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This wiki is a resource for people who do research in [[time_scale | time scale calculus]]. Time scale calculus is a unification and extension of differential and difference calculus in which one does calculus upon a set called a ''time scale'' $\mathbb{T}$. When [[Real_numbers | $\mathbb{T}=\mathbb{R}$]] the resulting theory is [http://en.wikipedia.org/wiki/Differential_calculus differential calculus] but when [[Multiples_of_integers | $\mathbb{T}=\mathbb{Z}$]] the resulting theory is [http://en.wikipedia.org/wiki/Difference_calculus difference calculus]. Time scales also include any closed subset of $\mathbb{R}$, so more exotic sets such as the [http://en.wikipedia.org/wiki/Cantor_set Cantor set] are also considered in the theory.
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This wiki is a resource for <strong>time scale calculus</strong>. Time scale calculus is a unification and extension of differential and difference calculus in which one does calculus upon a set $\mathbb{T}$ of real numbers called a [[time scale]]. When [[Real_numbers | $\mathbb{T}=\mathbb{R}$]] the resulting theory becomes [http://en.wikipedia.org/wiki/Differential_calculus differential calculus], when [[Multiples_of_integers | $\mathbb{T}=\mathbb{Z}$]] the resulting theory becomes [http://en.wikipedia.org/wiki/Difference_calculus difference calculus], and when [[Quantum q greater than 1 | $\mathbb{T}=\{1,q,q^2,\ldots\}, q>1$]], the resulting theory becomes the [https://en.wikipedia.org/wiki/Quantum_calculus $q$-calculus]. Time scales also include any closed subset of $\mathbb{R}$, so more exotic sets such as the [http://en.wikipedia.org/wiki/Cantor_set Cantor set] are also subsumed in the theory.
  
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==How to get access to edit this wiki==
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A result proven in time scale calculus implies the result for all choices of $\mathbb{T}$ so a result in time scale calculus immediately implies the result in differential calculus, the same result in difference calculus, the same result in $q$-calculus, the same result in calculus on the Cantor set, and countless others. For an example of this phenomenon, see the familiar properties of the [[delta_derivative | $\Delta$-derivative]] to classical differentiation or to taking a forward difference.
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In order to temper anonymous edits by web bots, I have restricted registration. Please send me an e-mail at tomcuchta.....at......gmail......dot.....com with the subject "Time scale wiki registration". When I receive the e-mail, I will enable registration for you.
 
