ECON6003_Mathematical Methods of Economic Analysis_2009 Semester 1_Final Practice

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Sciences物理海洋学Physical Oceanography海洋化学Marine Chemistry海洋生理学Marine Biology海洋地质学Marine Geology地球物理学Geophysics固体地球物理学Solid Earth Physics空间物理学Space Physics地质学Geology矿物学、岩石学、矿床学Mineralogy, Petrology, Mineral Deposit Geology地球化学Geochemistry古生物学与地层学（含古人类学）Paleontology and Stratigraphy《s d u d s gp SL e o a n -h r o p o o g s蔷睡旨渝犊structura一G e o o g y W旧^咨渝驱 QuaternaryGeoogy肝替犊E o o g y»售驱 Botany理替犊zooogy肝»驱p h y s o'o g y美 肝肝替^H l r o b o'o g y寿肝替喘M nr o b o'o g y#险府替^Neurobo'ogy 海流瓠Gen&cs滩珊府替犊Dffivoopmenta-E o o g y苗窗肝替瓠ce=E o og y府替^摩血冷*府替犊Bo'chem一%r yandM oe c u fil r B o'o g y 肝替替»琳 B o'p h y w c s 府辨嘿Ecoogy弁漆翠琳 systemsscs'nce渊^ffl ^s y a s m sTheory弁整牛M 4rr *aSystemsAnawsmandIntegrfilaon弟株»丹冷Estoryofscs'nceandTechnoogyEngseersg*株M *h a s.c s—漆*^血*琳融®GenerEandFimdamenta一 M*han_cs回皆*株S O -EMechaMcs盖寺*犊F c a Mechfi>n_csHM *4»E NGSEERSGM*HAN _CS$l M H ffiMechan-ca一 Engseeling ^M a j t ^M皿包^Mechanka- Manufacture and Au&ma£*<m^M t&f H ffi M e c h a rt>r o n n E n g 3$r 3g ^M ^¥JIr »^Mechan_ca- 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2019美国商科留学：杜克大学经济学项目介绍杜克大学经济学属于美国商科留学大类，相信不少童靴想了解，今天，小编就为大家揭开杜克大学经济学项目的面纱。

杜克大学位于美国北卡罗来纳州的杜伦市，是美国南部最好的私立大学，在2018年US News美国大学排名中位列第9。

虽然比起哈佛耶鲁这类常春藤名校来说，杜克大学不到二百年的校史略显“年轻”，但无论是学术水平还是其他方面都不逊于常春藤名校。

从杜克大学走出的名人遍布于各个行业，比如美国第37任总统理查德·尼克松，苹果公司现任CEO蒂姆·库克，以及2011年NBA状元秀凯里·欧文。

如果各位有意学商科的小僧想跟这些名人做校友的话，不妨跟小编一起来了解一下杜克大学的经济学项目。

项目简介杜克大学经济学项目的全称是MA degree in economics，由圣三一学院的艺术与科学学院下的经济学部开设【注意：该项目并不属于那个著名的“浮夸”(Fuqua)商学院】，学生可以通过该项目的学习对经济学的各个分支，包括计算经济学，经济分析和金融经济都有所掌握，进而提升在金融行业中找工作或者申请PhD的竞争力，项目鼓励学生去选择博士级别的相关课程。

