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
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
are semi-infinite considered as a subset of the integers; similarly, the
intervals and
and their closed counterparts are semi-infinite subsets of
if
is finite.
Half-spaces and
half-lines are sometimes described as semi-infinite regions.
Semi-infinite regions occur frequently in the study of
differential equations. For instance, one might study solutions of the heat equation in an idealised semi-infinite metal bar.
A semi-infinite
integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
is an
improper integral
In mathematical analysis, an improper integral is an extension of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral. In the context of Riemann integrals (or, equivalently, Darboux integral ...
over a semi-infinite interval. More generally, objects indexed or parametrised by semi-infinite sets may be described as semi-infinite.
Most forms of semi-infiniteness are
boundedness properties, not
cardinality
The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
or
measure properties: semi-infinite sets are typically infinite in cardinality and measure.
In optimization
Many
optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfiel ...
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.
[Reemsten, Rückmann]
Semi-infinite Programming
Kluwer Academic, 1998.
References
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