The methods of making diamond tools have undergone a remarkable development since the invention of synthetic diamond in the mid-1950s. The current review summarises key historical, recent and ongoing trends and developments in sintered diamond tools. The report concentrates on tools used within the stone and construction industries which, being the main consumers of metal-bonded diamond tools fabricated by powder metallurgy (P/M) technology, have always fuelled advances in tool design and production technologies.

Positively invariant sets play an important role in the theory and applications of dynamical systems. The stability in the sense of Lyapunov of the equilibrium x = 0 is equivalent to the existence of the ellipsoidal positively invariant sets. The constraints on the state and control vectors of dynamical systems can be formulated as polyhedral positively invariant sets in practical engineering problems. Numerical checking method of positive invariance of polyhedral sets is addressed in this paper. The validation of the positively invariant sets can be done by solving LPs which can be easily done numerically. It is illustrated by examples that our checking method is effective. Compared with the now existing algebraic methods, numerical checking method is an attractive method in that it’s easy to be implemented.