丘成桐数学竞赛(分析与方程)
丘成桐数学竞赛2020年笔试真题probability_and_statistics_20
S.-T.Yau College Student Mathematics Contests2020Probability and StatisticsSolve every problem.Part I:ProbabilityProblem2.Letλbe a positive number.Suppose that X is a random variable with E|X|<∞. Suppose thatλE f(X+1)=E{X f(X)}for all bounded smooth functions.Show that X has the Poisson distribution Poisson(λ).Problem3.Consider the random walkS n=a+X1+X2+···+X n,where a is a positive integer and{X i}are independent and identically distributed random variables with a common distributionP{X i=1}=p,P{X i=−1}=1−p.Letτ0=inf{n:S n=0}be thefirst time the random walk reaches the state x=0.For all p∈[0,1]find the probability P a{τ0<∞}that the random walk will eventually hit the state x=0.Problem4.Let Z=(X,Y)be an R2-valued random variable such that(1)X and Y are independent;(2)both X and Y have mean zero andfinite(nonvanishing)second moments;(3)the distribution of Z is invariant under the rotation counter-clockwise around the origin by an angleθnot a multiple of 90degrees.Show that X and Y must be normal random variables with the same variance.Part II:StatisticsThe following collection of questions concerns the design of a randomized experiment where the N units to be randomized to drug A or drug B are people,for whom we have a large number of background covariates,collectively labelled X(e.g.,age,sex,blood pressure,height,weight, occupational status,history of heart disease,family history of heart disease).The objective is to assign approximately half to drug A and half to drug B where the means of each of the X variables (and means of non-linear functions of them,such as squares or products)are close to equal in the two groups.Instead of using classical methods of design,such as blocking or stratification,the plan is to use modern computers to try many random allocations and discard those allocations that are considered unacceptable according to a pre-determined criterion for balanced X means,in particularan affinely invariant measure such as the Mahalanobis distance between the means of X in the two groups.After an acceptable allocation is found,outcome variables will be measured,and their means will be compared in group A and group B to estimate a treatment effect.Problem5.Prove that if the two groups are of the same size(i.e.,N/2for even N),this plan will result in unbiased estimates of the A versus B casual effect based on the sample means of Y in groups A and B,where Y is any linear function of X.Problem6.Provide a counter-example to the assertion that Problem5is true in small samples with odd N.。