也有很多学生跨部门选择一些诸如政治学，公共政策，金融，统计以及数学此类的课程。

课程设置此项目需修够至少30学分经济学或其相关领域的课程，其中至少15学分(5门课程)经济学部开设的课程(ECON打头)。

核心课程包括宏微观经济学课程，学生可从以下课程中任意选择，修满9学分即可。

Macroeconomic Theory微观经济学Advanced Microeconomic Analysis宏观经济学理论Advanced Macroeconomics II高级微观经济学分析Applied Econometrics in Microeconomics (Cannot double count to fulfill econometrics requirement.)微观应用计量经济学Game Theory with Applications of Economics and Other Social Sciences博弈论Economic Growth经济增长International Monetary Economics国际货币经济学Industrial Organization产业经济学Special Topics in Economics: General Equilibrium Theory and Financial Markets.专题：经济学中的一般均衡理论和金融市场Microeconomic Analysis I&II微观经济学分析Ⅰ和ⅡMacroeconomic Analysis I&II宏观经济学分析Ⅰ和Ⅱ和三门经导师许可后可选的专题课程：Special Topics in Applied Microeconomics应用微观经济学Special Topics in Macro International Finance宏观国际金融Special Topics in Economic Theory经济理论核心课程还包括9学分的计量经济学，计算方法，计算机科学，数学和/或统计学相关课程，学生须从以下课程中任选一门：Introduction to Econometrics计量经济学简介Time Series Econometrics时间序列计量经济学Applied Econometrics in Microeconomics应用计量经济学(微观)Empirical Methods in Financial Econometrics金融计量经济学实证方法Econometrics I&II计量经济学Ⅰ和ⅡSpecial Topics in Economics: Microeconometrics Tools专题：计量经济学工具Special Topics in Econometrics, with approval计量经济学专题(需经导师许可)接下来再从计量经济学，计算方法，计算机科学，数学和/或统计学课程中任选两门。

云南怒江水电开发中经济发展与生态保护的悖论及破解冯芸(昆明理工大学社会科学学院,云南昆明650224)摘要 深入分析了怒江水电开发中经济发展与生态保护两难悖论的典型性和特殊性,在研究经济发展与生态保护之间关系的基础上,提出了在保护的前提下实现怒江水能资源开发中经济发展与生态保护“双赢”的对策与建议。

关键词 怒江水电开发;经济发展;生态保护;悖论及破解中图分类号 X171 文献标识码 A 文章编号 0517-6611(2008)34-15188-02Pa ra do x b e tw e e n th e E c on om ic D e v e lo pm en t an d E c o lo g ic a l Pro te c tio n in th e H y dro pow e r D e v e lo pm e n t o f Nu jian g R iv e r o f Y un n a n Pro v in c e an d Its D e c la s s ific a tio nFENG Yun (S ch oo l o f S ocia l S cien ce s ,K u nm in g U n iv er sity o f S cien ce an d E n g in ee rin g ,K u nm in g ,Y u nn an 650224)A b s tra c t T h e typ ica lity and pa rticu la r ity o f tw o -d ifficu lt pa radox be tw eenth e econ om ic deve lopm en t and en v ironm en t pro te ction inth e h ydropow e r de-ve lopm en t o f N u jian g R ive r w ere deep ly an a ly zed .B ased onstudy i n gth e re la tion sh ip be tw eenth e econ om ic deve lopm en t an d en v ironm en t p ro tection,so m e cou n te rm ea su re s an d su gg estion s for rea lizin g th e dou b le w in s o f econ o m ic deve lopm en t an d en v ironm en t pro te ctioninth e w a te r en e rg y resou rce exp lo ita tion o f N u jian g R iv er un de r th e prem ise o f p ro tection w e re u t fo rw ard.K e y w o rd s H ydyopow e r dev e lopm en t o f N u jian g R ive r ;E con om ic dev e lopm en t ;E co lo g ica l pro te ction ;P a radox an d its de class ifica tion作者简介 冯芸(1972-),女,云南昆明人,讲师,从事区域可持续发展研究。

Econ1320期中攻略更新一：关于使用Lilliefors Test或K-S test检测正态性的问题，在将样本值调整到标准正态形态，使用x-μ/δ时，样本标准差δ的计算应是(x-xbar)^2/n-1，如是进行无偏估计，大家可回忆统计学或概率论内容。

此文原因有三：1、我好为人师；2、我闲得蛋疼；3、此科我已学过对我没有竞争性。

如怀疑此文动机，请参考如上三条，任何高尚特性，本人一概敬谢不敏，自甘鄙陋。

前言1320在排序上是1310的后续课程，对于华电2+2的同学而言，统计学或概率论及其数理统计课程和本课上的衔接是比较吃力的，甚或根本忘了，这并不稀奇。

不过相信5周的课程之后，大家基本能达到这样的共识：此科授课过程中对1310知识的假设不多，即不会突兀地冒出一个概念而不加以解释，譬如t-检验、p值的含义、z-分布的特性等等，课程中都作出了比较详细的解释。

另，此门课极重应用（p.s. 对于学过2300的同学，这一点和2300有差异，2300的最终考核中对数理考核较严而应用较松），于数理证明几乎不详细讨论，所以对号入座，确定题目的考核内容后，假设检验过程中死记五步法步骤而不乱，注意calculated值公式细节，注意critica值自由度、置信度，则不难获得高分。

