# Semi-infinite

In mathematics, semi-infinite objects are 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 ${\displaystyle (c,\infty )}$ and ${\displaystyle (-\infty ,c)}$ and their closed counterparts are semi-infinite subsets of ${\displaystyle \mathbb {R} }$ if ${\displaystyle c}$ is finite.[1] Half-spaces and half-lines are sometimes described as semi-infinite regions.

Semi-infinite regions occur frequently in the study of differential equations.[2][3] 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.[4]

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

## In optimization

Many optimization 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.[5]

## References

1. ^ Trench, William. Introduction to Real Analysis. p. 21. ISBN 0-13-045786-8.
2. ^ 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.
3. ^ Wolfram Demonstrations Project, Heat Diffusion in a Semi-Infinite Region (accessed November 2010).
4. ^ Cator, Pimentel, A shape theorem and semi-infinite geodesics for the Hammersley model with random weights, 2010.
5. ^ Reemsten, Rückmann, Semi-infinite Programming, Kluwer Academic, 1998. ISBN 0-7923-5054-5