlaps with that of integrable systems; there one has the idea of a normal form (dynamical systems).
=== Three dimensional geometry === In the study of manifolds in three dimensions, one has the first fundamental form, the second fundamental form and the third fundamental form.
=== Functional analysis ===
=== Classical logic ===
Negation normal form Conjunctive normal form Disjunctive normal form Algebraic normal form Prenex normal form Skolem normal form Blake canonical form, also known as the complete sum of prime implicants, the complete sum, or the disjunctive prime form
=== Set theory === Cantor normal form of an ordinal number
=== Game theory === Normal form game
=== Proof theory === Normal form (natural deduction)
=== 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. 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 if no beta reduction is possible; lambda calculus is a particular case of an abstract rewriting system. In the untyped lambda calculus, for example, the term
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x
.
(
x
x
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λ
x
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x
)
)
{\displaystyle (\lambda x.(xx)\;\lambda x.(xx))}
does not 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, 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 Canon(G) that is isomorphic 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: 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, the reduction of data to any kind of canonical form is commonly called data normalization. For instance, database normalization is the process of organizing the fields and tables of a relational database to minimize redundancy and dependency. In the field of software security, a common vulnerability is unchecked malicious input (see Code injection). 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. Other forms of data, typically associated with signal processing (including audio and imaging) or machine learning, can be normalized in order to provide a limited range of values. In content management, the concept of a single source of truth (SSOT) is applicable, just as it is in database normalization generally and in software development. Competent content management systems provide logical ways of obtaining it, such as transclusion.
== See also == Canonicalization Canonical basis Canonical class Normalization (disambiguation) Standardization
== Notes ==
== References == Shilov, Georgi E. (1977), Silverman, Richard A. (ed.), Linear Algebra, Dover, ISBN 0-486-63518-X. Hansen, Vagn Lundsgaard (2006), Functional Analysis: Entering Hilbert Space, World Scientific Publishing, ISBN 981-256-563-9.