Difference between revisions of "Square integers"

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(Created page with "The set $\mathbb{Z}^2 = \{0,1,4,9,16,\ldots\}$ of square integers is a time scale. {| class="wikitable" |+$\mathbb{T}=\mathbb{Z}^2$ |- |Generic element $t\in \mathbb{T}$:...")
 
 
(One intermediate revision by the same user not shown)
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|+$\mathbb{T}=\mathbb{Z}^2$
 
|+$\mathbb{T}=\mathbb{Z}^2$
 
|-
 
|-
−
|Generic element $t\in \mathbb{T}$:
+
|[[Forward jump]]:
−
|For some $n \in \mathbb{Z}, t=n^2$
+
|$\sigma(t)=t+2\sqrt{t}+1$
 +
|[[Derivation of forward jump for T=ZSquared|derivation]]
 
|-
 
|-
−
|Jump operator:
+
|[[Forward graininess]]:
−
|$\begin{array}{ll}
+
|$\mu(t)=2\sqrt{t}+1$
−
\sigma(t) &= (\sqrt{t}+1)^2 \\
+
|[[Derivation of forward graininess for T=ZSquared|derivation]]
−
&= t + 2\sqrt{t} + 1
 
−
\end{array}$
 
 
|-
 
|-
−
|Graininess operator:
+
|[[Backward jump]]:
−
|$\begin{array}{ll}
+
|$\rho(t)=$
−
\mu(t) &= (t+2\sqrt{t}+1)-t \\
+
|[[Derivation of backward jump for T=ZSquared|derivation]]
−
&= 2\sqrt{t}+1
 
−
\end{array}$
 
 
|-
 
|-
−
|[[Delta_derivative | $\Delta$-derivative:]]
+
|[[Backward graininess]]:
 +
|$\nu(t)=$
 +
|[[Derivation of backward graininess for T=ZSquared|derivation]]
 +
|-
 +
|[[Delta derivative | $\Delta$-derivative]]
 
|$f^{\Delta}(t)=\dfrac{f(t+2\sqrt{t}+1)-f(t)}{2\sqrt{t}-1}$
 
|$f^{\Delta}(t)=\dfrac{f(t+2\sqrt{t}+1)-f(t)}{2\sqrt{t}-1}$
 +
|[[Derivation of delta derivative for T=ZSquared|derivation]]
 +
|-
 +
|[[Nabla derivative | $\nabla$-derivative]]
 +
|$f^{\nabla}(t)=$
 +
|[[Derivation of nabla derivative for T=ZSquared|derivation]]
 +
|-
 +
|[[Delta integral | $\Delta$-integral]]
 +
|$\displaystyle\int_s^t f(\tau) \Delta \tau=$
 +
|[[Derivation of delta integral for T=ZSquared|derivation]]
 +
|-
 +
|[[Nabla integral | $\nabla$-integral]]
 +
|$\displaystyle\int_s^t f(\tau) \nabla \tau=$
 +
|[[Derivation of nabla integral for T=ZSquared|derivation]]
 +
|-
 +
|[[Delta hk|$h_k(t,s)$]]
 +
|$h_k(t,s)=$
 +
|[[Derivation of delta hk for T=ZSquared|derivation]]
 +
|-
 +
|[[Nabla hk|$\hat{h}_k(t,s)$]]
 +
|$\hat{h}_k(t,s)=$
 +
|[[Derivation of nabla hk for T=ZSquared|derivation]]
 +
|-
 +
|[[Delta gk|$g_k(t,s)$]]
 +
|$g_k(t,s)=$
 +
|[[Derivation of delta gk for T=ZSquared|derivation]]
 +
|-
 +
|[[Nabla gk|$\hat{g}_k(t,s)$]]
 +
|$\hat{g}_k(t,s)=$
 +
|[[Derivation of nabla gk for T=ZSquared|derivation]]
 +
|-
 +
|[[Delta exponential | $e_p(t,s)$]]
 +
|$e_p(t,s)=$
 +
|[[Derivation of delta exponential T=ZSquared|derivation]]
 +
|-
 +
|[[Nabla exponential | $\hat{e}_p(t,s)$]]
 +
|$\hat{e}_p(t,s)=$
 +
|[[Derivation of nabla exponential T=ZSquared|derivation]]
 +
|-
 +
|[[Gaussian bell]]
 +
|$\mathbf{E}(t)=$
 +
|[[Derivation of Gaussian bell for T=ZSquared|derivation]]
 +
|-
 +
|[[Delta sine | $\mathrm{sin}_p(t,s)=$]]
 +
|$\sin_p(t,s)=$
 +
|[[Derivation of delta sin sub p for T=ZSquared|derivation]]
 +
|-
 +
|$\mathrm{\sin}_1(t,s)$
 +
|$\sin_1(t,s)=$
 +
|[[Derivation of delta sin sub 1 for T=ZSquared|derivation]]
 +
|-
 +
|[[Nabla sine|$\widehat{\sin}_p(t,s)$]]
 +
|$\widehat{\sin}_p(t,s)=$
 +
|[[Derivation of nabla sine sub p for T=ZSquared|derivation]]
 +
|-
 +
|[[Delta cosine|$\mathrm{\cos}_p(t,s)$]]
 +
|$\cos_p(t,s)=$
 +
|[[Derivation of delta cos sub p for T=ZSquared|derivation]]
 +
|-
 +
|$\mathrm{\cos}_1(t,s)$
 +
|$\cos_1(t,s)=$
 +
|[[Derivation of delta cos sub 1 for T=ZSquared|derivation]]
 +
|-
 +
|[[Nabla cosine|$\widehat{\cos}_p(t,s)$]]
 +
|$\widehat{\cos}_p(t,s)=$
 +
|[[Derivation of nabla cos sub 1 for T=ZSquared|derivation]]
 +
|-
 +
|[[Delta sinh|$\sinh_p(t,s)$]]
 +
|$\sinh_p(t,s)=$
 +
|[[Derivation of delta sinh sub p for T=ZSquared|derivation]]
 +
|-
 +
|[[Nabla sinh|$\widehat{\sinh}_p(t,s)$]]
 +
|$\widehat{\sinh}_p(t,s)=$
 +
|[[Derivation of nabla sinh sub p for T=ZSquared|derivation]]
 +
|-
 +
|[[Delta cosh|$\cosh_p(t,s)$]]
 +
|$\cosh_p(t,s)=$
 +
|[[Derivation of delta cosh sub p for T=ZSquared|derivation]]
 +
|-
 +
|[[Nabla cosh|$\widehat{\cosh}_p(t,s)$]]
 +
|$\widehat{\cosh}_p(t,s)=$
 +
|[[Derivation of nabla cosh sub p for T=ZSquared|derivation]]
 +
|-
 +
|[[Gamma function]]
 +
|$\Gamma_{\mathbb{Z}^2}(x,s)=$
 +
|[[Derivation of gamma function for T=ZSquared|derivation]]
 +
|-
 +
|[[Euler-Cauchy logarithm]]
 +
|$L(t,s)=$
 +
|[[Derivation of Euler-Cauchy logarithm for T=ZSquared|derivation]]
 +
|-
 +
|[[Bohner logarithm]]
 +
|$L_p(t,s)=$
 +
|[[Derivation of the Bohner logarithm for T=ZSquared|derivation]]
 +
|-
 +
|[[Jackson logarithm]]
 +
|$\log_{\mathbb{Z}^2} g(t)=$
 +
|[[Derivation of the Jackson logarithm for T=ZSquared|derivation]]
 +
|-
 +
|[[Mozyrska-Torres logarithm]]
 +
|$L_{\mathbb{Z}^2}(t)=$
 +
|[[Derivation of the Mozyrska-Torres logarithm for T=ZSquared|derivation]]
 +
|-
 +
|[[Laplace transform]]
 +
|$\mathscr{L}_{\mathbb{Z}^2}\{f\}(z;s)=$
 +
|[[Derivation of Laplace transform for T=ZSquared|derivation]]
 
