Talk:Time-scale calculus

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untitled[edit]

The special issue, the first link under further reading, is a dead link. 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 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.Apophos (talk) 17:20, 9 October 2009 (UTC)

Riemann-Stieltjes integral[edit]

The Riemann-Stieltjes integral and the time scale delta-integral have been combined to get a Riemann-Stieltjes integral on time scales. The Riemann-Stieltjes integral on time scales.

The main point of Time-scale calculus is that difference equations and differential equation s can be treated on an equal footing. Can you say the Riemann-Stieltjes integral is useful for this ? 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 measures 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)? --Mathuvw (talk) 13:20, 27 October 2010 (UTC)