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3D surface classification is a fundamental problem in computer vision and computational geometry. Surfaces can be classified by different transformation groups. Traditional classification methods mainly use topological transformation groups and Euclidean transformation groups. We introduce a novel method to classify surfaces by conformal transformation groups. Conformal equivalent class is refiner than topological equivalent class and coarser than isometric equivalent class, making it suitable for practical classification purposes. For general surfaces, the gradient fields of conformal maps form a vector space, which has a natural structure invariant under conformal transformations. We present an algorithm to compute this conformal structure, which can be represented as matrices, and use it to classify surfaces. The result is intrinsic to the geometry, invariant to triangulation and insensitive to resolution. To the best

We prove that for spacetimes solving the EinsteinMaxwell (EM) equations, the electromagnetic field contributes at highest order to the nonlinear memory effect of gravitational waves. In Nonlinear nature of gravitation and gravitational-wave experiments, Christodoulou showed that gravitational waves have a nonlinear memory. He discussed how this effect can be measured as a permanent displacement of test masses in a laser interferometer gravitational-wave detector. Christodoulou derived a precise formula for this permanent displacement in the Einstein vacuum (EV) case. We prove in Theorem 2.6 that for the EM equations this permanent displacement exhibits a term coming from the electromagnetic field. This term is at the same highest order as the purely gravitational term that governs the EV situation. On the other hand, in Section 3, we show that to leading order, the presence of the electromagnetic field

It follows from work of Perelman that any finite-time singularity of the Ricci flow on a compact 3-manifold is modeled on an ancient ϰ-solution. We prove that every non-compact ancient ϰ-solution in dimension 3 is isometric to a family of shrinking cylinders (or a quotient thereof), or to the Bryant soliton. This confirms a conjecture of Perelman.

The sparsity-based approaches have demonstrated promising performance in image processing. In this paper, for better preservation of the salient edge structures of images, we propose an l_0 + l_2-norm based analysis model, which requires solving a challenging non-separable l0-norm related minimiza- tion problem, and we also propose an inexact augmented Lagrangian method with proven convergence to a local minimum. Extensive experiments in image smoothing, including texture removal and context smoothing, show that our method achieves better visual results over various sparsity-based models and the CNN method. Also, experiments on sparse view CT reconstruction further validate the advantage of the proposed method.