丘成桐大学生数学竞赛奖项设置
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第三届“丘成桐中学数学奖”获奖名单
第三届“丘成桐中学数学奖”获奖名单丘成桐中学数学奖金奖(1名)上海市市北中学参赛队员:陈波宇指导老师:金荣生论文题目:Weierstrass函数在不可列的稠密集上不可导的一种证明丘成桐中学数学奖银奖(1名)中国人民大学附属中学参赛队员:刘頔、于伦指导老师:阳庆节论文题目:对凸形内部一类特殊点的研究丘成桐中学数学奖铜奖(3名)杭州第二中学参赛队员:干悦、陈宇戈、孙璐璐指导老师:金洁论文题目:从画正多边形的铰链到连杆轨迹广东广雅中学参赛队员:吴俊熹、熊奥林、刘哲指导老师:杨志明论文题目:瓦西列夫不等式的推广、加强与类似美国Milton Academy参赛队员:Farzan Vafa 指导老师:Yong Lin论文题目:A new definition of distance for graphs丘成桐中学数学奖优胜奖(5名)绍兴市第一中学参赛队员:胡煜中、陈璐萍、朱晖指导老师:凌晓锋论文题目:关于凸四边形最小个数问题绍兴市第一中学参赛队员:卢尔涵、林志成指导老师:凌晓锋论文题目:正整系数多项式的首位数字分布中国人民大学附属中学参赛队员:靳兆融、曾力玮指导老师:唐晓苗论文题目:三角形内切椭圆及其性质的研究美国Hazleton Area High School参赛队员:Keenan Monks 指导老师:Kenneth G. Monks论文题目:On supersingular elliptic curves and hypergeometric functions美国Liberty High School参赛队员:Benjamin Kraft 指导老师:Gregory Minton 论文题目:Entries of random matrices丘成桐中学数学奖鼓励奖(8名)华东师范大学第二附属中学参赛队员:景琰杰、顾韬指导老师:施洪亮论文题目:高次剩余理论研究北师大实验中学参赛队员:施燕捷指导老师:黎栋材论文题目:汽车行驶最短时间武昌实验中学参赛队员:迟浩指导老师:瞿双清论文题目:关于京广铁路武汉枢纽列车最大通过能力的研究华中师范大学第一附属中学参赛队员:路康奕指导老师:王维佳论文题目:交通路口拥堵程度的量化与评价广东实验中学参赛队员:张舒瑶、张陈灵、陈明蕙指导老师:郭卫东论文题目:论Wilcoxon-Mann-Whitney检验的容许性华南师范大学附属中学参赛队员:霍泽恩指导老师:李维民论文题目:Heron 三角形的有理角平分线问题及与完全长方体的联系华南师范大学附属中学参赛队员:梁昊、张可天指导老师:黄毅文论文题目:金融投资中的复利期望模型深圳市外国语学校参赛队员:郭梦绮、袁可馨、萧桐桐指导老师:袁智斌论文题目:对选拔性考试量分法的实证性研究丘成桐中学应用数学科学奖金奖(1名)杭州外国语学校参赛队员:朱陶元敏、张允宜指导老师:潘俊论文题目:变速队伍中速度传递问题的研究及其应用丘成桐中学应用数学科学奖银奖(1名)新加坡NUS High School of Mathematics and Science参赛队员:Li Ang, Lim Sung Hyun, Wang Qi 指导老师:Chai Ming Huang 论文题目:Understanding flocking dynamics in nature丘成桐中学应用数学科学奖铜奖(2名)江门市第一中学参赛队员:张姝、李璟、李健斌指导老师:李凌山论文题目:Actuarial modeling on a children’s protection insurance广东广雅中学参赛队员:黎骁旸、蔡宇指导老师:杨志明论文题目:墨西哥湾原油泄漏事件在推广Fay公式基础上的建模丘成桐中学应用数学科学奖鼓励奖(2名)郑州市第八中学参赛队员:汪晗、李晨、宋晨指导老师:刘正峰论文题目:二次拟合灰色预测模型在黄河水质发展趋势预测中的应用越南High School for the Gifted- Vietnam University of Hochiminh City 参赛队员:Nguyen Manh Tien, Tu Nguyen Thai Son 指导老师:Tran Nam Dung 论文题目:A stone-picking game丘成桐中学数学奖组织奖(3名)中国人民大学附属中学绍兴市第一中学广东广雅中学第四届丘成桐中学数学奖获奖名单金奖学校:美国Sidwell Friends Upper School题目:Shock Profile For Gas Dynamics in Thermal Nonequilibrium 参赛队员:Wang Xie指导老师:Luo Tao银奖学校:深圳中学题目:存在一个非VAN