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Given a compact set of real numbers, a random $C^{m + \alpha}$ -diffeomorphism is constructed such that the image of any measure concentrated on the set and satisfying a certain condition involving a real number $s$ , almost surely has Fourier dimension greater than or equal to $s / (m + \alpha)$ . This is used to show that every Borel subset of the real numbers of Hausdorff dimension $s$ is $C^{m + \alpha}$ -equivalent to a set of Fourier dimension greater than or equal to $s / (m + \alpha )$ . In particular every Borel set is diffeomorphic to a Salem set, and the Fourier dimension is not invariant under $C^{m}$ -diffeomorphisms for any $m$ .

In this paper we present a new algorithm which turns an unstructured triangle mesh into a quad dominant mesh with edges well aligned to the principal directions of the underlying surface. Instead of computing a globally smooth parameterization or integrating curvature lines along a tangent vector field, we simply apply an iterative relaxation scheme which incrementally aligns the mesh edges to the principal directions. We further obtain the quad dominant mesh by dropping the not-aligned diagonal edges from the triangle mesh. A post-processing stage is introduced to further improve the results. The major advantage of our algorithm is its conceptual simplicity since it is merely based on elementary mesh operations such as edge collapse, flip, and split. Various results are presented in the paper; they show a good alignment to surface features and rather uniform distribution of mesh vertices. This makes them well suited, e.g., as CatmullClark Subdivision control meshes.

In this paper, we investigate the space of certain weak stability conditions on the triangulated category of D0-D2-D6 bound states on a smooth projective Calabi-Yau 3-fold. In the case of a quintic 3-fold, the resulting space is interpreted as a universal covering space of an infinitesimal neighborhood of the conifold point in the stringy K¨ahler moduli space. We then associate the DT type invariants counting semistable objects, which give new curve counting invariants on Calabi-Yau 3-folds. We also investigate the wall-crossing formula of our invariants and their interplay with the Seidel-Thomas twist.

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