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利用超曲面排列的方法探讨了前馈神经网络的VC维。Discussing the VC dimension of neural network with the hypersurface arrangement.

我们的结果推广了同样条件下M是极小超曲面的情形。Our result extends the case when M is a minimal hypersurface in the same condition.

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从零曲面方程出发，导出了黑洞视界面的普遍公式。The general event horizon formula of black hole is given from null hypersurface equation.

在超曲面几何学中，对主曲率的研究是至关重要的。It is essential to study the principal curvature of hypersurface in the hypersurface geometry.

并利用主曲率计算公式证明了W-超曲面的一个存在性定理。Using the principal curvature formula, we prove an existence theorem of Weingarten hypersurface.

这样就考虑了整个能量超曲面的形状，从而能保证收敛到体系的基态，或有选择地收敛到激发态。Thus the shape of the whole energy hypersurface is considered, and the convergence on the ground state of system is ensured.

通过对此超曲面主曲率的研究，给出了S的可能值存在第二空隙的两个充分条件，并分别估计了在这些条件下的空隙长度。It mainly deals with the properties for the principal curvatures of the hypersurface and gets two sufficient conditions under which S has a second gap.

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在这篇文章中，讨论了具有常数量曲率的拟紧致超曲面，并给出了它是全脐子流形的一个全脐条件。In this paper, an intrinsic condition for a Compact Space-like hypersurface with constant scalar curvature in a de Sitter space to be totally umbilical is obtained.

证明超曲面奇点的链环与亚历山大洛夫空间的奇点的方向空间有一些相似之处。It shows that there are similarities between the role of the link of a hypersurface singularity and the space of directions of a singularity in an Alexandrov space.

建立B样条推理规则，构造实现了B样条推理系统，并求出该系统的最终结果为B样条超曲面。B-spline inference rule was established and B-spline inference system was constructed, and then the final result of the system was calculated as a B-spline hypersurface.

在这个体系中，我们找到了甘氨酸和水作用的四种稳定构型，其中最稳定的构型是一个环状的双氢键结构。Four reasonable geometries on the potential energy hypersurface of glycine and water system are considered with the global minimum being a cyclic double-hydrogen bonded structure.

最后以插值形成三维空间规则数据场为例，证明利用超曲面样条函数法进行三维空间插值的可行性和准确性。Finally, it gives an example that generating regular 3D spatial dataset based on the interpolation of hypersurface spline, it testifies the feasibility and veracity of this method.

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最后研究了常平均曲率完备超曲面，得到了这类超曲面全脐的一个结果。At last the complete hypersurface with constant mean curvature in the quasi constant curvature space is investigated, some characterization of totally umbilical hypersurfaces are obtained.

本文给出实空间形式中超曲面的一个刚性定理，同时也得到求面中子流形的刚性结果。In this paper, a rigidity theorem of hypersurface in real space form will be given. In addition, we obtain rigidity theorems of submanifold in sphere which improve the result of Hou and Xu.

单位球面中极小浸入闭超曲面的第二基本形式S的可取值是否离散一直是受关注的问题。Let S be the second fundamental form of a closed hypersurface which is minimally immersed in a unit sphere. The question whether the value of S is discrete has been attractive for a long time.

通过对此超曲面主曲率的研究，给出了S的可能值存在第二空隙的两个充分条件，并分别估计了在这些条件下的空隙长度。It mainly deals with the properties for the principal curvatures of the hypersurface and gets two sufficient conditions under which S has a second gap. Also estimates for the gaps length are obtained.