余弦定理的推广

众所周知的余弦定理是指下面的数学命题: 设△ABC的三边的长分别为a、b、c,三个内角依次为A、B、C,则 当△ABC为直角三角形时,由余弦定理可得出勾股定理。 可是,在三维欧氏空间,对于四面体是否亦有类似定理呢?答案是肯定的。 事实上,我们有如下令人兴奋的结果:     

四面体中的“类正弦定理”

文给出了直角四面体类似于直角三角形的一些性质，文给出了四面体中的余弦定理．受此启发，经过研究，本文得到四面体中的“类正弦定理”．     

正弦定理与余弦定理的关系

一般中学教科书正弦定理与余弦定理都是分别加以证明的。这两个定理之间互有联系。如已证明正弦定理,余弦定理可成为正弦定理的推论。反之,如余弦定理先成立,正弦定理亦可成为余弦定理的推论。因此两者不是独立的。     

凸n边形中的正弦定理、余弦定理和射影定理

众所周知,正弦定理、余弦定理和射影定理是关于三角形基本元素(边和角)奖系的主要恒等式。它们在解三角形中扮演极为重要的角色。本文拟藉助于复数这个有力的工具将它们予以推广,得到凸n(≥3)边形中的正弦定理、余弦定理和射影定理。     

n维欧氏空间的Child不等式

ｎ维欧氏空间的Ｃｈｉｌｄ不等式马统一（甘肃省煤炭工业技校白银市７３０９１９）本刊文［１］中，杨世国先生用高等数学的方法将三角形的Ｃｈｉｌｄ不等式推广到三维欧氏空间的四面体中，获得了如下结果：定理１设Ｐ为四面体Ａ１Ａ２Ａ３Ａ４内部任一点，点Ｐ到各顶点Ａ...     

关于正弦定理余弦定理的注记

在试验修订本中，正弦定理和余弦定理是利用“向量”这个工具证明的，与传统方法相比，正弦定理的难度加大了，而余弦定理的证明则很简洁，这说明用“向量”这个工具解题，有可能简便，也可能复杂，因此在处理问题时要有所取舍。关于正弦定理、余弦定理，要注意以下几点：     

E~n中n维余弦定理和正弦定理的新证明

利用Grassmann代数的理论与方法,给出了n维欧氏空间En中n维单形第二余弦定理一种简单的证明.然后利用第二余弦定理给出了n维单形正弦定理一种新的简单证明.     

关于n维欧氏空间中Vasic的不等式

关于ｎ维欧氏空间中Ｖａｓｉｃ的不等式马统一（甘肃煤炭工业技校７３０９１９）最近，文［１］将欧氏平面上著名的Ｖａｓｉｃ不等式推广到三维欧氏空间中，获得了关于四面体二面角的Ｖａｓｉｃ不等式，同时还获得了比此更强的另一个不等式．该文的主要结果是：定理１设四...     

解三角形

1．本单元重、难点分析 本单元的重点：（1）正弦定理和余弦定理及其推导；（2）正弦定理和余弦定理的应用．     

解斜三角形

1本单元重难点分析 本章是在有了三角函数的基础知识之后,运用平面向量的思想推导出三角形的正弦定理和余弦定理,以及应用正、余弦定理求解三角形及有关实际问题.因而本章的重点是掌握正弦定理和余弦定理的推导及实际应用.难点有两个,一是理解用向量法推导正弦定理和余弦定理;二是在实际应用中如何建立相关的三角函数模型.本章运用的重要数学思想方法有数形结合思想、函数和方程的思想等.     

具有正Ricci曲率和体积Pinching流形的一个球定理

M~n是一个紧致无边单连通的n(≥3)维Riemannian流形,S~n为R~(n+1)中的单位球面.本文所关注的流形满足截面曲率K_M≤1,而Ricci曲率Ric(M)≥(n+2)/4以及体积V(M)≤3/2(1+η)V(S~(2n)),这里η是一个仅和维数n有关的常数.最终将给出一个具有正的Ricci曲率的球定理新证明.     

