en>Avsmal: Algorithm change (there were some useless variables that might have left from non-recursive implementation)

2013-12-11T08:35:59Z

Algorithm change (there were some useless variables that might have left from non-recursive implementation)

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In the branch of [[mathematics]] called [[potential theory]], a '''quadrature domain''' in two dimensional real Euclidean space is a domain D (an [[open set|open]] [[connected space|connected set]]) together with
a finite subset {''z''1, …, z''k''} of D such that, for every function ''u'' [[harmonic function|harmonic]] and integrable over D with respect to area measure, the integral of ''u'' with respect to this measure is given by a "quadrature formula"; that is,

:<math>
\iint_D u\, dx dy = \sum_{j=1}^k c_j u(z_j),
</math>

where the ''c''''j'' are nonzero complex constants independent of ''u''.

The most obvious example is when D is a circular disk: here ''k'' = 1, ''z''1 is the center of the circle, and ''c''1 equals the area of D. That quadrature formula expresses the [[Harmonic_function#The_mean_value_property|mean value property]] of harmonic functions with respect to disks.

It is known that quadrature domains exist for all values of ''k''. There is an analogous definition of quadrature domains in Euclidean space of dimension ''d'' larger than 2. There is also an alternative, [[electrostatic]] interpretation of quadrature domains: a domain D is a quadrature domain if a uniform distribution of electric charge on D creates the same electrostatic field outside D as does a ''k''-tuple of point charges at the points ''z''1, …, ''z''''k''.

Quadrature domains and numerous generalizations thereof (e.g., replace area measure by length measure on the boundary of D) have in recent years been encountered in various connections such as inverse problems of Newtonian [[gravitation]], [[Hele-Shaw flow]]s of viscous fluids, and purely mathematical isoperimetric problems, and interest in them seems to be steadily growing. They were the subject of an international conference at the University of California at Santa Barbara in 2003 and the state of the art as of that date can be seen in the proceedings of that conference, published by Birkhäuser Verlag.

==References==
*{{cite book|last = Ebenfelt|first = Peter|title = Quadrature Domains and Their Applications: The Harold S. Shapiro Anniversary Volume|publisher = Birkhäuser|date = 2005|isbn = 3-7643-7145-5|url = http://books.google.com/?id=XcdyCFSA54EC|accessdate = 2007-04-11}}
*{{cite journal|last=Aharonov|first = D.|coauthors = Shapiro, H.S.|title = Domains on which analytic functions satisfy quadrature identities|journal = J. Anal. Math.|volume = 30|date = 1976|pages = 39–73|doi=10.1007/BF02786704}}

[[Category:Potential theory]]

Hirschberg's algorithm - Revision history

en>Avsmal: Algorithm change (there were some useless variables that might have left from non-recursive implementation)