5,679 research outputs found
Enumeration and Asymptotic Formulas for Rectangular Partitions of the Hypercube
We study a two-parameter generalization of the Catalan numbers:
is the number of ways to subdivide the -dimensional hypercube into
rectangular blocks using orthogonal partitions of fixed arity . Bremner \&
Dotsenko introduced 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 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 -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
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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