P-stable group

2013-02-11T00:20:07Z

71.67.103.134: /* Definitions */

A '''kinetic smallest enclosing disk''' data structure is a [[kinetic data structure]] that maintains the [[Smallest circle problem|smallest enclosing disk]] of a set of moving points.

== 2D ==
In 2 dimensions, the best known kinetic smallest enclosing disk data structure uses the farthest point delaunay triangulation of the point set to maintain the smallest enclosing disk.<ref name="DEGS10" /> The farthest-point [[Delaunay triangulation]] is the [[Duality (projective geometry)|dual]] of the [[Voronoi diagram#Higher-order Voronoi diagrams|farthest-point Voronoi diagram]]. It is known that if the farthest-point delaunay triangulation of a point set contains an acute triangle, the [[circumcircle]] of this triangle is the smallest enclosing disk. Otherwise, the smallest enclosing disk has the diameter of the point set as its diameter. Thus, by maintaining the [[kinetic diameter]] of the point set, the farthest-point delaunay triangulation, and whether or not the farthest-point delaunay triangulation has an acute triangle, the smallest enclosing disk can be maintained.
This data structure is responsive and compact, but not local or efficient:<ref name="DEGS10" />
* '''[[Kinetic data structure#Performance|Responsiveness]]:''' This data structure requires <math>O(\log^2 n)</math> time to process each certificate failure, and thus is responsive.
* '''[[Kinetic data structure#Performance|Locality]]:''' A point can be involved in <math>\Theta(n)</math> certificates. Therefore, this data structure is not local.
* '''[[Kinetic data structure#Performance|Compactness]]:''' This data structure requires O(n) certificates total, and thus is compact.
* '''[[Kinetic data structure#Performance|Efficiency]]:''' This data structure has <math>O(n^{3+\epsilon})</math> events total.(for all <math>\epsilon>0</math> The best known lower bound on the number of changes to the smallest enclosing disk is <math>\Omega(n^2)</math>. Thus the efficiency of this data structure, the ratio of total events to external events, is <math>O(n^{1+\epsilon})</math>.
The existence of kinetic data structure that has <math>o(n^{3+\epsilon})</math> events is an open problem.<ref name="DEGS10" />

== Approximate 2D ==
The smallest enclosing disk of a set of n moving points can be [[Approximation_algorithm#Epsilon_terms|ε-approximated]] by a kinetic data structure that processes <math>O(1/\epsilon^{5/2})</math> events and requires <math>O((n/\sqrt{\epsilon})\log n)</math> time total.<ref name="AH01" />

== Higher dimensions ==
In dimensions higher than 2, efficiently maintaining the smallest enclosing sphere of a set of moving points is an open problem.<ref name="DEGS10" />

== References ==
{{Reflist| refs=
<ref name ="DEGS10">
Erik D. Demaine, Sarah Eisenstat, Leonidas J. Guibas, André Schulz, Kinetic Minimum Spanning Circle, 2010. [http://www.ams.sunysb.edu/~jsbm/fwcg10/papers/25.pdf]
</ref>
<ref name="AH01">
Pankaj K. Agarwal and Sariel Hal-Peled. Maintaining approximate extent measures of moving points. In SODA '01: Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms, pages 148–157, Philadelphia, PA, USA, 2001. Society for Industrial and Applied Mathematics.
</ref>
}}

[[Category:Kinetic data structures]]
[[Category:Computational geometry]]

Mathematical formulation of quantum mechanics

2012-08-30T11:38:53Z

71.67.113.0: /* The "new quantum theory" */

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P-stable group

Mathematical formulation of quantum mechanics