对于较细的概念题目，我也见过，我会在最后的给出一个例子，如若嫌麻烦，舍弃亦是简便之法。

以下内容中红色的部分我建议进入考场能开始写时（也就是听instruction，收到命令填签到卡、填姓名学号等，之后，听到命令开始Perusal 可以开始在草稿纸上写时写在草稿纸上，以提醒自己）；蓝色的部分是我自己的一些数理认识，有需求的同学可以看，也可以跟我讨论（我可能理解错误），而想免除麻烦的同学跳过即可。

正文首先，要读懂Kadd的Output！否则给出来了不认得不会用就麻烦了....在复习时，我曾经的Pass leader给出这样的建议：要总结检验名称、作用、假设（假设主要考察一些概念型选择或简答）、五步法（从H0、H1，到Decision Rule[hereafter DR]，Test Statistics，Decision，Conclusion）的写法。

⿇省理⼯化⼯数值分析第六课10.34, Numerical Methods Applied to Chemical EngineeringProfessor William H. Green Lecture #6: Modern Methods for Solving Nonlinear Equations.1D-Problemunknown: T of reactor f(x) = 0Q rxn exp(-Ea /RT ) + h(T – T a ) + c(T 4 – T a 4) = 0heat of reaction convection radiation (+) (-) (-)steady state temperatures Make a plot with MATLAB *nethe a t.m* function qdot = netheat(T) % computes the net heating rate of a reactor % qdot = 0 at the steady state qdot = Q.*exp(-Ea/(R.*T)) + h.*(T-Ta) + c.*(T.^4-Ta.^4);Figure 2. Professor Green modified variables Q and c until the plot looked likethe one above. Increased Q and decreased c.T o solve for steady state zerosf(T Figure 1. 1D problem Q = -2e-5; Ea = 5000; R = 1.987; h = 3; Ta = 300; c = 1e-8; Tvec = linspace(300,3000)qdot = netheat(Tvec) plot(Tvec,qdot) Figure 3. Have computer bracket in and find smallrange where plot goes from negative to positive.Bisection10.34 Numerical Methods Applied to Chemical EngineeringLecture 6 Prof. William GreenPage 2 of 4start a,b such that f(a)<0 and f(b) < 0 2b a x +=Figure 4. Funif f(x) · f(a) > 0 a = xelse b = xThis is a problem of TOLERANCEif((b-a) < tol) stopTypes of tolerance Absolute tolerance Relative tolerance atol: has unitsif |f(x)| < atol·f rtol: if(b-a) < rtol*|a| has to be BIG numberIn MATLAB while abs(b-a) > atolx x = (a+b)/2 if f(x)·f(a) > 0 a = x else b = x end *bisect.m* function x = bisect(f,a,b,atolx,rtolx, atolf) %solves f(x) = 0 while abs(b-a) > atolx x = 0.5*(b+a); if((feval(f,x)*feval(f,a))>0) a =x; else b=x; end endCommand Window x = bisect(@netheat,300,2000,0.1,0,0) x = 1.2373e+003CHECK: netheat(1237) = -1.0474 í closeKeep in mind: never get actual solution, but can come closeWe can change tolerances to improve results. ? while(abs(b-a)>atolx)&&(abs(b-a)>(rtolx*abs(a)))x = 0.5*(b+a); AND: must satisfy both conditions if(a bs(fev a l(f,x))x = 1.2363e+003 looser tolerance gives less accurate answerBisection cuts interval by 2 each timeEvery time we cut 3 times, we lose a sig figIn bisection, time grows linearly with the number of significant figures.a < x true < bx true = x soln ± b-a/2Newton’s Method (1-D)evaluates slope of f(x)next guess is the x new that satisfies f(x new)=0for a line from f(x guess) with the slope at f(x guess)Figure 5. Newton’s Method.For a good guess Newton’s method doublesthe number of significant figures after everyiteration; however, we lose robustness ifguess is poorf(x) = f(x0)+f’(x0)*(x-x0)+O(Δx2)0 = f(x guess)+f’(x guess)*(x-x guess)If f’(x guess) ≈ 0 -- doesn’t workx new = x guess – f(x guess)/f’(x guess)Figure 6.NO intersectionAnother drawback is one needs a derivative of the function. Secant Methodsame as Newton’s, but uses f’(x) approximate]1[][]1[][)()()('=kkkkapproxxxxfxfxfBisection method works only for 1D problems, but Newton/Secant can be used for problems with greater dimension 10.34 Numerical Methods Applied to Chemical Engineering Lecture 6 Prof. William Green Page 3 of 4 Broyden’s Method (Multi-dimensional) F(x) = F(x 010.34 Numerical Methods Applied to Chemical EngineeringLecture 6 Prof. William GreenPage 4 of 4f(x) = 0 approx J = B 2][1||||x B BΔ+=+k ][k Outer Product:ΔΔΔΔΔΔ (32221)2312111x F x F x F x F x F x FNewton’s Method (Multi-dimensional)O = F(x 0)+J(x 0)·(x-x 0)J*Δx = -F(x 0) B [k]Δx = -FLU LU [k+1] without redoing factorization Done in detail in homework problem.。

