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 $x_0,\dots ,x_n$ together with $t_0,\dots ,t_{n-1}$ is a tagged partition of $[a,b]$, and that $y_0,\dots ,y_m$ together with $s_0,\dots ,s_{m-1}$ is another tagged partition of $[a,b]$. We say that $y_0,\dots ,y_m$ and $s_0,\dots ,s_{m-1}$ together is a refinement of a tagged partition $x_0,\dots ,x_n$ together with $t_0,\dots ,t_{n-1}$ if for each integer $i$ with $0\le i\le n$, there is an integer $r(i)$ such that $x_i=y_{r(i)}$ and such that $t_i=s_j$ for some $j$ with $r(i)\le j\le r(i+1)-1$. 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

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Further reading

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534