Difference between revisions of "Multiples of integers"

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The set $h\mathbb{Z}=\{\ldots,-2h,-h,0,h,2h,\ldots\}$ of multiples of the integers is a [[time scale]].
 
The set $h\mathbb{Z}=\{\ldots,-2h,-h,0,h,2h,\ldots\}$ of multiples of the integers is a [[time scale]].
 +
 +
[[File:HZtime scale.png|500px]]
  
 
{| class="wikitable"
 
{| class="wikitable"
 
|+$\mathbb{T}=h\mathbb{Z}$
 
|+$\mathbb{T}=h\mathbb{Z}$
 
|-
 
|-
−
|Generic element $t\in \mathbb{T}$:
+
|[[Forward jump]]:
−
|For some $n \in \mathbb{Z}, t =hn$
 
−
|-
 
−
|Jump operator:
 
 
|$\sigma(t)=t+h$
 
|$\sigma(t)=t+h$
 +
|[[Derivation of forward jump for T=hZ|derivation]]
 
|-
 
|-
−
|Graininess operator:
+
|[[Forward graininess]]:
 
|$\mu(t)=h$
 
|$\mu(t)=h$
 +
|[[Derivation of forward graininess for T=hZ|derivation]]
 +
|-
 +
|[[Backward jump]]:
 +
|$\rho(t)=t-h$
 +
|[[Derivation of backward jump for T=hZ|derivation]]
 +
|-
 +
|[[Backward graininess]]:
 +
|$\nu(t)=h$
 +
|[[Derivation of backward graininess for T=hZ|derivation]]
 
|-
 
|-
−
|[[Delta_derivative | $\Delta$-derivative:]]
+
|[[Delta derivative | $\Delta$-derivative]]
 
|$f^{\Delta}(t)=\dfrac{f(t+h)-f(t)}{h}$
 
|$f^{\Delta}(t)=\dfrac{f(t+h)-f(t)}{h}$
 +
|[[Derivation of delta derivative for T=hZ|derivation]]
 +
|-
 +
|[[Nabla derivative | $\nabla$-derivative]]
 +
|$f^{\nabla}(t)=\dfrac{f(t)-f(t-h)}{h}$
 +
|[[Derivation of nabla derivative for T=hZ|derivation]]
 
|-
 
|-
−
|[[Nabla derivative | $\nabla$-derivative:]]
+
|[[Delta integral | $\Delta$-integral]]
−
|$f^{\nabla}(t) = \dfrac{f(t)-f(t-h)}{h}$
+
|$\displaystyle\int_s^t f(\tau) \Delta \tau=$
 +
|[[Derivation of delta integral for T=hZ|derivation]]
 
|-
 
|-
−
|[[Delta_integral | $\Delta$-integral:]]
+
|[[Nabla integral | $\nabla$-integral]]
−
| $\displaystyle\int_s^t f(\tau) \Delta \tau = \displaystyle\sum_{k=\frac{s}{h}}^{\frac{t}{h}-1} hf(hk)$
+
|$\displaystyle\int_s^t f(\tau) \nabla \tau=$
 +
|[[Derivation of nabla integral for T=hZ|derivation]]
 
|-
 
|-
−
|[[Polynomials | $h_k(t,s)$]]
+
|[[Delta hk|$h_k(t,s)$]]
 
|$h_k(t,s) = \dfrac{1}{k!} \displaystyle\prod_{\ell=0}^{k-1}(t-\ell h-s)$
 
|$h_k(t,s) = \dfrac{1}{k!} \displaystyle\prod_{\ell=0}^{k-1}(t-\ell h-s)$
 +
|[[Derivation of delta hk for T=hZ|derivation]]
 +
|-
 +
|[[Nabla hk|$\hat{h}_k(t,s)$]]
 +
|$\hat{h}_k(t,s)=$
 +
|[[Derivation of nabla hk for T=hZ|derivation]]
 +
|-
 +
|[[Delta gk|$g_k(t,s)$]]
 +
|$g_k(t,s)=$
 +
|[[Derivation of delta gk for T=hZ|derivation]]
 
|-
 
|-
−
|[[Delta exponential | $\Delta$-Exponential function]]:
+
|[[Nabla gk|$\hat{g}_k(t,s)$]]
−
| $\begin{array}{ll}
+
|$\hat{g}_k(t,s)=$
−
e_p(t,s) &= \exp \left( \displaystyle\int_{s}^{t} \dfrac{1}{\mu(\tau)} \log(1 + hp(\tau)) \Delta \tau \right) \\
+
|[[Derivation of nabla gk for T=hZ|derivation]]
−
&= \exp \left( \displaystyle\sum_{k=\frac{s}{h}}^{\frac{t}{h}-1} \log(1+hp(hk))  \right) \\
 
−
&= \displaystyle\prod_{k=\frac{s}{h}}^{\frac{t}{h}-1} \left( 1+hp(hk) \right) \\
 
