Clifford's Theorem (other)
Clifford's theorem may refer to: *Clifford's theorem on special divisors *Clifford theory in representation theory *Hammersley–Clifford theorem in probability *Clifford's circle theorems In geometry, Clifford's theorems, named after the English geometer William Kingdon Clifford, are a sequence of theorems relating to intersections of circles. Statement The first theorem considers any four circles passing through a common poin ...

Clifford's Theorem On Special Divisors
In mathematics, Clifford's theorem on special divisors is a result of on algebraic curves, showing the constraints on special linear systems on a curve ''C''. Statement A divisor on a Riemann surface ''C'' is a formal sum \textstyle D = \sum_P m_P P of points ''P'' on ''C'' with integer coefficients. One considers a divisor as a set of constraints on meromorphic functions in the function field of ''C,'' defining L(D) as the vector space of functions having poles only at points of ''D'' with positive coefficient, ''at most as bad'' as the coefficient indicates, and having zeros at points of ''D'' with negative coefficient, with ''at least'' that multiplicity. The dimension of L(D) is finite, and denoted \ell(D). The linear system of divisors attached to ''D'' is the corresponding projective space of dimension \ell(D)-1. The other significant invariant of ''D'' is its degree ''d'', which is the sum of all its coefficients. A divisor is called ''special'' if ''ℓ''(''K''  ...
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Clifford Theory
In mathematics, Clifford theory, introduced by , describes the relation between representations of a group and those of a normal subgroup. Alfred H. Clifford Alfred H. Clifford proved the following result on the restriction of finite-dimensional irreducible representations from a group ''G'' to a normal subgroup ''N'' of finite index: Clifford's theorem Theorem. Let π: ''G'' → GL(''n'',''K'') be an irreducible representation with ''K'' a field. Then the restriction of π to ''N'' breaks up into a direct sum of irreducible representations of ''N'' of equal dimensions. These irreducible representations of ''N'' lie in one orbit for the action of ''G'' by conjugation on the equivalence classes of irreducible representations of ''N''. In particular the number of pairwise nonisomorphic summands is no greater than the index of ''N'' in ''G''. Clifford's theorem yields information about the restriction of a complex irreducible character of a finite group ''G'' to a normal subgroup ' ...
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Hammersley–Clifford Theorem
The Hammersley–Clifford theorem is a result in probability theory, mathematical statistics and statistical mechanics that gives necessary and sufficient conditions under which a strictly positive probability distribution (of events in a probability space) can be represented as events generated by a Markov network (also known as a Markov random field). It is the fundamental theorem of random fields. It states that a probability distribution that has a strictly positive mass or density satisfies one of the Markov properties with respect to an undirected graph ''G'' if and only if it is a Gibbs random field, that is, its density can be factorized over the cliques (or complete subgraphs) of the graph. The relationship between Markov and Gibbs random fields was initiated by Roland Dobrushin and Frank Spitzer in the context of statistical mechanics. The theorem is named after John Hammersley and Peter Clifford, who proved the equivalence in an unpublished paper in 1971. Simpler proof ...
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