DOUWEN极大几乎不相交族队员:何卓东指导老师:张承宇铜奖(共2项,排名不分先后)学校:北京十一学校题目:三次函数切割线的斜率关系参赛队员:曾文远指导老师:潘国双学校:安徽省合肥市168中学题目:裴蜀定理的加强证明参赛队员:丁宇堃指导老师:孙文海优胜奖(共5项,排名不分先后)学校:华东师范大学第二附属中学题目:Euler-Maclaurin公式的推广及其应用参赛队员:景琰杰指导老师:施洪亮学校:河南省实验中学题目:曲线的拆解参赛队员:汪晗刘婧孜宋晨指导老师:李新德学校:东北育才中学题目:经典趣题“老鼠与毒药问题”推广研究参赛队员:白天衣项思陶指导老师:张雷学校:辽宁省鞍山市第一中学题目:拓扑和的推广参赛队员:夏铭辰指导老师:张继红学校:广东实验中学题目:树的模染色数参赛队员:王家林源洁莹指导老师:郭卫东鼓励奖(共4项,排名不分先后) 学校:浙江大学附属中学题目:数独的计数、分类与图案设计参赛队员:李梦鸽指导老师:李刚豪学校:江苏省苏州第十中学校题目:关于错排问题的思考与讨论参赛队员:钱炘祺于浩佳指导老师:吉剑锋学校:郑州第八中学题目:树枝的分形参赛队员:王佳瑞李明哲梁栋指导老师:刘正峰学校:郑州一高题目:基于防雷接地需求的土壤结构模型研究参赛队员:李豆豆指导老师:曹恒阁获得组织奖的学校(共2项,排名不分先后)杭州外国语学校广东省实验中学第五届丘成桐中学数学奖获奖名单金奖学校:清华大学附属中学题目:论两个函数方程解析解的渐近性质参赛队员:邵城阳指导老师:杨利军银奖学校:杭州第二中学题目:数学物理中的一个丢番图问题参赛队员:任之杨东辰指导老师:斯理炯铜奖(共3项,排名不分先后)学校:杭州外国语学校题目:一类离散最值问题的探究参赛队员:蔡煜晟赵海洲指导老师:张传鹏学校:广东实验中学题目:含Euler数和Bernounlli数的恒等式新探参赛队员:魏锐波储岸均指导老师:张俊杰学校:华南师范大学附属中学题目:莫比乌斯带分割的结构与拓扑性质参赛队员:樊润竹李想指导老师:罗碎海优胜奖(共3项,排名不分先后)学校:清华大学附属中学题目:扫雷游戏中数字和的最大值探究参赛队员:张益深指导老师:王殿军学校:北京十一学校题目:差分方程与微分方程间的关系及其解的性质的研究指导老师:潘国双学校:华南师范大学附属中学题目:N线坐标体系及应用参赛队员:余欣航指导老师:李兴怀获得组织奖的学校(共3项,排名不分先后)华南师范大学附属中学清华大学附属中学杭州外国语学校第三届丘成桐中学应用数学科学奖获奖名单银奖学校:南京外国语学校题目:最优交通拥堵费定价研究队员:陈宗灿杜楠指导老师:龚强铜奖(共3项,排名不分先后)学校:南京外国语学校题目:一种工业上测量椭圆截面积和椭球体积的新方法参赛队员:尤宸超田汉指导老师:朱胜强学校:Weston High School, USA题目:Computationally Determining the Dimensions of the Homology Groups of Directed Graphs 参赛队员:Ariya R. Shajii指导老师:Gabor Lippner学校:清华大学附属中学题目:CG图和形独基本性质探究指导老师:杨青明优胜奖(共5项,排名不分先后)学校:江苏省锡山高级中学题目:基于整数型延时差分方程组的基因表达调控的数学模型研究参赛队员:高竹指导老师:高军晖学校:清华大学附属中学题目:太阳时钟——计算时间的方法参赛队员:薛宇皓指导老师:周建军学校:澳门培正中学题目:云深不知处——代数学在云端储存上的应用参赛队员:谭知微指导老师:黄灿霖学校:福建省厦门第一中学题目:用微积分和微分方程解决家用热水器节能问题参赛队员:林梓楠谭天琪钱坤儿指导老师:徐小平学校:NUS High School, Singapore题目:Generalized Quantum Tic-Tac-Toe参赛队员:Ananya Kumar, ?Ang Yan Sheng指导老师:Chai Ming Huang金奖学校:St. Gregory College Preparatory School, USA题目:3D Surface Fabrication using Conformal Geometry参赛队员:Yuanqi Zhang指导老师:Xianfeng Gu银奖(共2项,排名不分先后)学校:北京市中关村中学题目:足球弧线球的数学分析方法参赛队员:杨祚儒富宏远计润达指导老师:潘凤易学校:NUS High School, Singapore题目:A General Algorithm of Flattening Convex Prismatoids 参赛队员:Chenglei Li 、Jingqi Zhou指导老师:Minghuang Chai铜奖(共4项,排名不分先后)学校:北京市十一学校题目:一类Pell方程的可解性研究参赛队员:王嘉琦蔡立德指导老师:贾祥雪学校:South Brunswick High School, USA题目:q-Symmetric Polynomials and nilHecke Algebras参赛队员:Ritesh Ragavender指导老师:Alexander Ellis学校:吉林大学附属中学题目:Quantum Watermarking in M-band Wavelet Domain参赛队员:刘通徐旋指导老师:任玉莲学校:Arcadia High School, USA题目:Feature Identification for Colon Tumor Classification 参赛队员:Anthony Hou指导老师:Ernie Esser优胜奖(共4项,排名不分先后)学校:澳门培正中学题目:A New Secure Distributed Storage Scheme for Cloud- Geometric and Algebraic design