Mathematical Preliminaries

Abstract

We let Rⁿ denote Euclidean n-space, and let$R_{ + + }^\eta$denote the positive orthant. The notation “xεS” means “x is an element of set S.” A set is taken to be a collection of objects. For example, we express the positive orthant in set notation as$R_{ + + }^\eta = \left\{ {X \in R^\eta |X \geqslant 0} \right\}$. Throughout this study, for vectors x, yε R^η, “x≥y” means “x_i ≥y_i for all i and x_j > y_j for some j.” Similarly, for two sets S and V, “S c V” means that “xεS implies xεV;” that is, S is a subset of V. The notation S c V means every x in the set S is also contained in V, but there exists an element of V that is not contained in S; in other words, S is a proper subset of V.