  
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==Calculus on time scales==
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See the Python library [https://github.com/tomcuchta/timescalecalculus timescalecalculus] on GitHub and its [[timescalecalculus python library documentation|documentation]].
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*[[Abel's theorem]]
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*[[time_scale | List of time scales]]
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<b><u>Registration</u></b>
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*[[delta_derivative | $\Delta$-derivative]]
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Due to a resurgence of automated spam bots, account registration and anonymous editing is currently disabled. Please contact Tom Cuchta (tomcuchta@gmail.com) to gain access to edit the wiki.
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*[[nabla_derivative | $\nabla$-derivative]]
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*[[diamond derivative | $\Diamond$-derivative]]
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=Time scales calculus=
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*[[delta_integral | $\Delta$-integral]]
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<center>{{:Time scales footer}}</center>
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*[[nabla integral | $\nabla$-integral]]
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<center>{{:Delta special functions footer}}</center>
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*[[diamond integral | $\Diamond$-integral]]
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<center>{{:Hilger complex plane footer}}</center>
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*[[dynamic_equations | Dynamic Equations]]
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{{:Delta inequalities footer}}
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*[[complex_calculus | Complex calculus on time scales]]
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*[[Convergence of time scales]]
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[[Bilateral Laplace transform]]<br />
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*[[Function spaces]]
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[[Unilateral Laplace transform]]<br />
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*[[Laplace transform]]
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[[Cauchy function]]<br />
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*[[Mean value theorem]]
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[[Chain rule]]<br />
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*[[Fourier transform]]
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[[Unilateral convolution]]<br />
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*[[Regressive function]]
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[[Dense point]]<br />
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*[[Variation of parameters]]
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[[Disconjugate]]<br />
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*[[Wronskian]]
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[[Dynamic equation]]<br />
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===Probability Theory===
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[[Forward circle minus]]<br />
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[[Backward circle minus]]<br />
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[[Forward circle plus]]<br />
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[[Backward circle plus]]<br />
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[[Convergence of time scales]]<br />
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[[Dilation of time scales]]<br />
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[[Duality of delta and nabla | Duality of $\Delta$ and $\nabla$]]<br />
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[[Fractional calculus]]<br />
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[[Frequency roots]]<br />
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[[Generalized square]]<br />
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[[Generalized zero]]<br />
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[[Induction on time scales]]<br />
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[[L'Hospital's Rule]]<br />
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[[First mean value theorem]]<br />
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[[Pre-differentiable]]<br />
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[[Marks-Gravagne-Davis Fourier transform]]<br />
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[[Cuchta-Georgiev Fourier transform]]<br />
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[[rd-continuous]]<br />
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[[Forward regressive function]]<br />
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[[Regulated function]]<br />
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[[Riccati equation]]<br />
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[[Scattered point]]<br />
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[[Self-adjoint]]<br />
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[[Shifting problem]]<br />
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[[Variation of parameters]]<br />
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[[Wronskian]]<br />
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==$\Delta$-calculus==
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[[delta_derivative | $\Delta$-derivative]]<br />
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[[Delta heat equation | $\Delta$ heat equation]]<br />
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[[delta_integral | $\Delta$-integral]]<br />
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[[Delta Taylor's formula|$\Delta$-Taylor's formula]]<br />
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[[Delta wave equation | $\Delta$ wave equation]]<br />
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[[Directional Delta Derivative | Directional $\Delta$ derivative]]<br />
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[[Partial Delta Derivative | Partial $\Delta$ derivative]]<br />
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[[Partial Delta Dynamic Equations | Partial $\Delta$ dynamic equations]]<br />
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==$\nabla$-calculus==
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[[nabla_derivative | $\nabla$-derivative]]<br />
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[[nabla integral | $\nabla$-integral]]<br />
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==$\Diamond_{\alpha}$-calculus==
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[[diamond alpha derivative | $\Diamond_{\alpha}$-derivative]]<br />
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[[diamond alpha holder inequality | $\Diamond_{\alpha}$-Hölder inequality ]]<br />
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[[diamond alpha Jensen's inequality | $\Diamond_{\alpha}$-Jensen's inequality]]<br />
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[[diamond alpha Minkowski's inequality | $\Diamond_{\alpha}$-Minkowski's inequality]]<br />
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[[diamond integral | $\Diamond$-integral]]<br />
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==Probability Theory==
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*[[Cumulant generating function]]
 
*[[Cumulative distribution function]]
 
*[[Cumulative distribution function]]
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*[[Moments]]
 
 
*[[Probability density function]]
 
*[[Probability density function]]
 
*[[Joint time scales probability density function]]
 
*[[Joint time scales probability density function]]
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*[[Variance]]
 
*[[Variance]]
  
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==Examples of time scales==
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{{:Examples of time scales}}
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# The real line: [[Real_numbers | $\mathbb{R}$]]
 
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# The integers: [[Integers | $\mathbb{Z} = \{\ldots, -1,0,1,\ldots\}$]]
 
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# Multiples of integers: [[Multiples_of_integers | $h\mathbb{Z} = \{ht \colon t \in \mathbb{Z}\}$]]
 
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# Quantum numbers ($q>1$): [[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}$]]
 
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# Quantum numbers ($q<1$): [[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}$]]
 
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# Square integers: [[Square_integers | $\mathbb{Z}^2 = \{t^2 \colon t \in \mathbb{Z} \}$]]
 
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# Harmonic numbers: [[Harmonic_numbers | $\mathbb{H}=\left\{\displaystyle\sum_{k=1}^n \dfrac{1}{k} \colon n \in \mathbb{Z}^+ \right\}$]]
 
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# The closure of the unit fractions: [[Closure_of_unit_fractions | $\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$]]
 