MCM备用数学方法中英文对照MCM临近了，专业规范的表达是必不可少的，大家都来热热身吧！各位模友们自己有想到的数学方法或相关方面的专业词汇也可以贴出中英对照的版本！[B]计算数学方法：[/B]内插法：interpolation method有限元方法：method of finite element有限差分法：method of finite difference曲线拟合法：method of curve fitting迭代法：method of iteration误差分析法：error analysis样条函数法：method of spline function最小二乘法：method of least square[B]概率论方法：[/B]可靠性分析法：method of reliability analysis时间序列分析法： method of time-series analysis抽样调查法：method of sampling investigation非参数统计法：method of nonparametric statistics实验设计法：method of experiment design故障分析法：method of fault analysis相关分析法：method of correlation analysis测度论方法：method of measure theory统计假设检验法： method of statistical hypothesis testing随机服务法（排队论法）：stochastic service system数理统计法：method of mathematical statistics 蒙特卡洛法：Monte Carlo method[B]运筹学方法：[/B]共轭函数法：method of conjugate function动态规划法：method of dynamic programming网络法：method of network优选法：method of optimum seeking图论法：method of graph theory爬山法：method of climbing线性规划法：method of linear programming罚函数法： method of penalty function统筹法：method of overall planning乘子法：method of multiplier最速下降法：method of steepest descent整数规划法：method of integer programming。

essential mathematics for economicanalysisEssential Mathematics for Economic AnalysisMathematics is a crucial tool for understanding the complex relationships and patterns in economics. It helps economists to model the economy, make predictions, and measure the impact of policies and events.Essential Mathematics for Economic Analysis is a fundamental concept in economics that involves a range of mathematical techniques used to analyze economic data and model economic phenomena. This article will outline some of the essential mathematical tools used in economic analysis.1. CalculusCalculus is a core tool used in economics for modeling relationships between variables. It is used to describe how the economy changes over time and to analyze the effects of different policy interventions. Calculus is used to calculate derivatives and integrals, which are used to find rates of change, slopes, area under curves, and optimization.2. Linear AlgebraLinear algebra is used in economics to model and solve complex systems of equations. It is the study of linear equations and their properties, including matrices, vectors, determinants, and eigenvalues. Linear algebra is used in economic models, such as input-output modeling and general equilibrium modeling.3. StatisticsStatistics is used in economics to analyze data and test hypotheses. It is used to estimate parameters of economic models, such as the mean, variance, and correlation coefficients. Statistical methods are also used to make predictions and forecast future trends in the economy.4. Game TheoryGame Theory is used in economics to analyze strategic interactions among agents in the economy. It is amathematical framework used to study decision-making behavior, such as pricing decisions by firms or voting behavior by individuals. Game theory is used to model situations where individuals or firms may have conflicting goals or interests.5. Differential EquationsDifferential equations are used in economics to modelthe relationship between two or more variables. It is a mathematical tool used to describe dynamic processes in the economy, such as population growth, inflation, and economic growth. Differential equations are used in macroeconomic models to describe the relationship between output, consumption, and investment.In conclusion, essential mathematics for economicanalysis is a critical tool for understanding the economy and making sound economic decisions. It includes calculus, linear algebra, statistics, game theory, and differential equations, among others. These mathematical tools are essential for modeling economic phenomena and analyzing economic data. By using these tools, economists can make predictions and inform public policy decisions that can have a significant impact on society.。

