# 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

#### 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

#### 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
This article was adapted from an original article by A.A. Dezin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article