Difference between revisions of "Time scale"
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| − | 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 $\mathbb{T}^{\kappa} = \mathbb{T} \setminus \sup \mathbb{T}$. Given a time scale we define the ''jump operator'' $\sigma \colon \mathbb{T} \rightarrow \mathbb{T}$ by the formula | + | 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 $\mathbb{T}^{\kappa} = \mathbb{T} \setminus \sup \mathbb{T}$ (whenever $\mathbb{T}$ has a left-scattered maximum, otherwise it equals $\mathbb{T}$) or the set $\mathbb{T}^{\kappa}=\mathbb{T} \setminus \inf \mathbb{T}$ (whenever $\mathbb{T}$ has a right-scattered minimum, otherwise it equals $\mathbb{T}$). Given a time scale we define the ''jump operator'' $\sigma \colon \mathbb{T} \rightarrow \mathbb{T}$ by the formula |
$$\sigma(t) := \inf \left\{ x \in \mathbb{T} \colon x > t \right\}.$$ | $$\sigma(t) := \inf \left\{ x \in \mathbb{T} \colon x > t \right\}.$$ | ||
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 | 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 | ||
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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 | | + | 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 | $\Delta$-derivative]] and [[delta_integration | $\Delta$-integral]], however there are also different types of derivatives and integrals such as the [[nabla derivative | $\nabla$-derivative]] and the [[nabla integral | $\nabla$-integral]]. | ||
=The set of time scales= | =The set of time scales= | ||
Revision as of 16:46, 21 September 2014
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 $\mathbb{T}^{\kappa} = \mathbb{T} \setminus \sup \mathbb{T}$ (whenever $\mathbb{T}$ has a left-scattered maximum, otherwise it equals $\mathbb{T}$) or the set $\mathbb{T}^{\kappa}=\mathbb{T} \setminus \inf \mathbb{T}$ (whenever $\mathbb{T}$ has a right-scattered minimum, otherwise it equals $\mathbb{T}$). 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
- The real line: $\mathbb{R}$
- The integers: $\mathbb{Z} = \{\ldots, -1,0,1,\ldots\}$
- Multiples of integers: $h\mathbb{Z} = \{ht \colon t \in \mathbb{Z}\}$
- Quantum numbers ($q>1$): $\overline{q^{\mathbb{Z}}}$
- Quantum numbers ($q<1$): $\overline{q^{\mathbb{Z}}}$
- Square integers: $\mathbb{Z}^2 = \{t^2 \colon t \in \mathbb{Z} \}$
- Harmonic numbers: $\mathbb{H}=\left\{\displaystyle\sum_{k=1}^n \dfrac{1}{k} \colon n \in \mathbb{Z}^+ \right\}$
- The closure of the unit fractions: $\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$
- Isolated points: $\mathbb{T}=\{\ldots, t_{-1}, t_{0}, t_1, \ldots\}$
