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 $(c,\infty )$ and $(-\infty ,c)$ and their closed counterparts are semi-infinite subsets of $\mathbb {R}$ if $c$ 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

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

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[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

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v t e Infinity ( $\infty$ )
History	Ananta (infinite) Apeiron Controversy over Cantor's theory Galileo's paradox Hilbert's paradox of the Grand Hotel Infinity (philosophy) Paradoxes of infinity Paradoxes of set theory
Branches of mathematics	Complex analysis Internal set theory Nonstandard analysis Set theory Synthetic differential geometry
Formalizations of infinity	0.999... Absolute infinite Actual infinity Aleph number Beth number Cardinal numbers Cardinality of the continuum Dedekind-infinite set Directed infinity Division by zero (Complex infinity) Epsilon number Gimel function Hilbert space Hyperreal numbers Infinite set Infinitesimal Ordinal numbers Point at infinity Regular cardinal Sphere at infinity (Kleinian group) Supertask Surreal numbers Transfinite numbers
Geometries	Differential geometry of surfaces Möbius plane Möbius transformation Riemannian manifold
Mathematicians	Georg Cantor David Hilbert Gottfried Wilhelm Leibniz August Ferdinand Möbius Bernhard Riemann Abraham Robinson