timescalewiki timescalewiki http://timescalewiki.org/index.php/Main_Page MediaWiki 1.28.0 first-letter Media Special Talk User User talk Timescalewiki Timescalewiki talk File File talk MediaWiki MediaWiki talk Template Template talk Help Help talk Category Category talk Main Page 0 1 1791 1788 2018-02-26T03:30:39Z Tom 1 wikitext text/x-wiki This wiki is a resource for people who do research in <strong>time scale calculus</strong>. 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 [[Real_numbers | $\mathbb{T}=\mathbb{R}$]] the resulting theory becomes [http://en.wikipedia.org/wiki/Differential_calculus differential calculus] but when [[Multiples_of_integers | $\mathbb{T}=\mathbb{Z}$]] the resulting theory becomes [http://en.wikipedia.org/wiki/Difference_calculus difference calculus]. Time scales also include any closed subset of $\mathbb{R}$, so more exotic sets such as the [http://en.wikipedia.org/wiki/Cantor_set 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 | $\Delta$-derivative]] to classical differentiation or to taking a forward difference. See the Python library [https://github.com/tomcuchta/timescalecalculus timescalecalculus] on GitHub and its [[timescalecalculus python library documentation|documentation]]. <b><u>Registration</u></b> Due to a resurgence of automated spam bots, account registration and anonymous editing is currently disabled. Please contact Tom Cuchta (tomcuchta@gmail.com) to gain access to edit the wiki. =Calculus on time scales= <center>{{:Time scales footer}}</center> <center>{{:Delta special functions footer}}</center> [[Abel's theorem]]<br /> [[Bilateral Laplace transform]]<br /> [[Cauchy function]]<br /> [[Calculus of variations]]<br /> [[Chain rule]]<br /> [[Convolution]]<br /> [[Dense point]]<br /> [[Disconjugate]]<br /> [[Dynamic equation]]<br /> [[Circle minus]]<br /> [[Circle plus]]<br /> [[complex_calculus | Complex calculus on time scales]]<br /> [[Convergence of time scales]]<br /> [[Dilation of time scales]]<br /> [[Duality of delta and nabla | Duality of $\Delta$ and $\nabla$]]<br /> [[Fractional calculus]]<br /> [[Frequency roots]]<br /> [[Function spaces]]<br /> [[Generalized square]]<br /> [[Generalized zero]]<br /> [[Hilger alternating axis]]<br /> [[Hilger circle]]<br /> [[Hilger complex plane]]<br /> [[Hilger imaginary part]]<br /> [[Hilger pure imaginary]]<br /> [[Hilger real axis]]<br /> [[Hilger real part]]<br /> [[Induction on time scales]]<br /> [[Laplace transform]]<br /> [[L'Hospital's Rule]]<br /> [[Mean value theorem]]<br /> [[Pre-differentiable]]<br /> [[Fourier transform]]<br /> [[rd-continuous]]<br /> [[Regressive function]]<br /> [[Regulated function]]<br /> [[Riccati equation]]<br /> [[Riesz representation theorem]]<br /> [[Scattered point]]<br /> [[Self-adjoint]]<br /> [[Shifting problem]]<br /> [[Substitution]]<br /> [[Variation of parameters]]<br /> [[Wronskian]]<br /> ==$\Delta$-calculus== [[Completely delta differentiable]]<br /> [[Delta Bernoulli inequality | $\Delta$-Bernoulli inequality]]<br /> [[Delta Bihari inequality | $\Delta$-Bihari inequality]]<br /> [[Delta Cauchy-Schwarz inequality | $\Delta$-Cauchy-Schwarz inequality]]<br /> [[delta_derivative | $\Delta$-derivative]]<br /> [[Delta Gronwall inequality | $\Delta$-Gronwall inequality]]<br /> [[Delta heat equation | $\Delta$ heat