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# Isolated points: [[Isolated_points | $\mathbb{T}=\{\ldots, t_{-1}, t_{0}, t_1, \ldots\}$]]
 
  
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==Inequalities==
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=Special functions on time scales=
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*[[Bernoulli inequality]]
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[[Delta cpq|$\mathrm{c}_{pq}$]]<br />
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*[[Bihari inequality]]
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[[Delta chpq|$\mathrm{ch}_{pq}$]]<br />
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*[[Cauchy-Schwarz inequality]]
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[[Delta spq|$\mathrm{s}_{pq}$]]<br />
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*[[Gronwall inequality]]
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[[Delta shpq|$\mathrm{sh}_{pq}$]]<br />
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*[[Hölder inequality]]
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[[Gamma function]]<br />
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*[[Jensen inequality]]
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[[Euler-Cauchy logarithm]]<br />
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*[[Lyapunov inequality]]
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[[Bohner logarithm]]<br />
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*[[Minkowski inequality]]
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[[Jackson logarithm]]<br />
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*[[Opial inequality]]
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[[Mozyrska-Torres logarithm]]<br />
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*[[Wirtinger inequality]]
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[[gaussian_bell | Gaussian bell]]<br />
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[[Uniform distribution]]<br />
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[[Exponential distribution]]<br />
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[[Gamma distribution]]<br />
  
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==Special functions on time scales==
 
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*[[polynomials | Polynomials]]
 
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*[[Delta cosine | $\Delta$-$\cos_p$]]
 
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*[[Delta exponential | $\Delta$-$e_p$]]
 
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*[[Delta sine | $\Delta$-$\sin_p$]]
 
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*[[Gamma function]]
 
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*[[hyperbolic_functions | Hyperbolic functions]]
 
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*[[logarithms | Logarithms]]
 
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*[[Nabla cosine | $\nabla$-$\cos_p$]]
 
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*[[Nabla exponential | $\nabla$-$\hat{e}_p$]]
 
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*[[Nabla sine | $\nabla$-$\sin_p$]]
 
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*[[trig_functions | Trigonometric functions]]
 
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*[[gaussian_bell | Gaussian bell]]
 
  
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===Probability Distributions on time scales===
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==$\nabla$-calculus==
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*[[Uniform distribution]]
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[[Nabla cosine | $\nabla \widehat{\cos}_p$]]<br />
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*[[Exponential distribution]]
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[[Nabla cosh | $\nabla \widehat{\cosh}_p$]]<br />
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*[[Gamma distribution]]
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[[Nabla exponential | $\nabla \widehat{\exp}$]]<br />
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[[Nabla hk|$\nabla \hat{h}_k$]]<br />
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[[Nabla gk|$\nabla \hat{g}_k$]]<br />
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[[Nabla sine | $\nabla \widehat{\sin}_p$]]<br />
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[[Nabla sinh | $\nabla \widehat{\sinh}_p$]]<br />

Latest revision as of 01:55, 6 February 2023

This wiki is a resource for time scale calculus. Time scale calculus is a unification and extension of differential and difference calculus in which one does calculus upon a set $\mathbb{T}$ of real numbers called a time scale. When $\mathbb{T}=\mathbb{R}$ the resulting theory becomes differential calculus, when $\mathbb{T}=\mathbb{Z}$ the resulting theory becomes difference calculus, and when $\mathbb{T}=\{1,q,q^2,\ldots\}, q>1$, the resulting theory becomes the $q$-calculus. Time scales also include any closed subset of $\mathbb{R}$, so more exotic sets such as the Cantor set are also subsumed in the theory.

A result proven in time scale calculus implies the result for all choices of $\mathbb{T}$ so a result in time scale calculus immediately implies the result in differential calculus, the same result in difference calculus, the same result in $q$-calculus, the same result in calculus on the Cantor set, and countless others. For an example of this phenomenon, see the familiar properties of the $\Delta$-derivative to classical differentiation or to taking a forward difference.

See the Python library timescalecalculus on GitHub and its documentation.

Registration Due to a resurgence of automated spam bots, account registration and anonymous editing is currently disabled. Please contact Tom Cuchta (tomcuchta@gmail.com) to gain access to edit the wiki.

