Difference between revisions of "Closure of unit fractions"

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|[[Forward jump]]:
 
|[[Forward jump]]:
|$\sigma(t)=$
+
|$\sigma(t)=\left\{ \begin{array}{ll}
 +
0 &; t=0 \\
 +
\dfrac{1}{k-1} &; t=\dfrac{1}{k}, \quad k \in \{2,3,\ldots\}
 +
\end{array} \right.$
 
|[[Derivation of forward jump for T=Closure of unit fractions|derivation]]
 
|[[Derivation of forward jump for T=Closure of unit fractions|derivation]]
 
|-
 
|-

Latest revision as of 22:32, 23 February 2016

The set $\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}=\left\{ 0,1,\dfrac{1}{2},\dfrac{1}{3},\ldots \right\}$, where the $\overline{\mathrm{overline}}$ denotes topological closure of this set in the usual topology on $\mathbb{R}$ is a time scale.

Closureunitfractionstimescale.png

$\mathbb{T}=\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$
Forward jump: $\sigma(t)=\left\{ \begin{array}{ll} 0 &; t=0 \\ \dfrac{1}{k-1} &; t=\dfrac{1}{k}, \quad k \in \{2,3,\ldots\} \end{array} \right.$ derivation
Forward graininess: $\mu(t)=$ derivation
Backward jump: $\rho(t)=$ derivation
Backward graininess: $\nu(t)=$ derivation
$\Delta$-derivative $f^{\Delta}(t)=$ derivation
$\nabla$-derivative $f^{\nabla}(t)=$ derivation
$\Delta$-integral $\displaystyle\int_s^t f(\tau) \Delta \tau=$ derivation
$\nabla$-integral $\displaystyle\int_s^t f(\tau) \nabla \tau=$ 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 $\Gamma_{\overline{\left\{\frac{1}{n} \colon n \in \mathbb{Z}^+\right\}}}(x,s)=$ derivation
Euler-Cauchy logarithm $L(t,s)=$ derivation
Bohner logarithm $L_p(t,s)=$ derivation
Jackson logarithm $\log_{\overline{\left\{\frac{1}{n} \colon n \in \mathbb{Z}^+\right\}}} g(t)=$ derivation
Mozyrska-Torres logarithm $L_{\overline{\left\{\frac{1}{n} \colon n \in \mathbb{Z}^+\right\}}}(t)=$ derivation
Laplace transform $\mathscr{L}_{\overline{\left\{\frac{1}{n} \colon n \in \mathbb{Z}^+\right\}}}\{f\}(z;s)=$ derivation
Hilger circle derivation


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