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# B数学(带目录)

1. 绪论。（1）运筹学的起源、发展。（2）运筹学的研究方法。（3）运筹学的主要分支。 （4）运筹学与管理科学。 1. Introduction. (1) Historical origins and development of operations research. (2) Research approaches. (3) Main branches of operations research. (4) Operations research and management science. 2. 线性规划及单纯形法。 （1）一般线性规划问题的数学模型。 （2）线性规划问题的图解法。 （2）单纯形法原理。（3）单纯形法的计算步骤。 （4）单纯形法的进一步讨论。大 M 单纯 形法，两阶段法，退化问题，检验数的几种表示法。 （5）数据包络分析。 （6）计算机求解法。 2. Linear programming and simplex algorithm. (1) Mathematical models of normal linear programming problems. (2) Graphs of linear programming problems. (3) Principles of simplex algorithm. (4) Further discussion of simplex algorithm. M-method, two-phase, degeneracy, and denotations of test numbers. (5) Data Envelopment Analysis. (6) Computer solutions. 3. 线性规划的对偶理论与灵敏度分析。（1）对偶问题的提出。（2）原问题与对偶问题。 （3）对偶问题的基本性质。 （4）影子价格。（5）对偶单纯形法。 （6）灵敏度分析。对偶理 论在灵敏度分析上的应用。（7）参数线性规划。

3. The theory of duality and sensitivity analysis in linear programming. (1) Proposal of duality. (2) Original problem and duality problem. (3) Basic qualities of duality problems. (4) Shadow price. (5) Simplex algorithm in duality. (6) Sensitivity analysis. Application of duality theory in sensitivity analysis. (7) Parametric linear programming. 4. 运输和指派问题。（1）运输问题及数学模型。（2）表上作业法。 （3）产销不平衡的运 输问题及其应用。 （4）指派问题及求解方法。 4. Transport problems and assignment problems. (1) Transport problems and mathematical models. (2) Table-manipulation method. (3) Transport problems in unbalanced production-marketing and its application. (4) Assignment problems and the solutions. 5. 整数规划与分配问题。 （1）整数规划的特点及作用。 （2）分配问题与匈亚利法。 （3）分 支定界法。 （4）割平面法。 （5）应用举例。 5. Integer programming and assignment problem. (1) Characteristics and functions of integer programming. (2) Assignment problem and Hungarian algorithm. (3) The branch and bound methods. (4) The cutting plane algorithm. (5) Applications and instances. 6. 目标规划。 （1）问题的提出与目标规划的数学模型。 （2）目标规划的图解分析法。 （3 ） 用单纯型法求解目标规划。 （4）求解目标规划的层次算法。 6. Goal Programming. (1) Proposals and mathematical models. (2) Graph-analytic method in goal programming. (3) Application of simplex algorithm in goal programming. (4) Hierarchical algorithms of goal programming. 7. 图与网络分析。（1） 图的基本概念与模型。 （2）树图和图的最小部分树。（3）最短路 径问题。（4）网络的最大流。 （5）最小费用流。（6）应用举例。 7. Graph and network analysis. (1) Basic concepts of graphs. (2) Tree graph and the tree graph of the minimum part of the graph. (3) Shortest path problem. (4) Maximum network flow. (5) Minimum cost. (6) Applications and instances. 8. 计划评审方法和关键路线法动态规划。 （1）PERT 网络图。 （2）PERT 网络图的计算。 （3） 关键路线和网络计划优化。 8. Plan reviews and dynamic programming of critical path method. (1) PERT network planning. (2) Computation of PERT network planning. (3) Critical path and network planning optimization. 9. 动态规划。（1）多阶段决策问题。 （2）最优化原理与动态规划的数学模型。 （3）离散确 定性动态规划模型的求解。 （4）离散随机性动态规划模型的求解。（5）一般数学规划模型 的动态规划解法。 9. Dynamic programming. (1) Multi-stage decision process. (2) Principle of optimality and mathematical models of dynamic programming. (3) Solution of discrete determined dynamic programming models. (4) Solution of discrete random dynamic programming models. (5) Dynamic programming solution of normal mathematical programming models. 10. 存贮论。 （1）引言。存贮问题的产生，存贮模型的建立。 （2）经济订货批量的存贮模型。 （3）具有约束条件的存贮模型。 （4）具有价格折扣优惠的存贮模型。 （5）动态的存贮模型。 （6）单时期的随机存贮模型。 （7）多时期的随机存贮模型。 （8）确定性的多梯次存贮模型。 10. Inventory theory. (1) Introduction. The rise of inventory problems and the proposal of inventory models. (2) Inventory model with economic order quantity. (3) Inventory model with restraint conditions. (4) Inventory model with discounts. (5) Dynamic inventory model. (6) Single-period stochastic inventory model. (7) Multi-period stochastic inventory model. (8) Deterministic multi-echelon inventory model. 11. 排队论。（1）排队服务系统的基本概念。 （2）输入与服务时间的分布。 （3）生灭过程。

