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In mathematics, a partition of an interval ā» on the real line is a finite sequence x0, x1, x2, ā¦, xn of real numbers such that
- a = x0 < x1 < x2 < ā¦ < xn = b.
In other terms, a partition of a compact interval I is a strictly increasing sequence of numbers (belongingāāto the interval I itself) starting from the initial point of I and arriving at the final point of I.
Every interval of the form ā» is referredāāto as a subinterval of the partition x.
Refinement of a partitionā»
Another partition Q of the given interval ā» 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, āfinerā than P. Given two partitions, P and Q, one can always form their common refinement, denoted P āØ Q, which consists of all the points of P and Q, in increasing order.
Norm of a partitionā»
The norm (or mesh) of the partition
- x0 < x1 < x2 < ā¦ < xn
is the length of the longest of these subintervals
- max{|xi ā xiā1| : i = 1, ā¦ , n }.
Applicationsā»
Partitions are used in the theory of the Riemann integral, the RiemannāStieltjes 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.
Tagged partitionsā»
A tagged partition/Perron Partition is a partition of a given interval together with a finite sequence of numbers t0, ā¦, tn ā 1 subject to the conditions that for each i,
- xi ā¤ ti ā¤ 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.
Suppose that x0, ā¦, xn together with t0, ā¦, tn ā 1 is a tagged partition of ā», and that y0, ā¦, ym together with s0, ā¦, sm ā 1 is another tagged partition of ā». We say that y0, ā¦, ym together with s0, ā¦, sm ā 1 is a refinement of a tagged partition x0, ā¦, xn together with t0, ā¦, tn ā 1 if for each integer i with 0 ā¤ i ā¤ n, there is an integer r(i) such that xi = yr(i) and such that ti = sj for some j with r(i) ā¤ j ā¤ r(i + 1) ā 1. Said more simply, a refinement of a tagged partition takes the starting partition and "adds more tags." But does not take any away.
See alsoā»
Referencesā»
- ^ Brannan, "D." A. (2006). A First Course in Mathematical Analysis. Cambridge University Press. p. 262. ISBN 9781139458955.
- ^ Hijab, Omar (2011). Introduction to Calculus and Classical Analysis. Springer. p. 60. ISBN 9781441994882.
- ^ Zorich, Vladimir A. (2004). Mathematical Analysis II. Springer. p. 108. ISBN 9783540406334.
- ^ Ghorpade, Sudhir; Limaye, Balmohan (2006). A Course in Calculus and Real Analysis. Springer. p. 213. ISBN 9780387364254.
- ^ Dudley, Richard M.; NorvaiŔa, Rimas (2010). Concrete Functional Calculus. Springer. p. 2. ISBN 9781441969507.
Further readingā»
- Gordon, Russell A. (1994). The integrals of Lebesgue, Denjoy, Perron, and Henstock. Graduate Studies in Mathematics, 4. Providence, RI: American Mathematical Society. ISBN 0-8218-3805-9.