About: Time-scale calculus

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In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying hybrid systems. It has applications in any field that requires simultaneous modelling of discrete and continuous data. It gives a new definition of a derivative such that if one differentiates a function defined on the real numbers then the definition is equivalent to standard differentiation, but if one uses a function defined on the integers then it is equivalent to the forward difference operator.

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rdfs:label	時間尺度微分積分学 (ja) Przestrzeń czasowa (pl) Time-scale calculus (en) 时标微积分 (zh)
rdfs:comment	In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying hybrid systems. It has applications in any field that requires simultaneous modelling of discrete and continuous data. It gives a new definition of a derivative such that if one differentiates a function defined on the real numbers then the definition is equivalent to standard differentiation, but if one uses a function defined on the integers then it is equivalent to the forward difference operator. (en) 数学における時間尺度微分積分学（じかんしゃくどびぶんせきぶんがく、英: time-scale calculus）は、微分積分学と和分差分学とを統一するもので、微分方程式の理論と差分方程式の理論とを統合した（連続と離散の入り混じった）力学系の研究の方法論を提供する。時間尺度微分積分学は、離散および連続データを同時にモデリングすることが要求される任意の分野において応用を持つ。この分野における新たな微分の定義は、実数全体を引数とする函数に作用したとき通常の意味での微分と同値になり、整数全体を引数とする函数に作用したとき前進差分と同値となるようなものとして与えられる。 (ja) Przestrzeń czasowa – dowolny domknięty podzbiór liczb rzeczywistych oznaczany Na przestrzeniach czasowych można rozważać równania Δ-różniczkowe, które są unifikacją równań różniczkowych na i równań różnicowych na oraz uogólnieniem na przestrzeniach czasowych. (pl) 在数学中，时标微积分是差分方程和微分方程的一种统一。时标微积分最初由德国数学家Stefan Hilger发明，应用于需要同时包含离散和连续的情况的模型的领域中。它为导数赋予了新的定义，使得如果你对定义在实数中的闭区间上的函数进行求导，就等价于通常意义上的导数；然而如果你将这种新定义的导数作用于定义在整数集上的函数，则它就等价于前移差分算子。 (zh)
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Link from a Wikipage to another Wikipage	Calculus Method of averaging Integral equation Continuous function Analysis on fractals Mathematics Measure (mathematics) Quantum calculus Function (mathematics) Z-transform Krylov–Bogoliubov averaging method Banach space Lebesgue measure Recurrence relations Finite differences Fractional calculus Partial differential equation Differential equation Dirac delta Discrete time Radon–Nikodym derivative Population dynamics Hybrid system Calculus Dynamical systems Laplace transform Difference equation Differential calculus Integer Kronecker delta Cantor set Real number Set (mathematics) Stochastic differential equation Multiple-scale analysis Summation equation Real line Riemann–Stieltjes integral Forward difference Forward difference operator Calculus of finite differences Partial difference equation Lebesgue–Stieltjes integral Closed subset Multiple integration dbr:Stefan_Hilger
Link from a Wikipage to an external page	http://timescalewiki.org/index.php/Main_Page http://www.timescales.org http://www.hindawi.com/journals/ade/volume-2006/si.1.html http://web.mst.edu/~bohner/tisc.html http://www.e-ndst.kiev.ua/v9n1.htm
sameAs	Time-scale calculus Time-scale calculus Time-scale calculus Time-scale calculus Time-scale calculus Time-scale calculus Time-scale calculus
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has abstract	In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying hybrid systems. It has applications in any field that requires simultaneous modelling of discrete and continuous data. It gives a new definition of a derivative such that if one differentiates a function defined on the real numbers then the definition is equivalent to standard differentiation, but if one uses a function defined on the integers then it is equivalent to the forward difference operator. (en) 数学における時間尺度微分積分学（じかんしゃくどびぶんせきぶんがく、英: time-scale calculus）は、微分積分学と和分差分学とを統一するもので、微分方程式の理論と差分方程式の理論とを統合した（連続と離散の入り混じった）力学系の研究の方法論を提供する。時間尺度微分積分学は、離散および連続データを同時にモデリングすることが要求される任意の分野において応用を持つ。この分野における新たな微分の定義は、実数全体を引数とする函数に作用したとき通常の意味での微分と同値になり、整数全体を引数とする函数に作用したとき前進差分と同値となるようなものとして与えられる。 (ja) Przestrzeń czasowa – dowolny domknięty podzbiór liczb rzeczywistych oznaczany Na przestrzeniach czasowych można rozważać równania Δ-różniczkowe, które są unifikacją równań różniczkowych na i równań różnicowych na oraz uogólnieniem na przestrzeniach czasowych. (pl) 在数学中，时标微积分是差分方程和微分方程的一种统一。时标微积分最初由德国数学家Stefan Hilger发明，应用于需要同时包含离散和连续的情况的模型的领域中。它为导数赋予了新的定义，使得如果你对定义在实数中的闭区间上的函数进行求导，就等价于通常意义上的导数；然而如果你将这种新定义的导数作用于定义在整数集上的函数，则它就等价于前移差分算子。 (zh)
gold:hypernym	Unification
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is Link from a Wikipage to another Wikipage of	Real analysis Raegan Higgins Glossary of areas of mathematics Z-transform Discrete time and continuous time Discretization

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