（4）最简单排队系统模型。 （5）M/G/1 的排队系统。 （6）服务机构串联的排队系统。 （ 7） 具有优先服务权的排队模型。 （8）排队决策模型。 （9）排队系统的模拟。 11. Queuing theory. (1) Basic concepts of queuing service system. (2) Input and service time distribution. (3) Birth-and-death process. (4) A simplest queuing system model. (5) M/G/1 queuing system. (6) Service agencies tandem queuing system. (7) Priority queuing model. (8) Queuing decision model. (9) Simulation of queuing system. 12. 决策分析。 （1）引言。决策问题的背景，决策问题的提出。 （2）不确定型的决策分析。 （3）风险情况下的决策。 （4）贝叶斯决策。 （5）决策分析中的效用度量。 （6）Pareto 最优。 （7）层次分析法。 （8）多属性决策。 12. Decision analysis. (1) Introduction. Backgrounds and proposal of decision analysis. (2) Decision analysis of uncertainty. (3) Decision under risk. (4) Bayesian decision. (5) Utility measurement of decision analysis. (6) Pareto optimality. (7) The analytic hierarchy process. (8) The multiple attribute decision making. 13. 博弈论。 （1）引言。博弈论产生的背景，博弈模型的基本结构。 （2）完全信息静态博弈。 二人零和博弈模型，具有鞍点的博弈，混合策略，纳什均衡，用划线法求具有纯策略的纳什 均衡，混合策略下的纳什均衡。 （3）完全信息动态博弈。 （4）冲突分析简介。冲突分析，冲 突分析的简单模型。 13. Game theory. (1) Introduction. Background of the origin of game theory. The basic structure of game theory. (2) The static games of complete information. Two-person zero-sum game; fixed-point theorem; mixed strategy; Nash equilibrium; pure strategy Nash equilibrium by means of scoring method; mixed strategy Nash equilibrium. (3) The dynamic games of complete information. (4) An introduction to conflict analysis. Conflict analysis; simple model of conflict analysis.

《概率论与数理统计》 是高等学校经济管理类专业核心课程经济数学基础之一， 是研究随机 现象规律性的一门学科。通过本课程的学习，应使学生掌握概率论与数理统计的基本概念， 了解它的基本理论和方法， 从而使学生初步掌握处理随机现象的基本思想和方法， 培养学生 运用概率统计方法分析和解决实际问题的能力。 Probability Theory and Mathematical Statistics, which studies the regularity of random phenomenon, is part of economic mathematics, the core course for economy and management majors in higher education. The course is intended to acquaint the students with the basic concepts of probability theory and mathematical statistics, teach the students the principles and methodology, which equips the students with the necessary thinking and method to deal with random phenomenon and enables the students to apply the mathematical statistics to address practical problems. 该课程主要内容包括： This course mainly covers: 随机事件与概率 random events and their probability, 随机变量及其概率分布 random variable and its probability distribution, 多维随机 变量及其概率分布 multi-dimensional random variable and its probability distribution, 随机变量 的数字特征 the alphanumeric characters of random variable, 大数定律及中心极限定理 law of large numbers and central limit theorem, 数理统计的基本知识 the essential knowledge of mathematical statistics, 参数估计 parameter estimation, 假设检验 hypothesis testing, 回归分析 和方差分析 regression analysis and variance analysis. 1.随机事件及其概率。 （1）随机事件。 （2）随机事件的概率。 （3）古典概型与几何概型。 （4） 条件概率。 （5）事件的独立性。