and Implementation参赛队员:张颢霆谭知微指导老师:黄灿霖学校:广东广雅中学题目:700阶以内有限群单性的探究参赛队员:邵芷茵赵子琪谢瑞恒指导老师:徐敏学校:清华大学附属中学题目:关于“青蛙跳几次,一米徘徊”概率问题的研究参赛队员:齐天博张胤泰指导老师:李劲松学校:杭州外国语学校题目:高层建筑安全疏散问题的研究及泛函背景下疏散方案设计的尝试参赛队员:周大桐王雨菡鼓励奖(共8项,排名不分先后)学校:华东师范大学第二附属中学题目:形如ax+bx=cx的简单指数方程解的无理性判定参赛队员:钮敏学蔡偌箐指导老师:戴中元学校:南京外国语学校题目:基于基本体格检查提出综合肥胖指数的分析与应用参赛队员:崔珈铭王英之杰王子南指导老师:黄文龙学校:南京外国语学校题目:Leslie修正模型在南京市人口预测中的应用参赛队员:王伯文陈思齐楼嘉钰指导老师:王刚学校:华东师范大学第二附属中学题目:三角形中神奇的点参赛队员:潘星宇指导老师:戴中元学校:清华大学附属中学题目:地图投影变换在全球一张图量算中的应用参赛队员:马浩程李寅晓指导老师:周建军学校:福建省南安第一中学题目:Analysis and Comparison between the Algorithm Time Efficiency of Dijkstra and SPFA 参赛队员:谢新锋指导老师:林建源学校:广东实验中学题目:稳健的对数最优策略理论研究与实践参赛队员:郭屹峰黄辰光指导老师:刘江宁学校:广州市第六中学题目:“三国杀”中的数学问题分析参赛队员:杨卓潇李思聪伍思航获得组织奖的学校(共4项,排名不分先后)广东广雅中学南京外国语学校华东师范大学第二附属中学清华大学附属中学。
丘成桐英才班考试范围
丘成桐英才班考试范围全文共四篇示例，供读者参考第一篇示例：一、数学1. 初等数论：包括整数、有理数等基本概念的考察，以及一些中级数论题目的解答。
2. 数列与数学归纳法：包括等差数列、等比数列等常见数列的求和公式，以及数列与数学归纳法在解题中的应用。
3. 平面几何：包括角度、三角形、四边形、圆等几何图形的性质和相关定理的考察。
4. 进阶数学：包括微积分、线性代数等高阶数学概念和定理的考察。
5. 竞赛数学：包括奥赛数学中的高难度题目和解题技巧的练习。
二、物理1. 力学：包括牛顿三大定律、摩擦力、弹簧力等力学知识和题目。
2. 热学：包括热力学、温度、热平衡等热学基础知识和问题。
3. 电磁学：包括电场、磁场、电流等电磁学基础知识和问题。
4. 光学：包括光的传播、反射、折射等光学知识和问题。
5. 现代物理：包括相对论、量子力学等现代物理领域的知识。
三、信息学1. 基本算法：包括排序算法、查找算法等常见算法的实现和应用。
2. 数据结构：包括链表、树、图等数据结构的基本概念和应用。
3. 计算机原理：包括计算机组成原理、操作系统、编程语言等计算机基础知识。
4. 算法设计：包括贪心算法、动态规划、回溯法等高级算法设计和分析。
5. 程序设计：包括编程能力、程序调试、算法实现等计算机编程技能的练习。
以上是丘成桐英才班考试范围的主要内容，学生们需要在这些领域取得一定的基础才能进入这个特殊的班级学习。
通过参加丘成桐英才班的学习，学生们将能够更好地提高自己的数学、物理和信息学能力，为未来参加奥赛比赛和科研工作打下坚实基础。
希望学生们在这个班级的学习过程中，不断努力，不断挑战自己，取得更好的成绩。
【2000字】第二篇示例：【丘成桐英才班考试范围】丘成桐英才班作为国内著名的数学培训机构，向来以其严格的教学标准和高质量的教育服务而闻名。
对于学生来说，通过丘成桐英才班的培训，不仅可以提高数学水平，更可以为未来的学业和职业发展打下坚实的基础。
2016 丘成桐大学生数学竞赛获奖名单
2016丘成桐大学生数学竞赛获奖名单丘成桐大学生数学竞赛华罗庚奖，即分析与微分方程方向获奖者是:铜奖清华大学邵城阳铜奖清华大学朱晶泽铜奖复旦大学钱列铜奖清华大学李阳垟铜奖清华大学张志宇银奖清华大学王昊宇银奖清华大学徐凯银奖中国科技大学马明辉银奖北京大学李艺轩金奖复旦大学张页丘成桐大学生数学竞赛陈省身奖，即几何与拓扑方向获奖者是:铜奖清华大学王高明铜奖北京大学沈澈铜奖清华大学白少云铜奖清华大学邵城阳铜奖清华大学熊昊仁银奖北京大学李艺轩银奖清华大学王志涵银奖中国科技大学马翘楚金奖清华大学徐凯金奖北京大学黄开丘成桐大学生数学竞赛周炜良，即代数、数论与组合方向奖获奖者是:铜奖清华大学王浩旭铜奖中国科技大学钱舰铜奖清华大学张志宇铜奖台湾大学羅啟恒铜奖台湾大学趙庭偉银奖北京大学陈成银奖复旦大学孟凡君银奖台湾大学吴博生金奖北京大学吕世极金奖清华大学徐凯丘成桐大学生数学竞赛林家翘奖，即应用数学与计算数学方向获奖者是:铜奖清华大学刘冠华铜奖北京大学金晨子铜奖清华大学王昊宇银奖武汉大学黄旷银奖北京大学金辉金奖清华大学李阳垟丘成桐大学生数学竞赛许宝騄奖，即概率统计方向获奖者是:铜奖北京大学王飞骋铜奖北京大学付伟龙铜奖北京大学顾超铜奖复旦大学唐博浩银奖北京大学刘浩然金奖清华大学王昊宇丘成桐大学生数学竞赛丘成桐奖，即个人全能奖获奖者是:银奖清华大学张志宇银奖北京大学李艺轩银奖清华大学王昊宇金奖清华大学李阳垟金奖清华大学徐凯丘成桐大学生数学竞赛团体赛获奖者是:铜奖复旦大学钱列、周易铖、石佳、陈小帖、陈品翰铜奖复旦大学孟凡君、邹嘉骅、缪欣晨、金正中、唐博浩铜奖中国科技大学何声、马翘楚、马明辉、高英瓒、袁望钧铜奖北京大学王翔、沈澈、孙成章、金辉、肖非依银奖清华大学徐凯、王志涵、赵瑞屾、贾楸烨、李林骏银奖清华大学王浩旭、白少云、李阳垟、郭怡辰、徐则驰银奖北京大学顾超、黄开、李艺轩、袁宏霖、段雅琦金奖清华大学秦翊宸、王怡、邵城阳、杨羽轩、王昊宇。