两个单形混合顶点角的广义正弦定理及应用（英文）

Some related problems of two n-dimensional simplices which are on an(n- 1)-dimensional hypersphere are investigated and a sine theorem of the k-dimensional mixed vertex angles which are defined in this paper is given. This result is a generalization of the sine theorem established. By using the generalized sine theorem, we present some new interesting geometric inequalities involving the k-dimensional vertex angles of each n-simplex and the k-dimensional mixed vertex angle of two n-simplices. These results can improve some recent results.     

单形正弦定理的再推广

本文利用Grassmann代数建立n维欧氏空间中单形的k级n-k s维顶点角的概念，在此基础上对单形的正弦定理再作推广，并获得单形新的一类体积公式和一个几何不等式．     

Biharmonic Riemannian submersions from 3-manifolds

An important theorem about biharmonic submanifolds proved independently by Chen-Ishikawa (Kyushu J Math 52(1):167?C185, 1998) and Jiang (Chin Ann Math Ser. 8A:376?C383, 1987) states that an isometric immersion of a surface into 3-dimensional Euclidean space is biharmonic if and only if it is harmonic (i.e, minimal). In a later paper, Caddeo et?al. (Isr J Math 130:109?C123, 2002) showed that the theorem remains true if the target Euclidean space is replaced by a 3-dimensional hyperbolic space form. In this paper, we prove the dual results for Riemannian submersions, i.e., a Riemannian submersion from a 3-dimensional space form into a surface is biharmonic if and only if it is harmonic.     

A generalization of Bonnet’s theorem on Darboux surfaces

The well-known Bonnet theorem claims that, on a Darboux surface in three-dimensional Euclidean space, along each line of curvature, the corresponding principal curvature is proportional to the cube of another principal curvature. In the present paper, this theorem is generalized (with respect to dimension) to n-dimensional hypersurfaces of Euclidean spaces.     

Cones of Functions with Monotonicity Conditions for Generalized Bessel and Riesz Potentials

An embedding theorem for spaces of functions of positive smoothness defined on irregular domains of n-dimensional Euclidean space in Lebesgue spaces is proved. The statement of the theorem depends on the geometric parameters of the domains of the functions.     

Bernstein type theorem of minimal Lagrangian graphs in quaternion Euclidean space H^n

In this paper, a Bernstein type theorem for minimal Lagrangian submanifolds in quaternion Euclidean space Hn is studied.     

关于高维蝴蝶定理

利用距离几何方法将古典的蝴蝶定理推广到n维欧氏空间一般二次超曲面情形,同时也得到高维蝴蝶定理的逆定理.     

欧氏空间中完备极小子流形

在一个类似于稳定不等式的条件下,得到了欧氏空间中完备极小子流形的Bernstein型定理.我们的结果部分推广了Li H.Z.和Wei G.X.的定理.     

Hyperelliptic Jacobians as billiard algebra of pencils of quadrics: Beyond Poncelet porisms

The thirty years old programme of Griffiths and Harris of understanding higher-dimensional analogues of Poncelet-type problems and synthetic approach to higher genera addition theorems has been settled and completed in this paper. Starting with the observation of the billiard nature of some classical constructions and configurations, we construct the billiard algebra, that is a group structure on the set T of lines simultaneously tangent to d−1 quadrics from a given confocal family in the d-dimensional Euclidean space. Using this tool, the related results of Reid, Donagi and Knörrer are further developed, realized and simplified. We derive a fundamental property of T: any two lines from this set can be obtained from each other by at most d−1 billiard reflections at some quadrics from the confocal family. We introduce two hierarchies of notions: s-skew lines in T and s-weak Poncelet trajectories, s=−1,0,…,d−2. The interrelations between billiard dynamics, linear subspaces of intersections of quadrics and hyperelliptic Jacobians developed in this paper enabled us to obtain higher-dimensional and higher-genera generalizations of several classical genus 1 results: Cayley's theorem, Weyr's theorem, Griffiths-Harris theorem and Darboux theorem.