1. Random event and its probability. (1) Random event. (2) Probability of random event. (3) Classical model of probability and geometric probability model. (4) Conditional probability. (5) Independence of event. 2.随机变量的分布与数字特征。 （1）随机变量及其分布。 （2）随机变量的数字特征。数学期 望、方差、标准差的概念，期望与方差的初等性质。 （3）常用的离散型分布。 （4）常用的连 续型分布。 （5）随机变量函数的分布。 2. Distribution of extraneous variables and the numerical characteristics. (1) Extraneous variables and its distribution. (2) The numerical characteristics of extraneous variables. Concepts of mathematical expectation, variance, and standard deviation; the elementary properties of expectation and variance. (3) Normal discrete distribution. (4) Normal continuous distribution. (5) Distribution of functions of random variables. 3. 随机向量。 （1）随机向量的分布。 （2）条件分布与随机变量的独立性。 （3）随机向量的函 数的分布与数学期望。 （4）随机向量的数字特征。协方差，协方差矩阵，相关系数，条件数 学期望，条件期望的预测含义。 （5）大数定律与中心极限定理。 3. Random vector. (1) Distribution of random vector. (2) Conditional distribution and the independence of random vector. (3) The distribution of functions of random vector and its mathematical expectation. (4) The numerical characteristics of random vector. Covariance, covariance matrix, correlation coefficient, conditional mathematical expectation, and the prediction meaning of conditional expectation. (5) Law of large numbers and central limit theorem. 4. 统计量及其分布。 （1）总体与样本。 （2）统计量。 （3）常用的统计分布。 （4）抽样分布。 4. Statistics and its distribution. (1) Population and samples. (2) Statistic. (3) Normal statistical distribution. (4) Sampling distribution. 5. 参数估计。 （1）点估计概述。点估计及其它的无偏性、有效性和相合性。 （2）参数的最大 似然估计与矩估计。 （3）置信区间。 5. Parametric estimation. (1) An introduction to point estimation. Point estimation and its unbiasedness, validity, and consistency (2) Maximum likelihood and method of moments of parameters (3) Confidence intervals 6.假设检验。 （1）假设检验概述。假设检验问题的提出，假设检验的基本思想和原理，假设 检验的一般步骤，检验的显著性水平与两类错误。 （2）单正态总体的参数假设检验。 （3）双 正态总体的参数假设检验。 （4）一般总体的参数假设检验。 （5）拟合优度 ? 检验与独立性
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and independence testing.

7.回归分析。 （1）一元线性回归模型及其参数估计。 （2）一元线性回归模型的检验。 （3）一 元线性回归的残差分析。 （4）一元线性回归的预测与控制。 （5）一元非线性问题的线性化。 （6）多元线性回归分析。回归系数的最小二乘估计，回归方程的显著性检验，多元线性回 归模型的预测。 7. Regression analysis. (1) Unary linear regression model and its parameter estimation. (2)

Testing of unary linear regression model. (3) Residual analysis of unary linear regression. (4) Prediction and control of unary linear regression. (5) Linearization of unary nonlinear equations. (6) Multiple linear regressions. Least square estimation of regression coefficient; significance test of regression equation; prediction of multiple linear regressions 龙永红主编： 《概率论与数理统计》 ，高等教育出版社，2001 年版。 Probability Theory and Mathematical Statistics, LONG Yonghong et al ed., Beijing: Higher Education Press, 2001.

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to geometry, directional derivative and gradient, extreme value of multivariable functions and its derivation, Taylor's formula for binary functions, least square method, Exercise 8 重积分：二重积分的概念与性质，二重积分的计算法，三重积分，重积分的应用，含参变量 的积分，总习题九 Multiple integral: concept and properties of double integral, operation of double integral, triple integral, application of multiple integral, integral including parameters, Exercise 9 曲线积分与曲面积分：对弧长的曲线积分，对坐标的曲线积分，格林公式及其应用，对面积 的曲面积分，对坐标的曲面积分，高斯公式、通量与散度，斯托克斯公式、环流量与旋度， 总习题十 Line integral and surface integral: line integral over arc length, line integral over coordinates, Green's theorem and its application, surface integral over areas, integral over coordinates, gauss formulas and flux and divergence, Stokes' formula, circulation and curl, Exercise 10 无穷级数：常数项级数的概念和性质，常数项级数的审敛法，幂级数，函数展开成幂级数， 函数的幂级数展开式的应用， 函数项级数的一致收敛性及一致收敛级数的基本性质， 傅里叶 级数，一般周期函数的傅里叶级数，总习题十一 Infinite series, concept and properties of constant term series, testing method of constant term series, power series, expansion of a function to a power series, Fourier series of general periodic functions, application of power series expansion, uniform convergence of function series and basic properties of uniform convergence series, Fourier series, Fourier series of General periodic function, Exercise 11. 微分方程：微分方程的基本概念，可分离变量的微分方程，齐次方程，一阶线性微分方程， 全微分方程，可降阶的高阶微分方程，高阶线性微分方程，常系数齐次线性微分方程，常系 数非齐次线性微分方程，欧拉方程，微分方程的幂级数解法，常系数线性微分方程组解法举 例，总习题十二 Differential equations: basic concept of differential equation, differential equation of separable variables, homogeneous equations, linear first-order differential equation, total differential equation, reducible higher differential equations, higher order linear differential equations, homogeneous linear differential equations with constant coefficients, nonhomogeneous linear differential equation with constant coefficients, Eulerian equation, power series solution of differential equation, examples of solution to system of linear differential equations with constant coefficients, Exercise 12

