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==Differential operators==
 
==Differential operators==
An order relation formulated in terms of the [[characteristic polynomial]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d0338301.png" />. For example, if
+
An order relation formulated in terms of the [[characteristic polynomial]] $  P ( \xi ) $.  
 +
For example, if
 +
 
 +
$$
 +
{\widetilde{P}  } {}  ^ {2} ( \xi )  =  \sum _ {\alpha \geq  0 } | P ^ {( \alpha
 +
) } ( \xi ) |  ^ {2} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d0338302.png" /></td> </tr></table>
+
$$
 +
P ^ {( \alpha ) } ( \xi )  =
 +
\frac{\partial  ^ {| \alpha | } }{\partial  \xi _ {1} ^ {\alpha _ {1} } \dots \partial
 +
\xi _ {n} ^ {\alpha _ {n} } }
 +
P ( \xi ) \
 +
\equiv \
 +
i ^ {| \alpha | } D  ^  \alpha  P ( \xi ) ,\  \xi \in \mathbf R  ^ {n} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d0338303.png" /></td> </tr></table>
+
then  $  P ( D) $
 +
is stronger than  $  Q ( D) $
 +
if for any  $  \xi \in \mathbf R  ^ {n} $,
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d0338304.png" /> is stronger than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d0338305.png" /> if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d0338306.png" />,
+
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d0338307.png" /></td> </tr></table>
+
\frac{ {\widetilde{Q}  } ( \xi ) }{ {\widetilde{P}  } ( \xi ) }
 +
  < \textrm{ const } .
 +
$$
  
 
There also exist other definitions of domination; see [[#References|[1]]].
 
There also exist other definitions of domination; see [[#References|[1]]].
Line 14: Line 43:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> L. Hörmander, "Linear partial differential operators" , Springer (1963) {{MR|0161012}} {{ZBL|0108.09301}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> L. Hörmander, "Linear partial differential operators" , Springer (1963) {{MR|0161012}} {{ZBL|0108.09301}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
Line 24: Line 50:
  
 
==Theory of games==
 
==Theory of games==
A relation expressing the superiority of one object ([[strategy (in game theory)]]; [[sharing]]) over another. Domination of strategies: A strategy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d0338308.png" /> of player <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d0338309.png" /> dominates (strictly dominates) his strategy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383010.png" /> if his pay-off in any situation containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383011.png" /> is not smaller (is greater) than his pay-off in the situation comprising the same strategies of the other players and the strategy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383012.png" />. Domination of sharings (in a [[Cooperative game|cooperative game]]): A sharing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383013.png" /> dominates a sharing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383014.png" /> (denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383015.png" />) if there exists a non-empty [[Coalition|coalition]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383016.png" /> such that
+
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|cooperative game]]): A sharing $  x $
 +
dominates a sharing $  y $(
 +
denoted by $  x \succ y $)  
 +
if there exists a non-empty [[Coalition|coalition]] $  P \subset  \mathbf N $
 +
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383017.png" /></td> </tr></table>
+
$$
 +
\sum _ {i \in P } x _ {i}  \leq  v ( P)
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383018.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383019.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383020.png" /> is the characteristic function of the game).
+
and $  x _ {i} > y _ {i} $
 +
for $  i \in P $(
 +
where $  v $
 +
is the characteristic function of the game).
  
 
''I.N. Vrublevskaya''
 
''I.N. Vrublevskaya''
Line 39: Line 79:
  
 
==Potential theory==
 
==Potential theory==
An order relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383021.png" /> between functions, in particular between potentials of specific classes, i.e. a fulfillment of the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383022.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383023.png" /> in the common domain of definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383025.png" />. In various domination principles the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383026.png" /> is established as the result of the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383027.png" /> on some proper subsets in the domains of definition. The simplest Cartan domination principle is: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383028.png" /> be a non-negative superharmonic function (cf. [[Subharmonic function|Subharmonic function]]) on the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383030.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383031.png" /> be the Newton potential of a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383032.png" /> of finite energy (cf. [[Energy of measures|Energy of measures]]). Then, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383033.png" /> on some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383034.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383035.png" />, the domination <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383036.png" /> holds. See also [[Potential theory, abstract|Potential theory, abstract]].
+
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|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|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|Potential theory, abstract]].
  
 
====References====
 
====References====
Line 47: Line 103:
  
 
==Further concepts==
 
==Further concepts==
There are some more concepts in mathematics which involve the word dominant or domination. Thus, a sequence of constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383037.png" /> for a sequence of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383038.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383039.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383040.png" /> is called a dominant or majorant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383041.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383042.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383043.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383044.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383046.png" /> are both local rings contained in a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383047.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383048.png" /> dominates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383049.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383050.png" /> but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383051.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383052.png" /> is the maximal ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383053.png" />.
+
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|Representation of a Lie algebra]] and [[Representation with a highest weight vector|Representation with a highest weight vector]] for the notions of a dominant weight and a dominant linear form.
 
Finally, cf. [[Representation of a Lie algebra|Representation of a Lie algebra]] and [[Representation with a highest weight vector|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383054.png" /> be a real-valued function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383056.png" /> for a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383057.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383058.png" />. The kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383059.png" /> is said to satisfy the balayage principle, or sweeping-out principle, if for each compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383060.png" /> and measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383061.png" /> supported by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383062.png" /> there is a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383063.png" /> supported by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383064.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383065.png" /> quasi-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383067.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383068.png" />. The measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383069.png" /> is the balayage of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383070.png" />; cf. also [[Balayage method|Balayage method]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383071.png" /> be the support of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383072.png" />. Then the balayage principle implies the Cartan domination principle in the form that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383073.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383074.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383075.png" /> of finite energy and some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383076.png" />, then the same holds in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383077.png" />. (The measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383078.png" /> has finite energy if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383079.png" /> is finite.) The potential is said to satisfy the inverse domination principle if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383080.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383081.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383082.png" /> of finite energy and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383083.png" /> implies the same inequality in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383084.png" />.
+
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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383085.png" /> be a locally bounded [[Potential|potential]], harmonic on the open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383086.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383087.png" /> be a positive hyperharmonic function (cf. [[Poly-harmonic function|Poly-harmonic function]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383088.png" /> on the complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383089.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033830/d03383090.png" />. See [[#References|[a1]]] for a survey of related properties.
+
In abstract potential theory the Cartan domination principle simplifies to the "axiom of dominationaxiom of domination" . Let $  p $
 +
be a locally bounded [[Potential|potential]], harmonic on the open set $  U $,  
 +
and let $  u $
 +
be a positive hyperharmonic function (cf. [[Poly-harmonic function|Poly-harmonic function]]). If $  u \geq  p $
 +
on the complement of $  U $,  
 +
then $  u \geq  p $.  
 +
See [[#References|[a1]]] for a survey of related properties.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C. Constantinescu, A. Cornea, "Potential theory on harmonic spaces" , Springer (1972) {{MR|0419799}} {{ZBL|0248.31011}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C. Constantinescu, A. Cornea, "Potential theory on harmonic spaces" , Springer (1972) {{MR|0419799}} {{ZBL|0248.31011}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>

Latest revision as of 19:36, 5 June 2020


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