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This wiki is a resource for people who do research in <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] but when [[Multiples_of_integers | $\mathbb{T}=\mathbb{Z}$]] the resulting theory becomes [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 subsumed in the theory.
  
Consult the [//meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.
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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.
  
== Getting started ==
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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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<b><u>Registration</u></b>
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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.
* [//www.mediawiki.org/wiki/Localisation#Translation_resources Localise MediaWiki for your language]
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=Calculus on time scales=
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<center>{{:Time scales footer}}</center>
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<center>{{:Delta special functions footer}}</center>
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[[Abel's theorem]]<br />
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[[Bilateral Laplace transform]]<br />
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[[Cauchy function]]<br />
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[[Calculus of variations]]<br />
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[[Chain rule]]<br />
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[[Convolution]]<br />
 +
[[Dense point]]<br />
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[[Disconjugate]]<br />
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[[Dynamic equation]]<br />
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[[Circle minus]]<br />
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[[Circle plus]]<br />
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[[complex_calculus | Complex calculus on time scales]]<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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[[Function spaces]]<br />
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[[Generalized square]]<br />
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[[Generalized zero]]<br />
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[[Hilger alternating axis]]<br />
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[[Hilger circle]]<br />
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[[Hilger complex plane]]<br />
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[[Hilger imaginary part]]<br />
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[[Hilger pure imaginary]]<br />
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[[Hilger real axis]]<br />
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[[Hilger real part]]<br />
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[[Induction on time scales]]<br />
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[[Laplace transform]]<br />
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[[L'Hospital's Rule]]<br />
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[[Mean value theorem]]<br />
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[[Pre-differentiable]]<br />
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[[Fourier transform]]<br />
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[[rd-continuous]]<br />
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[[Regressive function]]<br />
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[[Regulated function]]<br />
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[[Riccati equation]]<br />
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[[Riesz representation theorem]]<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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[[Substitution]]<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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[[Completely delta differentiable]]<br />
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[[Delta Bernoulli inequality | $\Delta$-Bernoulli inequality]]<br />
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[[Delta Bihari inequality | $\Delta$-Bihari inequality]]<br />
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[[Delta Cauchy-Schwarz inequality | $\Delta$-Cauchy-Schwarz inequality]]<br />
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[[delta_derivative | $\Delta$-derivative]]<br />
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[[Delta Gronwall inequality | $\Delta$-Gronwall inequality]]<br />
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[[Delta heat equation | $\Delta$ heat equation]]<br />
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[[Delta Hölder inequality | $\Delta$-Hölder inequality]]<br />
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[[delta_integral | $\Delta$-integral]]<br />
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[[Delta Jensen inequality | $\Delta$-Jensen inequality]]<br />
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[[Delta Lyapunov inequality | $\Delta$-Lyapunov inequality]]<br />
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[[Delta Markov inequality | $\Delta$-Markov inequality]]<br />
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[[Delta Minkowski inequality | $\Delta$-Minkowski inequality]]<br />
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[[Delta Opial inequality | $\Delta$-Opial inequality]]<br />
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[[Delta Taylor's formula|$\Delta$-Taylor's formula]]<br />
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[[Delta Tschebycheff inequality | $\Delta$-Tschebycheff inequality]]<br />
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[[Delta Wirtinger inequality | $\Delta$-Wirtinger inequality]]<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]]
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*[[Cumulative distribution function]]
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*[[Moments]]
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*[[Probability density function]]
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*[[Joint time scales probability density function]]
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*[[Moment generating function]]
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*[[Expected value]]
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*[[Variance]]
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{{:Examples of time scales}}
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=Special functions on time scales=
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[[Delta cpq|$\mathrm{c}_{pq}$]]<br />
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[[Delta chpq|$\mathrm{ch}_{pq}$]]<br />
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[[Delta spq|$\mathrm{s}_{pq}$]]<br />
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[[Delta shpq|$\mathrm{sh}_{pq}$]]<br />
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[[Gamma function]]<br />
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[[hyperbolic_functions | Hyperbolic functions]]<br />
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[[Euler-Cauchy logarithm]]<br />
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[[Bohner logarithm]]<br />
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[[Jackson logarithm]]<br />
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[[Mozyrska-Torres logarithm]]<br />
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[[gaussian_bell | Gaussian bell]]<br />
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==$\nabla$-calculus==
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[[Nabla cosine | $\nabla \widehat{\cos}_p$]]<br />
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[[Nabla cosh | $\nabla \widehat{\cosh}_p$]]<br />
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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 />
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==$\Diamond_{\alpha}$-calculus==
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[[Diamond alpha cosine | $\Diamond_{\alpha}$-$\cos_p$]]<br />
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[[Diamond alpha cosh | $\Diamond_{\alpha}$-$\cosh_p$]]<br />
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[[Diamond exponential | $\Diamond_{\alpha}$-$e_p$]]<br />
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[[Diamond sine | $\Diamond$-$\sin_p$]]<br />
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[[Diamond sinh | $\Diamond$-$\sinh_p$]]<br />
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==Probability Distributions on time scales==
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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 cases of time scales=
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==Differential equations==
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[[Hypergeometric differential equation]] <br />
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[[Confluent hypergeometric differential equation]]<br />
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 +
==Difference equations==
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[[Difference equation of hypergeometric type]]<br />
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==$q$-difference equations==
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[[q-difference equation of hypergeometric type]]<br />

