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($\Diamond_{\alpha}$-calculus)
(Special functions on time scales)
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=Special functions on time scales=
 
=Special functions on time scales=
*[[Delta cpq|$\mathrm{c}_{pq}$]]
+
[[Delta cpq|$\mathrm{c}_{pq}$]]<br />
*[[Delta chpq|$\mathrm{ch}_{pq}$]]
+
[[Delta chpq|$\mathrm{ch}_{pq}$]]<br />
*[[Delta spq|$\mathrm{s}_{pq}$]]
+
[[Delta spq|$\mathrm{s}_{pq}$]]<br />
*[[Delta shpq|$\mathrm{sh}_{pq}$]]
+
[[Delta shpq|$\mathrm{sh}_{pq}$]]<br />
*[[Gamma function]]
+
[[Gamma function]]<br />
*[[hyperbolic_functions | Hyperbolic functions]]
+
[[hyperbolic_functions | Hyperbolic functions]]<br />
*[[Euler-Cauchy logarithm]]
+
[[Euler-Cauchy logarithm]]<br />
*[[Bohner logarithm]]
+
[[Bohner logarithm]]<br />
*[[Jackson logarithm]]
+
[[Jackson logarithm]]<br />
*[[Mozyrska-Torres logarithm]]
+
[[Mozyrska-Torres logarithm]]<br />
*[[gaussian_bell | Gaussian bell]]
+
[[gaussian_bell | Gaussian bell]]<br />
  
 
==$\nabla$-calculus==
 
==$\nabla$-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==
 
==$\Diamond_{\alpha}$-calculus==
*[[Diamond alpha cosine | $\Diamond_{\alpha}$-$\cos_p$]]
+
[[Diamond alpha cosine | $\Diamond_{\alpha}$-$\cos_p$]]<br />
*[[Diamond alpha cosh | $\Diamond_{\alpha}$-$\cosh_p$]]
+
[[Diamond alpha cosh | $\Diamond_{\alpha}$-$\cosh_p$]]<br />
*[[Diamond exponential | $\Diamond_{\alpha}$-$e_p$]]
+
[[Diamond exponential | $\Diamond_{\alpha}$-$e_p$]]<br />
*[[Diamond sine | $\Diamond$-$\sin_p$]]
+
[[Diamond sine | $\Diamond$-$\sin_p$]]<br />
*[[Diamond sinh | $\Diamond$-$\sinh_p$]]
+
[[Diamond sinh | $\Diamond$-$\sinh_p$]]<br />
  
 
==Probability Distributions on time scales==
 
==Probability Distributions on time scales==
*[[Uniform distribution]]
+
[[Uniform distribution]]<br />
*[[Exponential distribution]]
+
[[Exponential distribution]]<br />
*[[Gamma distribution]]
+
[[Gamma distribution]]<br />
  
 
=Differential equations=
 
=Differential equations=
 
[[Hypergeometric differential equation]] <br />
 
[[Hypergeometric differential equation]] <br />
 
[[Confluent hypergeometric differential equation]]<br />
 
[[Confluent hypergeometric differential equation]]<br />

Revision as of 01:07, 23 August 2016

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.

How to get access to edit this wiki

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

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
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

Differential equations

Hypergeometric differential equation
Confluent hypergeometric differential equation