Abstract. This paper gives a practical method of extending an n × r matrix P (z), r ≤ n, with Laurent polynomial entries in one complex variable z, to a square matrix also with Laurent polynomial entries. If P (z) has orthonormal columns when z is restricted to the torus T, it can be extended to a paraunitary matrix. If P (z) has rank r for each z ∈ T, it can be extended to a matrix with nonvanishing determinant on T. The method is easily implemented in the computer. It is applied to the construction of compactly supported wavelets and prewavelets from multiresolutions generated by several univariate scaling functions with an arbitrary dilation parameter.

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In this article we construct a lattice of points which lie on k + I pencils of hyperplanes in the projective k-space, where, with a suitable choice of coordinate system, simple equations of the hyperplanes are obtained. This enables us to construct an interpolation formula on the projective k-space from which interpolating polynomials on a general class of lattices in the Euclidean k-space are obtained via a projective transformation.

Synopsis The effect of the volumetric composition of cellular concrete, particularly water and air voids, on its compressive strength has been demonstrated to follow Feret's general formula. The increase in strength at all ages with a corresponding increase in water/cement ratio (opposite to that of mortar mixes) as obtained in the experiment has been shown to be consistent with Feret's formula. The inclusion of the degree of hydration in the modified form of Power's gel/space ratio further improves the correlation with strength when this is taken as the parameter.

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