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

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

The real line: $\mathbb{R}$
The integers: $\mathbb{Z} = \{\ldots, -1,0,1,\ldots\}$
Multiples of integers: $h\mathbb{Z} = \{ht \colon t \in \mathbb{Z}\}$
Quantum numbers ($q>1$): $\overline{q^{\mathbb{Z}}}$
Quantum numbers ($q<1$): $\overline{q^{\mathbb{Z}}}$
Square integers: $\mathbb{Z}^2 = \{t^2 \colon t \in \mathbb{Z} \}$
Harmonic numbers: $\mathbb{H}=\left\{\displaystyle\sum_{k=1}^n \dfrac{1}{k} \colon n \in \mathbb{Z}^+ \right\}$
The closure of the unit fractions: $\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$
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

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

@@ Line 1: / Line 1: @@
-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.
+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.
 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.
-=How to get access to edit this wiki=
+See the Python library [https://github.com/tomcuchta/timescalecalculus timescalecalculus] on GitHub and its [[timescalecalculus python library documentation|documentation]].
-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.
-=Calculus on time scales=
+<b><u>Registration</u></b>
+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=
 <center>{{:Time scales footer}}</center>
 <center>{{:Delta special functions footer}}</center>
+<center>{{:Hilger complex plane footer}}</center>
+{{:Delta inequalities footer}}
-*[[Abel's theorem]]
+[[Bilateral Laplace transform]]<br />
-*[[Bilateral Laplace transform]]
+[[Unilateral Laplace transform]]<br />
-*[[Cauchy function]]
+[[Cauchy function]]<br />
-*[[Calculus of variations]]
+[[Chain rule]]<br />
-*[[Chain rule]]
+[[Unilateral convolution]]<br />
-*[[Convolution]]
+[[Dense point]]<br />
-*[[Dense point]]
+[[Disconjugate]]<br />
-*[[Disconjugate]]
+[[Dynamic equation]]<br />
-*[[Dynamic equation]]
+[[Forward circle minus]]<br />
-*[[Circle minus]]
+[[Backward circle minus]]<br />
-*[[Circle plus]]
+[[Forward circle plus]]<br />
-*[[complex_calculus | Complex calculus on time scales]]
+[[Backward circle plus]]<br />
-*[[Convergence of time scales]]
+[[Convergence of time scales]]<br />
-*[[Dilation of time scales]]
+[[Dilation of time scales]]<br />
-*[[Duality of delta and nabla | Duality of $\Delta$ and $\nabla$]]
+[[Duality of delta and nabla | Duality of $\Delta$ and $\nabla$]]<br />
-*[[Fractional calculus]]
+[[Fractional calculus]]<br />
-*[[Function spaces]]
+[[Frequency roots]]<br />
-*[[Generalized square]]
+[[Generalized square]]<br />
-*[[Generalized zero]]
+[[Generalized zero]]<br />
-*[[Hilger alternating axis]]
+[[Induction on time scales]]<br />
-*[[Hilger circle]]
+[[L'Hospital's Rule]]<br />
-*[[Hilger complex plane]]
+[[First mean value theorem]]<br />
-*[[Hilger imaginary part]]
+[[Pre-differentiable]]<br />
-*[[Hilger pure imaginary]]
+[[Marks-Gravagne-Davis Fourier transform]]<br />
-*[[Hilger real axis]]
+[[Cuchta-Georgiev Fourier transform]]<br />
-*[[Hilger real part]]
+[[rd-continuous]]<br />
