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Difference between revisions of "Sharp form"

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$$  
 
$$  
| \omega |  ^  \srp   =  \sup \{ | \omega | _ {0} ,\  
+
| \omega |  ^  \sharp   =  \sup \{ | \omega | _ {0} ,\  
 
( r + 1 ) {\mathcal L} _ {0} ( \omega ) \}
 
( r + 1 ) {\mathcal L} _ {0} ( \omega ) \}
 
$$
 
$$
Line 59: Line 59:
 
where  $  \sigma _ {1} , \sigma _ {2} \dots $
 
where  $  \sigma _ {1} , \sigma _ {2} \dots $
 
is a sequence of simplices containing the point  $  p $,  
 
is a sequence of simplices containing the point  $  p $,  
with diameters tending to zero, and lying in the same plane. This correspondence is a one-to-one mapping of the space of cochains  $  C ^ {\srp r } ( R) $
+
with diameters tending to zero, and lying in the same plane. This correspondence is a one-to-one mapping of the space of cochains  $  C ^ {\sharp r } ( R) $
into the space  $  \Omega ^ {\srp r } $
+
into the space  $  \Omega ^ {\sharp r } $
 
of sharp forms; moreover:
 
of sharp forms; moreover:
  
Line 69: Line 69:
 
i.e. the Lipschitz constant of  $  X $;
 
i.e. the Lipschitz constant of  $  X $;
  
$  | \omega _ {X} |  ^  \srp = | X |  ^  \srp $,  
+
$  | \omega _ {X} |  ^  \sharp = | X |  ^  \sharp $,  
 
i.e. the [[Sharp norm|sharp norm]] of  $  X $;
 
i.e. the [[Sharp norm|sharp norm]] of  $  X $;
  
$  \Omega ^ {\srp r } $
+
$  \Omega ^ {\sharp r } $
 
is a Banach space.
 
is a Banach space.
  
 
In particular, there is a correspondence between zero-dimensional sharp cochains and sharp functions (bounded functions which satisfy a Lipschitz condition).
 
In particular, there is a correspondence between zero-dimensional sharp cochains and sharp functions (bounded functions which satisfy a Lipschitz condition).
  
The space  $  C _ {r}  ^  \srp ( R) $
+
The space  $  C _ {r}  ^  \sharp ( R) $
 
of  $  r $-
 
of  $  r $-
 
dimensional sharp chains  $  A $
 
dimensional sharp chains  $  A $
 
of finite [[Mass|mass]]  $  | A | $
 
of finite [[Mass|mass]]  $  | A | $
with the sharp norm  $  | A |  ^  \srp $
+
with the sharp norm  $  | A |  ^  \sharp $
is isomorphic to the space  $  \Gamma _ {r}  ^  \srp ( E  ^ {n} ) $
+
is isomorphic to the space  $  \Gamma _ {r}  ^  \sharp ( E  ^ {n} ) $
 
of additive set functions whose values are  $  r $-
 
of additive set functions whose values are  $  r $-
 
vectors  $  \gamma $,  
 
vectors  $  \gamma $,  
provided with the sharp norm  $  | \gamma |  ^  \srp $;  
+
provided with the sharp norm  $  | \gamma |  ^  \sharp $;  
 
this correspondence is defined by the formula:
 
this correspondence is defined by the formula:
  
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i.e. the complete variation of  $  \gamma _ {A} $;
 
i.e. the complete variation of  $  \gamma _ {A} $;
  
$  | \gamma _ {A} |  ^  \srp = | A |  ^  \srp $,  
+
$  | \gamma _ {A} |  ^  \sharp = | A |  ^  \sharp $,  
 
i.e. the sharp norm of the chain  $  A $.
 
i.e. the sharp norm of the chain  $  A $.
  