−
\end{array}$
 
 
|-
 
|-
−
| [[Nabla exponential | $\nabla$-Exponential function]]:
+
|[[Delta exponential | $e_p(t,s)$]]  
−
| $\hat{e}_p(t,s)=\left\{ \begin{array}{ll}
+
|$e_p(t,s)=\left\{ \begin{array}{ll}
−
\displaystyle\prod_{k=\frac{s}{h}}^{\frac{t}{h}-1} \dfrac{1}{1-hp(hk)} &; t \gt s \\
+
\displaystyle\prod_{k=\frac{t}{h}}^{\frac{s}{h}-1} \dfrac{1}{1+hp(hk)} &; t < s \\
 
1 &; t=s \\
 
1 &; t=s \\
−
\prod_{k=\frac{t}{h}}^{\frac{s}{h}-1} (1-hp(hk)) &; t \lt s
+
\displaystyle\prod_{k=\frac{s}{h}}^{\frac{t}{h}-1} 1+hp(hk) &; t>s.
 
\end{array} \right.$
 
\end{array} \right.$
 +
|[[Derivation of delta exponential T=hZ|derivation]]
 
|-
 
|-
−
|[[Hilger circle]]:
+
|[[Nabla exponential | $\hat{e}_p(t,s)$]]
−
| [[File:Hilgercircle%2CT%3DhZ.png|250px]]
+
|$\hat{e}_p(t,s)=\left\{ \begin{array}{ll}
 +
\displaystyle\prod_{k=\frac{t}{h}+1}^{\frac{s}{h}} (1-hp(hk)) &; t \lt s \\
 +
1 &; t=s \\
 +
\displaystyle\prod_{k=\frac{s}{h}+1}^{\frac{t}{h}} \dfrac{1}{1-hp(hk)} &; t \gt s.
 +
\end{array} \right.$
 +
|[[Derivation of nabla exponential T=hZ|derivation]]
 +
|-
 +
|[[Gaussian bell]]
 +
|$\mathbf{E}(t)=\left[(1+h)^{\frac{1}{h}} \right]^{\frac{-t(t-h)}{2}}$
 +
|[[Derivation of Gaussian bell for T=hZ|derivation]]
 +
|-
 +
|[[Delta sine | $\mathrm{sin}_p(t,s)=$]]
 +
|$\sin_p(t,s)=$
 +
|[[Derivation of delta sin sub p for T=hZ|derivation]]
 +
|-
 +
|$\mathrm{\sin}_1(t,s)$
 +
|$\sin_1(t,s)=$
 +
|[[Derivation of delta sin sub 1 for T=hZ|derivation]]
 +
|-
 +
|[[Nabla sine|$\widehat{\sin}_p(t,s)$]]
 +
|$\widehat{\sin}_p(t,s)=$
 +
|[[Derivation of nabla sine sub p for T=hZ|derivation]]
 +
|-
 +
|[[Delta cosine|$\mathrm{\cos}_p(t,s)$]]
 +
|$\cos_p(t,s)=$
 +
|[[Derivation of delta cos sub p for T=hZ|derivation]]
 +
|-
 +
|$\mathrm{\cos}_1(t,s)$
 +
|$\cos_1(t,s)=$
 +
|[[Derivation of delta cos sub 1 for T=hZ|derivation]]
 +
|-
 +
|[[Nabla cosine|$\widehat{\cos}_p(t,s)$]]
 +
|$\widehat{\cos}_p(t,s)=$
 +
|[[Derivation of nabla cos sub 1 for T=hZ|derivation]]
 +
|-
 +
|[[Delta sinh|$\sinh_p(t,s)$]]
 +
|$\sinh_p(t,s)=$
 +
|[[Derivation of delta sinh sub p for T=hZ|derivation]]
 +
|-
 +
|[[Nabla sinh|$\widehat{\sinh}_p(t,s)$]]
 +
|$\widehat{\sinh}_p(t,s)=$
 +
|[[Derivation of nabla sinh sub p for T=hZ|derivation]]
 +
|-
 +
|[[Delta cosh|$\cosh_p(t,s)$]]
 +
|$\cosh_p(t,s)=$
 +
|[[Derivation of delta cosh sub p for T=hZ|derivation]]
 +
|-
 +
|[[Nabla cosh|$\widehat{\cosh}_p(t,s)$]]
 +
|$\widehat{\cosh}_p(t,s)=$
 +
|[[Derivation of nabla cosh sub p for T=hZ|derivation]]
 +
|-
 +
|[[Gamma function]]
 +
|$\Gamma_{h\mathbb{Z}}(t;s)=h\displaystyle\sum_{k=0}^{\infty} \left( \displaystyle\prod_{j=s}^{k-1} \dfrac{j+x}{j+1} \right) \dfrac{1}{(1+h)^{k+1}}$
 +
|[[Derivation of gamma function for T=hZ|derivation]]
 +
|-
 +
|[[Euler-Cauchy logarithm]]
 +
|$L(t,s)=$
 +
|[[Derivation of Euler-Cauchy logarithm for T=hZ|derivation]]
 +
|-
 +
|[[Bohner logarithm]]
 +
|$L_p(t,s)=$
 +
|[[Derivation of the Bohner logarithm for T=hZ|derivation]]
 +
|-
 +
|[[Jackson logarithm]]
 +
|$\log_{h\mathbb{Z}} g(t)=$
 +
|[[Derivation of the Jackson logarithm for T=hZ|derivation]]
 +
|-
 +
|[[Mozyrska-Torres logarithm]]
 +
|$L_{h\mathbb{Z}}(t)=$
 +
|[[Derivation of the Mozyrska-Torres logarithm for T=hZ|derivation]]
 +
|-
 +
|[[Laplace transform]]
 +
|$\mathscr{L}_{h\mathbb{Z}}\{f\}(z;s)=$
 +
|[[Derivation of Laplace transform for T=hZ|derivation]]
 +
|-
 +
|[[Hilger circle]]  
 +
|[[File:Hilgercircle%2CT%3DhZ.png|250px]]
 +
|[[Derivation of Hilger circle for T=hZ|derivation]]
 
