Difference between revisions of "Main Page"
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*[[logarithms | Logarithms]] | *[[logarithms | Logarithms]] | ||
*[[Nabla cosine | $\nabla$-$\cos_p$]] | *[[Nabla cosine | $\nabla$-$\cos_p$]] | ||
| − | *[[Nabla exponential | $\nabla$- | + | *[[Nabla exponential | $\nabla$-$\hat{e}_p$]] |
*[[Nabla sine | $\nabla$-$\sin_p$]] | *[[Nabla sine | $\nabla$-$\sin_p$]] | ||
*[[trig_functions | Trigonometric functions]] | *[[trig_functions | Trigonometric functions]] | ||
*[[gaussian_bell | Gaussian bell]] | *[[gaussian_bell | Gaussian bell]] | ||
Revision as of 02:36, 21 October 2014
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.
Contents
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
- Abel's theorem
- List of time scales
- $\Delta$-derivative
- $\nabla$-derivative
- $\Diamond$-derivative
- $\Delta$-integral
- $\nabla$-integral
- $\Diamond$-integral
- Dynamic Equations
- Complex calculus on time scales
- Convergence of time scales
- Function spaces
- Laplace transform
- Mean value theorem
- Fourier transform
- Regressive function
- Variation of parameters
- Wronskian
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: $\mathbb{T}=\{\ldots, t_{-1}, t_{0}, t_1, \ldots\}$
Inequalities
- Bernoulli inequality
- Bihari inequality
- Cauchy-Schwarz inequality
- Gronwall inequality
- Hölder inequality
- Jensen inequality
- Lyapunov inequality
- Minkowski inequality
- Opial inequality
- Wirtinger inequality
