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''in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027440/c0274401.png" /> along a non-closed simple arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027440/c0274402.png" />''
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The removal of the points of the arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027440/c0274403.png" /> from the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027440/c0274404.png" />, that is, replacing the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027440/c0274405.png" /> by the domain (or domains) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027440/c0274406.png" />, as well as the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027440/c0274407.png" /> itself. Here it is assumed that either the whole arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027440/c0274408.png" /> or the whole arc except the initial or end point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027440/c0274409.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027440/c02744010.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027440/c02744011.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027440/c02744012.png" /> belong to the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027440/c02744013.png" />. To each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027440/c02744014.png" /> of the cut <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027440/c02744015.png" />, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027440/c02744016.png" />, there correspond two prime ends of the part of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027440/c02744017.png" /> which belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027440/c02744018.png" />; the left and right prime ends (cf. [[Limit elements|Limit elements]]). The union of these prime ends form the left and right-hand sides of the cut <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027440/c02744019.png" />.
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''in a domain  $  D \subset  \mathbf C $
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along a non-closed simple arc  $  \gamma = \{ {z ( t) } : {0 \leq  t \leq  1 } \} $''
  
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The removal of the points of the arc  $  \gamma $
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from the domain  $  D $,
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that is, replacing the domain  $  D $
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by the domain (or domains)  $  D \setminus  \gamma $,
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as well as the set  $  \gamma $
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itself. Here it is assumed that either the whole arc  $  \gamma $
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or the whole arc except the initial or end point  $  z ( 0), z ( 1) $
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belongs to  $  D $
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and that  $  z ( 0) $
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or  $  z ( 1) $
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belong to the boundary  $  \partial  D $.
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To each point  $  z ( t) $
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of the cut  $  \gamma $,
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when  $  0 < t < 1 $,
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there correspond two prime ends of the part of the domain  $  D $
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which belongs to  $  \gamma $;
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the left and right prime ends (cf. [[Limit elements|Limit elements]]). The union of these prime ends form the left and right-hand sides of the cut  $  \gamma $.
  
 
====Comments====
 
====Comments====
 
A cut is also called a slit.
 
A cut is also called a slit.
  
One also speaks of boundary elements instead of limit elements or prime ends. These notions do not coincide in general, but for  "nice"  domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027440/c02744020.png" /> (e.g. with Jordan boundary) they can be identified. A related notion is that of a crosscut: an open simple arc that begins and ends at two different points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027440/c02744021.png" />, cf. [[#References|[a1]]], especially Chapt. 3.
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One also speaks of boundary elements instead of limit elements or prime ends. These notions do not coincide in general, but for  "nice"  domains $  D $(
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e.g. with Jordan boundary) they can be identified. A related notion is that of a crosscut: an open simple arc that begins and ends at two different points of $  \partial  D $,  
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cf. [[#References|[a1]]], especially Chapt. 3.
  
 
The word  "cut"  also occurs in several more meanings in various parts of mathematics. Thus, there is the notion of a [[Dedekind cut|Dedekind cut]] in the real or rational numbers, cf. [[Real number|Real number]]. The notion cut point on a [[Geodesic line|geodesic line]]; and a cut or cutset in a graph or transportation network (cf. [[Flow in a network|Flow in a network]] for the latter). Finally there are the [[Cut locus|cut locus]] and the [[Cutting problem|cutting problem]].
 
The word  "cut"  also occurs in several more meanings in various parts of mathematics. Thus, there is the notion of a [[Dedekind cut|Dedekind cut]] in the real or rational numbers, cf. [[Real number|Real number]]. The notion cut point on a [[Geodesic line|geodesic line]]; and a cut or cutset in a graph or transportation network (cf. [[Flow in a network|Flow in a network]] for the latter). Finally there are the [[Cut locus|cut locus]] and the [[Cutting problem|cutting problem]].

Latest revision as of 17:31, 5 June 2020


in a domain $ D \subset \mathbf C $ along a non-closed simple arc $ \gamma = \{ {z ( t) } : {0 \leq t \leq 1 } \} $

The removal of the points of the arc $ \gamma $ from the domain $ D $, that is, replacing the domain $ D $ by the domain (or domains) $ D \setminus \gamma $, as well as the set $ \gamma $ itself. Here it is assumed that either the whole arc $ \gamma $ or the whole arc except the initial or end point $ z ( 0), z ( 1) $ belongs to $ D $ and that $ z ( 0) $ or $ z ( 1) $ belong to the boundary $ \partial D $. To each point $ z ( t) $ of the cut $ \gamma $, when $ 0 < t < 1 $, there correspond two prime ends of the part of the domain $ D $ which belongs to $ \gamma $; the left and right prime ends (cf. Limit elements). The union of these prime ends form the left and right-hand sides of the cut $ \gamma $.

Comments

A cut is also called a slit.

One also speaks of boundary elements instead of limit elements or prime ends. These notions do not coincide in general, but for "nice" domains $ D $( e.g. with Jordan boundary) they can be identified. A related notion is that of a crosscut: an open simple arc that begins and ends at two different points of $ \partial D $, cf. [a1], especially Chapt. 3.

The word "cut" also occurs in several more meanings in various parts of mathematics. Thus, there is the notion of a Dedekind cut in the real or rational numbers, cf. Real number. The notion cut point on a geodesic line; and a cut or cutset in a graph or transportation network (cf. Flow in a network for the latter). Finally there are the cut locus and the cutting problem.

References

[a1] M. Ohtsuka, "Dirichlet problem, extremal length and prime ends" , v. Nostrand-Reinhold (1970)
How to Cite This Entry:
Cut. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cut&oldid=13864
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article