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We attach the degenerate signature (n,0,1) to the dual Grassmann algebra of projective space to obtain a real Clifford algebra which provides a powerful, efficient model for euclidean geometry. We avoid problems with the degenerate metric by constructing an algebra isomorphism between the Grassmann algebra and its dual that yields non-metric meet and join operators. We focus on the cases of n=2...

We study the Bäcklund transformations of integrable geometric curve flows in certain geometries. These curve flows include the KdV and Camassa-Holm flows in the two-dimensional centro-equiaffine geometry, the mKdV and modified Camassa-Holm flows in the two-dimensional Euclidean geometry, the Schrödinger and extended Harry-Dym flows in the three-dimensional Euclidean geometry and the Sawada-Kote...

We study a general nonconvex formulation for low-rank minimization problems. use recent results on non-Euclidean first-order methods to provide efficient and scalable algorithms. Our approach uses the geometry induced by Bregman divergence of well-chosen kernel functions; unconstrained problems, we introduce novel family Gram quartic kernels that improve numerical performance. Numerical experim...

We classify gamma-ray bursts (GRBs) according to their observed durations and physical properties of their spectra. We find that long/hard bursts (of duration T90 > 2.5 s, and typical photon energy Ep > ∼ 0.8 MeV corresponding to BATSE’s energy fluence hardness H 32 > 3) show the strongest deviation from the three-dimensional Euclidean brightness distribution. The majority of GRBs, i.e., short ...

We study the thermodynamics of de Sitter black holes with a conformally coupled scalar field. The geometry is that of the “lukewarm” Reissner-Nordström-de Sitter black holes, with the event and cosmological horizons at the same temperature. This means that the region between the event and cosmological horizons can form a regular Euclidean instanton. The entropy is modified by the non-minimal co...