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The term '''''semi-infinite''''' has several related meanings in various branches of pure and applied [[mathematics]].  It typically describes objects which are [[Infinity|infinite]] or [[unbounded]] in some but not all possible ways.
 
==In ordered structures and Euclidean spaces==
 
Generally, a semi-infinite set is [[bounded]] in one direction, and [[unbounded]] in another.  For instance, the [[natural numbers]] are semi-infinite considered as a subset of the integers; similarly, the [[interval (mathematics)|intervals]] <math>(c,\infty)</math> and <math>(-\infty,c)</math> and their closed counterparts are semi-infinite subsets of <math>\R</math>. [[Half-space (geometry)|Half-space]]s are sometimes described as semi-infinite regions.
 
Semi-infinite regions occur frequently in the study of [[differential equations]].<ref>Bateman, [http://projecteuclid.org/euclid.bams/1183492736 Transverse seismic waves on the surface of a semi-infinite solid composed of heterogeneous material], Bull. Amer. Math. Soc. Volume 34, Number 3 (1928), 343–348.</ref><ref>Wolfram Demonstrations Project, [http://demonstrations.wolfram.com/HeatDiffusionInASemiInfiniteRegion/ Heat Diffusion in a Semi-Infinite Region] (accessed November 2010).</ref> For instance, one might study solutions of the heat equation in an idealised semi-infinite metal bar.
 
A semi-infinite [[integral]] is an [[improper integral]] over a semi-infinite interval.  More generally, objects indexed or parametrised by semi-infinite sets may be described as semi-infinite.<ref>Cator, Pimentel, [http://arxiv.org/abs/1001.4706v3 A shape theorem and semi-infinite geodesics for the Hammersley model with random weights], 2010.</ref>
 
Most forms of semi-infiniteness are [[bounded]]ness properties, not [[cardinality]] or [[measure (mathematics)|measure]] properties: semi-infinite sets are typically infinite in cardinality and measure.
 
==In optimisation==
{{Main|Semi-infinite programming}}
 
Many [[optimisation (mathematics)|optimisation]] problems involve some set of variables and some set of constraints.  A problem is called semi-infinite if one (but not both) of these sets is finite. The study of such problems is known as [[semi-infinite programming]].<ref>Reemsten, Rückmann, [http://books.google.ca/books?id=sJgX5jQZnQcC&lpg=PP1&ots=gvz6-MY_t1&dq=semi-infinite%20programming&pg=PP1#v=onepage&q&f=false Semi-infinite Programming], Kluwer Academic, 1998. ISBN 0-7923-5054-5</ref>
 
==References==
<references />
 
 
 
 
 
 
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[[Category:Infinity]]

Revision as of 23:13, 13 October 2013

The term semi-infinite has several related meanings in various branches of pure and applied mathematics. It typically describes objects which are infinite or unbounded in some but not all possible ways.

In ordered structures and Euclidean spaces

Generally, a semi-infinite set is bounded in one direction, and unbounded in another. For instance, the natural numbers are semi-infinite considered as a subset of the integers; similarly, the intervals (c,∞) and (−∞,c) and their closed counterparts are semi-infinite subsets of ℝ. Half-spaces are sometimes described as semi-infinite regions.

Semi-infinite regions occur frequently in the study of differential equations.[1][2] For instance, one might study solutions of the heat equation in an idealised semi-infinite metal bar.

A semi-infinite integral is an improper integral over a semi-infinite interval. More generally, objects indexed or parametrised by semi-infinite sets may be described as semi-infinite.[3]

Most forms of semi-infiniteness are boundedness properties, not cardinality or measure properties: semi-infinite sets are typically infinite in cardinality and measure.

In optimisation

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Many optimisation problems involve some set of variables and some set of constraints. A problem is called semi-infinite if one (but not both) of these sets is finite. The study of such problems is known as semi-infinite programming.[4]

References

  1. ↑ Bateman, Transverse seismic waves on the surface of a semi-infinite solid composed of heterogeneous material, Bull. Amer. Math. Soc. Volume 34, Number 3 (1928), 343–348.
  2. ↑ Wolfram Demonstrations Project, Heat Diffusion in a Semi-Infinite Region (accessed November 2010).
  3. ↑ Cator, Pimentel, A shape theorem and semi-infinite geodesics for the Hammersley model with random weights, 2010.
  4. ↑ Reemsten, Rückmann, Semi-infinite Programming, Kluwer Academic, 1998. ISBN 0-7923-5054-5




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