The resemblance of the Goursat problem for the hyperbolic partial differential equations to the initial value problem for the ordinary differential equations has suggested the extension of many well known numerical methods existing for (1.2) to the numerical treatment of (1.1). Day [2] discusses the quadrature methods while Diaz [3] generalizes the simple Euler-method. Moore [6] gives an analogue to the fourth order Runge-Kutta-method and Tornig [7] generalizes the explicit and implicit Adams-methods.

Let Xi, i = 1,2,3,… be a sequence of independent and identically distributed random variables and write S0 = 0, Sn = ∑ni=1Xi, n ≧ 1. Let In(0), In(1), …, In (n) be that unique permutaion of 1, 2, …, n such that SIn(0) ≦ SIn(1) ≦ … ≦ SIn(n) and such that if Si = Sk with i < k then In(k) < In(j). Thus, In(j) is an index of the j-th largest partial sum.

The aim of this note is to evaluate an integral involving the product of two H-functions.

In this note some rudimentary results about the characteristic multipliers of periodic solutions of differential equations are given which supplement those given by Poincaré [2], Chapitre IV, and by Wintner [4].

Beniamino Segre, in his memorial lecture of 1958 [5], [6], inaugurated the study of non-linear geometry in three dimensions over a division ring. In his treatment of sections of quadrics by planes, he is naturally led to consider conics and the problem of tangency. Now in the commutative case the locus of intersection of a quadric and a plane containing a generator is the line-pair consisting of this generator and one from the other family. Such a plane is then the tangent plane of the point of intersection of the two generators. Segre extends this notion to the non-commutative case, where the locus of intersection is not always a line-pair. He joins up the remaining points of intersection in pairs, and calls the points where the lines so formed cut the base generator, the ‘points of contact’ of the plane (π) and the quadric (Q). A line in π is called a ‘tangent’ if it passes through a point of contact, but does not contain any of the points of intersection of Q and π.