Difference between revisions of "Isolated points"

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We say that a time scale $\mathbb{T}$ is a time scale of isolated points if there exists $\epsilon > 0$ such that for all $t \in \mathbb{T}$, $\mu(t) \geq \epsilon$. Let $\mathbb{T}=\{\ldots,t_{-1},t_0,t_1,\ldots\}$ be a [[time_scale | time scale]] of isolated points with $t_k > t_n$ iff $k>n$. Define the bijection $\pi \colon \mathbb{T} \rightarrow \mathbb{Z}$, $\pi(t_k)=k$.
+
Let $\mathbb{T}$ be a [[time_scale | time scale]]. We say that $\mathbb{T}$ is a time scale of isolated points if there exists $\epsilon > 0$ such that for all $t \in \mathbb{T}$, $\mu(t) \geq \epsilon$. Let $\mathbb{T}=\{\ldots,t_{-1},t_0,t_1,\ldots\}$ be a [[time_scale | time scale]] of isolated points with $t_k > t_n$ iff $k>n$. Define the [[bijection]] $\pi \colon \mathbb{T} \rightarrow \mathbb{Z}$, $\pi(t_k)=k$.
  
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The set $h\mathbb{Z}=\{\ldots,-2h,-h,0,h,2h,\ldots\}$ of multiples of the integers is a [[time scale]].
+
{| class="wikitable"
 +
|+$\mathbb{T}=\mathbb{T}_{\mathrm{iso}}$
 +
|-
 +
|Generic element $t \in \mathbb{T}$:
 +
|for some $n \in \mathbb{Z}$, $t=t_n$
 +
|-
 +
|[[Forward jump]]:
 +
|$\sigma(t_n)=t_{n+1}$
 +
|[[Derivation of forward jump for T=isolated points|derivation]]
 +
|-
 +
|[[Forward graininess]]:
 +
|$\mu(t_n)=t_{n+1}-t_n$
 +
|[[Derivation of forward graininess for T=isolated points|derivation]]
 +
|-
 +
|[[Backward jump]]:
 +
|$\rho(t_n)=t_{n-1}$
 +
|[[Derivation of backward jump for T=isolated points|derivation]]
 +
|-
 +
|[[Backward graininess]]:
 +
|$\nu(t_n)=t_{n}-t_{n-1}$
 +
|[[Derivation of backward graininess for T=isolated points|derivation]]
 +
|-
 +
|[[Delta derivative | $\Delta$-derivative]]
 +
|$f^{\Delta}(t_n)=\dfrac{f(t_{n+1})-f(t_n)}{t_{n+1}-t_n}$
 +
|[[Derivation of delta derivative for T=isolated points|derivation]]
 +
|-
 +
|[[Nabla derivative | $\nabla$-derivative]]
 +
|$f^{\nabla}(t_n)=\dfrac{f(t_n)-f(t_{n-1})}{t_{n}-t_{n-1}}$
 +
|[[Derivation of nabla derivative for T=isolated points|derivation]]
 +
|-
 +
|[[Delta integral | $\Delta$-integral]]
 +
|$\displaystyle\int_{s}^{t} f(\tau) \Delta \tau= \left\{ \begin{array}{ll}
 +
-\displaystyle\sum_{k=\pi(t)}^{\pi(s)-1} \mu(t_k)f(t_k) &; t<s, \\
 +
0 &; t=s, \\
 +
\displaystyle\sum_{k=\pi(s)}^{\pi(t)-1} \mu(t_k)f(t_k) &; t>s
 +
\end{array}\right.$
  
