5,679 research outputs found

    Enumeration and Asymptotic Formulas for Rectangular Partitions of the Hypercube

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    We study a two-parameter generalization of the Catalan numbers: Cd,p(n)C_{d,p}(n) is the number of ways to subdivide the dd-dimensional hypercube into nn rectangular blocks using orthogonal partitions of fixed arity pp. Bremner \& Dotsenko introduced Cd,p(n)C_{d,p}(n) in their work on Boardman--Vogt tensor products of operads; they used homological algebra to prove a recursive formula and a functional equation. We express Cd,p(n)C_{d,p}(n) as simple finite sums, and determine their growth rate and asymptotic behaviour. We give an elementary proof of the functional equation, using a bijection between hypercube decompositions and a family of full pp-ary trees. Our results generalize the well-known correspondence between Catalan numbers and full binary trees

    Implementation of multigrid methods for solving Navier-Stokes equations on a multiprocessor system

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    Presented are schemes for implementing multigrid algorithms on message based MIMD multiprocessor systems. To address the various issues involved, a nontrivial problem of solving the 2-D incompressible Navier-Stokes equations is considered as the model problem. Three different multigrid algorithms are considered. Results from implementing these algorithms on an Intel iPSC are presented
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