ECON6003 Mathematical Methods of Economic AnalysisSemester 1, 2009 Unit of Study OutlineInstructor: Dr. Jian HongOffice: Room 438 Merewether BuildingPhone: 9351 6943Email: j.hong@.auOffice Hours: Wednesday, 3:30-4:30PM or by appointmentClasses: Monday 5-8pm at Merewether Lecture Room 41. Unit of study information1.1. Faculty Handbook descriptionThis unit is an introduction to mathematical economics. It has three purposes. First, to introduce students to the mathematical concepts and methods that are central to moderneconomics. Second, to give a set of economic applications of the mathematical methods.Third, to develop the students' ability to formulate logical arguments with the degree ofprecision and rigour demanded in modern economics. The mathematical topics coveredinclude introductory analysis and topology, convex analysis, linear algebra, calculus offunctions of several variables, optimisation, and introduction to dynamic programming and dynamical systems. The particular economic applications presented may vary from year toyear, but usually include demand theory, production theory, and growth theory.1.2. Aims and contextThe key purpose of this unit of study is to provide the mathematical tools necessary forMasters students to be able to read current advanced research in economics. It is a core unitin the Master of Economics degree and is a complement to ECON6001 and ECON6002.2. Learning outcomes, teaching and learning activities & assessment2.1. Intended learning outcomes1. Apply economic theories and concepts to problems and practice.2. Critically evaluate underlying theories, concepts, assumptions, limitations and argumentsin economics and related fields of study in business.3. Develop coherent arguments when recommending solutions and critically evaluatingtheories in major fields of study.2.2. Learning and teaching activitiesIn this unit, topics are presented in the first two-hours of lectures. The third hour is a workshop in which assignment from the previous week are discussed. There are 12 weeks of lectures and 12weeks of workshops. Workshops begin in week 2; especially, the midterm will be review right after the exam.Textbook: Mathematics for economists, by Simon and Blume, Norton, 1994.The textbook is available from the University Co-operative Bookshop located on the ground floor of the Noel Martin Sports Complex. This text is highly recommended.Blackboard: Blackboard will be used to disseminate all information concerning this unit. Workshop questions will be uploaded after each lecture.2.3. AssessmentAssessment task Weighting Date Learning outcomes Midterm 2-hour in class 25% 20 April 1, 2 and 3Assignments 25% 10 weeks 1, 2 and 3Final Exam 2-hour 50% TBA 1, 2 and 32.4. Assessment detailsMidterm – Topics covered up to the end of week 6 and the associated workshop material will be examined in a 2-hour exam, in class.Assignment – There are 10 assignments and only 4 of them will be randomly chosen for grading. The assignment will be collected at the beginning of class if it has been chosen. Late submission is not accepted.Final Exam – All topics covered in all lectures and workshops will be examined in a two hour exam.2.5. Workload and minimum requirementsThe Academic Board student workload for six credit point units equates to an average of 9-12 hours of student effort per week per 13-week semester. This is what is expected of you.2.6. FeedbackThe midterm will be marked as soon as possible after the exam and the marks placed on blackboard. The exam will handed back and brief answers placed on blackboard. It will also be discussed in a workshop.Assignments will be marked as soon as possible and the marks placed on blackboard. The assignments will be returned before the end of semester and will be discussed in a workshopThe final exam will be marked as soon as possible after the exam and results will be available on the web according to university requirements.2.7. Academic honesty and plagiarismCommencing students should complete the academic honesty module available viaBlackboard before their first assessment submission. Students should refer to theUniversity’s policies on academic honesty and plagiarism (.au/policy), the Faculty’s procedures for dealing with allegations (.au/sio) and use the assessment cover sheet.Academic honesty is important to protect students' right to receive due credit for work submitted for assessment. It is clearly unfair for students to submit work for assessment that dishonestly represents the work of others as their own and gain marks and degrees, which are not based on their own efforts and abilities. Deliberate breaches of academic honesty constitute academic misconduct. These breaches include: plagiarism, fabrication of data, recycling previously submitted material, engaging someone else to complete an assessment on one’s behalf and misconduct during supervised assessments.The penalties for academic misconduct may include: a mark of zero on the assessment; a fail grade in the unit of study, additional assessment (including an unseen exam), and reference of the matter to the University Registrar.3. Student evaluation and feedbackStudent feedback on this unit of study will be obtained via a Unit of Study Evaluation survey administered near the end of the semester. This is the first time I have taught this unit of study so I have used student feedback obtained from other units I have taught in designing this unit. In the past, students have requested more questions be set and I have responded by doing so in the workshop program.4. University policies and servicesAll students must comply with all Faculty and University policies and procedures.Faculty policies are contained in the Administration Manual for Students athttps://.au/StudentManual/ (e.g. special consideration, appeals, late submission, feedback mechanisms, academic honesty and plagiarism). Assistance is available from the Faculty’s Student Information Office at https://.au/sio/University policies at .au/policy/ include: academic honesty.au/ab/policies/Academic_Honesty_Cwk.pdf , plagiarism.au/senate/policies/Plagiarism.pdf . Assistance is available from the University’s Student Centre .au/studentcentre/The code of conduct is an important policy which outlines the University’s expectations about treating all staff employees and students with respect, dignity, impartiality, courtesy and sensitivity and refrain from acts of discrimination, harassment or bullying..au/ab/policies/Student_code_conduct.pdfLinks to other student services and resources are included on Blackboard and on theFaculty’s Learning and Teaching website .au/students/learningsupport/ (e.g. counselling services)Special Consideration:Special consideration will be granted where a student misses an exam and submits a Special Consideration Form that complies with the requirements set out in the Administration Manual for Students (including the requirement that the form be submitted within a week of the assessment). Special consideration will not be granted where a student attempts an assessment and submits a Special Consideration Form for that assessment, unless the student can demonstrate to the satisfaction of the Chair of Discipline that their serious illness or misadventure occurred during the exam.If you experience serious illness or misadventure that would impact on your performance in an exam, you should not sit the exam and submit a Special Consideration Form.If you are in doubt about these procedures, please contact your lecturer.Special ArrangementsAdministration Manual for Students outlines the limited set of circumstances for which students may be granted special arrangements for assessments and examinations (e.g., for students with special needs), and explains the procedures to follow to apply for Special Arrangements.5. Week-by-Week Topic Guide (Tentative, subject to change)1. Introduction and Foundations, Simon and Blume, Chs 1-2.2. Univariate Calculus and Applications, Simon and Blume, Chs 3-4.3. Linear System and Linear Algebra, Simon and Blume, Chs. 6-7, 10-11.4. Linear Algebra, Vector Space and Matrix, Simon and Blume, Chs. 8-11.5. Multivariate Calculus, Simon and Blume, Chs 13-14.6. Implicit Functions, Simon and Blume, Ch 15.AVCC Common Week: 12-17 April, No ClassMidterm Exam: 20 April, in class7. Quadratic Forms, Simon and Blume, Ch. 16.8. Unconstrained Optimization, Simon and Blume, Ch. 17.9. Optimization with Equality Constraints, Simon and Blume, Chs. 18-19.10. Optimization with Inequality Constraints, Simon and Blume, Chs. 18-19.11. Dynamics, Simon and Blume, Chs. 23-24.12. Review of Assignments and TopicsStudy Vacation: 8-12 June, No ClassFinal Exam: 15 July at Quad Room S421。

