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(Calculus on time scales)
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*[[Completely delta differentiable]]
[[Completely delta differentiable]]<br />
*[[Delta Bernoulli inequality | $\Delta$-Bernoulli inequality]]
[[Delta Bernoulli inequality | $\Delta$-Bernoulli inequality]]<br />
*[[Delta Bihari inequality | $\Delta$-Bihari inequality]]
[[Delta Bihari inequality | $\Delta$-Bihari inequality]]<br />
*[[Delta Cauchy-Schwarz inequality | $\Delta$-Cauchy-Schwarz inequality]]
[[Delta Cauchy-Schwarz inequality | $\Delta$-Cauchy-Schwarz inequality]]<br />
*[[delta_derivative | $\Delta$-derivative]]
[[delta_derivative | $\Delta$-derivative]]<br />
*[[Delta Gronwall inequality | $\Delta$-Gronwall inequality]]
[[Delta Gronwall inequality | $\Delta$-Gronwall inequality]]<br />
*[[Delta heat equation | $\Delta$ heat equation]]
[[Delta heat equation | $\Delta$ heat equation]]<br />
*[[Delta Hölder inequality | $\Delta$-Hölder inequality]]
[[Delta Hölder inequality | $\Delta$-Hölder inequality]]<br />
*[[delta_integral | $\Delta$-integral]]
[[delta_integral | $\Delta$-integral]]<br />
*[[Delta Jensen inequality | $\Delta$-Jensen inequality]]
[[Delta Jensen inequality | $\Delta$-Jensen inequality]]<br />
*[[Delta Lyapunov inequality | $\Delta$-Lyapunov inequality]]
[[Delta Lyapunov inequality | $\Delta$-Lyapunov inequality]]<br />
*[[Delta Markov inequality | $\Delta$-Markov inequality]]
[[Delta Markov inequality | $\Delta$-Markov inequality]]<br />
*[[Delta Minkowski inequality | $\Delta$-Minkowski inequality]]
[[Delta Minkowski inequality | $\Delta$-Minkowski inequality]]<br />
*[[Delta Opial inequality | $\Delta$-Opial inequality]]
[[Delta Opial inequality | $\Delta$-Opial inequality]]<br />
*[[Delta Taylor's formula|$\Delta$-Taylor's formula]]
[[Delta Taylor's formula|$\Delta$-Taylor's formula]]<br />
*[[Delta Tschebycheff inequality | $\Delta$-Tschebycheff inequality]]
[[Delta Tschebycheff inequality | $\Delta$-Tschebycheff inequality]]<br />
*[[Delta Wirtinger inequality | $\Delta$-Wirtinger inequality]]
[[Delta Wirtinger inequality | $\Delta$-Wirtinger inequality]]<br />
*[[Delta wave equation | $\Delta$ wave equation]]
[[Delta wave equation | $\Delta$ wave equation]]<br />
*[[Directional Delta Derivative | Directional $\Delta$ derivative]]
[[Directional Delta Derivative | Directional $\Delta$ derivative]]<br />
*[[Partial Delta Derivative | Partial $\Delta$ derivative]]
[[Partial Delta Derivative | Partial $\Delta$ derivative]]<br />
*[[Partial Delta Dynamic Equations | Partial $\Delta$ dynamic equations]]
[[Partial Delta Dynamic Equations | Partial $\Delta$ dynamic equations]]<br />

Revision as of 01:06, 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

Real numbers
Multiples of integers
Square integers
Harmonic numbers
Isolated points
nth root numbers
Evenly spaced intervals
Quantum, $q>1$
Quantum, $q<1$
$\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$
Closure of unit fractions
Cantor set
$\Delta$-special functions on time scales








Abel's theorem
Bilateral Laplace transform
Cauchy function
Calculus of variations
Chain rule
Dense point
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
Fourier transform
Regressive function
Regulated function
Riccati equation
Riesz representation theorem
Scattered point
Shifting problem
Variation of parameters


Completely delta differentiable
$\Delta$-Bernoulli inequality
$\Delta$-Bihari inequality
$\Delta$-Cauchy-Schwarz inequality
$\Delta$-Gronwall inequality
$\Delta$ heat equation
$\Delta$-Hölder inequality
$\Delta$-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



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



Probability Distributions on time scales

Differential equations

Hypergeometric differential equation
Confluent hypergeometric differential equation