A set of irregularly distributed points is fitted to a Least-Squares approximating surface constructed as linear combination of wavelets. The construction proceeds adaptively on a coarse-to-fine refinement. We discuss by numerical experiments the efficiency of the method in relation to iterative solution procedures. Furthermore, we present extensions of the method that allow to cope with wrong ...

For any $k in mathbb{N}$, the $k$-subdivision of graph $G$ is a simple graph $G^{frac{1}{k}}$, which is constructed by replacing each edge of $G$ with a path of length $k$. In [Moharram N. Iradmusa, On colorings of graph fractional powers, Discrete Math., (310) 2010, No. 10-11, 1551-1556] the $m$th power of the $n$-subdivision of $G$ has been introduced as a fractional power of $G$, denoted by ...

Mader conjectured that every C4-free graph has a subdivision of a clique of order linear in its average degree. We show that every C6-free graph has such a subdivision of a large clique. We also prove the dense case of Mader’s conjecture in a stronger sense, i.e. for every c, there is a c′ such that every C4-free graph with average degree cn 1/2 has a subdivision of a clique K` with ` = bc′n1/2...

This paper describes a successful approach to improving the robustness and speed of Sandia National Laboratory’s 3-D MEMS geometry modeler through a combination of mask subdivision and code parallelization. Symptoms of the robustness problems experienced with the original modeler that suggested the need for the subdivision strategy pursued here are explained. The basic elements of the subdivisi...

Subdivision rules formesheswith boundary are essential for practical applications of subdivision surfaces. These rules have to result in piecewise C -continuous boundary limit curves and ensure C -continuity of the surface itself. Extending the theory of Zorin (Constr Approx 16(3):359–397, 2000), we present in this paper general necessary and sufficient conditions for C -continuity of subdivisi...

We construct a C multiscale approximation scheme for functions defined on the (Riemann) sphere. Based on a three-directional box-spline, a flexible C scheme over a valence 3 extraordinary vertex can be constructed. Such a flexible C subdivision scheme is known to be impossible for arbitrary valences. The subdivision scheme can be used to model spherical surfaces based on a recursively subdivide...

The volume enclosed by subdivision surfaces, such as Doo-Sabin, Catmull-Clark, and Loop has recently been derived. Moments of higher degree d are more challenging because of the growing number of coefficients in the d + 3-linear forms. We derive the intrinsic symmetries of the tensors, and thereby reduce the complexity of the problem. Our framework allows to compute the 4-linear forms that de...

We show that for every positive integer k, any tournament with minimum out-degree at least (2+o(1))k2 contains a subdivision of the complete directed graph on k vertices, where each path has length most 3. This result is best possible condition (up to multiplicative factor 8), and it tight respect paths. It may be viewed as analogue theorem proved by Bollobás Thomason, independently Komlós Szem...