Fix a finite set of points in Euclidean n-space En, thought of as a point-cloud sampling of a certain domainD ⊂ En. The Rips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easily-computed but high-dimensional approximation to the homotopy type of D. There is a natural “shadow” projection map from the Rips complex to En that has as its image a mor...

Fix a finite set of points in Euclidean n-space En, thought of as a point-cloud sampling of a certain domain D ⊂ En. The VietorisRips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easily-computed but high-dimensional approximation to the homotopy type of D. There is a natural “shadow” projection map from the Vietoris-Rips complex to En that has ...

We consider a d-dimensional well-formed weighted projective space P(w¯) as toric variety associated with fan ?(w¯) in Nw¯?R whose 1-dimensional cones are spanned by primitive vectors v0,v1,…,vd?Nw¯ generating lattice Nw¯ and satisfying the linear relation ?iwivi=0. For any fixed dimension d, there exist only finitely many weight w¯=(w0,…,wd) such that contains quasi-smooth Calabi–Yau hypersurfa...

This is the part II of our series of two papers, “Clemens’ conjecture: part I”, “Clemens’ conjecture: part II”. Continuing from part I, in this paper we turn out attention to general quintic threefolds. In a universal quintic threefold X, we construct a family of quasi-regular deformation Bb such that the generic member in this family is non-deviated, but some special member is deviated. By the...

In this paper we study critial isometric and minimal isometric embeddings of classes of Riemannian metrics which we call quasi-κ-curved metrics. Quasi-κ-curved metrics generalize the metrics of space forms. We construct explicit examples and prove results about existence and rigidity. Introduction Definition: Let (M, g̃) be a Riemannian manifold. We will say g̃ is a quasi-κcurved metric if there ...