We prove that the standard double bubble provides the least-area way to enclose and separate two regions of prescribed volume in R 3. 1.

this article we shall develop a class of algorithms to capture the behavior of multiphase b ubbles and drops in two and three space dimensions. We include some very interesting and recently analyzed steady state cases---e.g. [8], where "double bubbles minimize." This general class of problems has recently received a lot of attention [19]. 1 Research supported by ARPA/ONR N00014-92-J-1890, NSF DMS94-04942, and ARO DAAH04-95-1-0155 2 H. K. Zhao is currently at: Mathematics Department, Stanford University, Stanford, CA 94305-2125. 495 0021-9991/98 $25.00 Cop yright c 1998 by Academic Press All rights of reproduction in any form reserved. 496 ZHAO ET AL. W e shall use the level set method, first developed in [15], which has been successfully applied to a variety of problems, in order to capture the evolution of complex interf aces in fluid dynamics and elsewhere (see, e.g., [13, 14] and the references therein). Topological changes and the development of singularities pose no difficulties for this method. As initially designed in [15], the level-set method applied naturally to problems in which there is a clear distinction between "inside" and "outside" of a (possibly multiply connected) re gion. Two phase flow problems coupling the motion to the full Navier--Stokes of Euler equations [2, 21] or to heat release [3], as well as unstable vortex sheet motion [5] and other unstable fronts [6], were recently solved using this method, extending its utility beyond the geometry-dri

This chapter provides a general introduction to interval computations, especially to interval computations as an important part of granular computing.

We prove that the standard double bubble is the least-area way to enclose and separate two regions of equal volume in H 3, and in S 3 when the exterior is at least ten percent of S 3. 2000 Mathematics Subject Classification: 49Q10, 53A10. 1. Introduction. The