Difference between revisions of "Quantum q greater than 1"

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Let $q>1$. The set $\overline{q^{\mathbb{Z}}}=\{0, \ldots, q^{-2}, q^{-1}, 1, q, q^2, \ldots \}$ of quantum numbers is a [[time scale]].
 
Let $q>1$. The set $\overline{q^{\mathbb{Z}}}=\{0, \ldots, q^{-2}, q^{-1}, 1, q, q^2, \ldots \}$ of quantum numbers is a [[time scale]].
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{| class="wikitable"
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|+$\mathbb{T}=\overline{q^{\mathbb{Z}}}$
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|-
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|[[Forward jump]]:
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|$\sigma(t)=qt$
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|[[Derivation of forward jump for T=Quantum q greater than 1|derivation]]
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|-
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|[[Forward graininess]]:
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|$\mu(t)=t(q-1)$
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|[[Derivation of forward graininess for T=Quantum q greater than 1|derivation]]
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|-
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|[[Backward jump]]:
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|$\rho(t)=$
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|[[Derivation of backward jump for T=Quantum q greater than 1|derivation]]
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|-
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|[[Backward graininess]]:
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|$\nu(t)=$
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|[[Derivation of backward graininess for T=Quantum q greater than 1|derivation]]
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|-
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|[[Delta derivative | $\Delta$-derivative]]
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|$f^{\Delta}(t)= \left\{ \begin{array}{ll}
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\dfrac{f(qt)-f(t)}{t(q-1)} &; t\neq 0 \\
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\displaystyle\lim_{h \rightarrow 0} \dfrac{f(h)-f(0)}{h} &; t=0
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\end{array} \right.$
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|[[Derivation of delta derivative for T=Quantum q greater than 1|derivation]]
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|-
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|[[Nabla derivative | $\nabla$-derivative]]
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|$f^{\nabla}(t)=$
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|[[Derivation of nabla derivative for T=Quantum q greater than 1|derivation]]
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|-
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|[[Delta integral | $\Delta$-integral]]
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|$\begin{array}{ll}
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\displaystyle\int_s^t f(\tau) \Delta \tau &= \displaystyle\sum_{k=\log_q(s)}^{\log_q(t)-1} q^{k} (q-1) f(q^k) \\
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\end{array}$
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|[[Derivation of delta integral for T=Quantum q greater than 1|derivation]]
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|-
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|[[Nabla integral | $\nabla$-integral]]
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|$\displaystyle\int_s^t f(\tau) \nabla \tau=$
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|[[Derivation of nabla integral for T=Quantum q greater than 1|derivation]]
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|-
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|[[Delta hk|$h_k(t,s)$]]
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|$h_k(t,s)=\displaystyle\prod_{n=0}^{k-1} \dfrac{t-q^ns}{\sum_{i=0}^n q^i}$
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|[[Derivation of delta hk for T=Quantum q greater than 1|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=Quantum q greater than 1|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=Quantum q greater than 1|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=Quantum q greater than 1|derivation]]
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|-
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|[[Delta exponential | $e_p(t,s)$]]
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|$e_p(t,s)=\displaystyle\prod_{k=\log_q(s)}^{\log_q(t)-1} 1 + p(q^k)q^{k}(q-1)$
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|[[Derivation of delta exponential T=Quantum q greater than 1|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=Quantum q greater than 1|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=Quantum q greater than 1|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=Quantum q greater than 1|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=Quantum q greater than 1|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=Quantum q greater than 1|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=Quantum q greater than 1|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=Quantum q greater than 1|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=Quantum q greater than 1|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=Quantum q greater than 1|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=Quantum q greater than 1|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=Quantum q greater than 1|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=Quantum q greater than 1|derivation]]
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|-
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|[[Gamma function]]
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|$\Gamma_{\overline{q^{\mathbb{Z}}}}(x,s)=$
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|[[Derivation of gamma function for T=Quantum q greater than 1|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=Quantum q greater than 1|derivation]]
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|-
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|[[Bohner logarithm]]
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|$L_p(t,s)=$
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|[[Derivation of the Bohner logarithm for T=Quantum q greater than 1|derivation]]
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|-
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|[[Jackson logarithm]]
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|$\log_{\overline{q^{\mathbb{Z}}}} g(t)=$
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|[[Derivation of the Jackson logarithm for T=Quantum q greater than 1|derivation]]
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|-
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|[[Mozyrska-Torres logarithm]]
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|$L_{\overline{q^{\mathbb{Z}}}}(t)=$
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|[[Derivation of the Mozyrska-Torres logarithm for T=Quantum q greater than 1|derivation]]
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|-
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|[[Laplace transform]]
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|$\mathscr{L}_{\overline{q^{\mathbb{Z}}}}\{f\}(z;s)=$
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|[[Derivation of Laplace transform for T=Quantum q greater than 1|derivation]]
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|-
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|[[Hilger circle]]
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|
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|[[Derivation of Hilger circle for T=Quantum q greater than 1|derivation]]
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|-
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|}
  
 
{| class="wikitable"
 
{| class="wikitable"

Revision as of 03:37, 10 August 2015

Let $q>1$. The set $\overline{q^{\mathbb{Z}}}=\{0, \ldots, q^{-2}, q^{-1}, 1, q, q^2, \ldots \}$ of quantum numbers is a time scale.


