Gauss积分及其简单应用

提供几种计算G auss积分的方法,并由此介绍一下它们在积分计算中的简单应用.     

向量的夹角概念在高维欧氏空间E^n中的推广

本文给出了Ｅ＾ｎ中不同维数平面之间多维角、单维角的定义，它们是向量的夹角概念的推广其几何意义非常明显，在此基础上还给出了ＱＡＢＰ＾－１Ａ的特征多项式各项系数的几何意义，这些结论对于研究各类曲率非常有用。     

关于rectifiable空间中的局部(序列)连通性的几个注记

给出连通的rectifiable空间是局部序列连通(或局部连通)的刻画,推广了拓扑群中的相应结果;利用rectifiable空间G中e的局部邻域基给出G是局部连通(或局部序列连通)的刻画;证明了若A是rectifiable空间G中的序列开子集,那么H=A是G的序列开rectifiable子空间.     

张量空间对称化算子的指标

设G≤S_m,f:G→C为任意函数,我们给出了对称化算子θ(G,f)=sum from σ∈G f(σ)P(σ)指标的一个公式,并给出了其它的指标等式。     

基于Gauss公式的GM(1,1)模型的背景值构造新方法与应用

分析了GM(1,1)模型中的背景值,改进了相关文献的方法,提出用数值积分中的G auss公式重构模型中的背景值,可以有效地提高模型的预测精度和适用性,并将此方法应用到我国城市内分泌、营养和代谢疾病及免疫病致死人数占死亡总人数的百分比的有效预测,理论分析和应用实例表明了文章所提的方法的有效性.     

Banach空间逼近紧性与预报算子连续性的几何条件

首先给出 Banach空间为逼近紧的几何特征 .然后用随机变量组成的 Banach空间及空间中预报集的几何性质给出预报算子连续的条件 .     

3阶齐次线性常微分方程的球面解曲线

首先给出3阶齐次线性常微分方程的球面解曲线的定义,然后讨论其球面解曲线与系数函数的关系,最后给出了球面解曲线的几何性质.     

Banach空间中的广义Dunkl-Williams常数及其性质

本文给出了广义Dunkl-Williams常数与一些著名几何常数例如凸系数、光滑系数、James常数之间的关系,从而得到一些蕴含不动点性质的充分条件,另外通过广义Dunkl-Williams常数的上下界的估计给出了Banach空间一致非方的刻画.     

球几何三维流形到透镜空间的映射度

本文讨论球几何三维流形M=S~3/G,即S~3在一群G自由作用下的轨道空间.所谓球几何是指S~3上被赋予的标准的度量,其等距变换群是SO(4),而上述G就是SO(4)的离散子群.主要结果是利用Z在ZG模上的投射预解以及群G的上同调和流形K(G,1)的上同调的关系,计算出流形M的系数为Z_m(m不必为素数)的上同调环,以及Bockstein同态H~n(M,Z_m)→H~(n+1)(M,Z_m).利用上述结果进而计算出任一球几何三维流形到三维透镜空间的映射的映射度,最后可以判断一类映射是否具有值为1的映射度.     

G框架下的欧式期权定价公式

用G几何布朗运动描述标的资产的价格变动,得到了欧式看涨期权定价的动态公式,并给出了动态复制策略的显示表达.     

A spherical Bernstein theorem for minimal submanifolds of higher codimension

Combining the tools of geometric analysis with properties of Jordan angles and angle space distributions, we derive a spherical and a Euclidean Bernstein theorem for minimal submanifolds of arbitrary dimension and codimension, under the condition that the Gauss image is contained in some geometrically defined closed region of a Grassmannian manifold. The proof depends on the subharmoncity of an auxiliary function, the Codazzi equations and geometric measure theory.     

一元函数的极值及其奇异性

针对一元函数的极值理论,为学生介绍这一理论的进一步发展即奇点理论的一些基本概念和理论.指出二者之间的密切联系.通过对平面曲线的高斯映射以及奇点的介绍,向学生展示高斯曲率与函数的二阶导数以及高斯映射的奇点与函数的拐点之间的奇妙关系.并指出它们的几何意义.同时强调在学习高等数学的相关内容时,应该养成思考它们的几何背景及意义...     

