Optimal partitions for triples

Optimal Partitions for Eigenvalues

We introduce a new numerical method to approximate partitions of a domain minimizing the sum of Dirichlet-Laplacian eigenvalues of any order. First we prove the equivalence of the original problem and a relaxed formulation based on measures. Using this result, we build a numerical algorithm to approximate optimal configurations. We describe numerical experiments aimed at studying the asymptotic...

Partitions for Optimal Approximations

The Riemann integral can be approximated using partitions and a rule for assigning weighted sums of the function at points determined by the partition. Approximation methods commonly used include endpoint rules, the midpoint rule, the trapezoid rule, Simpson’s rule, and other quadrature methods. The rate of approximation depends to a large degree on the rule being used and the smoothness of the...

On Optimal Guillotine Partitions Approximating Optimal D-box Partitions

Given a set of n points, P, in E d (the plane when d = 2) that lie inside a d-box (rectangle when d = 2) R, we study the problem of partitioning R into d-boxes by introducing a set of orthogonal hyperplane segments (line segments when d = 2) whose total (d?1)-volume (length when d = 2) is the least possible. In a valid d-box partition, each point in P must be included in at least one partitioni...

M-partitions: optimal partitions of weight for one scale pan

An M -partition of a positive integer m is a partition with as few parts as possible such that any positive integer less than m has a partition made up of parts taken from that partition of m. This is equivalent to partitioning a weight m so as to be able to weigh any integer weight l < m with as few weights as possible and only one scale pan. We show that the number of parts of an M -partition...

Optimal rectangular partitions