Riemann sum

Definite integrals often can be graphically evaluated as so-called Riemann sums by partitioning the area under the curve into rectangles, the horizontal width of which... 

Written in terms of continuous functions, a functional integral represents a formal writing. The calculation of such an integral requires to perform a discretization of the space to reduce the integral to Riemann sums. We may imagine that the sampling points form the sites of a square lattice of lattice... 

In mathematical jargon, this is called a Riemann sum. As we divide the area into a greater and greater number of narrower strips, n oo and Ax 0. The limiting process defines the definite integral (also called a Riemann integral) ... 

Recall that the Riemann integral of a continuous function g on [0, t] is the limit of Riemann sums constructed as follows partition the interval [0, t] into K subintervals each of width St = r/v, choose an arbitrary tk from the Mi subinterval, 1 < A < v, and evaluate the integral as a limit of the sum of products ... 

Next, unlike in the case of the Riemann sums, it turns out that the result depends on where we place the points tk. Using the properties of the Wiener process we may easily show that the choice 4 = kSt (left endpoint) and the choice tk = (k+ l/2)St yield different formulas for the integral. The first choice is commonly referred to as ltd integration, and the second as Stratonovich integration. As an example, consider the case where g(t) = W(t). We suppose this to be approximated by a sum... 

Definite Integral Limit of a Riemann sum as the number of terms approaches infinity. 

Riemann Sum Sum of the products of functional values and the lengths of the subintervals over which the function is defined. 

Each of these sums is called a Riemann sum, / is called the integrand, and a and b are called the limits of integration. 

Z) is a triangle. The integral in (4,75) is, in Riemann s definition, an improper integral. It can be approximated by a finite sum because the integrand is monotone, which implies, furthermore, that is bounded. I skip the details because considerations of this type are standard in the discussion of improper Riemannian integrals, Equations (4,73) and (4.75), in the notation of (4.56), imply... 

If a series is conditionally convergent, its sums can be made to have any arbitrary value by a suitable rearrangement of the series it can in fact be made divergent or oscillatory (Riemanns theorem). 

In the Fourier method each path contributing to Eq. (4.13) is expanded in a Fourier series and the sum over all contributing paths is replaced by an equivalent Riemann integration over all Fourier coefficients. This method was first introduced by Feynman and Hibbs to determine analytic expressions for the harmonic oscillator propagator and has been used by Miller in the context of chemical reaction dynamics. We have further developed the approach for use in finite-temjjerature Monte Carlo studies of quantum sys-tems, and we have found the method to be very useful in the cluster studies discussed in this chapter. 

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 ... 

From the definition of the Riemann integral, the quantity can be approximated by a sum over discrete events (configurations)... 

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 ... 

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