Matrix majorization Geir Dahl

Majorization for partially ordered sets

We generalize the classical notion of majorization in Rn to a majorization order for functions defined on a partially ordered set P . In this generalization we use inequalities for partial sums associated with ideals in P . Basic properties are established, including connections to classical majorization. Moreover, we investigate transfers (given by doubly stochastic matrices), complexity issue...

Majorization, Polyhedra and Statistical Testing Problems. Majorization, Polyhedra and Statistical Testing Problems

There are important connections between majorization and convex polyhedra. Both weak majorization and majorization are preorders related to certain simple convex cones. We investigate the facial structure of a polyhedral cone C associated with a layered directed graph. A generalization of weak majorization based on C is introduced. It de nes a preorder of matrices. An application in statistical...

Some constrained partitioning problems and majorization

We consider some constrained partitioning problems for a finite set of objects of different types. We look for partitions that are sizeand type-similar, and, in addition, for a pair of such partitions that are “very different” in a certain sense. The motivation stems from a problem involving the partitioning of a set of students into smaller groups. We give these problems precise mathematical f...

Integral majorization Polytopes

Let n be a positive integer. We may write (or split) n as sums of n nonincreasing nonnegative integers p1, p2, . . . , pn in different ways (or partitions; see Section 2). For example, 3 = 3 + 0 + 0 = 2 + 1 + 0 = 1 + 1 + 1. If we denote by P (n) the number of different partitions of n, then P (3) = 3. One may check that P (5) = 7. As n gets large, P (n) increases rapidly. It is astounding that ...

A note on nonnegative diagonally dominant matrices