Showing papers in "Communications on Pure and Applied Mathematics in 1975"

Reflection of singularities of solutions to systems of differential equations

Abstract: where P EPS(0) is a pseudo-differential operator of order zero. We make the assumption that P(y,x,q,t)=det(qiG, (y ,x , t ) ) is real and has simple characteristics. Then, as is well known (see [ 1 I), singularities of solutions to (1.1) propagate along the null bicharacteristic strips of p in the interior of 9. Actually, the reference does not quite apply, since a/ayG is not a pseudodifferential operator on 9 (see the appendix). Suppose ( x o , t o ) E T*( a s2) 0 and that j null-bicharacteristic strips of p pass over (xo,to). That means there a r e j real solutions q,,. . . ,q, of p ( O , ~ ~ , q , [ ~ ) = 0 . The associated bicharacteristics y, ( t) = (y (t), x ( t ) , (t), t( t)) solve the equations

On the Existence of a Capillary Surface with Prescribed Contact Angle

Abstract: In this note we consider the problem of the existence and regularity of an equilibrium-free surface S of a liquid that partially fills a cylindrical container, as determined by surface forces, gravitational forces and boundary adhesion. Let D denote 'a right cross section of the cylinder and aD its boundary. We represent the capillary surface S by a height function u(x, y), (x, y)~ D. The form of the capillary surface S as determined by the principle of virtual work (cf. [l]) leads to an equation of the form

The succession problem

Abstract: There are numerous physical situations in which a sequence of changes of state occurs, in an apparently random manner. Examples range from the reigning party in a multiparty political system to the sequence of tissue types in a linear tissue sample of a disorganized tissue mass. The question arises whether the two successive states are really correlated. The subject considered in this paper is the following: given the number of occurrences of each event in a sequence of trials, what is the probability distribution of the number of transitions. A weighted Markov transition probability is found. The associated generating function is evaluated explicitly by contour integration, and the desired probability distribution is extracted. Its long-chain asymptotic value is also obtained. (RWR)