一个几何概念──长度伸缩率

本文对一元函数在一点处导数绝对值的几何意义进行了讨论，指出在极限的意义下它实际上是映射在一点局部的一种长度的伸缩率。     

基于曲线导数的二元函数微分中值定理

给定二元函数,文献[1]定义了其在光滑曲线上的方向导数(简称为曲线导数).本文主要利用曲线导数建立二元函数的微分中值定理,比如罗尔定理,拉格朗日中值定理,柯西中值定理.这些中值定理可视作一元函数微分中值定理在二维情形的推广.     

二阶可导函数的一个定理及应用

对数学分析中一类与二阶导数有关的等式或不等式问题的解法进行了优化，利用二阶泰勒公式，建立了[a，b]上与二阶可导函数f（x）及曲线弦斜率有关的一个公式，利用它方便地饵决了一类与二阶导数有关的等式或不等式问题     

方向导数的应用

本文将一元函数的高阶导数对应为多元函数的高阶导数，用方向导数表达泰勒公式，使之与一元函数的泰勒公式有统，的形式。又引入方向单调性、方向极值等概念，使多元函数的极值判别法基于一元函数极值判别法，此法不但直观又解法了判别式Δ＝０时的不确定性。限于篇幅只讨论二元函数。     

读“关于二阶导数的教学”一文后

《数学通报》1984年第1期登载叶余本同志所译小寺裕“关于二阶导数的教学”(下简称[1])一文。该文先导出曲线y=f(x)当f′(a)=0时,在x=a口处的曲率是k=|f″(a)|;然后借助坐标轴的平移旋转,来研究f′(a)≠0的点     

非光滑函数的凸性

本文借助于一元函数左、右导数的定义及其性质 ,将多元函数的方向导数转化为一元函数的左、右导数 ,并利用一元函数的凸性判别准则给出并证明了判别多元函数凸性的充分必要条件 .     

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...     

凸轮曲线的磨光问题

为了改善内燃机配气凸轮曲线加速度的光滑性、本文讨论了四次磨光函数及其二阶导数的保凸性,估计了误差,用四次磨光函数对组合式的凸轮曲线及已有凸轮的实测的离散数据进行了全局拟合,使其加速度成为光滑的按段三次方曲线.由于本文所进行的凸轮曲线拟合是为了使凸轮曲线的二阶导数即加速度光滑,这在凸轮曲线拟合方面还是     

一个二阶非线性方程边值问题解的存在性研究

在工程设计的拱、曲梁或壳体中常遇到所谓“合理曲线”问题:即在一定荷重作用下,在没有轴向压缩时,不产生弯曲的轴线。在曲率较小的情况下,工程中常用二次导数代替曲率,但当曲率较大时,必须采用曲率的精确表达式,在一维结构或柱壳中,合理曲线的基本方程为:     

扩展的Bianchi恒等式及其在几何流演化方程中的应用

在初始版本的第一,二Bianchi恒等式的基础上,利用二阶或三阶协变导数引申出扩展的二阶协变和三阶协变Bianchi恒等式.这类二阶协变Bianchi恒等式在黎曼曲率张量沿着两类特殊的几何流-里奇(Ricci)流和双曲几何流的演化方程中有一定的应用.给出这方面的应用例子并加以阐述.     

A priori Estimates and Existence for a Class of Fully Nonlinear Elliptic Equations in Conformal Geometry

In this paper we prove the interior gradient and second derivative estimates for a class of fully nonlinear elliptic equations determined by symmetric functions of eigenvalues of the Ricci or Schouten tensors. As an application we prove the existence of solutions to the equations when the manifold is locally conformally flat or the Ricci curvature is positive.     

Complete stable CMC surfaces with empty singular set in Sasakian sub-Riemannian 3-manifolds

For constant mean curvature surfaces of class C ² immersed inside Sasakian sub-Riemannian 3-manifolds we obtain a formula for the second derivative of the area which involves horizontal analytical terms, the Webster scalar curvature of the ambient manifold, and the extrinsic shape of the surface. Then we prove classification results for complete surfaces with empty singular set which are stable, i.e., second order minima of the area under a volume constraint, inside the 3-dimensional sub-Riemannian space forms. In the first Heisenberg group we show that such a surface is a vertical plane. In the sub-Riemannian hyperbolic 3-space we give an upper bound for the mean curvature of such surfaces, and we characterize the horocylinders as the unique ones with squared mean curvature 1. Finally we deduce that any complete surface with empty singular set in the sub-Riemannian 3-sphere is unstable.     

