Partition of an interval
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In mathematics, a partition of an interval Template:Math on the real line is a finite sequence Template:Math of real numbers such that
In other terms, a partition of a compact interval Template:Mvar is a strictly increasing sequence of numbers (belonging to the interval Template:Mvar itself) starting from the initial point of Template:Mvar and arriving at the final point of Template:Mvar.
Every interval of the form Template:Math is referred to as a subinterval of the partition x.
Refinement of a partition
Another partition Template:Mvar of the given interval [a, b] is defined as a refinement of the partition Template:Mvar, if Template:Mvar contains all the points of Template:Mvar and possibly some other points as well; the partition Template:Mvar is said to be “finer” than Template:Mvar. Given two partitions, Template:Mvar and Template:Mvar, one can always form their common refinement, denoted Template:Math, which consists of all the points of Template:Mvar and Template:Mvar, in increasing order.[1]
Norm of a partition
The norm (or mesh) of the partition
is the length of the longest of these subintervals[2][3]
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 or Perron Partition is a partition of a given interval together with a finite sequence of numbers Template:Math subject to the conditions that for each Template:Mvar,
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.[5]
See also
References
Further reading
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