|-
 
|-
−
|[[Gamma function]]:
 
−
| $\Gamma_{h\mathbb{Z}}(t;s)=h\displaystyle\sum_{k=0}^{\infty} \left( \displaystyle\prod_{j=s}^{k-1} \dfrac{j+x}{j+1} \right) \dfrac{1}{(1+h)^{k+1}}$
 
 
|}
 
|}
 +
  
 
=References=
 
=References=
 +
*{{PaperReference|A generalized Fourier transform and convolution on time scales|2008|Robert J. Marks II|author2=Ian A. Gravagne|author3=John M. Davis|prev=Real numbers|next=}}: Section 2.1(a)
 +
* {{PaperReference|Partial dynamic equations on time scales|2006|Billy Jackson||prev=Quantum q greater than 1|next=Forward jump}}: Appendix
 
[http://pi.unl.edu/~apeterson1/higgins-peterson.pdf Cauchy Functions and Taylor's Formula for Time Scales T]
 
[http://pi.unl.edu/~apeterson1/higgins-peterson.pdf Cauchy Functions and Taylor's Formula for Time Scales T]
 +
 +
<center>{{:Time scales footer}}</center>

Latest revision as of 15:55, 15 January 2023

The set $h\mathbb{Z}=\{\ldots,-2h,-h,0,h,2h,\ldots\}$ of multiples of the integers is a time scale.

HZtime scale.png

$\mathbb{T}=h\mathbb{Z}$
Forward jump: $\sigma(t)=t+h$ derivation
Forward graininess: $\mu(t)=h$ derivation
Backward jump: $\rho(t)=t-h$ derivation
Backward graininess: $\nu(t)=h$ derivation
$\Delta$-derivative $f^{\Delta}(t)=\dfrac{f(t+h)-f(t)}{h}$ derivation
$\nabla$-derivative $f^{\nabla}(t)=\dfrac{f(t)-f(t-h)}{h}$ derivation
$\Delta$-integral $\displaystyle\int_s^t f(\tau) \Delta \tau=$ derivation
$\nabla$-integral $\displaystyle\int_s^t f(\tau) \nabla \tau=$ derivation
$h_k(t,s)$ $h_k(t,s) = \dfrac{1}{k!} \displaystyle\prod_{\ell=0}^{k-1}(t-\ell h-s)$ 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
$e_p(t,s)$ $e_p(t,s)=\left\{ \begin{array}{ll} \displaystyle\prod_{k=\frac{t}{h}}^{\frac{s}{h}-1} \dfrac{1}{1+hp(hk)} &; t < s \\ 1 &; t=s \\ \displaystyle\prod_{k=\frac{s}{h}}^{\frac{t}{h}-1} 1+hp(hk) &; t>s. \end{array} \right.$ derivation
$\hat{e}_p(t,s)$ $\hat{e}_p(t,s)=\left\{ \begin{array}{ll} \displaystyle\prod_{k=\frac{t}{h}+1}^{\frac{s}{h}} (1-hp(hk)) &; t \lt s \\ 1 &; t=s \\ \displaystyle\prod_{k=\frac{s}{h}+1}^{\frac{t}{h}} \dfrac{1}{1-hp(hk)} &; t \gt s. \end{array} \right.$ derivation
Gaussian bell $\mathbf{E}(t)=\left[(1+h)^{\frac{1}{h}} \right]^{\frac{-t(t-h)}{2}}$ 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 $\Gamma_{h\mathbb{Z}}(t;s)=h\displaystyle\sum_{k=0}^{\infty} \left( \displaystyle\prod_{j=s}^{k-1} \dfrac{j+x}{j+1} \right) \dfrac{1}{(1+h)^{k+1}}$ derivation
Euler-Cauchy logarithm $L(t,s)=$ derivation
Bohner logarithm $L_p(t,s)=$ derivation
Jackson logarithm $\log_{h\mathbb{Z}} g(t)=$ derivation
Mozyrska-Torres logarithm $L_{h\mathbb{Z}}(t)=$ derivation
Laplace transform $\mathscr{L}_{h\mathbb{Z}}\{f\}(z;s)=$ derivation
Hilger circle Hilgercircle,T=hZ.png derivation


References

Cauchy Functions and Taylor's Formula for Time Scales T

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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