有关于数学归纳法的英文文献Mathematical induction is a powerful and fundamental proof technique used in mathematics to establish the truth of statements or propositions that involve natural numbers. It is a deductive reasoning method that allows one to prove that a given statement is true for all natural numbers, starting from a base case and then showing that if the statement is true for a particular natural number, then it is also true for the next natural number.The essence of mathematical induction lies in the fact that the natural numbers form an ordered set with a well-defined starting point, the number 1, and a clear successor relationship, where each natural number has a unique next number. This structure allows us to build up proofs by starting with the base case and then demonstrating that the statement holds for the next number, and so on, until we can conclude that the statement is true for all natural numbers.The general structure of a mathematical induction proof consists of two main steps:1. Base case: Establish that the statement is true for the first natural number, typically 1.2. Inductive step: Assume that the statement is true for a particular natural number k, and then show that it is also true for the next natural number, k+1.Once these two steps are completed, the principle of mathematical induction allows us to conclude that the statement is true for all natural numbers.One of the key advantages of mathematical induction is its ability to handle statements that involve an infinite number of cases, such as those related to sequences, series, or properties of natural numbers. By proving the statement for the base case and then showing that it holds for the next number, we can establish the truth of the statement for all natural numbers, even though there are infinitely many of them.Mathematical induction has a wide range of applications in various branches of mathematics, including number theory, combinatorics, algorithm analysis, and even in some areas of computer science and physics. It is a fundamental tool for proving the correctness of algorithms, establishing properties of recursive data structures, and solving problems that involve the manipulation of natural numbers.One classic example of the application of mathematical induction is the proof of the formula for the sum of the first n positive integers, which states that the sum of the first n positive integers is equal to n(n+1)/2. To prove this formula using induction, we first establish the base case by showing that the statement is true for n=1, as 1(1+1)/2 = 1. Then, we assume that the statement is true for some natural number k, and we show that it is also true for the next natural number, k+1, by using the following steps:Assume the statement is true for n=k:Sum of the first k positive integers = k(k+1)/2Now, consider the sum of the first k+1 positive integers:Sum of the first k+1 positive integers = 1 + 2 + 3 + ... + k + (k+1)= (k(k+1)/2) + (k+1)= (k(k+1) + 2(k+1))/2= (k^2 + 3k + 2)/2= (k+1)(k+2)/2Therefore, the statement is also true for n=k+1. By the principle of mathematical induction, we can conclude that the formula for the sum of the first n positive integers is true for all natural numbers n.Another important application of mathematical induction is in thefield of algorithm analysis, where it is used to prove the correctness and runtime complexity of algorithms. For example, when analyzing the time complexity of a recursive algorithm, we can use induction to show that the algorithm's running time grows in a certain way as the input size increases.Consider the classic Fibonacci sequence, where each number in the sequence is the sum of the two preceding numbers, starting with 0 and 1. We can use mathematical induction to prove that the nth Fibonacci number can be calculated using the formula F(n) = (phi^n - psi^n) / sqrt(5), where phi = (1 + sqrt(5))/2 and psi = (1 - sqrt(5))/2.To prove this formula, we first establish the base cases for n=0 and n=1, where the formula holds true. Then, we assume that the formula holds for some natural number k, and we show that it also holds for the next number, k+1, by using the recursive definition of the Fibonacci sequence and some algebraic manipulations.Through this inductive argument, we can conclude that the formula for the nth Fibonacci number is valid for all natural numbers n.In addition to its applications in mathematics and computer science, mathematical induction has also found uses in other fields, such as physics and engineering. For example, in the study of the properties of materials, induction can be used to prove that a certain materialproperty holds true for all atoms or molecules in a system, starting from the base case of a single atom or molecule and then showing that the property extends to the next atom or molecule.Overall, mathematical induction is a powerful and versatile proof technique that allows us to establish the truth of statements involving natural numbers. Its ability to handle infinite cases and its applications in various domains make it an essential tool in the mathematician's and scientist's toolkit.。

