Difference between revisions of "Time scale"

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The ''graininess operator'' is the function $\mu \colon \mathbb{T} \rightarrow \mathbb{R}^+ \cup \{0\}$ is defined by the formula
 
The ''graininess operator'' is the function $\mu \colon \mathbb{T} \rightarrow \mathbb{R}^+ \cup \{0\}$ is defined by the formula
 
$$\mu(t) := \sigma(t)-t.$$
 
$$\mu(t) := \sigma(t)-t.$$
 +
To every time scale we have a standard [[delta_derivative | differentiation operator]] and [[delta_integration | integration operator]].
  
 
== Examples of time scales ==
 
== Examples of time scales ==

Revision as of 02:48, 18 May 2014

A time scale is a set $\mathbb{T} \subset \mathbb{R}$ which is closed under the standard topology of $\mathbb{R}$. Given a time scale we define the jump operator $\sigma \colon \mathbb{T} \rightarrow \mathbb{T}$ by the formula $$\sigma(t) := \inf \left\{ x \in \mathbb{T} \colon x > t \right\}.$$ The graininess operator is the function $\mu \colon \mathbb{T} \rightarrow \mathbb{R}^+ \cup \{0\}$ is defined by the formula $$\mu(t) := \sigma(t)-t.$$ To every time scale we have a standard differentiation operator and integration operator.

Examples of time scales

  1. The real line: $\mathbb{R}$
  2. The integers: $\mathbb{Z} = \{\ldots, -1,0,1,\ldots\}$
  3. Multiples of integers: $h\mathbb{Z} = \{ht \colon t \in \mathbb{Z}\}$
  4. Harmonic numbers: $\mathbb{H}=\left\{\displaystyle\sum_{k=1}^n \dfrac{1}{k} \colon n \in \mathbb{Z}^+ \right\}$
  5. The closure of the unit fractions: $\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$