Latest revision as of 16:26, 22 February 2021


An $ r $- dimensional differential form $ \omega $ on an open subset $ R \subset E ^ {n} $ such that the co-mass (cf. Mass and co-mass) $ | \omega | _ {0} $ and the co-mass Lipschitz constant

$$ {\mathcal L} _ {0} ( \omega ) = \sup \frac{| \omega ( p) - \omega ( q) | }{| p - q | } , $$

where $ p , q \in R $ and $ | p - q | $ is the length of the vector $ p - q $, are finite. The number

$$ | \omega | ^ \sharp = \sup \{ | \omega | _ {0} ,\ ( r + 1 ) {\mathcal L} _ {0} ( \omega ) \} $$

is known as the sharp norm of the form $ \omega $.

Whitney's theorem. To each $ r $- dimensional sharp cochain $ X $ in $ R $ corresponds a unique $ r $- dimensional sharp form $ \omega _ {X} $ for which

$$ X \sigma ^ {r} = \int\limits _ {\sigma ^ {r} } \omega _ {X} $$

for all $ r $- dimensional oriented simplices $ \sigma ^ {r} $; $ \omega _ {X} ( p) $ is defined by the formula

$$ \omega _ {X} ( p) = \lim\limits \frac{X \sigma _ {i} }{| \sigma _ {i} | } , $$

where $ \sigma _ {1} , \sigma _ {2} \dots $ is a sequence of simplices containing the point $ p $, with diameters tending to zero, and lying in the same plane. This correspondence is a one-to-one mapping of the space of cochains $ C ^ {\sharp r } ( R) $ into the space $ \Omega ^ {\sharp r } $ of sharp forms; moreover:

$ | \omega _ {X} | _ {0} = | X | $, i.e. the co-mass of $ X $;

$ {\mathcal L} ( \omega _ {X} ) = {\mathcal L} ( X) $, i.e. the Lipschitz constant of $ X $;

$ | \omega _ {X} | ^ \sharp = | X | ^ \sharp $, i.e. the sharp norm of $ X $;

$ \Omega ^ {\sharp r } $ is a Banach space.

In particular, there is a correspondence between zero-dimensional sharp cochains and sharp functions (bounded functions which satisfy a Lipschitz condition).

The space $ C _ {r} ^ \sharp ( R) $ of $ r $- dimensional sharp chains $ A $ of finite mass $ | A | $ with the sharp norm $ | A | ^ \sharp $ is isomorphic to the space $ \Gamma _ {r} ^ \sharp ( E ^ {n} ) $ of additive set functions whose values are $ r $- vectors $ \gamma $, provided with the sharp norm $ | \gamma | ^ \sharp $; this correspondence is defined by the formula:

$$ \tag{* } X A = \int\limits _ {E ^ {n} } \omega _ {X} d {\gamma _ {A} } = [ \omega \cdot \gamma ] ( E ^ {n} ) $$

for any cochain $ X $, where $ \omega _ {X} $ is the $ r $- dimensional sharp form corresponding to the cochain $ X $, and

$ \gamma _ {A} ( E ^ {n} ) = \{ A \} $, i.e. the covector of the chain $ A $;

$ | A | = | \gamma _ {A} | $, i.e. the complete variation of $ \gamma _ {A} $;

$ | \gamma _ {A} | ^ \sharp = | A | ^ \sharp $, i.e. the sharp norm of the chain $ A $.

Thus, (*) is a generalization of the ordinary Lebesgue–Stieltjes integral. In particular, the Lebesgue-measurable summable function $ \alpha ( p) $ associated with $ A $( cf. Flat form), i.e.

$$ X \cdot A = \int\limits _ {E ^ {n} } \omega _ {X} \cdot \alpha ( p) dp $$

for any cochain $ X $, exists for $ A $ if and only if $ \gamma _ {A} $ is absolutely continuous.

If $ \omega _ {A} $ is a regular form and $ X $ is a sharp cochain, then there exists a form $ \omega _ {dX } = d \omega _ {X} $, and Stokes' formula

$$ \int\limits _ {\partial \sigma } \omega _ {X} = \int\limits _ \sigma d \omega $$

applies. Other results established for regular forms can be generalized in an analogous manner.

For references see Sharp norm.

How to Cite This Entry:
Sharp form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sharp_form&oldid=48680
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article