-*[[Induction on time scales]]
+[[Forward regressive function]]<br />
-*[[Laplace transform]]
+[[Regulated function]]<br />
-*[[L'Hospital's Rule]]
+[[Riccati equation]]<br />
-*[[Mean value theorem]]
+[[Scattered point]]<br />
-*[[Pre-differentiable]]
+[[Self-adjoint]]<br />
-*[[Fourier transform]]
+[[Shifting problem]]<br />
-*[[rd-continuous]]
+[[Variation of parameters]]<br />
-*[[Regressive function]]
+[[Wronskian]]<br />
-*[[Regulated function]]
-*[[Riccati equation]]
-*[[Riesz representation theorem]]
-*[[Scattered point]]
-*[[Self-adjoint]]
-*[[Shifting problem]]
-*[[Substitution]]
-*[[Variation of parameters]]
-*[[Wronskian]]
 ==$\Delta$-calculus==
-*[[Completely delta differentiable]]
+[[delta_derivative | $\Delta$-derivative]]<br />
-*[[Delta Bernoulli inequality | $\Delta$-Bernoulli inequality]]
+[[Delta heat equation | $\Delta$ heat equation]]<br />
-*[[Delta Bihari inequality | $\Delta$-Bihari inequality]]
+[[delta_integral | $\Delta$-integral]]<br />
-*[[Delta Cauchy-Schwarz inequality | $\Delta$-Cauchy-Schwarz inequality]]
+[[Delta Taylor's formula|$\Delta$-Taylor's formula]]<br />
-*[[delta_derivative | $\Delta$-derivative]]
+[[Delta wave equation | $\Delta$ wave equation]]<br />
-*[[Delta Gronwall inequality | $\Delta$-Gronwall inequality]]
+[[Directional Delta Derivative | Directional $\Delta$ derivative]]<br />
-*[[Delta heat equation | $\Delta$ heat equation]]
+[[Partial Delta Derivative | Partial $\Delta$ derivative]]<br />
-*[[Delta Hölder inequality | $\Delta$-Hölder inequality]]
+[[Partial Delta Dynamic Equations | Partial $\Delta$ dynamic equations]]<br />
-*[[delta_integral | $\Delta$-integral]]
-*[[Delta Jensen inequality | $\Delta$-Jensen inequality]]
-*[[Delta Lyapunov inequality | $\Delta$-Lyapunov inequality]]
-*[[Delta Markov inequality | $\Delta$-Markov inequality]]
-*[[Delta Minkowski inequality | $\Delta$-Minkowski inequality]]
-*[[Delta Opial inequality | $\Delta$-Opial inequality]]
-*[[Delta Taylor's formula|$\Delta$-Taylor's formula]]
-*[[Delta Tschebycheff inequality | $\Delta$-Tschebycheff inequality]]
-*[[Delta Wirtinger inequality | $\Delta$-Wirtinger inequality]]
-*[[Delta wave equation | $\Delta$ wave equation]]
-*[[Directional Delta Derivative | Directional $\Delta$ derivative]]
-*[[Partial Delta Derivative | Partial $\Delta$ derivative]]
-*[[Partial Delta Dynamic Equations | Partial $\Delta$ dynamic equations]]
 ==$\nabla$-calculus==
-*[[nabla_derivative | $\nabla$-derivative]]
+[[nabla_derivative | $\nabla$-derivative]]<br />
-*[[nabla integral | $\nabla$-integral]]
+[[nabla integral | $\nabla$-integral]]<br />
 ==$\Diamond_{\alpha}$-calculus==
-*[[diamond alpha derivative | $\Diamond_{\alpha}$-derivative]]
+[[diamond alpha derivative | $\Diamond_{\alpha}$-derivative]]<br />
-*[[diamond alpha holder inequality | $\Diamond_{\alpha}$-Hölder inequality ]]
+[[diamond alpha holder inequality | $\Diamond_{\alpha}$-Hölder inequality ]]<br />
-*[[diamond alpha Jensen's inequality | $\Diamond_{\alpha}$-Jensen's inequality]]