Singularities of lightcone Gauss images of spacelike hypersurfaces in de Sitter space

We define the notions of lightcone Gauss images of spacelike hypersurfaces in de Sitter space. We investigate the relationships between singularities of these maps and geometric properties of spacelike hypersurfaces as an application of the theory of Legendrian singularities. We classify the singularities and give some examples in the generic case in de Sitter 3-space.     

The Gauss map of a hypersurface in Euclidean sphere and the spherical Legendrian duality

We investigate the Gauss map of a hypersurface in Euclidean n-sphere as an application of the theory of Legendrian singularities. We can interpret the image of the Gauss map as the wavefront set of a Legendrian immersion into a certain contact manifold. We interpret the geometric meaning of the singularities of the Gauss map from this point of view.     

Generalized surfaces with constant <Emphasis Type="Italic">H</Emphasis>/<Emphasis Type="Italic">K</Emphasis> in Euclidean three-space

Surfaces in Euclidean three-space with constant ratio of mean curvature to Gauss curvature arise naturally as the parallel surfaces to minimal surfaces. They might possess singularities which occur naturally as focal points of minimal surfaces. We study geometric properties and the singularities of such surfaces, prove some global results about them, and provide a Björling formula to construct such surfaces with prescribed point or curve singularities.     

广义de Sitter空间中法向Lorentzian超曲面的nullcone Gauss映射的奇点

利用奇点理论研究了广义de Sitter空间中具有Lorentzian法空间的一类超曲面.介绍了这类超曲面的局部微分几何,定义了nullcone Gauss映射及nullcone高度函数族,进而研究了nullcone高度函数族的性质及nullcone高斯映射的几何意义,最后研究了这类超曲面的通有性质.     

Spacelike surfaces in anti de Sitter four-space from a contact viewpoint

We define the notions of (S _t ¹ × S _s ²)-nullcone Legendrian Gauss maps and S ₊²-nullcone Lagrangian Gauss maps on spacelike surfaces in anti de Sitter 4-space. We investigate the relationships between singularities of these maps and geometric properties of surfaces as an application of the theory of Legendrian/Lagrangian singularities. By using S ₊²-nullcone Lagrangian Gauss maps, we define the notion of S ₊²-nullcone Gauss-Kronecker curvatures and show a Gauss-Bonnet type theorem as a global property. We also introduce the notion of horospherical Gauss maps which have geometric properties different from those of the above Gauss maps. As a consequence, we can say that anti de Sitter space has much richer geometric properties than the other space forms such as Euclidean space, hyperbolic space, Lorentz-Minkowski space and de Sitter space.     

广义de Sitter空间中的类时超曲面

利用奇点理论研究广义de Sitter空间中的类时超曲面.介绍类时超曲面的局部微分几何,定义了广义de Sitter高斯像及广义de Sitter高度函数,研究广义deSitter高度函数族的性质及广义de Sitter高斯像的几何意义,介绍了一种证明高度函数为Morse族的新方法.最后研究了类时超曲面的通有性质.     

Singularities of Anti de Sitter torus Gauss maps

We study timelike surfaces in Anti de Sitter 3-space as an application of singularity theory. We define two mappings associated to a timelike surface which are called Anti de Sitter nullcone Gauss image and Anti de Sitter torus Gauss map. We also define a family of functions named Anti de Sitter null height function on the timelike surface. We use this family of functions as a basic tool to investigate the geometric meanings of singularities of the Anti de Sitter nullcone Gauss image and the Anti de Sitter torus Gauss map.     

On some conjectures on generalized quadratic Gauss sums and related problems

The main purpose of this article is to study higher power mean values of generalized quadratic Gauss sums using estimates for character sums, analytic methods and algebraic geometric methods. We prove two conjectures which were proposed recently by the above authors in a previous article (2022). Here we obtain an asymptotic formula for arbitrary power means of generalized quadratic Gauss sums and one corresponding power moment of a character sum.