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| In [[mathematics]], '''time-scale calculus''' is a unification of the theory of [[difference equation]]s with that of [[differential equation]]s, unifying integral and differential [[calculus]] with the [[calculus of finite differences]], offering a formalism for studying hybrid discrete–continuous [[dynamical system]]s. 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 which acts on the real numbers then the definition is equivalent to standard differentiation, but if one uses a function acting on the integers then it is equivalent to the forward difference operator.
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| ==History==
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| Time-scale calculus was introduced in 1988 by the German mathematician [[Stefan Hilger]].<ref name=hilger>{{cite journal| last = Hilger | first = Stefan | authorlink = Stefan Hilger |title = Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten |publisher = Universität Würzburg | year = 1998}}</ref> However, similar ideas have been used before and go back at least to the introduction of the [[Riemann–Stieltjes integral]] which unifies sums and integrals.
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| ==Dynamic equations==
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| Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their [[continuous function|continuous]] counterparts.<ref name=bp>{{cite book | author=Martin Bohner & Allan Peterson | title=Dynamic Equations on Time Scales | publisher=Birkhäuser | year=2001 | isbn=978-0-8176-4225-9 | url = http://www.springer.com/west/home/birkhauser?SGWID=4-40290-22-2117582-0 }}</ref> The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice — once for differential equations and once again for difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown [[function (mathematics)|function]] is a so-called time scale (also known as a time-set), which may be an arbitrary closed subset of the reals. In this way, results apply not only to the [[Set (mathematics)|set]] of [[real number]]s or set of [[integer]]s but to more general time scales such as a [[Cantor set]].
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| The three most popular examples of [[calculus]] on time scales are [[differential calculus]], [[finite differences|difference calculus]], and [[quantum calculus]]. Dynamic equations on a time scale have a potential for applications, such as in [[population dynamics]]. For example, they can model insect populations that evolve continuously while in season, die out in winter while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a non–overlapping population.
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| ==Formal definitions==
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| A '''time scale''' (or '''measure chain''') is a [[closed subset]] of the [[real line]] <math>\mathbb{R}</math>. The common notation for a general time scale is <math>\mathbb{T}</math>.
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| The two most commonly encountered examples of time scales are the real numbers <math>\mathbb{R}</math> and the [[Discrete time|discrete]] time scale <math>h\mathbb{Z}</math>.
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| A single point in a time scale is defined as:
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| :<math>t:t\in\mathbb{T}</math>
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| === Operations on time scales===
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| [[File:Timescales jump operators.png|thumb|upright=2.0|The forward jump, backward jump, and graininess operators on a discrete time scale]]
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| The ''forward jump'' and ''backward jump'' operators represent the closest point in the time scale on the right and left of a given point <math>t</math>, respectively. Formally:
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| :<math>\sigma(t) = \inf\{s \in \mathbb{T} : s>t\}</math> (forward shift operator / forward jump operator)
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| :<math>\rho(t) = \sup\{s \in \mathbb{T} : s<t\}</math> (backward shift operator / backward jump operator)
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| <br /> | |
| The ''graininess'' <math>\mu</math> is the distance from a point to the closest point on the right and is given by:
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| :<math>\mu(t) = \sigma(t) -t.</math>
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| <br />
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| For a right-dense <math>t</math>, <math>\sigma(t)=t</math> and <math>\mu(t)=0</math>.<br />
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| For a left-dense <math>t</math>, <math>\rho(t)=t.</math>
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| ===Classification of points===
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| [[File:Timescales point classifications.png|thumb|upright=2.0|Several points on a time scale with different classifications]] | |
| For any <math>t\in\mathbb{T}</math>, <math>t</math> is:
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| * ''left dense'' if <math>\rho(t) =t</math>
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| * ''right dense'' if <math>\sigma(t) =t</math>
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| * ''left scattered'' if <math>\rho(t)< t</math>
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| * ''right scattered'' if <math>\sigma(t) > t</math>
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| * ''dense'' if both left dense and right dense
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| * ''isolated'' if both left scattered and right scattered
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| <br />
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| As illustrated by the figure at right:
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| * Point <math>t_1</math> is ''dense''
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| * Point <math>t_2</math> is ''left dense'' and ''right scattered''
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| * Point <math>t_3</math> is ''isolated''
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| * Point <math>t_4</math> is ''left scattered'' and ''right dense''
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| ===Continuity===
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| [[Continuous function|Continuity]] on a time scale is redefined as equivalent to density. A time scale is said to be ''right-continuous at point <math>t</math>'' if it is right dense at point <math>t</math>. Similarly, a time scale is said to be ''left-continuous at point <math>t</math>'' if it is left dense at point <math>t</math>.
