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.
@@ Line 8: / Line 8: @@
 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=
+=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]]<br />
 [[Bilateral Laplace transform]]<br />
+[[Unilateral Laplace transform]]<br />
 [[Cauchy function]]<br />
-[[Calculus of variations]]<br />
 [[Chain rule]]<br />
-[[Convolution]]<br />
+[[Unilateral convolution]]<br />
 [[Dense point]]<br />
 [[Disconjugate]]<br />
 [[Dynamic equation]]<br />
-[[Circle minus]]<br />
+[[Forward circle minus]]<br />
-[[Circle plus]]<br />
+[[Backward circle minus]]<br />
-[[complex_calculus | Complex calculus on time scales]]<br />
+[[Forward circle plus]]<br />
+[[Backward circle plus]]<br />
 [[Convergence of time scales]]<br />
 [[Dilation of time scales]]<br />
@@ Line 29: / Line 31: @@
 [[Fractional calculus]]<br />
 [[Frequency roots]]<br />
-[[Function spaces]]<br />
 [[Generalized square]]<br />
 [[Generalized zero]]<br />
-[[Hilger alternating axis]]<br />
-[[Hilger circle]]<br />
-[[Hilger complex plane]]<br />
-[[Hilger imaginary part]]<br />
-[[Hilger pure imaginary]]<br />
-[[Hilger real axis]]<br />
-[[Hilger real part]]<br />
 [[Induction on time scales]]<br />
-[[Laplace transform]]<br />
 [[L'Hospital's Rule]]<br />
-[[Mean value theorem]]<br />
+[[First mean value theorem]]<br />
 [[Pre-differentiable]]<br />
-[[Fourier transform]]<br />
+[[Marks-Gravagne-Davis Fourier transform]]<br />
+[[Cuchta-Georgiev Fourier transform]]<br />
 [[rd-continuous]]<br />
-[[Regressive function]]<br />
+[[Forward regressive function]]<br />
 [[Regulated function]]<br />
 [[Riccati equation]]<br />
-[[Riesz representation theorem]]<br />
 [[Scattered point]]<br />
 [[Self-adjoint]]<br />
 [[Shifting problem]]<br />
-[[Substitution]]<br />
 [[Variation of parameters]]<br />
 [[Wronskian]]<br />
 ==$\Delta$-calculus==
-[[Completely delta differentiable]]<br />
-[[Delta Bernoulli inequality | $\Delta$-Bernoulli inequality]]<br />
-[[Delta Bihari inequality | $\Delta$-Bihari inequality]]<br />
-[[Delta Cauchy-Schwarz inequality | $\Delta$-Cauchy-Schwarz inequality]]<br />
 [[delta_derivative | $\Delta$-derivative]]<br />
-[[Delta Gronwall inequality | $\Delta$-Gronwall inequality]]<br />
 [[Delta heat equation | $\Delta$ heat equation]]<br />
-[[Delta Hölder inequality | $\Delta$-Hölder inequality]]<br />
 [[delta_integral | $\Delta$-integral]]<br />
-[[Delta Jensen inequality | $\Delta$-Jensen inequality]]<br />
-[[Delta Lyapunov inequality | $\Delta$-Lyapunov inequality]]<br />
-[[Delta Markov inequality | $\Delta$-Markov inequality]]<br />
-[[Delta Minkowski inequality | $\Delta$-Minkowski inequality]]<br />
-[[Delta Opial inequality | $\Delta$-Opial inequality]]<br />
 [[Delta Taylor's formula|$\Delta$-Taylor's formula]]<br />
-[[Delta Tschebycheff inequality | $\Delta$-Tschebycheff inequality]]<br />
-[[Delta Wirtinger inequality | $\Delta$-Wirtinger inequality]]<br />
 [[Delta wave equation | $\Delta$ wave equation]]<br />
 [[Directional Delta Derivative | Directional $\Delta$ derivative]]<br />
@@ Line 94: / Line 73: @@
 *[[Cumulant generating function]]
 *[[Cumulative distribution function]]
-*[[Moments]]
 *[[Probability density function]]
 *[[Joint time scales probability density function]]
@@ Line 109: / Line 87: @@
 [[Delta shpq|$\mathrm{sh}_{pq}$]]<br />
 [[Gamma function]]<br />
-[[hyperbolic_functions | Hyperbolic functions]]<br />
 [[Euler-Cauchy logarithm]]<br />
 [[Bohner logarithm]]<br />
@@ Line 115: / Line 92: @@
 [[Mozyrska-Torres logarithm]]<br />
 [[gaussian_bell | Gaussian bell]]<br />
+[[Uniform distribution]]<br />
+[[Exponential distribution]]<br />
+[[Gamma distribution]]<br />
 ==$\nabla$-calculus==
@@ Line 124: / Line 105: @@
 [[Nabla sine | $\nabla \widehat{\sin}_p$]]<br />
 [[Nabla sinh | $\nabla \widehat{\sinh}_p$]]<br />
-==$\Diamond_{\alpha}$-calculus==
-[[Diamond alpha cosine | $\Diamond_{\alpha}$-$\cos_p$]]<br />
-[[Diamond alpha cosh | $\Diamond_{\alpha}$-$\cosh_p$]]<br />
-[[Diamond exponential | $\Diamond_{\alpha}$-$e_p$]]<br />
-[[Diamond sine | $\Diamond$-$\sin_p$]]<br />
-[[Diamond sinh | $\Diamond$-$\sinh_p$]]<br />
-==Probability Distributions on time scales==
-[[Uniform distribution]]<br />
-[[Exponential distribution]]<br />
-[[Gamma distribution]]<br />
-=Special cases of time scales=
-==Differential equations==
-[[Hypergeometric differential equation]] <br />
-[[Confluent hypergeometric differential equation]]<br />
-==Difference equations==
-[[Difference equation of hypergeometric type]]<br />
-==$q$-difference equations==
-[[q-difference equation of hypergeometric type]]<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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