Classification theorem

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 related issues to classification are the following.

  • The equivalence problem is "given two objects, determine if they are equivalent".
  • A complete set of invariants, together with which invariants are realizable, solves the classification problem, and is often a step in solving it.
  • A computable complete set of invariants (together with which invariants are realizable) solves both the classification problem and the equivalence problem.
  • A canonical form solves the classification problem, and is more data: it not only classifies every class, but gives a distinguished (canonical) element of each class.

There exist many classification theorems in mathematics, as described below.

Contents

Geometry

Algebra

Linear algebra

Complex analysis


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Classification of finite simple groups — Group theory Group theory …   Wikipedia

  • Classification of electromagnetic fields — In differential geometry and theoretical physics, the classification of electromagnetic fields is a pointwise classification of bivectors at each point of a Lorentzian manifold. It is used in the study of solutions of Maxwell s equations and has… …   Wikipedia

  • Classification problem — In science or mathematics, a classification problem for objects in a particular domain is the problem of separating these objects into smaller classes, and giving criteria for determining whether a particular object in the domain is in a… …   Wikipedia

  • Classification of manifolds — In mathematics, specifically geometry and topology, the classification of manifolds is a basic question, about which much is known, and many open questions remain. Contents 1 Main themes 1.1 Overview 1.2 Different categories and additional… …   Wikipedia

  • Theorem — The Pythagorean theorem has at least 370 known proofs[1] In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements …   Wikipedia

  • Classification rule — See also: Statistical classification and Classification in machine learning Given a population whose members can be potentially separated into a number of different sets or classes, a classification rule is a procedure in which the elements… …   Wikipedia

  • Classification of Clifford algebras — In mathematics, in particular in the theory of nondegenerate quadratic forms on real and complex vector spaces, the finite dimensional Clifford algebras have been completely classified in terms of isomorphisms that preserve the Clifford product.… …   Wikipedia

  • Classification of discontinuities — Jump point redirects here. For the science fiction concept, see jump drive. Continuous functions are of utmost importance in mathematics and applications. However, not all functions are continuous. If a function is not continuous at a point in… …   Wikipedia

  • Classification of Fatou components — In mathematics, if f = P(z) / Q(z) is a rational function defined in the extended complex plane, and if then for a periodic component U of the Fatou set, exactly one of the following holds: U contains an attracting periodic point U is parabolic U …   Wikipedia

  • Petrov classification — In differential geometry and theoretical physics, the Petrov classification describes the possible algebraic symmetries of the Weyl tensor at each event in a Lorentzian manifold.It is most often applied in studying exact solutions of Einstein s… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”