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>
-[[time_scale | List of time scales]]
+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.
-*[[Abel's theorem]]
+=Time scales calculus=
-*[[Calculus of variations]]
+<center>{{:Time scales footer}}</center>
-*[[dynamic_equations | Dynamic Equations]]
+<center>{{:Delta special functions footer}}</center>
-*[[complex_calculus | Complex calculus on time scales]]
+<center>{{:Hilger complex plane footer}}</center>
-*[[Convergence of time scales]]
+{{:Delta inequalities footer}}
-*[[Fractional calculus]]
-*[[Function spaces]]
-*[[Laplace transform]]
-*[[L'Hospital's Rule]]
-*[[Mean value theorem]]
-*[[Fourier transform]]
-*[[Regressive function]]
-*[[Taylor's formula]]
-*[[Variation of parameters]]
-*[[Wronskian]]
-===$\Delta$-calculus===
+[[Bilateral Laplace transform]]<br />
-*[[Delta Bernoulli inequality | $\Delta$-Bernoulli inequality]]
+[[Unilateral Laplace transform]]<br />
-*[[Delta Bihari inequality | $\Delta$-Bihari inequality]]
+[[Cauchy function]]<br />
-*[[Delta Cauchy-Schwarz inequality | $\Delta$-Cauchy-Schwarz inequality]]
+[[Chain rule]]<br />
-*[[delta_derivative | $\Delta$-derivative]]
+[[Unilateral convolution]]<br />
-*[[Delta Gronwall inequality | $\Delta$-Gronwall inequality]]
+[[Dense point]]<br />
-*[[Delta Hölder inequality | $\Delta$-Hölder inequality]]
+[[Disconjugate]]<br />
-*[[delta_integral | $\Delta$-integral]]
+[[Dynamic equation]]<br />
-*[[Delta Jensen inequality | $\Delta$-Jensen inequality]]
+[[Forward circle minus]]<br />
-*[[Delta Lyapunov inequality | $\Delta$-Lyapunov inequality]]
+[[Backward circle minus]]<br />
-*[[Delta Markov inequality | $\Delta$-Markov inequality]]
+[[Forward circle plus]]<br />
-*[[Delta Minkowski inequality | $\Delta$-Minkowski inequality]]
+[[Backward circle plus]]<br />
-*[[Delta Opial inequality | $\Delta$-Opial inequality]]
+[[Convergence of time scales]]<br />
-*[[Delta Tschebycheff inequality | $\Delta$-Tschebycheff inequality]]
+[[Dilation of time scales]]<br />
-*[[Delta Wirtinger inequality | $\Delta$-Wirtinger inequality]]
+[[Duality of delta and nabla | Duality of $\Delta$ and $\nabla$]]<br />
+[[Fractional calculus]]<br />
+[[Frequency roots]]<br />
+[[Generalized square]]<br />
+[[Generalized zero]]<br />
+[[Induction on time scales]]<br />
+[[L'Hospital's Rule]]<br />
+[[First mean value theorem]]<br />
+[[Pre-differentiable]]<br />
+[[Marks-Gravagne-Davis Fourier transform]]<br />
+[[Cuchta-Georgiev Fourier transform]]<br />
+[[rd-continuous]]<br />
+[[Forward regressive function]]<br />
+[[Regulated function]]<br />
+[[Riccati equation]]<br />
+[[Scattered point]]<br />
+[[Self-adjoint]]<br />
+[[Shifting problem]]<br />
+[[Variation of parameters]]<br />
+[[Wronskian]]<br />
-===$\nabla$-calculus===
+==$\Delta$-calculus==
-*[[nabla_derivative | $\nabla$-derivative]]
+[[delta_derivative | $\Delta$-derivative]]<br />
-*[[nabla integral | $\nabla$-integral]]
+[[Delta heat equation | $\Delta$ heat equation]]<br />
+[[delta_integral | $\Delta$-integral]]<br />
+[[Delta Taylor's formula|$\Delta$-Taylor's formula]]<br />
+[[Delta wave equation | $\Delta$ wave equation]]<br />
+[[Directional Delta Derivative | Directional $\Delta$ derivative]]<br />
+[[Partial Delta Derivative | Partial $\Delta$ derivative]]<br />
+[[Partial Delta Dynamic Equations | Partial $\Delta$ dynamic equations]]<br />
-===$\Diamond_{\alpha}$-calculus===
+==$\nabla$-calculus==
-*[[diamond alpha derivative | $\Diamond_{\alpha}$-derivative]]
+[[nabla_derivative | $\nabla$-derivative]]<br />
-*[[diamond alpha holder inequality | $\Diamond_{\alpha}$-Hölder inequality ]]
+[[nabla integral | $\nabla$-integral]]<br />
-*[[diamond alpha Jensen's inequality | $\Diamond_{\alpha}$-Jensen's inequality]]
-*[[diamond alpha Minkowski's inequality | $\Diamond_{\alpha}$-Minkowski's inequality]]