−
{| class="wikitable"
+
|[[Derivation of delta integral for T=isolated points|derivation]]
−
|+$\mathbb{T}=\{\ldots,t_{-1},t_0,t_1,\ldots\}, isolated points$
+
|-
 +
|[[Nabla integral | $\nabla$-integral]]
 +
|$\displaystyle\int_s^t f(\tau) \nabla \tau=\left\{ \begin{array}{ll}
 +
-\displaystyle\sum_{k=\pi(t)+1}^{\pi(s)} \nu(t_k)f(t_k) &; t<s, \\
 +
0&; t=s, \\
 +
\displaystyle\sum_{k=\pi(s)+1}^{\pi(t)} \nu(t_k)f(t_k) &; t>s
 +
\end{array} \right.$
 +
|[[Derivation of nabla integral for T=isolated points|derivation]]
 +
|-
 +
|[[Delta hk|$h_k(t,s)$]]
 +
|$h_k(t,s)=$
 +
|[[Derivation of delta hk for T=isolated points|derivation]]
 +
|-
 +
|[[Nabla hk|$\hat{h}_k(t,s)$]]
 +
|$\hat{h}_k(t,s)=$
 +
|[[Derivation of nabla hk for T=isolated points|derivation]]
 +
|-
 +
|[[Delta gk|$g_k(t,s)$]]
 +
|$g_k(t,s)=$
 +
|[[Derivation of delta gk for T=isolated points|derivation]]
 +
|-
 +
|[[Nabla gk|$\hat{g}_k(t,s)$]]
 +
|$\hat{g}_k(t,s)=$
 +
|[[Derivation of nabla gk for T=isolated points|derivation]]
 +
|-
 +
|[[Delta exponential | $e_p(t,s)$]]
 +
|$e_p(t,s)=$
 +
|[[Derivation of delta exponential T=isolated points|derivation]]
 +
|-
 +
|[[Nabla exponential | $\hat{e}_p(t,s)$]]
 +
|$\hat{e}_p(t,s)=$
 +
|[[Derivation of nabla exponential T=isolated points|derivation]]
 +
|-
 +
|[[Gaussian bell]]
 +
|$\mathbf{E}(t)=$
 +
|[[Derivation of Gaussian bell for T=isolated points|derivation]]
 +
|-
 +
|[[Delta sine | $\mathrm{sin}_p(t,s)=$]]
 +
|$\sin_p(t,s)=$
 +
|[[Derivation of delta sin sub p for T=isolated points|derivation]]
 +
|-
 +
|$\mathrm{\sin}_1(t,s)$
 +
|$\sin_1(t,s)=$
 +
|[[Derivation of delta sin sub 1 for T=isolated points|derivation]]
 +
|-
 +
|[[Nabla sine|$\widehat{\sin}_p(t,s)$]]
 +
|$\widehat{\sin}_p(t,s)=$
 +
|[[Derivation of nabla sine sub p for T=isolated points|derivation]]
 +
|-
 +
|[[Delta cosine|$\mathrm{\cos}_p(t,s)$]]
 +
|$\cos_p(t,s)=$
 +
|[[Derivation of delta cos sub p for T=isolated points|derivation]]
 +
|-
 +
|$\mathrm{\cos}_1(t,s)$
 +
|$\cos_1(t,s)=$
 +
|[[Derivation of delta cos sub 1 for T=isolated points|derivation]]
 +
|-
 +
|[[Nabla cosine|$\widehat{\cos}_p(t,s)$]]
 +
|$\widehat{\cos}_p(t,s)=$
 +
|[[Derivation of nabla cos sub 1 for T=isolated points|derivation]]
 +
|-
 +
|[[Delta sinh|$\sinh_p(t,s)$]]
 +
|$\sinh_p(t,s)=$
 +
|[[Derivation of delta sinh sub p for T=isolated points|derivation]]
 +
|-
 +
|[[Nabla sinh|$\widehat{\sinh}_p(t,s)$]]
 +
|$\widehat{\sinh}_p(t,s)=$
 +
|[[Derivation of nabla sinh sub p for T=isolated points|derivation]]
 +
|-
 +
|[[Delta cosh|$\cosh_p(t,s)$]]
 +
|$\cosh_p(t,s)=$
 +
|[[Derivation of delta cosh sub p for T=isolated points|derivation]]
 +
|-
 +
|[[Nabla cosh|$\widehat{\cosh}_p(t,s)$]]
 +
|$\widehat{\cosh}_p(t,s)=$
 +
|[[Derivation of nabla cosh sub p for T=isolated points|derivation]]
 +
|-
 +
|[[Gamma function]]
 +
|$\Gamma_{\mathbb{T}_{\mathrm{iso}}}(x,s)=$
 +
|[[Derivation of gamma function for T=isolated points|derivation]]
 +
|-
 +
|[[Euler-Cauchy logarithm]]
 +
|$L(t,s)=$
 +
|[[Derivation of Euler-Cauchy logarithm for T=isolated points|derivation]]
 