equation]]<br /> [[Delta Hölder inequality | $\Delta$-Hölder inequality]]<br /> [[delta_integral | $\Delta$-integral]]<br /> [[Delta Jensen inequality | $\Delta$-Jensen inequality]]<br /> [[Delta Lyapunov inequality | $\Delta$-Lyapunov inequality]]<br /> [[Delta Markov inequality | $\Delta$-Markov inequality]]<br /> [[Delta Minkowski inequality | $\Delta$-Minkowski inequality]]<br /> [[Delta Opial inequality | $\Delta$-Opial inequality]]<br /> [[Delta Taylor's formula|$\Delta$-Taylor's formula]]<br /> [[Delta Tschebycheff inequality | $\Delta$-Tschebycheff inequality]]<br /> [[Delta Wirtinger inequality | $\Delta$-Wirtinger inequality]]<br /> [[Delta wave equation | $\Delta$ wave equation]]<br /> [[Directional Delta Derivative | Directional $\Delta$ derivative]]<br /> [[Partial Delta Derivative | Partial $\Delta$ derivative]]<br /> [[Partial Delta Dynamic Equations | Partial $\Delta$ dynamic equations]]<br /> ==$\nabla$-calculus== [[nabla_derivative | $\nabla$-derivative]]<br /> [[nabla integral | $\nabla$-integral]]<br /> ==$\Diamond_{\alpha}$-calculus== [[diamond alpha derivative | $\Diamond_{\alpha}$-derivative]]<br /> [[diamond alpha holder inequality | $\Diamond_{\alpha}$-Hölder inequality ]]<br /> [[diamond alpha Jensen's inequality | $\Diamond_{\alpha}$-Jensen's inequality]]<br /> [[diamond alpha Minkowski's inequality | $\Diamond_{\alpha}$-Minkowski's inequality]]<br /> [[diamond integral | $\Diamond$-integral]]<br /> ==Probability Theory== *[[Cumulant generating function]] *[[Cumulative distribution function]] *[[Moments]] *[[Probability density function]] *[[Joint time scales probability density function]] *[[Moment generating function]] *[[Expected value]] *[[Variance]] {{:Examples of time scales}} =Special functions on time scales= [[Delta cpq|$\mathrm{c}_{pq}$]]<br /> [[Delta chpq|$\mathrm{ch}_{pq}$]]<br /> [[Delta spq|$\mathrm{s}_{pq}$]]<br /> [[Delta shpq|$\mathrm{sh}_{pq}$]]<br /> [[Gamma function]]<br /> [[hyperbolic_functions | Hyperbolic functions]]<br /> [[Euler-Cauchy logarithm]]<br /> [[Bohner logarithm]]<br /> [[Jackson logarithm]]<br /> [[Mozyrska-Torres logarithm]]<br /> [[gaussian_bell | Gaussian bell]]<br /> ==$\nabla$-calculus== [[Nabla cosine | $\nabla \widehat{\cos}_p$]]<br /> [[Nabla cosh | $\nabla \widehat{\cosh}_p$]]<br /> [[Nabla exponential | $\nabla \widehat{\exp}$]]<br /> [[Nabla hk|$\nabla \hat{h}_k$]]<br /> [[Nabla gk|$\nabla \hat{g}_k$]]<br /> [[Nabla sine | $\nabla \widehat{\sin}_p$]]<br /> [[Nabla sinh | $\nabla \widehat{\sinh}_p$]]<br /> ==$\Diamond_{\alpha}$-calculus== [[Diamond alpha cosine | $\Diamond_{\alpha}$-$\cos_p$]]<br /> [[Diamond alpha cosh | $\Diamond_{\alpha}$-$\cosh_p$]]<br /> [[Diamond exponential | $\Diamond_{\alpha}$-$e_p$]]<br /> [[Diamond sine | $\Diamond$-$\sin_p$]]<br /> [[Diamond sinh | $\Diamond$-$\sinh_p$]]<br /> ==Probability Distributions on time scales== [[Uniform distribution]]<br /> [[Exponential distribution]]<br /> [[Gamma distribution]]<br /> =Special cases of time scales= ==Differential equations== [[Hypergeometric differential equation]] <br /> [[Confluent hypergeometric differential equation]]<br /> ==Difference equations== [[Difference equation of hypergeometric type]]<br /> ==$q$-difference equations== [[q-difference equation of hypergeometric type]]<br /> latcs60sikle3xqp4jvr6nzltbo955k