-*[[diamond integral | $\Diamond$-integral]]
-===Probability Theory===
+==$\Diamond_{\alpha}$-calculus==
+[[diamond alpha derivative | $\Diamond_{\alpha}$-derivative]]<br />
+[[diamond alpha holder inequality | $\Diamond_{\alpha}$-Hölder inequality ]]<br />
+[[diamond alpha Jensen's inequality | $\Diamond_{\alpha}$-Jensen's inequality]]<br />
+[[diamond alpha Minkowski's inequality | $\Diamond_{\alpha}$-Minkowski's inequality]]<br />
+[[diamond integral | $\Diamond$-integral]]<br />
+==Probability Theory==
 *[[Cumulant generating function]]
 *[[Cumulative distribution function]]
-*[[Moments]]
 *[[Probability density function]]
 *[[Joint time scales probability density function]]
@@ Line 62: / Line 79: @@
 *[[Variance]]
-==Examples of time scales==
+{{:Examples of time scales}}
-# The real line: [[Real_numbers | $\mathbb{R}$]]
-# The integers: [[Integers | $\mathbb{Z} = \{\ldots, -1,0,1,\ldots\}$]]
-# Multiples of integers: [[Multiples_of_integers | $h\mathbb{Z} = \{ht \colon t \in \mathbb{Z}\}$]]
-# Quantum numbers ($q>1$): [[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}$]]
-# Quantum numbers ($q<1$): [[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}$]]
-# Square integers: [[Square_integers | $\mathbb{Z}^2 = \{t^2 \colon t \in \mathbb{Z} \}$]]
-# Harmonic numbers: [[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: [[Closure_of_unit_fractions | $\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$]]
-# Isolated points: [[Isolated_points | $\mathbb{T}=\{\ldots, t_{-1}, t_{0}, t_1, \ldots\}$]]
-==Inequalities==
-*[[Bihari inequality]]
-*[[Cauchy-Schwarz inequality]]
-*[[Gronwall inequality]]
-*[[Hölder inequality]]
-*[[Jensen inequality]]
-*[[Lyapunov inequality]]
-*[[Markov inequality]]
-*[[Minkowski inequality]]
-*[[Opial inequality]]
-*[[Tschebycheff inequality]]
-*[[Wirtinger inequality]]
-==Special functions on time scales==
-*[[polynomials | Polynomials]]
-*[[Gamma function]]
-*[[hyperbolic_functions | Hyperbolic functions]]
-*[[logarithms | Logarithms]]
-*[[trig_functions | Trigonometric functions]]
-*[[gaussian_bell | Gaussian bell]]
-===$\Delta$-calculus===
-*[[Delta cosine | $\Delta$-$\cos_p$]]
-*[[Delta cosh | $\Delta$-$\cosh_p$]]
-*[[Delta exponential | $\Delta$-$e_p$]]
-*[[Delta hk | $\Delta$-$h_k$]]
-*[[Delta gk | $\Delta$-$g_k$]]
-*[[Delta sine | $\Delta$-$\sin_p$]]
-*[[Delta sinh | $\Delta$-$\sinh_p$]]
-===$\nabla$-calculus===
+=Special functions on time scales=
-*[[Nabla cosine | $\nabla$-$\widehat{\cos}_p$]]
+[[Delta cpq|$\mathrm{c}_{pq}$]]<br />
-*[[Nabla cosh | $\nabla$-$\widehat{\cosh}_p$]]
+[[Delta chpq|$\mathrm{ch}_{pq}$]]<br />
-*[[Nabla exponential | $\nabla$-$\hat{e}_p$]]
+[[Delta spq|$\mathrm{s}_{pq}$]]<br />
-*[[Nabla hk|$\nabla$-$\hat{h}_k$]]
+[[Delta shpq|$\mathrm{sh}_{pq}$]]<br />
-*[[Nabla gk|$\nabla$-$\hat{g}_k$]]
+[[Gamma function]]<br />
-*[[Nabla sine | $\nabla$-$\widehat{\sin}_p$]]
+[[Euler-Cauchy logarithm]]<br />
-*[[Nabla sinh | $\nabla$-$\widehat{\sinh}_p$]]
+[[Bohner logarithm]]<br />
+[[Jackson logarithm]]<br />
+[[Mozyrska-Torres logarithm]]<br />
+[[gaussian_bell | Gaussian bell]]<br />
+[[Uniform distribution]]<br />
+[[Exponential distribution]]<br />
+[[Gamma distribution]]<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===
+==$\nabla$-calculus==
-*[[Uniform distribution]]
+[[Nabla cosine | $\nabla \widehat{\cos}_p$]]<br />
-*[[Exponential distribution]]
+[[Nabla cosh | $\nabla \widehat{\cosh}_p$]]<br />
-*[[Gamma distribution]]
+[[Nabla exponential | $\nabla \widehat{\exp}$]]<br />
+[[Nabla hk|$\nabla \hat{h}_k$]]<br />
+[[Nabla gk|$\nabla \hat{g}_k$]]<br />
+[[Nabla sine | $\nabla \widehat{\sin}_p$]]<br />
+[[Nabla sinh | $\nabla \widehat{\sinh}_p$]]<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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