+[[diamond alpha Jensen's inequality | $\Diamond_{\alpha}$-Jensen's inequality]]<br />
-*[[diamond alpha Minkowski's inequality | $\Diamond_{\alpha}$-Minkowski's inequality]]
+[[diamond alpha Minkowski's inequality | $\Diamond_{\alpha}$-Minkowski's inequality]]<br />
-*[[diamond integral | $\Diamond$-integral]]
+[[diamond integral | $\Diamond$-integral]]<br />
 ==Probability Theory==
 *[[Cumulant generating function]]
 *[[Cumulative distribution function]]
-*[[Moments]]
 *[[Probability density function]]
 *[[Joint time scales probability density function]]
@@ Line 101: / Line 82: @@
 =Special functions on time scales=
-*[[Gamma function]]
+[[Delta cpq|$\mathrm{c}_{pq}$]]<br />
-*[[hyperbolic_functions | Hyperbolic functions]]
+[[Delta chpq|$\mathrm{ch}_{pq}$]]<br />
-*[[Euler-Cauchy logarithm]]
+[[Delta spq|$\mathrm{s}_{pq}$]]<br />
-*[[Bohner logarithm]]
+[[Delta shpq|$\mathrm{sh}_{pq}$]]<br />
-*[[Jackson logarithm]]
+[[Gamma function]]<br />
-*[[Mozyrska-Torres logarithm]]
+[[Euler-Cauchy logarithm]]<br />
-*[[gaussian_bell | Gaussian bell]]
+[[Bohner logarithm]]<br />
+[[Jackson logarithm]]<br />
-==$\Delta$-calculus==
+[[Mozyrska-Torres logarithm]]<br />
-<center>{{:Delta special functions footer}}</center>
+[[gaussian_bell | Gaussian bell]]<br />
+[[Uniform distribution]]<br />
+[[Exponential distribution]]<br />
+[[Gamma distribution]]<br />
-*[[Delta hk | $\Delta h_k$]]
-*[[Delta gk | $\Delta g_k$]]
 ==$\nabla$-calculus==
-*[[Nabla cosine | $\nabla \widehat{\cos}_p$]]
+[[Nabla cosine | $\nabla \widehat{\cos}_p$]]<br />
-*[[Nabla cosh | $\nabla \widehat{\cosh}_p$]]
+[[Nabla cosh | $\nabla \widehat{\cosh}_p$]]<br />
-*[[Nabla exponential | $\nabla \widehat{\exp}$]]
+[[Nabla exponential | $\nabla \widehat{\exp}$]]<br />
-*[[Nabla hk|$\nabla \hat{h}_k$]]
+[[Nabla hk|$\nabla \hat{h}_k$]]<br />
-*[[Nabla gk|$\nabla \hat{g}_k$]]
+[[Nabla gk|$\nabla \hat{g}_k$]]<br />
-*[[Nabla sine | $\nabla \widehat{\sin}_p$]]
+[[Nabla sine | $\nabla \widehat{\sin}_p$]]<br />
-*[[Nabla sinh | $\nabla \widehat{\sinh}_p$]]
+[[Nabla sinh | $\nabla \widehat{\sinh}_p$]]<br />
-==$\Diamond_{\alpha}$-calculus==
-*[[Diamond alpha cosine | $\Diamond_{\alpha}$-$\cos_p$]]
-*[[Diamond alpha cosh | $\Diamond_{\alpha}$-$\cosh_p$]]
-*[[Diamond exponential | $\Diamond_{\alpha}$-$e_p$]]
-*[[Diamond sine | $\Diamond$-$\sin_p$]]
-*[[Diamond sinh | $\Diamond$-$\sinh_p$]]
-==Probability Distributions on time scales==
-*[[Uniform distribution]]
-*[[Exponential distribution]]
-*[[Gamma distribution]]
-=Differential equations=
-[[Hypergeometric differential equation]] <br />
-[[Confluent hypergeometric differential equation]]<br />

Difference between revisions of "Main Page"

Latest revision as of 01:55, 6 February 2023

Contents

Time scales calculus

Examples of time scales

$\Delta$-special functions on time scales

Hilger complex plane and friends

$\Delta$-Inequalities

$\Delta$-calculus

$\nabla$-calculus

$\Diamond_{\alpha}$-calculus

Probability Theory

Examples of time scales

Special functions on time scales

$\nabla$-calculus

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