Time scales calculus

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

$\Delta$-special functions on time scales


$\cos_p$

$\cosh_p$

$e_p$

$g_k$

$h_k$

$\sin_p$

$\sinh_p$

Hilger complex plane and friends

$\Huge\mathbb{A}_h$
Hilger alternating axis
$\Huge\mathbb{I}_h$
Hilger circle
$\Huge\mathbb{C}_h$
Hilger complex plane
$\Huge\mathrm{Im}_h$
Hilger imaginary part
$\Huge\mathring{\iota}$
Hilger pure imaginary
$\Huge\mathbb{R}_h$
Hilger real axis
$\Huge\mathrm{Re}_h$
Hilger real part

$\Delta$-Inequalities

Bernoulli Bihari Cauchy-Schwarz Gronwall Hölder Jensen Lyapunov Markov Minkowski Opial Tschebycheff Wirtinger

Bilateral Laplace transform
Unilateral Laplace transform
Cauchy function
Chain rule
Unilateral convolution
Dense point
Disconjugate
Dynamic equation
Forward circle minus
Backward circle minus
Forward circle plus
Backward circle plus
Convergence of time scales
Dilation of time scales
Duality of $\Delta$ and $\nabla$
Fractional calculus
Frequency roots
Generalized square
Generalized zero
Induction on time scales
L'Hospital's Rule
First mean value theorem
Pre-differentiable
Marks-Gravagne-Davis Fourier transform
Cuchta-Georgiev Fourier transform
rd-continuous
Forward regressive function
Regulated function
Riccati equation
Scattered point
Self-adjoint
Shifting problem
Variation of parameters
Wronskian

$\Delta$-calculus

$\Delta$-derivative
$\Delta$ heat equation
$\Delta$-integral
$\Delta$-Taylor's formula
$\Delta$ wave equation
Directional $\Delta$ derivative
Partial $\Delta$ derivative
Partial $\Delta$ dynamic equations

$\nabla$-calculus

$\nabla$-derivative
$\nabla$-integral

$\Diamond_{\alpha}$-calculus

$\Diamond_{\alpha}$-derivative
$\Diamond_{\alpha}$-Hölder inequality
$\Diamond_{\alpha}$-Jensen's inequality
$\Diamond_{\alpha}$-Minkowski's inequality
$\Diamond$-integral

Probability Theory

Examples of time scales

  1. The real line: $\mathbb{R}$
  2. The integers: $\mathbb{Z} = \{\ldots, -1,0,1,\ldots\}$
  3. Multiples of integers: $h\mathbb{Z} = \{ht \colon t \in \mathbb{Z}\}$
  4. Quantum numbers ($q>1$): $\overline{q^{\mathbb{Z}}}$
  5. Quantum numbers ($q<1$): $\overline{q^{\mathbb{Z}}}$
  6. Square integers: $\mathbb{Z}^2 = \{t^2 \colon t \in \mathbb{Z} \}$
  7. Harmonic numbers: $\mathbb{H}=\left\{\displaystyle\sum_{k=1}^n \dfrac{1}{k} \colon n \in \mathbb{Z}^+ \right\}$
  8. The closure of the unit fractions: $\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$
  9. Isolated points: $\mathbb{T}=\{\ldots, t_{-1}, t_{0}, t_1, \ldots\}$

Special functions on time scales

$\mathrm{c}_{pq}$
$\mathrm{ch}_{pq}$
$\mathrm{s}_{pq}$
$\mathrm{sh}_{pq}$
Gamma function
Euler-Cauchy logarithm
Bohner logarithm
Jackson logarithm
Mozyrska-Torres logarithm
Gaussian bell
Uniform distribution
Exponential distribution
Gamma distribution


$\nabla$-calculus

$\nabla \widehat{\cos}_p$
$\nabla \widehat{\cosh}_p$
$\nabla \widehat{\exp}$
$\nabla \hat{h}_k$
$\nabla \hat{g}_k$
$\nabla \widehat{\sin}_p$
$\nabla \widehat{\sinh}_p$

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