Riemann integration

This result, called the Riemann Integral, can be applied to unsteady isentropic compression waves as well as to expansion waves. By defining a Riemann function ... 

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 ... 

Exercise 1.22 Consider an equivalence class c of functions as defined in Section 1.7. Show that if any one element of c is Riemann integrable on an interval [a, b] C R, then every element of c is Riemann integrable on [a, b]. Show that the value of the definite integral does not depend on the choice of function in the equivalence class. Hence the real number c is well defined. 

INTRODUCTION TO ANALYSIS, Maxwell Rosenlicht. Unusually clear, accessible coverage of set theory, real number system, metric spaces, continuous functions, Riemann integration, multiple integrals, more. Wide range of problems. Undergraduate level. Bibliography. 254pp. 5X x 8X. 65038-3 Pa. 7.00... 

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. 

Given a bounded function y(x) defined on a bounded interval [a,b], the definite Riemann integral of fix) is defined as... 

Note that here we only consider functions on the usual three-dimensional coordinate space TZ . The letter L refers to Lebesque integration, a feature that assures that the function spaces are complete (complete normed spaces are also called Banach spaces). We will, however, not go into the detailed mathematics and refer the interested reader to the literature [4]. We just note that for continuous functions the integral is equivalent to the usual (Riemann) integral. Equation (16) defines a norm on the space If and we see from equation (10) that the density belongs to L1. From the condition of finite kinetic energy and the use of a Sobolev inequality one can show that [1]... 

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]... 

The symbol J does not denote an ordinary (Riemann) integral, but... 

The DPI representation of the path integral that was developed in the preceding section is not unique. Another path-integral representation is often used that has come to be known as the Fourier representation [33,34,36-42,44,85]. Like the DPI representation, the Fourier representation transforms the path integral into an infinite-dimensional Riemann integral. In this formalism, we consider the paths to be periodic signals that can be represented as a Fourier series. Consider the density matrix p(x, x j8). Since the partition function is the trace of the density matrix, we have... 

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) ... 

The line integral (10.66) reduces to a Riemann integral when the path of integration is parallel to either coordinate axis. For example, along the linear path y = JO = const, we obtain... 

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 ... 

Here both integrals can be interpreted as ordinary Riemann integrals is the nonspecified fluctuating parameter). 

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

Supposey(j ) is a Riemann integrable function that is defined on —it < x < it. Then one can define the kth partial stun... 

The interpretation of probability as an area, when suitably generalized, gives a universal framework. The formulation given here, in terms of Riemann integrals, is sufficient for many appHcations. Consider a nonnegative function f x) defined on the real fine with the property that its integral over the entire line is equal to one (Fig. 21.3.)... 

In this formulation by density functions and (Riemann) integrals, the basic events are the intervals [a,b] and countable unions of these disjoint intervals. It is easy to see that this approach satisfies aU of the axioms for a probability space listed earlier. This formulation lets us apply aU of the techniques of calculus and analysis to probability. This feature is demonstrated repeatedly in the sections to follow. 

One last comment should be made. Riemann integration and continuous (or piecewise continuous) density functions are sufficient for a considerable amount of applied probability, but they cannot handle all of the topics in probability because of the highly discontinuous nature of some probability distributions. A generalization of Riemann integration (measure theory and general integration), however, does handle all aspects of probability. 

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