Revision as of 06:39, 17 June 2015

A time scale is a set $\mathbb{T} \subset \mathbb{R}$ which is closed under the standard topology of $\mathbb{R}$. Sometimes we deal with the set $\mathbb{T}^{\kappa} = \mathbb{T} \setminus \left\{ \sup \mathbb{T} \right\}$ (if $\sup \mathbb{T}=\infty$ then $\mathbb{T}^{\kappa}=\mathbb{T}$). Given a time scale we define the jump operator $\sigma \colon \mathbb{T} \rightarrow \mathbb{T}$ by the formula $$\sigma(t) := \inf \left\{ x \in \mathbb{T} \colon x > t \right\}.$$ Let $f \colon \mathbb{T} \rightarrow \mathbb{R}$. The following is a common notation: $f^{\sigma} \colon \mathbb{T}^{\kappa} \rightarrow \mathbb{R}$ is given by the formula $f^{\sigma}(t)=f(\sigma(t))$. A similar operator, the backward jump operator $\rho \colon \mathbb{T}\rightarrow \mathbb{T}$ is defined by the formula $$\rho(t) = \sup \{ x \in \mathbb{T} \colon x<t\}.$$ Let $t \in \mathbb{T}$. We say that $t$ is right-scattered if $\sigma(t)>t$ (left-scattered if $\rho(t)<t$) and that $t$ is right-dense if $\sigma(t)=t$ (left-dense if $\rho(t)=t$).

The graininess operator is the function $\mu \colon \mathbb{T} \rightarrow \mathbb{R}^+ \cup \{0\}$ is defined by the formula $$\mu(t) := \sigma(t)-t.$$ The backwards graininess operator is the function $\nu \colon \mathbb{T} \rightarrow \mathbb{R}^+ \cup \{0\}$ is defined by the formula $$\nu(t) := t - \rho(t).$$ To every time scale we have a standard calculus operators: the $\Delta$-derivative and $\Delta$-integral, however there are also different types of derivatives and integrals such as the $\nabla$-derivative and the $\nabla$-integral.

The set of time scales

Let $\mathcal{H} = \{\mathbb{T} \subset \mathbb{R} \colon \mathbb{T}$ is a closed set $\}$. A set like this can be given a standard topological structure making it the hyperspace $\mathcal{H}=\mathrm{CL}(\mathbb{R})$. We can characterize time scales using the Cantor-Bendixson derivative -- a time scale $\mathbb{T}$ is the union of a perfect set and a countable set.

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. Quantum numbers ($q>1$): $\overline{q^{\mathbb{Z}}}$
5. Quantum numbers ($q<1$): $\overline{q^{\mathbb{Z}}}$
6. Square integers: $\mathbb{Z}^2 = \{t^2 \colon t \in \mathbb{Z} \}$
7. Harmonic numbers: $\mathbb{H}=\left\{\displaystyle\sum_{k=1}^n \dfrac{1}{k} \colon n \in \mathbb{Z}^+ \right\}$
8. The closure of the unit fractions: $\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$
9. Isolated points: $\mathbb{T}=\{\ldots, t_{-1}, t_{0}, t_1, \ldots\}$

Applications of time scales

1. Control theory, see this and this and this
2. Economics, see this and this
3. Ecology, see this
4. Possible application to geophysics here
5. Biological systems here
6. Population model for flies here
7. this