数学公式

Events,Operation and Relation of Sets, Classical Probability, Geometrical Probability , Statistical Stability of a Frequency, Axioms of Probability, Conditional Probability, Total Probability Theorem, Bayes' Rule,Independent Events,Independent Repeated Trials, One Dimensional Random Variables, Discrete Random Variables, Distribution Function of a Random Variables , Continuous Random Variables, Normal Distribution, Distribution of a Function of a Random Variable, Multidimensional Random Variables, Joint Distribution Function, Marginal Distribution Function,Discrete Two—Dimensional Random Variables,Continuous Two—Dimensional Random Variables, Independent Random Variables, Distribution of Functions of Random Variables,Expectation,Variance, Covariance, Coefficient of Correlation, Bivariate Normal Distribution, Law of Large Numbers, The Central Limit Theorems, Sample and Population ,Chi—Squared, T and F Distributions , Sampling Distributions , Point Estimation , Interval Estimation , Testing Hypotheses , A Test of Significance for Parameters in a Single Sample From a Normally Distributed Population , A Test of Significance for Parameters in Two Sample From Normally Distributed Populations .

本课程的主要内容：概率的概念与运算、随机变量及其分布、随机变量的数字特征与极限定理、数理统计的基本概念、估计和检验的基本方法，随机事件与概率随机事件、事件的关系与运算、几何概率、统计概率等，条件概率、全概率公式、贝叶斯公式、事件的独立性、二项概率公式，随机变量的概念、离散型随机变量、随机变量的分布函数、连续型随机变量、随机变量函数的分布，多维随机变量及其分布函数、边缘分布函数、随机变量的独立性、二维随机变量函数的分布，数学期望、方差、协方差和相关系数、大数定律、中心极限定理，总体与样本， X 2-分布、 t-分布和 F-分布，统计量及抽样分布，假设检验的基本概念、单个正态总体参数的显著性检验、两个正态总体参数的显著性检验。

The paper uses precise step to produce the continuity function of water hammer mathematic model, not only gains more precise water hammer pressure function and water hammer wave speed functionbut also founds correct water hammer mathematic modelContinuity EquationMovement EquationAfter ignoring subordinate facts, function (1) and (2) translate into the current applied water hammer pressure function and water hammer wave speed function.

本论文对水击数学模型中的连续性方程重新进行了严谨的推导，不仅得到了更加准确的水击压强和水击波速的计算公式也建立了正确的水击数学模型连续性方程运动方程当忽略次要因素以后，公式(1)和(2)即可转化为当前应用的水击压强计算公式和水击波速计算公式。

Main creative points of the thesis are as following:1. It is the first time to apply 2-D and 3-D turbulent model of the air-water two phases flow to the study on hydraulic properties of groyne. It makes description of the turbulent flow field to be possible precisely. This offers a more accurate method to determine the water surface in channel improvement.2. Based on the results of experiment and simulation for 3-D turbulent model of air-water two phases flow, a new formula to compute the height of damming is built.3. Based on the 2-D mode and the formula to compute the height of damming, the subdivision method of 2-D computation for water surface at upstream and downstream.

本文的创新工作主要是：首次将三维水气两相流紊流数学模型应用于丁坝水动力学特性研究，使丁坝附近流场和紊动场的精细描述成为可能，为航道整治中水面线确定提供了一个更为准确的方法；(2)基于水槽实验和三维水气两相流紊流数学模型数值模拟结果，建立了新的丁坝前奎水高度计算公式；(3)基于上述奎水高度计算公式和本文二维水面计算数学模型，首次提出了丁坝上下游二维水面分段计算方法。

Professor Chen Xingshen achievements in order to affirm and encourage the development of our young mathematicians to contribute to the cause of China's Mathematical What he has in-depth introduction to the mathematics outside the field of theoretical physics to become an important tool G introduced geometric structure of the equivalent problem, a 20th century one of the major developments of mathematics, Integral geometry for the generalized basis, access to basic kinematics formulas.

