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We present a method for the precise determination of the volume of subsets of R d which are bounded by a hypersurface parametrized by a set of refinable functions. The derivation is based on the linear refinement equations rather than on closed form expressions of these functions, which may not be available. In particular, our approach makes it possible to compute the area of planar domains bounded by subdivision curves or the volume of spatial domains bounded by subdivision surfaces. [ABSTRACT FROM AUTHOR]

We suggest a novel method to determine the joint spectral radius of finite sets of matrices by validating the finiteness property. It is based on finding a certain finite tree with nodes representing sets of matrix products. Our approach accounts for cases where one or several matrix products satisfy the finiteness property. Moreover, is potentially functional even for reducible sets of matrices. [ABSTRACT FROM AUTHOR]

Abstract: We investigate polynomial approximation of functions in Sobolev spaces on connected domains which are bounded by a family of diffeomorphic images of graphs of continuous functions. Constants and approximating polynomials are specified explicitly. [Copyright &y& Elsevier]

A new set of tools for verifying smoothness of surfaces generated by stationary subdivision algorithms is presented. The main challenge here is the verification of injectivity of the characteristic map. The tools are sufficiently versatile and easy to wield to allow, as an application, a full analysis of algorithms generalizing biquadratic and bicubic B-spline subdivision. In the case of generalized biquadratic subdivision the analysis yields a hitherto unknown sharp bound strictly less than 1 on the second largest eigenvalue of any smoothly converging subdivision. [ABSTRACT FROM AUTHOR]

Abstract: We show that the interpolatory four-point-scheme with tension parameter generates -limit curves if and only if , where is the unique real solution of the cubic equation . [Copyright &y& Elsevier]