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| ==Derivative==
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| Take a function:
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| :<math>f: \mathbb{T} \rightarrow \mathbb{R}</math>,
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| (where R could be any normed [[Banach space]], but set it to be the real line for simplicity).
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| Definition: The ''delta derivative'' (also Hilger derivative) <math>f^{\Delta}(t)</math> exists if and only if:
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| For every <math>\epsilon > 0</math> there exists a neighborhood <math>U</math> of <math>t</math> such that:
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| :<math>|f(\sigma(t))-f(s)- f^{\Delta}(t)(\sigma(t)-s)|\le \varepsilon|\sigma(t)-s|</math>
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| for all <math>s</math> in <math>U</math>.
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| Take <math>\mathbb{T} =\mathbb{R}.</math> Then <math>\sigma(t) = t</math>, <math>\mu(t) = 0</math>, <math>f^{\Delta} = f'</math>; is the derivative used in standard [[calculus]]. If <math>\mathbb{T} = \mathbb{Z}</math> (the [[integer]]s), <math>\sigma(t) = t + 1</math>, <math>\mu(t)=1</math>, <math>f^{\Delta} = \Delta f</math> is the [[forward difference operator]] used in difference equations.
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| ==Integration==
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| The ''delta integral'' is defined as the antiderivative with respect to the delta derivative. If <math>F(t)</math> has a continuous derivative <math>f(t)=F^\Delta(t)</math> one sets
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| :<math>\int_r^s f(t) \Delta(t) = F(s) - F(r).</math>
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| ==Laplace transform and z-transform==
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| A [[Laplace transform]] can be defined for functions on time scales, which uses the same table of transforms for any arbitrary time scale. This transform can be used to solve dynamic equations on time scales. If the time scale is the non-negative integers then the transform is equal<ref name=bp/> to a modified [[Z-transform]]:
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| <math>\mathcal{Z}'\{x[z]\}=\frac{\mathcal{Z}\{x[z+1]\}}{z+1}</math>
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| ==Partial differentiation==
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| [[Partial differential equation]]s and [[partial difference equation]]s are unified as partial dynamic equations on time scales.<ref>[http://dx.doi.org/10.1016/S0377-0427(01)00434-4 Partial differential equations on time scales], Calvin D. Ahlbrandt, Christina Morian</ref><ref>[http://marksmannet.com/TimeScales/Papers/partial.pdf Partial dynamic equations on time scales], B Jackson – Journal of Computational and Applied Mathematics, 2006</ref><ref>[http://web.mst.edu/~bohner/papers/pdots.pdf Partial differentiation on time scales], M Bohner, GS Guseinov, Dynamic Systems and Applications 13 (2004) 351–379</ref>
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| ==Multiple integration==
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| [[Multiple integration]] on time scales is treated in Bohner (2005).<ref>{{cite journal | id = {{citeseerx|10.1.1.79.8824}} | title = Multiple integration on time scales | first = M | last1 = Bohner | first2 = GS | last2 = Guseino | journal = Dynamic Systems and Applications | year = 2005 }}</ref>
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| ==Stochastic dynamic equations on time scales==
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| [[Stochastic differential equation]]s and stochastic difference equations can be generalized to stochastic dynamic equations on time scales.<ref>[http://scholarsmine.mst.edu/thesis/pdf/Sanyal_09007dcc80519030.pdf STOCHASTIC DYNAMIC EQUATIONS], SUMAN SANYAL, 2008</ref>
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| ==Measure theory on time scales==
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| Associated with every time scale is a natural [[Measure (mathematics)|measure]]<ref>{{cite journal | doi = 10.1016/S0022-247X(03)00361-5 | title = Integration on time scales | first = GS | last = Guseinov | journal = J. Math. Anal. Appl. | volume = 285 | year = 2003 | pages = 107–127 }}</ref><ref>{{cite web | url = http://library.iyte.edu.tr/tezler/master/matematik/T000568.pdf | title = Measure theory on time scales | first = A | last = Deniz | year = 2007 }}</ref> defined via
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| :<math>\mu^\Delta(A) = \lambda(\rho^{-1}(A)),</math>
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| where <math>\lambda</math> denotes [[Lebesgue measure]] and <math>\rho</math> is the backward shift operator defined on <math>\mathbb{R}</math>. The delta integral
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| turns out to be the usual [[Lebesgue–Stieltjes integral]] with respect to this measure
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| :<math>\int_r^s f(t) \Delta t = \int_{[r,s)} f(t) d\mu^\Delta(t)</math>
