Difference between revisions of "Main Page"

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($\Delta$-calculus)
($\nabla$-calculus)
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===$\nabla$-calculus===
 
===$\nabla$-calculus===
*[[Nabla cosine | $\nabla$-$\cos_p$]]
+
*[[Nabla cosine | $\nabla$-$\widehat{\cos}_p$]]
*[[Nabla cosh | $\nabla$-$\cosh_p$]]
+
*[[Nabla cosh | $\nabla$-$\widehat{\cosh}_p$]]
 
*[[Nabla exponential | $\nabla$-$\hat{e}_p$]]
 
*[[Nabla exponential | $\nabla$-$\hat{e}_p$]]
*[[Nabla sine | $\nabla$-$\sin_p$]]
+
*[[Nabla sine | $\nabla$-$\widehat{\sin}_p$]]
*[[Nabla sinh | $\nabla$-$\sinh_p$]]
+
*[[Nabla sinh | $\nabla$-$\widehat{\sinh}_p$]]
  
 
===$\Diamond_{\alpha}$-calculus===
 
===$\Diamond_{\alpha}$-calculus===

Revision as of 07:47, 23 March 2015

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

List of time scales

$\Delta$-calculus

$\nabla$-calculus

$\Diamond_{\alpha}$-calculus

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: $\mathbb{T}=\{\ldots, t_{-1}, t_{0}, t_1, \ldots\}$

Inequalities

Special functions on time scales

$\Delta$-calculus

$\nabla$-calculus

$\Diamond_{\alpha}$-calculus

Probability Distributions on time scales