# Advanced methods in applied mathematics, lecture course by Courant R.

By Courant R.

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1 have coordinates x =--= 'V' , y === 'V". * The proof is given in Sec. 2. 66434, etc. It can be proved* that since the abscissae and ordinates of these points are independent the probability for one such point to fall into any region inside the square is equal to the area of this region. The points drawn in Fig. 0077), ... 5 2 One of the points fell outside the square and was rejected. As follows from Eq. 04. * The proof is given in Sec. 3. CHAPTER II Examples of Application of the Monte Carlo Method Sec.

We should hope that in the nearest future such calculations will become a routine business, and the probabilistic characteristics of the elements will invari ahl y be supplied by their manufacturers. Sec. 7. ) with matter are known. Usually it is needed to find the macroscopic characteristics of processes in which the number of participating particles is enormous, such as densities, fluxes and so on. This situation is quite similar to that encountered in Sees. 5 and 6, and offers a very convenient object for the Monte Carlo calculations.

Such methods of calculation are extremely helpful in designing economic units: expensive (and sometimes even impossible) actual experiment is substituted by a simulated computer experiment in which various versions of work organization or equipment operation are simulated. Sec. 6. 1. The Simplest Scheme of Computing Quality. Let us consider a device S consisting of a certain (possibly large) number of elements. If, for instance, S is an electric apparatus, then its elements can be resistors (Ron), capacitors (C(k»)' electron tubes, etc.