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| and the delta derivative turns out to be the [[Radon–Nikodym derivative]] with respect to this measure<ref>{{cite journal | arxiv = 1102.2511 | title = On the connection between the Hilger and Radon–Nikodym derivatives | first1 = J | last1 =Eckhardt | authorlink2 = Gerald Teschl | first2 = G | last2 = Teschl | journal = J. Math. Anal. Appl. | volume = 385 | year = 2012 | pages = 1184–1189 }}</ref>
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| :<math>f^\Delta(t) = \frac{df}{d\mu^\Delta}(t).</math>
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| ==Distributions on time scales==
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| The [[Dirac delta]] and [[Kronecker delta]] are unified on time scales as the ''Hilger delta'':<ref>[http://www.marksmannet.com/RobertMarks/REPRINTS/2007_TheLaplaceTransformOnTimeScales.pdf The Laplace transform on time scales revisited], John M. Davis, Ian A. Gravagne , Billy J. Jackson ,
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| Robert J. Marks II , Alice A. Ramos, J. Math. Anal. Appl. 332 (2007) 1291–1307</ref><ref>[http://marksmannet.com/RobertMarks/REPRINTS/short/BLaplaceOct2009.pdf Bilateral Laplace Transforms on Time Scales: Convergence, Convolution, and the Characterization of Stationary Stochastic Time Series], John M. Davis, Ian A. Gravagne and Robert J. Marks II</ref>
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| : <math>\delta_{a}^{\mathbb{H}}(t) = \begin{cases} \frac{1}{\mu(a)}, & t = a \\ 0, & t \neq a \end{cases}</math>
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| ==Integral equations on time scales==
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| [[Integral equation]]s and [[summation equation]]s are unified as integral equations on time scales.<ref>[http://web.maths.unsw.edu.au/~cct/tis-tomasia-IJDE-rev.pdf Volterra integral equations on time scales: Basic qualitative and quantitative results with applications to initial value problems on unbounded domains], Tomasia Kulik and Christopher C. Tisdell, 2007</ref>
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| ==Fractional calculus on time scales==
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| [[Fractional calculus]] on time scales is treated in Bastos, Mozyrska, and Torres.<ref>{{cite paper | arxiv = 1012.1555 | title = Fractional Derivatives and Integrals on Time Scales via the Inverse Generalized Laplace Transform | first1 = Nuno R. O. | last1 = Bastos | first2 = Dorota | last2 = Mozyrska | first3 = Delfim F. M. | last3 = Torres }}</ref>
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| ==See also==
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| *[[Analysis on fractals]] for dynamic equations on a [[Cantor set]].
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *[http://web.mst.edu/~bohner/papers/deotsas.pdf Dynamic equations on time scales: a survey], Ravi Agarwal, Martin Bohner, Donal O’Regan, Allan Peterson, Journal of Computational and Applied Mathematics 141 (2002) 1–26
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| ==Further reading==
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| * [http://www.timescales.org The Baylor University Time Scales Group]
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| * [http://web.mst.edu/~bohner/tisc.html Dynamic Equations on Time Scales] Special issue of ''Journal of Computational and Applied Mathematics'' (2002)
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| * [http://www.hindawi.com/journals/ade/volume-2006/si.1.html Dynamic Equations And Applications] Special Issue of ''Advances in Difference Equations'' (2006)
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| * [http://www.e-ndst.kiev.ua/v9n1.htm Dynamic Equations on Time Scales: Qualitative Analysis and Applications] Special issue of ''Nonlinear Dynamics And Systems Theory'' (2009)
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| {{DEFAULTSORT:Time Scale Calculus}}
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| [[Category:Dynamical systems]]
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| [[Category:Calculus]]
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| [[Category:Recurrence relations]]
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System tray icon makes it effortless to launch the program plus displays "clean" status or the number of errors in the last scan. The ability to obtain and remove the Invalid class keys and shell extensions is regarded as the primary advantages of the program. That is not routine function for the different Registry Cleaners. Class keys plus shell extensions which are not working could seriously slow down your computer. RegCure scans to find invalid entries and delete them.
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Reboot PC - Just reboot a PC to see if the error is gone. Often, rebooting the PC readjusts the internal settings plus software plus hence fixes the problem. If it doesn't then move on to follow the instructions under.
All of these difficulties is easily solved by the clean registry. Installing the registry cleaner usually allow we to employ a PC without worries behind. You may capable to use you program without being afraid that it's going to crash inside the center. Our registry cleaner might fix a host of errors on a PC, identifying lost, invalid or corrupt settings inside a registry.