Partition of an interval

From formulasearchengine
Jump to navigation Jump to search
A partition of an interval being used in a Riemann sum. The partition itself is shown in grey at the bottom, with one subinterval indicated in red.

In mathematics, a partition, P, of an interval [a, b] on the real line is a finite sequence of the form

a = x0 < x1 < x2 < ... < xn = b.

Every interval of the form [xi,xi+1] is referred to as a sub-interval.

Refinement of a partition

Another partition of the given interval, Q, is defined as a refinement of the partition, P, when it contains all the points of P and possibly some other points as well; the partition Q is said to be “finer” than P. Given two partitions, P and Q, one can always form their common refinement, denoted P ∨ Q, which consists of all the points of P and Q, re-numbered in order.[1]

Norm of a partition

The norm (or mesh) of the partition

x0 < x1 < x2 < ... < xn

is the length of the longest of these subintervals,[2][3] that is

max{ |xixi−1| : i = 1, ..., n }.

Applications

Partitions are used in the theory of the Riemann integral, the Riemann–Stieltjes integral and the regulated integral. Specifically, as finer partitions of a given interval are considered, their mesh approaches zero and the Riemann sum based on a given partition approaches the Riemann integral.[4]

Tagged partitions

A tagged partition[5] is a partition of a given interval together with a finite sequence of numbers t0, ..., tn−1 subject to the conditions that for each i,

xi ≤ ti ≤ xi+1.

In other words, a tagged partition is a partition together with a distinguished point of every subinterval: its mesh is defined in the same way as for an ordinary partition. It is possible to define a partial order on the set of all tagged partitions by saying that one tagged partition is bigger than another if the bigger one is a refinement of the smaller one.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

Suppose that together with is a tagged partition of , and that together with is another tagged partition of . We say that and together is a refinement of a tagged partition together with if for each integer with , there is an integer such that and such that for some with . Said more simply, a refinement of a tagged partition takes the starting partition and adds more tags, but does not take any away.

See also

References

  1. {{#invoke:citation/CS1|citation |CitationClass=book }}
  2. {{#invoke:citation/CS1|citation |CitationClass=book }}
  3. {{#invoke:citation/CS1|citation |CitationClass=book }}
  4. {{#invoke:citation/CS1|citation |CitationClass=book }}
  5. {{#invoke:citation/CS1|citation |CitationClass=book }}

Further reading

  • {{#invoke:citation/CS1|citation

|CitationClass=book }}