# Domination

## Differential operators

An order relation formulated in terms of the characteristic polynomial $ P ( \xi ) $. For example, if

$$ {\widetilde{P} } {} ^ {2} ( \xi ) = \sum _ {\alpha \geq 0 } | P ^ {( \alpha ) } ( \xi ) | ^ {2} , $$

$$ P ^ {( \alpha ) } ( \xi ) = \frac{\partial ^ {| \alpha | } }{\partial \xi _ {1} ^ {\alpha _ {1} } \dots \partial \xi _ {n} ^ {\alpha _ {n} } } P ( \xi ) \ \equiv \ i ^ {| \alpha | } D ^ \alpha P ( \xi ) ,\ \xi \in \mathbf R ^ {n} , $$

then $ P ( D) $ is stronger than $ Q ( D) $ if for any $ \xi \in \mathbf R ^ {n} $,

$$ \frac{ {\widetilde{Q} } ( \xi ) }{ {\widetilde{P} } ( \xi ) } < \textrm{ const } . $$

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 $ s $ of player $ i $ dominates (strictly dominates) his strategy $ t $ if his pay-off in any situation containing $ s $ is not smaller (is greater) than his pay-off in the situation comprising the same strategies of the other players and the strategy $ t $. Domination of sharings (in a cooperative game): A sharing $ x $ dominates a sharing $ y $( denoted by $ x \succ y $) if there exists a non-empty coalition $ P \subset \mathbf N $ such that

$$ \sum _ {i \in P } x _ {i} \leq v ( P) $$

and $ x _ {i} > y _ {i} $ for $ i \in P $( where $ v $ 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 $ v _ {1} \geq v _ {2} $ between functions, in particular between potentials of specific classes, i.e. a fulfillment of the inequality $ v _ {1} ( x) \geq v _ {2} ( x) $ for all $ x $ in the common domain of definition of $ v _ {1} $ and $ v _ {2} $. In various domination principles the relation $ v _ {1} \geq v _ {2} $ is established as the result of the inequality $ v _ {1} ( x) \geq v _ {2} ( x) $ on some proper subsets in the domains of definition. The simplest Cartan domination principle is: Let $ v = v ( x) $ be a non-negative superharmonic function (cf. Subharmonic function) on the Euclidean space $ \mathbf R ^ {n} $, $ n \geq 3 $, and let $ U _ \mu = U _ \mu ( x) $ be the Newton potential of a measure $ \mu \geq 0 $ of finite energy (cf. Energy of measures). Then, if $ v ( x) \geq U _ \mu ( x) $ on some set $ A \subset \mathbf R ^ {n} $ such that $ \mu ( CA) = 0 $, the domination $ v \geq U _ \mu $ 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 $ M _ {n} $ for a sequence of functions $ \{ f _ {n} \} $ such that $ | f _ {n} ( x) | \leq M _ {n} $ for all $ x $ is called a dominant or majorant of $ \{ f _ {n} \} $.

In algebraic geometry one speaks of a dominant morphism $ \phi : X \rightarrow Y $ if $ \phi ( X) $ is dense in $ Y $.

In the theory of commutative local rings, if $ R $, $ S $ are both local rings contained in a field $ K $, then $ S $ dominates $ R $ if $ R \subseteq S $ but $ \mathfrak m _ {S} \cap R = \mathfrak m _ {R} $, where $ \mathfrak m _ {R} $ is the maximal ideal of $ R $.

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 $ \Phi ( x , y ) $ be a real-valued function on $ \Omega \times \Omega $, $ \Phi ( x , \nu ) = \int \Phi ( x , y ) d \nu ( y) $ for a measure $ \nu $ on $ \Omega $. The kernel $ \Phi $ is said to satisfy the balayage principle, or sweeping-out principle, if for each compact set $ K $ and measure $ \mu $ supported by $ K $ there is a measure $ \nu $ supported by $ K $ such that $ \Phi ( x , \nu ) = \Phi ( x , \mu ) $ quasi-everywhere on $ K $ and $ \Phi ( x , \nu ) \leq \Phi ( y , \mu ) $ in $ \Omega $. The measure $ \nu $ is the balayage of $ \mu $; cf. also Balayage method. Let $ S _ \mu $ be the support of $ \mu $. Then the balayage principle implies the Cartan domination principle in the form that if $ \Phi ( x , \mu ) < \Phi ( x , \nu ) $ on $ S _ \mu $ for some $ \mu $ of finite energy and some $ \nu $, then the same holds in $ \Omega $. (The measure $ \mu $ has finite energy if $ ( \mu , \mu ) = \int \Phi ( x , \mu ) d \mu ( x) $ is finite.) The potential is said to satisfy the inverse domination principle if $ \Phi ( x , \mu ) < \Phi ( x , \nu ) $ on $ S _ \nu $ for $ \mu $ of finite energy and any $ \nu $ implies the same inequality in $ \Omega $.

In abstract potential theory the Cartan domination principle simplifies to the "axiom of dominationaxiom of domination" . Let $ p $ be a locally bounded potential, harmonic on the open set $ U $, and let $ u $ be a positive hyperharmonic function (cf. Poly-harmonic function). If $ u \geq p $ on the complement of $ U $, then $ u \geq p $. 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://encyclopediaofmath.org/index.php?title=Domination&oldid=46764