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Abstract. We present a module based criterion, i.e. a sufficient condition based on the absolute value of the matrix coefficients, for the convergence of Gauss-Seidel method (GSM) for a square system of linear algebraic equations, the Generalized Line Criterion (GLC).We prove GLC to be the "most general" module based criterion and derive, as GLC corollaries, some previously know and also some new criteria for GSM convergence. Although far more general than the previously known results, the proof of GLC is simpler. The results used here are related to recent research in stability of dynamical systems and control of manufacturing systems.Mathematical subject classification: 65F10, 65F35, 15A09.

We provide a simple and explicit example of the influence of the kinetic energy in the stability of the equilibrium of classical Hamiltonian systems of the type H (q, p)=< B(q)p; p >+ (q). We construct a potential energy of class C k with a critical point at 0 and two different positive defined matrices B 1 andB 2 , both independent of q, and show that the equilibrium (0, 0) is stable according to Lyapunov for the Hamiltonian H 1 = < B 1 (q)p; p > + (q), while for H 2 = < B 2 (q)p; p > + (q) the equilibrium is unstable. Moreover, we give another example showing that even in the analytical situation the kinetic energy has influence in the stability, in the sense that there is an analytic potential energy and two kinetic energies, also analytic, T 1 and T 2 such that the attractive basin of (0, 0) is a two-dimensional manifold in the system of Hamiltonian + T 1 and a one-dimensional manifold in the system of Hamiltonian + T 2 .

Abstract. Sharifrtia, Caramanis, and Gershwin [1991] introduced a class of policies for manufacturing systems, called by them linear corridor policies. They proved that their stability can be discussed by the study of a simpler subset of such policies (cone policies). This paper revisits their work presenting a different description of the dynamics of the systems under study and explores it to device a necessary and sufficient condition for stability, obtained by the strengthening of the assumptions in Sharifnia et al. (1991). This condition is shown to be simply tested (M -1 >-0) and valid for various realizations.

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