Difference between revisions of "Time scale"
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# The integers: [[Integers | $\mathbb{Z} = \{\ldots, -1,0,1,\ldots\}$]] | # The integers: [[Integers | $\mathbb{Z} = \{\ldots, -1,0,1,\ldots\}$]] | ||
# Multiples of integers: [[Multiples_of_integers | $h\mathbb{Z} = \{ht \colon t \in \mathbb{Z}\}$]] | # Multiples of integers: [[Multiples_of_integers | $h\mathbb{Z} = \{ht \colon t \in \mathbb{Z}\}$]] | ||
| + | # Quantum numbers ($q>1$): [[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q>1$]] | ||
| + | # Quantum numbers ($q<1$): [[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q<1$]] | ||
# Harmonic numbers: [[Harmonic_numbers | $\mathbb{H}=\left\{\displaystyle\sum_{k=1}^n \dfrac{1}{k} \colon n \in \mathbb{Z}^+ \right\}$]] | # Harmonic numbers: [[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: [[Closure_of_unit_fractions | $\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$]] | # The closure of the unit fractions: [[Closure_of_unit_fractions | $\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$]] | ||
Revision as of 04:33, 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
- 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 ($q>1$): $\overline{q^{\mathbb{Z}}}, q>1$
- Quantum numbers ($q<1$): $\overline{q^{\mathbb{Z}}}, q<1$
- 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: $\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$
