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Calcium is widely used for improving the castability of liquid steel, as well as for improvement of steel cleanliness and inclusion modification for better quality steel. Calcium modifies solid alumina inclusions, arising out of deoxidation of liquid steel, into liquid calcium aluminate. Depending upon the steel composition, calcium sulfide (CaS) and/or various forms of calcium aluminates may form. Sulfides are often associated with the oxide phase, which is typically known as oxide-sulfide duplex inclusion. Formation of solid calcium sulfide must be avoided during the ladle treatment of liquid steel, since it is detrimental to the castability of steel. In the present work a thermodynamic model has been developed for predicting the formation of oxide-sulfide duplex inclusions arising out of competitive reactions between [O], [S] and [Ca] in Al-killed steel. The model predictions of the present work were compared with those reported in literature, as well as with the types of inclusions observed in steel samples collected from the plant. Reasonably good agreements amongst them were observed. The results indicated that in order to achieve completely liquid calcium aluminate without forming any sulfides the sulfur content of liquid steel must be sufficiently low. With increasing S content of liquid steel, complete modification of alumina inclusions into liquid calcium aluminate becomes difficult. The maximum sulfur content to avoid formation of CaS depends upon the steel composition, principally aluminum. The sulfide inclusions are often a solid solution of CaS and MnS. Thermodynamic analysis for this system was also carried out. Based on the analysis in the present work, it is possible to predict the influence of sulfide composition on formation of duplex inclusions.

The present paper establishes quantitative estimates on the rate of diophantine approximation in homogeneous varieties of semisimple algebraic groups. The estimates established generalize and improve previous ones, and are sharp in a number of cases. We show that the rate of diophantine approximation is controlled by the spectrum of the automorphic representation, and thus subject to the generalised Ramanujan conjectures.

Abstract This paper establishes upper and lower bounds on the speed of approximation in a wide range of natural Diophantine approximation problems. The upper and lower bounds coincide in many cases, giving rise to optimal results in Diophantine approximation which were inaccessible previously. Our approach proceeds by establishing, more generally, upper and lower bounds for the rate of distribution of dense orbits of a lattice subgroup Γ in a connected Lie (or algebraic) group G, acting on suitable homogeneous spaces G/H. The upper bound is derived using a quantitative duality principle for homogeneous spaces, reducing it to a rate of convergence in the mean ergodic theorem for a family of averaging operators supported on H and acting on G/Γ. In particular, the quality of the upper bound on the rate of distribution we obtain is determined explicitly by the spectrum of H in the automorphic representation on L^{2} (Γ \setminus G). We show that the rate is best possible when the representation in question is tempered, and show that the latter condition holds in a wide range of examples.

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