$\mathbb{T}=\overline{q^{\mathbb{Z}}}$
Forward jump: $\sigma(t)=qt$ derivation
Forward graininess: $\mu(t)=t(q-1)$ derivation
Backward jump: $\rho(t)=$ derivation
Backward graininess: $\nu(t)=$ derivation
$\Delta$-derivative $f^{\Delta}(t)= \left\{ \begin{array}{ll} \dfrac{f(qt)-f(t)}{t(q-1)} &; t\neq 0 \\ \displaystyle\lim_{h \rightarrow 0} \dfrac{f(h)-f(0)}{h} &; t=0 \end{array} \right.$ derivation
$\nabla$-derivative $f^{\nabla}(t)=$ derivation
$\Delta$-integral $\begin{array}{ll} \displaystyle\int_s^t f(\tau) \Delta \tau &= \displaystyle\sum_{k=\log_q(s)}^{\log_q(t)-1} q^{k} (q-1) f(q^k) \\ \end{array}$ derivation
$\nabla$-integral $\displaystyle\int_s^t f(\tau) \nabla \tau=$ derivation
$h_k(t,s)$ $h_k(t,s)=\displaystyle\prod_{n=0}^{k-1} \dfrac{t-q^ns}{\sum_{i=0}^n q^i}$ 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
$e_p(t,s)$ $e_p(t,s)=\displaystyle\prod_{k=\log_q(s)}^{\log_q(t)-1} 1 + p(q^k)q^{k}(q-1)$ 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 $\Gamma_{\overline{q^{\mathbb{Z}}}}(x,s)=$ derivation
Euler-Cauchy logarithm $L(t,s)=$ derivation
Bohner logarithm $L_p(t,s)=$ derivation
Jackson logarithm $\log_{\overline{q^{\mathbb{Z}}}} g(t)=$ derivation
Mozyrska-Torres logarithm $L_{\overline{q^{\mathbb{Z}}}}(t)=$ derivation
Laplace transform $\mathscr{L}_{\overline{q^{\mathbb{Z}}}}\{f\}(z;s)=$ derivation
Hilger circle derivation
$\mathbb{T}=\overline{q^{\mathbb{Z}}}, q>1$
Generic element $t\in \mathbb{T}$: For some $n \in \mathbb{Z}, t =q^n$
Jump operator: $\sigma(t)=qt$
Graininess operator: $\begin{array}{ll} \mu(t)&=qt-t\\ &=t(q-1) \end{array}$
$\Delta$-derivative: $f^{\Delta}(t)= \left\{ \begin{array}{ll} \dfrac{f(qt)-f(t)}{t(q-1)} &; t\neq 0 \\ \displaystyle\lim_{h \rightarrow 0} \dfrac{f(h)-f(0)}{h} &; t=0 \end{array} \right.$
$\Delta$-integral: $\begin{array}{ll} \displaystyle\int_s^t f(\tau) \Delta \tau &= \displaystyle\sum_{k=\log_q(s)}^{\log_q(t)-1} q^{k} (q-1) f(q^k) \\ \end{array}$
$h_k(t,s)$ $\displaystyle\prod_{n=0}^{k-1} \dfrac{t-q^ns}{\sum_{i=0}^n q^i}$
Exponential function: $\begin{array}{ll} e_p(t,s) &= \exp \left( \displaystyle\int_{s}^{t} \dfrac{1}{\mu(\tau)} \log( 1 + p(\tau) \mu(\tau) ) \Delta \tau \right) \\ &= \exp \left( \displaystyle\sum_{k=\log_q(s)}^{\log_q(t)-1} \mu(q^k) \dfrac{1}{\mu(q^k)} \log(1 + p(q^k)\mu(q^k)) \right) \\ &= \exp \left( \displaystyle\sum_{k=\log_q(s)}^{\log_q(t)-1} \log(1 + p(q^k)\mu(q^k)) \right) \\ &= \displaystyle\prod_{k=\log_q(s)}^{\log_q(t)-1} 1 + p(q^k)q^{k}(q-1) \end{array}$

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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