Improving directions of negative curvature in an efficient manner

In order to converge to second-order KKT points, second derivative information has to be taken into account. Therefore, methods for minimization satisfying convergence to second-order KKT points must, at least implicitly, compute a direction of negative curvature of an indefinite matrix. An important issue is to determine the quality of the negative curvature direction. This problem is closely related to the symmetric eigenvalue problem. More specifically we want to develop algorithms that improve directions of negative curvature with relatively little effort, extending the proposals by Boman and Murray. This paper presents some technical improvements with respect to their work. In particular, we study how to compute “good” directions of negative curvature. In this regard, we propose a new method and we present numerical experiments that illustrate its practical efficiency compared to other proposals.     

Differential operators associated with holomorphic mappings

Two differential operators which act on holomorphic mappings to complex projective space are studied. One operator is of second order and characterizes projective linear mappings. The other operator is of third order and may be viewed as a curvature. The two operators together play a role analogous to the Schwarzian derivative.A canonical approximation to a holomorphic mapping is defined, and a relationship between the approximation and the operators is derived. In the one variable case, this reduces to a classical result relating the Schwarzian derivative and the best Möbius approximation to a holomorphic function.     

Covariant derivative of the curvature tensor of pseudo-Kählerian manifolds

It is well known that the curvature tensor of a pseudo-Riemannian manifold can be decomposed with respect to the pseudo-orthogonal group into the sum of the Weyl conformal curvature tensor, the traceless part of the Ricci tensor and of the scalar curvature. A similar decomposition with respect to the pseudo-unitary group exists on a pseudo-Kählerian manifold; instead of the Weyl tensor one obtains the Bochner tensor. In the present paper, the known decomposition with respect to the pseudo-orthogonal group of the covariant derivative of the curvature tensor of a pseudo-Riemannian manifold is refined. A decomposition with respect to the pseudo-unitary group of the covariant derivative of the curvature tensor for pseudo-Kählerian manifolds is obtained. This defines natural classes of spaces generalizing locally symmetric spaces and Einstein spaces. It is shown that the values of the covariant derivative of the curvature tensor for a non-locally symmetric pseudo-Riemannian manifold with an irreducible connected holonomy group different from the pseudo-orthogonal and pseudo-unitary groups belong to an irreducible module of the holonomy group.     

Asymptotic Behaviors of Hyperbolic Hypersurfaces with Constant Affine Gauss�CKronecker Curvature

Considering two linked Monge?CAmpère equations that correspond to hyperbolic hypersurfaces with constant affine Gauss?CKronecker curvature, we obtain some asymptotic behaviors of affine hypersurfaces. In this paper we also extend Loewner?CNirenberg??s sharp second order derivative estimates for hyperbolic affine spheres to higher dimensions.     

Gradient estimates and the first Neumann eigenvalue on manifolds with boundary

By studying the local time of reflecting diffusion processes, explicit gradient estimates of the Neumann heat semigroup on non-convex manifolds are derived from a recent derivative formula established by Hsu. As an application, an explicit lower bound of the first Neumann eigenvalue is presented via dimension, radius and bounds of the curvature and the second fundamental form. Finally, some new estimates are also presented for the strictly convex case.     

Accelerations for global optimization covering methods using second derivatives

Two improvements for the algorithm of Breiman and Cutler are presented. Better envelopes can be built up using positive quadratic forms. Better utilization of first and second derivative information is attained by combining both global aspects of curvature and local aspects near the global optimum. The basis of the results is the geometric viewpoint developed by the first author and can be applied to a number of covering type methods. Improvements in convergence rates are demonstrated empirically on standard test functions.Partially supported by an University of Canterbury Erskine grant.     

Ruled surfaces with vanishing second Gaussian curvature

For a surface free of points of vanishing Gaussian curvature in Euclidean space the second Gaussian curvature is defined formally. It is first pointed out that a minimal surface has vanishing second Gaussian curvature but that a surface with vanishing second Gaussian curvature need not be minimal. Ruled surfaces for which a linear combination of the second Gaussian curvature and the mean curvature is constant along the rulings are then studied. In particular the only ruled surface in Euclidean space with vanishing second Gaussian curvature is a piece of a helicoid.