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(Calculus on time scales)
(Special functions on time scales)
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==Special functions on time scales==
 
==Special functions on time scales==
 
*[[polynomials | Polynomials]]
 
*[[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 cosine | $\Delta$-$\cos_p$]]
 
*[[Delta cosh | $\Delta$-$\cosh_p$]]
 
*[[Delta cosh | $\Delta$-$\cosh_p$]]
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*[[Delta sine | $\Delta$-$\sin_p$]]
 
*[[Delta sine | $\Delta$-$\sin_p$]]
 
*[[Delta sinh | $\Delta$-$\sinh_p$]]
 
*[[Delta sinh | $\Delta$-$\sinh_p$]]
*[[Gamma function]]
+
 
*[[hyperbolic_functions | Hyperbolic functions]]
+
===$\nabla$-calculus===
*[[logarithms | Logarithms]]
 
 
*[[Nabla cosine | $\nabla$-$\cos_p$]]
 
*[[Nabla cosine | $\nabla$-$\cos_p$]]
 
*[[Nabla cosh | $\nabla$-$\cosh_p$]]
 
*[[Nabla cosh | $\nabla$-$\cosh_p$]]
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*[[Nabla sine | $\nabla$-$\sin_p$]]
 
*[[Nabla sine | $\nabla$-$\sin_p$]]
 
*[[Nabla sinh | $\nabla$-$\sinh_p$]]
 
*[[Nabla sinh | $\nabla$-$\sinh_p$]]
*[[trig_functions | Trigonometric functions]]
+
 
*[[gaussian_bell | Gaussian bell]]
+
===$\Diamond_{\alpha}$-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===
 
===Probability Distributions on time scales===

Revision as of 08:10, 10 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 called a time scale $\mathbb{T}$. When $\mathbb{T}=\mathbb{R}$ the resulting theory is differential calculus but when $\mathbb{T}=\mathbb{Z}$ the resulting theory is difference calculus. Time scales also include any closed subset of $\mathbb{R}$, so more exotic sets such as the Cantor set are also considered in the theory.

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Calculus on time scales

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