Logarithmic derivative

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 < ... < x_n = b.

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.

The norm (or mesh) of the partition

x₀ < x₁ < x₂ < ... < x_n

is the length of the longest of these subintervals, that is

max{ |x_i − x_i−1| : i = 1, ..., n }.

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.

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

x_i ≤ t_i ≤ x_i+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 $\scriptstyle x_{0},\ldots ,x_{n}$ together with $\scriptstyle t_{0},\ldots ,t_{n-1}$ is a tagged partition of $[a,b]$, and that $\scriptstyle y_{0},\ldots ,y_{m}$ together with $\scriptstyle s_{0},\ldots ,s_{m-1}$ is another tagged partition of $[a,b]$. We say that $\scriptstyle y_{0},\ldots ,y_{m}$ and $\scriptstyle s_{0},\ldots ,s_{m-1}$ together is a refinement of a tagged partition $\scriptstyle x_{0},\ldots ,x_{n}$ together with $\scriptstyle t_{0},\ldots ,t_{n-1}$ if for each integer $i$ with $\scriptstyle 0\leq i\leq n$, there is an integer $r(i)$ such that $\scriptstyle x_{i}=y_{r(i)}$ and such that $t_{i}=s_{j}$ for some $j$ with $\scriptstyle r(i)\leq j\leq r(i+1)-1$. Said more simply, a refinement of a tagged partition takes the starting partition and adds more tags, but does not take any away.

See also

• Regulated integral
• Riemann integral
• Riemann–Stieltjes integral
• Partition of a set

References

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