In this paper, we deal with a simple geometric problem: Is it possible to partition rectangle into k non-congruent rectangles of equal area? This problem is motivated by the so-called ‘Mondrian art problem’ that asks similar question for dissections integer sides. Here, generalize Mondrian allowing real case, show minimum value have ‘perfect partition’ (that is, equal-area rectangles) seven. Mo...

A growth model for topological trees is formulated as a generalization of the terminal and segmental growth model. For this parameterized growth model, expressions are derived for the partition probabilities (probabilities of subtree pairs of certain degrees). The probabilities of complete trees are easily derived from these partition probabilities. 1. Introduction. In the study of growth of br...

Partition functions of probability distributions are important quantities for model evaluation and comparisons. We present a new method to compute partition functions of complex and multimodal distributions. Such distributions are often sampled using simulated tempering, which augments the target space with an auxiliary inverse temperature variable. Our method exploits the multinomial probabili...

The minimum message length principle is an information theoretic criterion that links data compression with statistical inference. This paper studies the strict minimum message length (SMML) estimator for d-dimensional exponential families with continuous sufficient statistics, for all d. The partition of an SMML estimator is shown to consist of convex polytopes (i.e. convex polygons when d = 2...

the probability of losing a customer in m/g/n/0 and gi/m/n/0 loss queuing systems withheterogeneous servers is minimized. the first system uses a queue discipline in which a customer who arriveswhen there are free servers chooses any one of them with equal probability, but is lost otherwise. providedthat the sum of the servers rates are fixed, loss probability in this system attains minimum val...

Given a set of integers W , the Partition problem determines whether W can be divided into two disjoint subsets with equal sums. We model the Partition problem as a system of polynomial equations, and then investigate the complexity of a Hilbert’s Nullstellensatz refutation, or certificate, that a given set of integers is not partitionable. We provide an explicit construction of a minimum-degre...

An M-partition of a positive integer m is a partition of m with as few parts as possible such that every positive integer less than m can be written as a sum of parts taken from the partition. This type of partition is a variation of MacMahon’s perfect partition, and was recently introduced and studied by O’Shea, who showed that for half the numbers m, the number of M-partitions of m is equal t...