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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. Often, it is one which provides the simplest representation of an object and allows it to be identified in a unique way.
Canonical ensemble, in statistical mechanics, is a statistical ensemble representing a probability distribution of microscopic states of the system; Canonical quantum gravity, an attempt to quantize the canonical formulation of general relativity; Canonical stress–energy tensor, a conserved current associated with translations through space ...
Rigor is a cornerstone quality of mathematics, and can play an important role in preventing mathematics from degenerating into fallacies. well-behaved An object is well-behaved (in contrast with being Pathological ) if it satisfies certain prevailing regularity properties, or if it conforms to mathematical intuition (even though intuition can ...
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 Kronecker delta .
Every vector a in three dimensions is a linear combination of the standard basis vectors i, j and k.. In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as or ) is the set of vectors, each of whose components are all zero, except one that equals 1. [1]
It is difficult to choose a canonical rule in this case. "Natural" is a well-defined mathematical notion, but it does not ensure uniqueness. "Canonical" does, but generally is more or less conventional. A consistent choice of canonical equivalences is an inevitable component of equivalent definitions of mathematical structures.
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 ...
The De Morgan dual is the canonical conjunctive normal form , maxterm canonical form, or Product of Sums (PoS or POS) which is a conjunction (AND) of maxterms. These forms can be useful for the simplification of Boolean functions, which is of great importance in the optimization of Boolean formulas in general and digital circuits in particular.