Riemann-Stieltjes integral

Although in many homogeneous systems fluorophores have distinct and discrete decay constants for their fluorescence, in heterogeneous systems the luminescent molecules have different environments and consequently different energy levels and also pathways for the energy dissipation. Moreover, in the RET processes the distance between the donor and acceptor is not constant but may vary slightly. Then it can be expected that the lifetimes are not sharply defined but they are actually continuously distributed. Mathematically this means that instead of the sum in (40), we have to use a Riemann-Stieltjes integral ... 

Equations 9 and 10 can be solved for arbitrary strain (stress) histories to obtain the material stress (strain) response. Solution for the step-deformations requires use of the unit Heaviside function and is discussed in detail by Tschoegl (10) and by Findley, Onaran and co-workers. (21) Wineman and Rajagopal (22) deal with the step-strains, using Riemann-Stieltjes integrals. Other histories are more directly solvable. Also, in the linear theory the limits on the integral can be written from 0 to rather than from -00 to t (9,10,21,22). 

This theorem has been rigorously proved by Riemann and Stiel es. The integral is called the Riemann-Stieltjes integral. The term h itself is also a function of x. If /(x) and h x) are both bounded on a closed internal [a, b], the Riemann-Stiel es integral is then in the form jj /(x) Ah x). 

After the divisimi of the interval [0, t onto disjoint, contiguous subintervals, X(t) can be written down as the Riemann-Stieltjes sum. The limit, in the mean-square sense, of the sequence of such sums, is the mean-square Riemann-Stieltjes integral with respect to the counting process N t) or the stochastic integral ... 

Moments of this conditional distribution can be written as standard Riemann Integrals of the pdf fx(z (N)) or as Stieltjes integrals of the cdf Fx(zf(N)) For example, the conditional expectation is written ... 

For multiplicative noise the determination of these moments requires a more detailed consideration of the stochastic integral since white noise is too irregular for Riemann integrals to be applied. Application of Stieltjes integration yields a dependence of the moments on how the limit to white noise is taken. If t) is the limit of the Ornstein-Uhlenbeck -process with r —> 0 (Stratonovich sense) the coefficients read [50]... 

Using the Riemann-Stieltjes theorem, we can convert the Fourier series into Fourier integrals. Fourier integrals express/(x) in terms of sine and cosine functions of x in the form... 

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