The problem of determining if a graph is 2-colourable (i.e., bipartite) has long been known to have a simple polynomial time algorithm. Being 2-colourable is equivalent to having a bipartition of the vertex set where each cell is &-free. We extend this notion to determining if there exists a bipartition where each cell is G-free for some fixed graph G. One might expect that for some graphs othe...

We consider the following problem: given a forest of gene family trees on a set of genomes, find a first speciation which splits these genomes into two subsets and minimizes the number of gene duplications that happened before this speciation. We call this problem the Minimum Duplication Bipartition Problem. Using a generalization of the Minimum Edge-Cut Problem, known as Submodular Function Mi...

We consider the following problem: given a set of gene family trees, spanning a given set of species, find a first speciation which splits these species into two subsets and minimizes the number of gene duplications that happened before this speciation. We call this problem the Minimum Duplication Bipartition Problem. Using a generalization of the Minimum Edge-Cut Problem, we propose a polynomi...

Let A = (ai,j) , i = 1, 2, . . . , j = 0, 1, 2, . . . , be an infinite matrix with elements ai,j = 0 or 1; p (n, k;A) the number of partitions of n into k parts whose number yi of parts which are equal to i belongs to the set Yi = {j : ai,j = 1} , i = 1, 2, . . . . The universal theorem on partitions states that 1 X n=0 n X k=0 p (n, k;A)ut = 1 Y i=1 0 @ 1 X j=0 ai,ju t 1 A . In this paper, we ...

We present an algorithm which partitions a data set in two parts with equal size and experimentally nearly the same distribution measured through the likelihood of a Parzen kernel density estimator. The generation of the partition takes O( 1 2 N(N − 1)) operations (N number of data) and is 2 orders of magnitude faster than the state of the art. 1 Generating an equi-distributed bipartition 1.1 P...

Unextendible product basis is an important object in quantum information theory and features a broad spectrum of applications, ranging bound entangled states, nonlocality without entanglement, Bell inequalities with no violation. A generalized concept called uncompletable also attracts much attention. In this paper, we find some unextendible bases that are every bipartition, which answers 19 ye...

In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: a b s t r a c t The present paper is focused on the mathematical modeling of the charged particle transport in nonuniform me...