Finite

Another way to phrase the pigeonhole principle for finite sets is similar to the principle that finite sets are Dedekind finite : Let A and B be finite sets.

Ind-finite groups There is a notion of ind-finite group, which is the concept dual to profinite groups; i.e. a group G is ind-finite if it is the direct limit of an inductive system of finite groups.

A polytope is finite if it is defined in terms of a finite number of objects, e.g., as an intersection of a finite number of half-planes.

Finite volume method main Similar to the finite difference method or finite element method, values are calculated at discrete places on a meshed geometry.

From the quantum phenomena it appears to follow with certainty that a finite system of finite energy can be completely described by a finite set of numbers (quantum numbers).

Intuitively, it cannot be recognized with a finite automaton, since a finite automaton has finite memory and it cannot remember the exact number of a's.

Most of the elementary and special functions are D-finite, and the integral of a D-finite function is also a D-finite function.

Numerical methods to solve PDEs The three most widely used numerical methods to solve PDEs are the finite element method (FEM), finite volume methods (FVM) and finite difference methods (FDM).

Roughly, this means that every finitely generated group can be constructed from finite and one-ended groups via a finite number of amalgamated free products and HNN extensions over finite subgroups.

Sigma-finite measures main A measure space (X, Σ, μ) is called finite if μ(X) is a finite real number (rather than ∞).

The dual space of a finite-dimensional vector space is again a finite-dimensional vector space of the same dimension, and these are thus isomorphic, since dimension is the only invariant of finite-dimensional vector spaces over a given field.

The Galois group of every finite extension of a finite field is finite and cyclic, with an iterate of the Frobenius endomorphism as its generator.

There are no non-commutative finite division rings: Wedderburn's little theorem states that all finite division rings are commutative, hence finite fields.

There are a finite number of words in our language and only a finite number of ways to describe one’s characteristics and ideologies.

A characterization of finite dimensionality is that a Hausdorff TVS is locally compact if and only if it is finite-dimensional (therefore isomorphic to some Euclidean space).

A finite language shouldn't be confused with a (usually infinite) language generated by a finite automaton.

A finite projective plane will produce a finite affine plane when one of its lines and the points on it are removed.

A leaky mode or tunneling mode in an optical fiber or other waveguide is a mode having an electric field that decays monotonically for a finite distance in the transverse direction but becomes oscillatory everywhere beyond that finite distance.

A logic has the finite model property (FMP) if it is complete with respect to a class of finite frames.

Alternatively, one may be able to take the trace of operators on an infinite-dimensional space; in this case a (finite) trace is defined, even though no (finite) dimension exists, and gives a notion of "dimension of the operator".