Latest revision as of 03:30, 26 February 2018

This wiki is a resource for people who do research in 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 but when $\mathbb{T}=\mathbb{Z}$ the resulting theory becomes difference 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.

Calculus on time scales

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

$\cos_p$

$\cosh_p$

$e_p$

$g_k$

$h_k$

$\sin_p$

$\sinh_p$

Abel's theorem
Bilateral Laplace transform
Cauchy function
Calculus of variations
Chain rule
Convolution
Dense point
Disconjugate
Dynamic equation
Circle minus
Circle plus
Complex calculus on time scales
Convergence of time scales
Dilation of time scales
Duality of $\Delta$ and $\nabla$
Fractional calculus
Frequency roots
Function spaces
Generalized square
Generalized zero
Hilger alternating axis
Hilger circle
Hilger complex plane
Hilger imaginary part
Hilger pure imaginary
Hilger real axis
Hilger real part
Induction on time scales
Laplace transform
L'Hospital's Rule
Mean value theorem
Pre-differentiable
Fourier transform
rd-continuous
Regressive function
Regulated function
Riccati equation
Riesz representation theorem
Scattered point
Self-adjoint
Shifting problem
Substitution
Variation of parameters
Wronskian

$\Delta$-calculus

Completely delta differentiable
$\Delta$-Bernoulli inequality
$\Delta$-Bihari inequality
$\Delta$-Cauchy-Schwarz inequality
$\Delta$-derivative
$\Delta$-Gronwall inequality
$\Delta$ heat equation
$\Delta$-Hölder inequality
$\Delta$-integral
$\Delta$-Jensen inequality
$\Delta$-Lyapunov inequality
$\Delta$-Markov inequality
$\Delta$-Minkowski inequality
$\Delta$-Opial inequality
$\Delta$-Taylor's formula
$\Delta$-Tschebycheff inequality
$\Delta$-Wirtinger inequality
$\Delta$ 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
Hyperbolic functions
Euler-Cauchy logarithm
Bohner logarithm
Jackson logarithm
Mozyrska-Torres logarithm
Gaussian bell

$\nabla$-calculus

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

$\Diamond_{\alpha}$-calculus

$\Diamond_{\alpha}$-$\cos_p$
$\Diamond_{\alpha}$-$\cosh_p$
$\Diamond_{\alpha}$-$e_p$
$\Diamond$-$\sin_p$
$\Diamond$-$\sinh_p$

Probability Distributions on time scales

Uniform distribution
Exponential distribution
Gamma distribution

Special cases of time scales

Differential equations

Hypergeometric differential equation
Confluent hypergeometric differential equation

Difference equations

Difference equation of hypergeometric type

$q$-difference equations

q-difference equation of hypergeometric type