Partition of an interval

In today's world, Partition of an interval has become a topic of great relevance and interest to a wide audience. Since its emergence, Partition of an interval has captured the attention of experts and fans alike, generating passionate debates and conflicting opinions. Its impact has extended to various areas, from politics and society to popular culture and entertainment. In this article, we will explore different aspects related to Partition of an interval, analyzing its evolution over time, its implications in the current world and future prospects. Partition of an interval continues to be a topic in constant evolution, and its influence never ceases to surprise those who delve into its complexity.

This article is about grouping elements of an interval using a sequence. For grouping elements of a set using a set of sets, see Partition of a set.
A partition of an interval being used in a Riemann sum. The partition itself is shown in grey at the bottom, with the norm of the partition indicated in red.

In mathematics, a partition of an interval on the real line is a finite sequence x0, x1, x2, …, xn of real numbers such that

a = x0 < x1 < x2 < … < xn = b.

In other terms, a partition of a compact interval I is a strictly increasing sequence of numbers (belonging to the interval I itself) starting from the initial point of I and arriving at the final point of I.

Every interval of the form is referred to as a subinterval of the partition x.

Refinement of a partition

Another partition Q of the given interval is defined as a refinement of the partition P, if Q 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, in increasing 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]

max{|xi − xi−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 or Perron Partition 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.[5]

See also

References

  1. ^ Brannan, D. A. (2006). A First Course in Mathematical Analysis. Cambridge University Press. p. 262. ISBN 9781139458955.
  2. ^ Hijab, Omar (2011). Introduction to Calculus and Classical Analysis. Springer. p. 60. ISBN 9781441994882.
  3. ^ Zorich, Vladimir A. (2004). Mathematical Analysis II. Springer. p. 108. ISBN 9783540406334.
  4. ^ Ghorpade, Sudhir; Limaye, Balmohan (2006). A Course in Calculus and Real Analysis. Springer. p. 213. ISBN 9780387364254.
  5. ^ Dudley, Richard M.; Norvaiša, Rimas (2010). Concrete Functional Calculus. Springer. p. 2. ISBN 9781441969507.

Further reading