外国文学作选读Selected Reading of Foreign Literature现代企业管理概论Introduction to Modern Enterprise Managerment电力电子技术课设计Power Electronics Technology Design计算机动画设计3D Animation Design中国革命史China’s Revolutionary History中国社会主义建设China Socialist Construction集散控制DCS Distributed Control计算机控制实现技术Computer Control Realization Technology计算机网络与通讯Computer Network and CommunicationERP/WEB应用开发Application ＆ Development of ERP/WEB数据仓库与挖掘Data Warehouse and Data Mining物流及供应链管理Substance and Supply Chain Management成功心理与潜能开发Success Psychology & Potential Development信息安全技术Technology of Information Security图像通信Image Communication金属材料及热加工Engineering Materials & Thermo-processing机械原理课程设计Course Design for Principles of Machine机械设计课程设计Course Design for Mechanical Design机电系统课程设计Course Design for Mechanical and Electrical System。

受限非线性最优控制问题的三角正交函数求解方法李树荣【摘要】提出一种求解带有状态和控制不等式约束的非线性最优控制问题的数值求解方法.该方法通过三角正交函数对状态变量和控制变量的逼近,进而对目标函数、不等式约束和终端约束进行逼近,最终将原最优控制问题转化为非线性规划问题进行迭代求解.仿真算例结果验证了该算法的有效性.%A numerical method for solving nonlinear optimal control problems including terminal state constraints,state and control inequality constraints was proposed.The method is based on triangular orthogonal functions.By approximating the dynamic systems,performance index and boundary conditions into triangular orthogonal series,the optimal control problem is converted into algebraic equations with unknown coefficients.The results of illustrative examples demonstrate the accuracy and applicability of the proposed method.【期刊名称】《中国石油大学学报（自然科学版）》【年(卷),期】2013(037)003【总页数】5页(P172-176)【关键词】最优控制;三角正交函数;直接法;不等式约束;非线性【作者】李树荣【作者单位】中国石油大学信息与控制工程学院,山东青岛266580【正文语种】中文【中图分类】TP301在求解最优控制问题的方法中，直接法始终是一种被广泛应用的方法［1-2］。