为了肯定陈省身教授的功绩，激励我国中青年数学工作者对发展我国的数学事业做出贡献他引入的东西已深入到数学之外的领域，成为理论物理的重要工具，引进了G 几何的结构，研究其等价问题，成为20世纪后数学的主要发展之一，为广义的积分几何奠定基础，获得基本运动学公式。

Moreover,we also gave the proofs of generalization formula of integration by parts,the Green s formula of first type and the Green s formula of second type by using the divergence theorem .

首先指出数学分析中反映二重积分和第二类平面线积分联系的Green公式可以看成是散度定理的一种特殊情形，然后利用散度定理给出广义分部积分公式、第一Green公式和第二Green公式的证明

In this paper, the Lagrange interpolation formula asked a class function expression question, the determination function to the higher mathematics in the elementary mathematics in some value scope question, the equality proof question, the multinomial factorisation question, to ask to have the Finite series to pass the formula question, the certificate any n function at will n+1 different spots function value maximum value minimal problem and so on six aspects should serve as the simple introduction, and has separately made the comparison with these six aspects elementary mathematics method, thus solved some operation problems in elementary mathematics.

摘要本文对高等数学中Lagrange插值公式在初等数学中的求一类函数表达式问题、确定函数在某点的取值范围问题、恒等式的证明问题、多项式的因式分解问题、求有穷数列的通项公式问题、证明任一n次函数的随意的n+1个两两不同点的函数值的最大值的最小值问题等六个方面的应用作了简单介绍，并与这六个方面的初等数学方法分别作了比较，从而解决一些初等数学中的运算问题。

Theory; the spatial meshing theory includes comparative motion, comparative differential and conjugative curved surface; and the application of meshing theory includes gear drive and worm drive. The spatial meshing theory is the main part of the course.

课程的重点讲解内容有曲线的参数方程、切线、法面、弧长、曲率、空间曲线的基本公式、挠率及平面曲线的基本公式等；曲面第一和第二基本公式，法曲率、主方向和主曲率、欧拉公式、短程挠率、欧拉公式和贝特朗公式推广、相对法曲率和相对短程挠率等；刚体的绝对和相对运动速度、相对微分和绝对微分、相对速度和相对微分、轨迹曲面的法曲率和短程挠率等；空间共轭曲面的啮合条件、诱导法曲率、两类界限点、等距共轭曲面、空间啮合的二次接触原理等；蜗杆传动的数学模型建立及啮合特性分析方法等。

With the theory of mode split, the distortion of resonance curve of hydrophone is explained and math formulas describing the distortion of resonance curve are given, and then the interference process of hydrophone after mode split is simulated by MathCAD and the theoretical curve is plotted.

用模式分裂的原理解释了水听器的谐振曲线畸变现象，提出了描述谐振曲线畸变的数学公式，用MathCAD软件仿真了模式分裂后水听器的干涉过程并绘制了理论曲线。

E. textures that are created by a mathematical formula, images, or environment maps to create the impression of reflections and refractions.

这可能是程序纹理即通过数学公式创建的纹理，图片或者环境映射来制造对反射和折射的效果。

A rational explanation is given by application of modified Langmuir adsorption theory based on the experimental results.

对以上实验结果用修正的Langmuir吸附理论提出合理的解释，建立了理论模型，并推导出了数学公式。

I'm strongly against the death penalty — it's an eye for an eye.

我不赞成死刑——这是以牙还牙的报复行为。

And to get you the support you need, we're enlisting all elements of our national power: our diplomacy and development, our economic might and our moral suasion, so that you and the rest of our military do not bear the burden of our security alone.

并给你们所须的支援，我们正徵召国家所有各种的力量：我们的外交及发展，我们的经济力量与道德劝说，所以你们与其他军人不须要孤独地负起国家安全的责任。