In this paper we present a factorization framework for Hermite subdivision schemes refining function values and first derivatives, which satisfy a spectral condition of high order. In particular we show that spectral order d allows for d factorizations of the subdivision operator with respect to the Gregory operators: A new sequence of operators we define using Stirling numbers and Gregory coefficients. We further prove that the d-th factorization provides a "convergence from contractivity" method for showing C d -convergence of the associated Hermite subdivision scheme. The power of our factorization framework lies in the reduction of computational effort for large d: In order to prove C d -convergence, up to now, d factorization steps were needed, while our method requires only one step, independently of d. Furthermore, in this paper, we show by an example that the spectral condition is not equivalent to the reproduction of polynomials.

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We analyse the convergence of nonlinear Riemannian analogues of linear subdivision processes operating on data in the sphere. We show how for curve subdivision rules we can derive bounds guaranteeing convergence if the density of input data is below that threshold. Previous results only yield thresholds that are several magnitudes smaller and are thus useless for a priori checking of convergence. It is the first time that such a result has been shown for a geometry with positive curvature and for subdivision rules not enjoying any special properties like being interpolatory or having non-negative mask.

This paper studies well-defindness and convergence of subdivision schemes which operate on Riemannian manifolds with nonpositive sectional curvature. These schemes are constructed from linear ones by replacing affine averages by the Riemannian center of mass. In contrast to previous work, we consider schemes without any sign restriction on the mask, and our results apply to all input data. We also analyse the Hölder continuity of the resulting limit curves. Our main result states that convergence is implied by contractivity of a derived scheme, resp. iterated derived scheme. In this way we establish that convergence of a linear subdivision scheme is almost equivalent to convergence of its nonlinear manifold counterpart.

The distinguishing number D(G) of a graph G is the smallest number of colors that is needed to color G such that the only color preserving automorphism is the identity. We give a complete classification for all connected graphs G of maximum valence (G) = 3 and distinguishing number D(G) = 3. As one of the consequences we get that all infinite connected graphs with (G) = 3 are 2-distinguishable. * Supported by the Austrian Science Fund FWF W1230 and P24028 and by Award 317689 from the Simons Foundation.1 If α is smaller than the first uncountable inaccessible cardinal, then there also exists a coloring with a finite number of colors that is only preserved by the identity endomorphism.