Difference between revisions of "Isolated points"

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Let $X \subset \mathbb{R}$. We say a point $x \in X$ is an isolated point if there exists a $\delta > 0$ such that $(t-\delta,t+\delta) \cap X = \emptyset$. It is known that for any such $X$, the set of isolated points of $X$ is [http://math.stackexchange.com/questions/402827/a-question-on-countability-of-isolated-points-of-a-subset-of-r at most countable]. Moreover a set of isolated points is closed in $\mathbb{R}$ because its complement is a union of open intervals.
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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]] $\pi \colon \mathbb{T} \rightarrow \mathbb{Z}$, $\pi(t_k)=k$.
  
 +
{| class="wikitable"
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|+$\mathbb{T}=\mathbb{T}_{\mathrm{iso}}$
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|-
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|Generic element $t \in \mathbb{T}$:
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|for some $n \in \mathbb{Z}$, $t=t_n$
 +
|-
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|[[Forward jump]]:
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|$\sigma(t_n)=t_{n+1}$
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|[[Derivation of forward jump for T=isolated points|derivation]]
 +
|-
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|[[Forward graininess]]:
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|$\mu(t_n)=t_{n+1}-t_n$
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|[[Derivation of forward graininess for T=isolated points|derivation]]
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|-
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|[[Backward jump]]:
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|$\rho(t_n)=t_{n-1}$
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|[[Derivation of backward jump for T=isolated points|derivation]]
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|-
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|[[Backward graininess]]:
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|$\nu(t_n)=t_{n}-t_{n-1}$
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|[[Derivation of backward graininess for T=isolated points|derivation]]
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|-
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|[[Delta derivative | $\Delta$-derivative]]
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|$f^{\Delta}(t_n)=\dfrac{f(t_{n+1})-f(t_n)}{t_{n+1}-t_n}$
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|[[Derivation of delta derivative for T=isolated points|derivation]]
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|-
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|[[Nabla derivative | $\nabla$-derivative]]
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|$f^{\nabla}(t_n)=\dfrac{f(t_n)-f(t_{n-1})}{t_{n}-t_{n-1}}$
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|[[Derivation of nabla derivative for T=isolated points|derivation]]
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|-
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|[[Delta integral | $\Delta$-integral]]
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|$\displaystyle\int_{s}^{t} f(\tau) \Delta \tau= \left\{ \begin{array}{ll}
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-\displaystyle\sum_{k=\pi(t)}^{\pi(s)-1} \mu(t_k)f(t_k) &; t<s, \\
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0 &; t=s, \\
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\displaystyle\sum_{k=\pi(s)}^{\pi(t)-1} \mu(t_k)f(t_k) &; t>s
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\end{array}\right.$
  
−
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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|[[Derivation of delta integral for T=isolated points|derivation]]
−
 
+
|-
−
The set $h\mathbb{Z}=\{\ldots,-2h,-h,0,h,2h,\ldots\}$ of multiples of the integers is a [[time scale]].
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|[[Nabla integral | $\nabla$-integral]]
−
 