|-
 
|-
−
|Generic element $t\in \mathbb{T}$:
+
|[[Bohner logarithm]]
−
|For some $n \in \mathbb{Z}, t=t_n$
+
|$L_p(t,s)=$
 +
|[[Derivation of the Bohner logarithm for T=isolated points|derivation]]
 
|-
 
|-
−
|Jump operator:
+
|[[Jackson logarithm]]
−
|$\sigma(t)=\sigma(t_n)=t_{n+1}$
+
|$\log_{\mathbb{T}_{\mathrm{iso}}} g(t)=$
 +
|[[Derivation of the Jackson logarithm for T=isolated points|derivation]]
 
|-
 
|-
−
|Graininess operator:
+
|[[Mozyrska-Torres logarithm]]
−
|$\mu(t)=\mu(t_n)=t_{n+1}-t_n$
+
|$L_{\mathbb{T}_{\mathrm{iso}}}(t)=$
 +
|[[Derivation of the Mozyrska-Torres logarithm for T=isolated points|derivation]]
 
|-
 
|-
−
|[[Delta_derivative | $\Delta$-derivative:]]
+
|[[Laplace transform]]
−
|$f^{\Delta}(t)=f^{\Delta}(t_n) = \dfrac{f(t_{n+1})-f(t_n)}{t_{n+1}-t_n}$
+
|$\mathscr{L}_{\mathbb{T}_{\mathrm{iso}}}\{f\}(z;s)=$
 +
|[[Derivation of Laplace transform for T=isolated points|derivation]]
 
|-
 
|-
−
|[[Delta_integral | $\Delta$-integral:]]
+
|[[Hilger circle]]  
−
|  
+
|
 +
|[[Derivation of Hilger circle for T=isolated points|derivation]]
 
|-
 
|-
−
|[[Exponential_functions | Exponential function]]:
 
−
|
 
 
|}
 
|}
 +
 +
== Examples of time scales of isolated points ==
 +
*[[Integers | $\mathbb{Z}$]]
 +
*[[Multiples_of_integers | $h\mathbb{Z}$]]
 +
*[[Square_integers | $\mathbb{Z}^2$]]
 +
*[[Harmonic_numbers | $\mathbb{H}$]]
 +
 +
<center>{{:Time scales footer}}</center>

Latest revision as of 23:20, 9 June 2015

Let $\mathbb{T}$ be a time scale. We say that $\mathbb{T}$ is a time scale of isolated points if there exists $\epsilon > 0$ such that for all $t \in \mathbb{T}$, $\mu(t) \geq \epsilon$. Let $\mathbb{T}=\{\ldots,t_{-1},t_0,t_1,\ldots\}$ be a time scale of isolated points with $t_k > t_n$ iff $k>n$. Define the bijection $\pi \colon \mathbb{T} \rightarrow \mathbb{Z}$, $\pi(t_k)=k$.

