MAXIMAL CLOSED SUBROOT SYSTEMS OF REAL AFFINE ROOT SYSTEMS

Extended Affine Root Systems

There are two notions of the extended affine root systems in the literature which both are introduced axiomatically. One, extended affine root system (SAERS for short), consists only of nonisotropic roots, while the other, extended affine root system (EARS for short), contains certain isotropic roots too. We show that there is a one to one correspondence between (reduced) SEARSs and EARSs. Name...

Polynomial Approximation of Closed-form MPC for Piecewise Affine Systems

This paper addresses the issue of the practical implementation of closed-form Model Predictive Controllers (MPC) to processes with very short sampling times. Such questions come in consideration when the solution to MPC problems is expressed in a so-called parametric or closed-form fashion. The underlying idea of this paper is to approximate the optimal control law defined over state space regi...

Characterizing the Multiplicative Group of a Real Closed Field in Terms of its Divisible Maximal Subgroup

Parametrizations of Infinite Biconvex Sets in Affine Root Systems

We investigate in detail relationships between the set B of all infinite “biconvex” sets in the positive root system ∆+ of an arbitrary untwisted affine Lie algebra g and the set W of all infinite “reduced word” of the Weyl group of g. The study is applied to the classification of “convex orders” on ∆+ (cf. [Ito]), which are indispensable to construct “convex bases” of PoincaréBirkhoff-Witt typ...

The Classification of Convex Orders on Affine Root Systems

We classify all total orders having a certain convexity property on the positive root system of an arbitrary untwisted affine Lie algebra g. Such total orders are called convex orders and were used to construct Poincaré-BirkhoffWitt type bases of the upper triangular subalgebra of the quantized enveloping algebra of g which are called convex bases.