|-
 
|-
−
|[[Delta_integral | $\Delta$-integral:]]
+
|[[Hilger circle]]  
−
| $\displaystyle\int_s^t f(t) \Delta t = \displaystyle\sum_{k=\sqrt{s}}^{\sqrt{t}-1} (2k+1)f(k^2)$
+
|
 +
|[[Derivation of Hilger circle for T=ZSquared|derivation]]
 
|-
 
|-
−
|[[Exponential_functions | 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=\sqrt{s}}^{\sqrt{t}-1} \mu(k^2) \dfrac{1}{\mu(k^2)} \log ( 1 + p(k^2)\mu(k^2)) \right) \\
 
−
&= \exp \left( \displaystyle\sum_{k=\sqrt{s}}^{\sqrt{t}-1} \log ( 1 + p(k^2)\mu(k^2)) \right) \\
 
−
&= \displaystyle\prod_{k=\sqrt{s}}^{\sqrt{t}-1} 1 + p(k^2)(2k+1)
 
−
\end{array}$
 
 
|}
 
|}
 +
 +
<center>{{:Time scales footer}}</center>

Latest revision as of 01:27, 22 May 2015

The set $\mathbb{Z}^2 = \{0,1,4,9,16,\ldots\}$ of square integers is a time scale.

$\mathbb{T}=\mathbb{Z}^2$
Forward jump: $\sigma(t)=t+2\sqrt{t}+1$ derivation
Forward graininess: $\mu(t)=2\sqrt{t}+1$ derivation
Backward jump: $\rho(t)=$ derivation
Backward graininess: $\nu(t)=$ derivation
$\Delta$-derivative $f^{\Delta}(t)=\dfrac{f(t+2\sqrt{t}+1)-f(t)}{2\sqrt{t}-1}$ 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 $\Gamma_{\mathbb{Z}^2}(x,s)=$ derivation
Euler-Cauchy logarithm $L(t,s)=$ derivation
Bohner logarithm $L_p(t,s)=$ derivation
Jackson logarithm $\log_{\mathbb{Z}^2} g(t)=$ derivation
Mozyrska-Torres logarithm $L_{\mathbb{Z}^2}(t)=$ derivation
Laplace transform $\mathscr{L}_{\mathbb{Z}^2}\{f\}(z;s)=$ 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
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