$\mathbb{T}=\mathbb{T}_{\mathrm{iso}}$
Generic element $t \in \mathbb{T}$: for some $n \in \mathbb{Z}$, $t=t_n$
Forward jump: $\sigma(t_n)=t_{n+1}$ derivation
Forward graininess: $\mu(t_n)=t_{n+1}-t_n$ derivation
Backward jump: $\rho(t_n)=t_{n-1}$ derivation
Backward graininess: $\nu(t_n)=t_{n}-t_{n-1}$ derivation
$\Delta$-derivative $f^{\Delta}(t_n)=\dfrac{f(t_{n+1})-f(t_n)}{t_{n+1}-t_n}$ derivation
$\nabla$-derivative $f^{\nabla}(t_n)=\dfrac{f(t_n)-f(t_{n-1})}{t_{n}-t_{n-1}}$ derivation
$\Delta$-integral $\displaystyle\int_{s}^{t} f(\tau) \Delta \tau= \left\{ \begin{array}{ll} -\displaystyle\sum_{k=\pi(t)}^{\pi(s)-1} \mu(t_k)f(t_k) &; t<s, \\ 0 &; t=s, \\ \displaystyle\sum_{k=\pi(s)}^{\pi(t)-1} \mu(t_k)f(t_k) &; t>s \end{array}\right.$ derivation
$\nabla$-integral $\displaystyle\int_s^t f(\tau) \nabla \tau=\left\{ \begin{array}{ll} -\displaystyle\sum_{k=\pi(t)+1}^{\pi(s)} \nu(t_k)f(t_k) &; t<s, \\ 0&; t=s, \\ \displaystyle\sum_{k=\pi(s)+1}^{\pi(t)} \nu(t_k)f(t_k) &; t>s \end{array} \right.$ derivation
$h_k(t,s)$ $h_k(t,s)=$ derivation
$\hat{h}_k(t,s)$ $\hat{h}_k(t,s)=$ derivation
$g_k(t,s)$ $g_k(t,s)=$ derivation
$\hat{g}_k(t,s)$ $\hat{g}_k(t,s)=$ derivation
$e_p(t,s)$ $e_p(t,s)=$ derivation
$\hat{e}_p(t,s)$ $\hat{e}_p(t,s)=$ derivation
Gaussian bell $\mathbf{E}(t)=$ derivation
$\mathrm{sin}_p(t,s)=$ $\sin_p(t,s)=$ derivation
$\mathrm{\sin}_1(t,s)$ $\sin_1(t,s)=$ derivation
$\widehat{\sin}_p(t,s)$ $\widehat{\sin}_p(t,s)=$ derivation
$\mathrm{\cos}_p(t,s)$ $\cos_p(t,s)=$ derivation
$\mathrm{\cos}_1(t,s)$ $\cos_1(t,s)=$ derivation
$\widehat{\cos}_p(t,s)$ $\widehat{\cos}_p(t,s)=$ derivation
$\sinh_p(t,s)$ $\sinh_p(t,s)=$ derivation
$\widehat{\sinh}_p(t,s)$ $\widehat{\sinh}_p(t,s)=$ derivation
$\cosh_p(t,s)$ $\cosh_p(t,s)=$ derivation
$\widehat{\cosh}_p(t,s)$ $\widehat{\cosh}_p(t,s)=$ derivation
Gamma function $\Gamma_{\mathbb{T}_{\mathrm{iso}}}(x,s)=$ derivation
Euler-Cauchy logarithm $L(t,s)=$ derivation
Bohner logarithm $L_p(t,s)=$ derivation
Jackson logarithm $\log_{\mathbb{T}_{\mathrm{iso}}} g(t)=$ derivation
Mozyrska-Torres logarithm $L_{\mathbb{T}_{\mathrm{iso}}}(t)=$ derivation
Laplace transform $\mathscr{L}_{\mathbb{T}_{\mathrm{iso}}}\{f\}(z;s)=$ derivation
Hilger circle derivation

Examples of time scales of isolated points

Examples of time scales
$\Huge\mathbb{R}$
Real numbers
$\Huge\mathbb{Z}$
Integers
$\Huge{h\mathbb{Z}}$
Multiples of integers
$\Huge\mathbb{Z}^2$
Square integers
$\Huge\mathbb{H}$
Harmonic numbers
$\Huge\mathbb{T}_{\mathrm{iso}}$
Isolated points
$\Huge\sqrt[n]{\mathbb{N}_0}$
nth root numbers
$\Huge\mathbb{P}_{a,b}$
Evenly spaced intervals
$\huge\overline{q^{\mathbb{Z}}}$
Quantum, $q>1$
$\huge\overline{q^{\mathbb{Z}}}$
Quantum, $q<1$
$\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$
Closure of unit fractions
$\Huge\mathcal{C}$
Cantor set
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