Difference between revisions of "Real numbers"

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The set $\mathbb{R}$ of real numbers is a [[time scale]].
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The set $\mathbb{R}$ of real numbers is a [[time scale]]. In this time scale, all derivatives reduce to the clasical derivative and the integrals reduce to the [http://en.wikipedia.org/wiki/Riemann_integral Riemann integral].
  
 
{| class="wikitable"
 
{| class="wikitable"
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|$\mu(t)=0$
 
|$\mu(t)=0$
 
|-
 
|-
|[[Delta_derivative | $\Delta$-derivative:]]
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|[[Delta derivative | $\Delta$-derivative]]
|$f^{\Delta}(t)=\displaystyle\lim_{h\rightarrow 0} \dfrac{f(t+h)-f(t)}{h}$
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|$$f^{\Delta}(t)=\lim_{h\rightarrow 0} \dfrac{f(t+h)-f(t)}{h}=f'(t)$$
 
|-
 
|-
|[[Delta_integral | $\Delta$-integral:]]
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|[[Nabla derivative | $\nabla$-derivative]]
| $\displaystyle\int_s^t f(\tau) \Delta \tau = \displaystyle\int_s^t f(\tau) d\tau$ is the [http://en.wikipedia.org/wiki/Riemann_integral Riemann integral]
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|$$f^{\nabla}(t) =\lim_{h \rightarrow 0} \dfrac{f(t)-f(t-h)}{h}= f'(t)$$
 
|-
 
|-
|[[Exponential_functions | Exponential function]]:
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|[[Delta integral | $\Delta$-integral]]
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| $$\int_s^t f(\tau) \Delta \tau = \int_s^t f(\tau) d\tau$$
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|-
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|[[Nabla derivative | $\nabla$-derivative]]
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|$$\int_s^t f(\tau) \nabla \tau = \int_s^t f(\tau) d\tau$$
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|-
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|[[Delta exponential | $\Delta$-exponential]]
 
| $\begin{array}{ll}
 
| $\begin{array}{ll}
 
e_p(t,s) &= \exp \left( \displaystyle\int_s^t \displaystyle\lim_{h \rightarrow 0} \dfrac{1}{h} \log(1 + hp(\tau)) d\tau \right) \\
 
e_p(t,s) &= \exp \left( \displaystyle\int_s^t \displaystyle\lim_{h \rightarrow 0} \dfrac{1}{h} \log(1 + hp(\tau)) d\tau \right) \\
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&= \exp \left( \displaystyle\int_s^t p(\tau) d \tau \right)
 
&= \exp \left( \displaystyle\int_s^t p(\tau) d \tau \right)
 
\end{array}$
 
\end{array}$
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|-
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|[[Nabla exponential | $\nabla$-exponential]]
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|
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|-
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|[[Trig functions | $\mathrm{sin}_p(t,s)$]]
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|-
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|$\mathrm{\sin}_1(t,0)$
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|
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|-
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|$\mathrm{\cos}_p(t,0)$
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|
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|-
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|$\mathrm{\cos}_1(t,0)$
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|
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|-
 
|}
 
|}

Revision as of 21:21, 20 October 2014

The set $\mathbb{R}$ of real numbers is a time scale. In this time scale, all derivatives reduce to the clasical derivative and the integrals reduce to the Riemann integral.

$\mathbb{T}=\mathbb{R}$
Generic element $t \in \mathbb{T}$: $t=t$
Jump operator: $\sigma(t)=t$
Graininess operator: $\mu(t)=0$
$\Delta$-derivative $$f^{\Delta}(t)=\lim_{h\rightarrow 0} \dfrac{f(t+h)-f(t)}{h}=f'(t)$$
$\nabla$-derivative $$f^{\nabla}(t) =\lim_{h \rightarrow 0} \dfrac{f(t)-f(t-h)}{h}= f'(t)$$
$\Delta$-integral $$\int_s^t f(\tau) \Delta \tau = \int_s^t f(\tau) d\tau$$
$\nabla$-derivative $$\int_s^t f(\tau) \nabla \tau = \int_s^t f(\tau) d\tau$$
$\Delta$-exponential $\begin{array}{ll} e_p(t,s) &= \exp \left( \displaystyle\int_s^t \displaystyle\lim_{h \rightarrow 0} \dfrac{1}{h} \log(1 + hp(\tau)) d\tau \right) \\ &\hspace{-10pt} \stackrel{\mathrm{L'Hôpital}}{=} \exp \left( \displaystyle\int_s^t \displaystyle\lim_{h \rightarrow 0} \dfrac{1}{1+hp(\tau)} p(\tau) d\tau \right) \\ &= \exp \left( \displaystyle\int_s^t p(\tau) d \tau \right) \end{array}$
$\nabla$-exponential
$\mathrm{sin}_p(t,s)$
$\mathrm{\sin}_1(t,0)$
$\mathrm{\cos}_p(t,0)$
$\mathrm{\cos}_1(t,0)$