We prove the Bershadsky-Cecotti-Ooguri-Vafa's conjecture for all genus Gromov-Witten potentials of the quintic threefolds, by identifying the Feynman graph sum with the $\nmsp$ stable graph sum via an R-matrix action. The Yamaguchi-Yau functional equations and the formulas of $F_1,F_2$, are direct consequences of the BCOV Feynman sum rule.

In this paper, we consider the L p dual Minkowski problem by geometric variational method. Using anisotropic Gauss–Kronecker curvature flows, we establish the existence of smooth solutions of the L p dual Minkowski problem when pq ≥ 0 and the given data is even. If f ≡ 1, we show under some restrictions on p and q that the only even, smooth, uniformly convex solution is the unit ball.

We propose a novel framework for reconstructing lightweight polygonal surfaces from point clouds. Unlike traditional methods that focus on either extracting good geometric primitives or obtaining proper arrangements of primitives, the emphasis of this work lies in intersecting the primitives (planes only) and seeking for an appropriate combination of them to obtain a manifold polygonal surface model without boundary. We show that reconstruction from point clouds can be cast as a binary labeling problem. Our method is based on a hypothesizing and selection strategy. We first generate a reasonably large set of face candidates by intersecting the extracted planar primitives. Then an optimal subset of the candidate faces is selected through optimization. Our optimization is based on a binary linear programming formulation under hard constraints that enforce the final polygonal surface model to be manifold and watertight. Experiments on point clouds from various sources demonstrate that our method can generate lightweight polygonal surface models of arbitrary piecewise planar objects. Besides, our method is capable of recovering sharp features and is robust to noise, outliers, and missing data.

Some filtrations of the tensor product of a highest weight module and a lowest weight module over quantum group Uq(g)are constructed in[1]and one can use them to define a two-sided ideal of the modified quantized enveloping algebra. It is shown that the quotient algebra inherits a canonical basis from the modified quantized enveloping algebra and is dual to the quantum coordinate algebra defined by Kashiwara for a symmetrizable Kac–Moody algebra g.

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