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This paper is concerned with refinement equations of the type [This symbol cannot be presented in ASCII format] where f is the unknown function defined on the s-dimensional Euclidean space Rs, a is a finitely supported sequence on Zs, and M is an s × s dilation matrix with m := | det M|. The solution of a refinement equation can be obtained by using the subdivision scheme associated with the mask. In this paper we give a characterization for the convergence of the subdivision scheme when the mask is nonnegative. Our method is to relate the problem of convergence to m column-stochastic matrices induced by the mask. In this way, the convergence of the subdivision scheme can be determined in a finite number of steps by checking whether each finite product of those column-stochastic matrices has a positive row. As a consequence of our characterization, we show that the convergence of the subdivision scheme with a nonnegative mask depends only on the location of its positive coefficients. Several examples are provided to demonstrate the power and applicability of our approach. [ABSTRACT FROM AUTHOR]

In this paper linear periodic systems of difference equations are classified with respect to periodic similarity and kinematic similarity. Complete sets of invariants of periodic difference equations relative to such similarity transformations are given, and corresponding canonical forms are described. Also the irreducible periodic difference equations, i.e., those that cannot be reduced by such similarities to a nontrivial direct sum, are identified. [ABSTRACT FROM AUTHOR]

For nonsingular n×n matrices A, φ(A) = AοA-T(ο denotes the Hadamard product and A-T the inverse transpose, (A-1)T , of A) arises in mathematical control theory associated with chemical engineering design problems and in a matrix theoretic problem involving the relation between the diagonal entries and eigenvalues. For any positive integer t, we show that the equation [This symbol cannot be presented in ASCII format] has at least 2t-1and 2t mutually φ-distinct real solutions for the case n = 4 and n > 4, respectively, and that these solutions can be determined by an inverted iteration. We also prove that the above equation has only one φ-distinct real solution (1¹ -1¹) for t = 1 and has no real solutions for t ≥ 2 when n = 2. The above results answer the problem that has been proposed by Johnson and Shapiro in [SIAM J. Algebraic Discrete Methods, 7 (1986), pp. 627–644]. When does the equation [This symbol cannot be presented in ASCII format] have a real solution? [ABSTRACT FROM AUTHOR]