+
|$\displaystyle\int_s^t f(\tau) \nabla \tau=\left\{ \begin{array}{ll}
−
{| class="wikitable"
+
-\displaystyle\sum_{k=\pi(t)+1}^{\pi(s)} \nu(t_k)f(t_k) &; t<s, \\
−
|+$\mathbb{T}=\{\ldots,t_{-1},t_0,t_1,\ldots\}$
+
0&; t=s, \\
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\displaystyle\sum_{k=\pi(s)+1}^{\pi(t)} \nu(t_k)f(t_k) &; t>s
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\end{array} \right.$
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|[[Derivation of nabla integral for T=isolated points|derivation]]
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|-
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|[[Delta hk|$h_k(t,s)$]]
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|$h_k(t,s)=$
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|[[Derivation of delta hk for T=isolated points|derivation]]
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|-
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|[[Nabla hk|$\hat{h}_k(t,s)$]]
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|$\hat{h}_k(t,s)=$
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|[[Derivation of nabla hk for T=isolated points|derivation]]
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|-
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|[[Delta gk|$g_k(t,s)$]]
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|$g_k(t,s)=$
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|[[Derivation of delta gk for T=isolated points|derivation]]
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|-
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|[[Nabla gk|$\hat{g}_k(t,s)$]]
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|$\hat{g}_k(t,s)=$
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|[[Derivation of nabla gk for T=isolated points|derivation]]
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|-
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|[[Delta exponential | $e_p(t,s)$]]
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|$e_p(t,s)=$
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|[[Derivation of delta exponential T=isolated points|derivation]]
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|-
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|[[Nabla exponential | $\hat{e}_p(t,s)$]]
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|$\hat{e}_p(t,s)=$
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|[[Derivation of nabla exponential T=isolated points|derivation]]
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|-
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|[[Gaussian bell]]
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|$\mathbf{E}(t)=$
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|[[Derivation of Gaussian bell for T=isolated points|derivation]]
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|-
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|[[Delta sine | $\mathrm{sin}_p(t,s)=$]]
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|$\sin_p(t,s)=$
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|[[Derivation of delta sin sub p for T=isolated points|derivation]]
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|-
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|$\mathrm{\sin}_1(t,s)$
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|$\sin_1(t,s)=$
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|[[Derivation of delta sin sub 1 for T=isolated points|derivation]]
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|-
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|[[Nabla sine|$\widehat{\sin}_p(t,s)$]]
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|$\widehat{\sin}_p(t,s)=$
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|[[Derivation of nabla sine sub p for T=isolated points|derivation]]
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|-
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|[[Delta cosine|$\mathrm{\cos}_p(t,s)$]]
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|$\cos_p(t,s)=$
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|[[Derivation of delta cos sub p for T=isolated points|derivation]]
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|-
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|$\mathrm{\cos}_1(t,s)$
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|$\cos_1(t,s)=$
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|[[Derivation of delta cos sub 1 for T=isolated points|derivation]]
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|-
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|[[Nabla cosine|$\widehat{\cos}_p(t,s)$]]
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|$\widehat{\cos}_p(t,s)=$
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|[[Derivation of nabla cos sub 1 for T=isolated points|derivation]]
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|-
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|[[Delta sinh|$\sinh_p(t,s)$]]
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|$\sinh_p(t,s)=$
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|[[Derivation of delta sinh sub p for T=isolated points|derivation]]
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|-
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|[[Nabla sinh|$\widehat{\sinh}_p(t,s)$]]
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|$\widehat{\sinh}_p(t,s)=$
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|[[Derivation of nabla sinh sub p for T=isolated points|derivation]]
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|-
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|[[Delta cosh|$\cosh_p(t,s)$]]
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|$\cosh_p(t,s)=$
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|[[Derivation of delta cosh sub p for T=isolated points|derivation]]
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|-
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|[[Nabla cosh|$\widehat{\cosh}_p(t,s)$]]
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|$\widehat{\cosh}_p(t,s)=$
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|[[Derivation of nabla cosh sub p for T=isolated points|derivation]]
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|-
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|[[Gamma function]]
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|$\Gamma_{\mathbb{T}_{\mathrm{iso}}}(x,s)=$
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|[[Derivation of gamma function for T=isolated points|derivation]]
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|-
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|[[Euler-Cauchy logarithm]]
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|$L(t,s)=$
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|[[Derivation of Euler-Cauchy logarithm for T=isolated points|derivation]]
 
|-
 
|-
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|Generic element $t\in \mathbb{T}$:
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|[[Bohner logarithm]]
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|For some $n \in \mathbb{Z}, t=t_n$
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|$L_p(t,s)=$
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|[[Derivation of the Bohner logarithm for T=isolated points|derivation]]
 
|-
 
|-
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|Jump operator:
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|[[Jackson logarithm]]
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|$\sigma(t)=\sigma(t_n)=t_{n+1}$
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|$\log_{\mathbb{T}_{\mathrm{iso}}} g(t)=$
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|[[Derivation of the Jackson logarithm for T=isolated points|derivation]]
 
|-
 
|-
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|Graininess operator:
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|[[Mozyrska-Torres logarithm]]
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|$\mu(t)=\mu(t_n)=t_{n+1}-t_n$
+
|$L_{\mathbb{T}_{\mathrm{iso}}}(t)=$
 +
|[[Derivation of the Mozyrska-Torres logarithm for T=isolated points|derivation]]
 
|-
 
|-
−
|[[Delta_derivative | $\Delta$-derivative:]]
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|[[Laplace transform]]
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|$f^{\Delta}(t)=f^{\Delta}(t_n) = \dfrac{f(t_{n+1})-f(t_n)}{t_{n+1}-t_n}$
+
|$\mathscr{L}_{\mathbb{T}_{\mathrm{iso}}}\{f\}(z;s)=$
 +
|[[Derivation of Laplace transform for T=isolated points|derivation]]
 
|-
 
|-
−
|[[Delta_integral | $\Delta$-integral:]]
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|[[Hilger circle]]  
−
|  
+
|
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|[[Derivation of Hilger circle for T=isolated points|derivation]]
 
|-
 
|-
−
|[[Exponential_functions | Exponential function]]:
 
−
|
 
 
|}
 
|}
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 +
== Examples of time scales of isolated points ==
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*[[Integers | $\mathbb{Z}$]]
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*[[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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