? ? ? ? ? ? ? ? ? ? ? 研究函数与极限的基本方法 Basic approaches for studying functions and limits 一元函数微分法及其应用 Single variable differential calculus and its application 一元函数积分法及其应用 Single variable integral calculus and its application 多元函数微分法及其应用 Multivariable differential calculus and its application 法及其应用 Multivariable integral calculus and its application 级数的判敛、求和及展开法 Convergence checking summation and expansion of series 几类常微分方程的求解法 Method of solving several types of ordinary differential equation 高等数学中的方法综述 An overview of methods used in higher mathematics 数学建模方法 Mathematical modeling method 数值计算方法 Numerical calculation Method 近代分析概念简介 Introduction to the concept of modern analysis

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higher difficulty and guiding meaning, in order to offer students some self-learning materials, and to provide worthy reference information for teachers for their review and preparation to set papers for the examination. 行列式: 二阶和三阶行列式, n 阶行列式定义，行列式性质，行列式按行展开定理，克莱姆法 则，习题一 Determinant: the second-order determinant, the third-order determinant, the definition of the nth-order determinant, properties of determinants, Cramer's rule, theorem of expansion of a determinant in row, Exercise 1 矩阵：高斯消元法与矩阵的初等变化,矩阵的运算，特殊矩阵，逆矩阵，分块矩阵，利用初 等变换求矩阵，矩阵的秩 Matrix: Gaussian elimination and elementary transformation of matrix, matrix operation, special matrices, inverse matrix, partitioned matrix, elementary transformation, solution of matrix by elementary transformation. rank of matrix 线性方程组：n 维向量及其线性运算，向量组的线性相关性，向量组的秩，矩阵的秩与向量 组秩的关系，齐次线性方程，线性方程组解的结构，习题三 System of linear equations: n dimensional vector and its linear operation, linear correlation of vector group, rank of vector group, correlation between rank of matrix and rank of vector group 矩阵的对角化：矩阵的特征值和特征向量，相似矩阵和矩阵对角化，向量的内积和施密特正 交化，实对称矩阵的对角化 Diagonalization of matrix: eigenvalue and eigenvector of matrix, similar matrix and diagonalization of matrix, inner product of vector and schmidt orthogonalization 二次型：二次型极其矩阵表示，化二次型为标准型，惯性定理与正定二次型，习题五 Quadratic form: quadratic form and its matrix representation, converting quadratic form into standard form, inertial theorem and positive definite quadratic form, Exercise 5 线性空间与线性变换：线性空间的定义与性质，基、坐标及其变换，线性空间的子空间，线 性变换，线性变换的矩阵表示 Linear space and linear transformation: definition and properties of linear space, base, coordinate and their transformation, subspace of linear space, linear transformation, matrix representation of linear transformation 欧式空间与酉空间：向量的内积与欧式空间，标准正交基，正交变换，向量到子空间的距离. 最小二乘法，酉空间介绍 Euclid space and unitary space: inner product of vector and Euclid space, orthonormal basis. orthogonal transformation, distance from vector to subspace, method of the least squares, introduction to unitary space

Ordinary Differential Equations by Wang Gaoxiong, Zhou Zhiming & Wang Shousong, 3rd edition, Beijing: Higher Education Press, Jul. 2006(reprinting in 2009).