# Domination

## Contents

## Differential operators

An order relation formulated in terms of the characteristic polynomial . For example, if

then is stronger than if for any ,

There also exist other definitions of domination; see [1].

#### References

[1] | L. Hörmander, "Linear partial differential operators" , Springer (1963) MR0161012 Zbl 0108.09301 |

#### Comments

#### References

[a1] | L.V. Hörmander, "The analysis of linear partial differential operators" , 2 , Springer (1983) pp. §10.4 MR0717035 MR0705278 Zbl 0521.35002 Zbl 0521.35001 |

## Theory of games

A relation expressing the superiority of one object (strategy (in game theory); sharing) over another. Domination of strategies: A strategy of player dominates (strictly dominates) his strategy if his pay-off in any situation containing is not smaller (is greater) than his pay-off in the situation comprising the same strategies of the other players and the strategy . Domination of sharings (in a cooperative game): A sharing dominates a sharing (denoted by ) if there exists a non-empty coalition such that

and for (where is the characteristic function of the game).

*I.N. Vrublevskaya*

#### Comments

Instead of sharing the terms imputation and pay-off vector are also used (see also Gain function).

#### References

[a1] | G. Owen, "Game theory" , Acad. Press (1982) MR0697721 Zbl 0544.90103 |

## Potential theory

An order relation between functions, in particular between potentials of specific classes, i.e. a fulfillment of the inequality for all in the common domain of definition of and . In various domination principles the relation is established as the result of the inequality on some proper subsets in the domains of definition. The simplest Cartan domination principle is: Let be a non-negative superharmonic function (cf. Subharmonic function) on the Euclidean space , , and let be the Newton potential of a measure of finite energy (cf. Energy of measures). Then, if on some set such that , the domination holds. See also Potential theory, abstract.

#### References

[1] | M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959) MR0106366 Zbl 0084.30903 |

[2] | M. Brelot, "On topologies and boundaries in potential theory" , Springer (1971) MR0281940 Zbl 0222.31014 |

*E.D. Solomentsev*

## Further concepts

There are some more concepts in mathematics which involve the word dominant or domination. Thus, a sequence of constants for a sequence of functions such that for all is called a dominant or majorant of .

In algebraic geometry one speaks of a dominant morphism if is dense in .

In the theory of commutative local rings, if , are both local rings contained in a field , then dominates if but , where is the maximal ideal of .

Finally, cf. Representation of a Lie algebra and Representation with a highest weight vector for the notions of a dominant weight and a dominant linear form.

The Cartan domination principle is also called Cartan's maximum principle. Let be a real-valued function on , for a measure on . The kernel is said to satisfy the balayage principle, or sweeping-out principle, if for each compact set and measure supported by there is a measure supported by such that quasi-everywhere on and in . The measure is the balayage of ; cf. also Balayage method. Let be the support of . Then the balayage principle implies the Cartan domination principle in the form that if on for some of finite energy and some , then the same holds in . (The measure has finite energy if is finite.) The potential is said to satisfy the inverse domination principle if on for of finite energy and any implies the same inequality in .

In abstract potential theory the Cartan domination principle simplifies to the "axiom of dominationaxiom of domination" . Let be a locally bounded potential, harmonic on the open set , and let be a positive hyperharmonic function (cf. Poly-harmonic function). If on the complement of , then . See [a1] for a survey of related properties.

#### References

[a1] | C. Constantinescu, A. Cornea, "Potential theory on harmonic spaces" , Springer (1972) MR0419799 Zbl 0248.31011 |

[a2] | R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 MR0463157 Zbl 0367.14001 |

**How to Cite This Entry:**

Domination.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Domination&oldid=37421