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 $\mathbb{R}$. Given a time scale we define the ''jump operator'' $\sigma \colon \mathbb{T} \rightarrow \mathbb{T}$ by the formula
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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 $\mathbb{T}^{\kappa} = \mathbb{T} \setminus \left\{ \sup \mathbb{T} \right\}$ (if $\sup \mathbb{T}=\infty$ then $\mathbb{T}^{\kappa}=\mathbb{T}$). For some set $X$, let $f \colon \mathbb{T} \rightarrow X$. The following is a common notation using the [[forward jump]] operator: $f^{\sigma} \colon  \mathbb{T}^{\kappa} \rightarrow X$ is given by the formula $f^{\sigma}(t)=f(\sigma(t))$. Similarly the [[backward jump]] is used to define the function $f^{\rho}$.
$$\sigma(t) := \inf \left\{ x \in \mathbb{T} \colon x > t \right\}.$$
 
The ''graininess operator'' is the function $\mu \colon \mathbb{T} \rightarrow \mathbb{R}^+ \cup \{0\}$ is defined by the formula
 
$$\mu(t) := \sigma(t)-t.$$
 
  
== Examples of time scales ==
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To every time scale we have "standard" calculus operators: the [[delta_derivative | $\Delta$-derivative]] and [[delta_integration | $\Delta$-integral]], however there are also different types of other derivatives and integrals such as the [[nabla derivative | $\nabla$-derivative]] and the [[nabla integral | $\nabla$-integral]].
# The real line: $\mathbb{R}$
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# The integers: $\mathbb{Z} = \{\ldots, -1,0,1,\ldots\}$
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=The set of time scales=
# Multiples of integers: $h\mathbb{Z} = \{ht \colon t \in \mathbb{Z}\}$
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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 [http://hyperspacewiki.org/index.php/Hyperspace hyperspace] $\mathcal{H}=\mathrm{CL}(\mathbb{R})$. We can characterize time scales using the [http://en.wikipedia.org/wiki/Derived_set_%28mathematics%29 Cantor-Bendixson derivative] -- a time scale $\mathbb{T}$ is the union of a [https://proofwiki.org/wiki/Definition:Perfect_Set perfect set] and a countable set.
# Harmonic numbers: $\mathbb{H}=\left\{\displaystyle\sum_{k=1}^n \dfrac{1}{k} \colon n \in \mathbb{Z}^+ \right\}$
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# The closure of the unit fractions: $\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$
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=Examples of time scales=
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# The real line: [[Real_numbers | $\mathbb{R}$]]
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# The integers: [[Integers | $\mathbb{Z} = \{\ldots, -1,0,1,\ldots\}$]]
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# Multiples of integers: [[Multiples_of_integers | $h\mathbb{Z} = \{ht \colon t \in \mathbb{Z}\}$]]
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# Quantum numbers ($q>1$): [[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}$]]
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# Quantum numbers ($q<1$): [[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}$]]
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# Square integers: [[Square_integers | $\mathbb{Z}^2 = \{t^2 \colon t \in \mathbb{Z} \}$]]
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# Harmonic numbers: [[Harmonic_numbers | $\mathbb{H}=\left\{\displaystyle\sum_{k=1}^n \dfrac{1}{k} \colon n \in \mathbb{Z}^+ \right\}$]]
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# The closure of the unit fractions: [[Closure_of_unit_fractions | $\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$]]
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# Isolated points: [[Isolated_points | $\mathbb{T}=\{\ldots, t_{-1}, t_{0}, t_1, \ldots\}$]]
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=Applications of time scales=
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#Control theory, see [http://www.kirj.ee/public/Phys_Math/2007/issue_3/phys-2007-3-3.pdf this] and [http://arxiv.org/pdf/0805.0274.pdf this] and [http://www.nt.ntnu.no/users/skoge/prost/proceedings/ifac11-proceedings/data/html/papers/1815.pdf this]
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#Economics, see [http://web.mst.edu/~BOHNER/tss-10/aaotste.pdf this] and [http://www.janlibich.com/LibichStehlik_scales2.pdf this]
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#Ecology, see [http://www.ecologyandsociety.org/vol14/iss2/art21/ this]
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#Possible application to geophysics [http://meetingorganizer.copernicus.org/EGU2013/EGU2013-4225.pdf here]
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#Biological systems [http://www.advancesindifferenceequations.com/content/pdf/s13662-015-0383-0.pdf here]
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#Population model for flies [http://campus.mst.edu/ijde/contents/v8n2p1.pdf here]
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#[http://www.sciencedirect.com/science/article/pii/S089396591100468X this]
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=References=
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* {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|next=Forward jump}}: Theorem 1.16 (i)
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* {{PaperReference|Square Integrability of Gaussian Bells on Time Scales|2008|Lynn Erbe|author2=Allan Peterson|author3=Moritz Simon|prev=findme|next=findme}}: Definition $2.30$
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* {{PaperReference|Functional series on time scales|2008|Dorota Mozyrska|author2=Ewa Pawluszewicz|next=Forward jump}}

Revision as of 14:44, 21 October 2017

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 $\mathbb{T}^{\kappa}=\mathbb{T}$). For some set $X$, let $f \colon \mathbb{T} \rightarrow X$. The following is a common notation using the forward jump operator: $f^{\sigma} \colon \mathbb{T}^{\kappa} \rightarrow X$ is given by the formula $f^{\sigma}(t)=f(\sigma(t))$. Similarly the backward jump is used to define the function $f^{\rho}$.

To every time scale we have "standard" calculus operators: the $\Delta$-derivative and $\Delta$-integral, however there are also different types of other 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

References