View source for Talk:Time-scale calculus

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==untitled==
The special issue, the first link under further reading, is a dead link. [[User:RhysU|RhysU]] ([[User talk:RhysU|talk]]) 00:20, 3 April 2009 (UTC)
:The broken link has been fixed. The university's address had changed from web.umr.edu to web.mst.edu [[User:Charvest|Charvest]] ([[User talk:Charvest|talk]]) 10:05, 3 April 2009 (UTC)

I have added Hilger's PhD thesis as a reference, but I was unable to locate an electronic copy of it to link.  If anyone can provide a link to a copy of it (or even just a doi number), it would be much appreciated.[[User:Apophos|Apophos]] ([[User talk:Apophos|talk]]) 17:20, 9 October 2009 (UTC)

== Riemann-Stieltjes integral ==

The [[Riemann-Stieltjes integral]] and the time scale delta-integral have been combined to get a Riemann-Stieltjes integral on time scales. [http://arxiv.org/abs/0903.1224 The Riemann-Stieltjes integral on time scales].

The main point of Time-scale calculus is that [[difference equation]]s and [[differential equation]] s can be treated on an equal footing. Can you say the Riemann-Stieltjes integral is useful for this ? [[Special:Contributions/89.241.225.148|89.241.225.148]] ([[User talk:89.241.225.148|talk]]) 07:52, 27 October 2010 (UTC)

:I am not an expert on this, but take for example [[Sturm-Liouville problems]]. People have considered coefficients which are measures. That is, you look at the ode in a weak sense after performing the Lebesgue-Stieltjes integral on both sides and, voila, you can treat differential and difference equations at the same time if you take either a purely absolutely continuous or a pure point measure for the weight, respectively. See for example the books by Atkinson from 1964 or by Mingarelli on ''Volterra-Stieltjes Integral Equations and Generalized Ordinary Differential Expressions'' (Springer LNM 989) from 1983. Moreover, consider the measure which is [[Lebesgue measure]] restricted to the dense points plus a sum of [[Dirac measure]]s at all isolated points with weight given by the graininess. Then it seems to me that the Hilger derivative is just the [[Radon-Nikodym derivative]] with respect to this measure (i.e. the inverse operation to the corresponding Lebesgue-Stieltjes integral)? --[[User:Mathuvw|Mathuvw]] ([[User talk:Mathuvw|talk]]) 13:20, 27 October 2010 (UTC)