丘成桐领军计划数学水平测试申请理由
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2011年丘成桐大学生数学竞赛——计算与概率卷
S.-T.Yau College Student Mathematics Contests 2011Applied Math.,Computational Math.,Probability and StatisticsIndividual6:30–9:00pm,July 9,2011(Please select 5problems to solve)1.Given a weight function ρ(x )>0,let the inner-product correspond-ing to ρ(x )be defined as follows:(f,g ):= baρ(x )f (x )g (x )d x,and let f :=(f,f ).(1)Define a sequence of polynomials as follows:p 0(x )=1,p 1(x )=x −a 1,p n (x )=(x −a n )p n −1(x )−b n p n −2(x ),n =2,3,···wherea n =(xp n −1,p n −1)(p n −1,p n −1),n =1,2,···b n =(xp n −1,p n −2)(p n −2,p n −2),n =2,3,···.Show that {p n (x )}is an orthogonal sequence of monic polyno-mials.(2)Let {q n (x )}be an orthogonal sequence of monic polynomialscorresponding to the ρinner product.(A polynomial is called monic if its leading coefficient is 1.)Show that {q n (x )}is unique and it minimizes q n amongst all monic polynomials of degree n .(3)Hence or otherwise,show that if ρ(x )=1/√1−x 2and [a,b ]=[−1,1],then the corresponding orthogonal sequence is the Cheby-shev polynomials:T n (x )=cos(n arccos x ),n =0,1,2,···.and the following recurrent formula holds:T n +1(x )=2xT n (x )−T n −1(x ),n =1,2,···.(4)Find the best quadratic approximation to f (x )=x 3on [−1,1]using ρ(x )=1/√1−x 2.1Applied Math.Prob.Stat.,2011-Individual 22.If two polynomials p (x )and q (x ),both of fifth degree,satisfyp (i )=q (i )=1i,i =2,3,4,5,6,andp (1)=1,q (1)=2,find p (0)−q (0)y aside m black balls and n red balls in a jug.Supposes 1≤r ≤k ≤n .Each time one draws a ball from the jug at random.1)If each time one draws a ball without return,what is the prob-ability that in the k -th time of drawing one obtains exactly the r -th red ball?2)If each time one draws a ball with return,what is the probability that in the first k times of drawings one obtained totally an odd number of red balls?4.Let X and Y be independent and identically distributed random variables.Show thatE [|X +Y |]≥E [|X |].Hint:Consider separately two cases:E [X +]≥E [X −]and E [X +]<E [X −].5.Suppose that X 1,···,X n are a random sample from the Bernoulli distribution with probability of success p 1and Y 1,···,Y n be an inde-pendent random sample from the Bernoulli distribution with probabil-ity of success p 2.(a)Give a minimum sufficient statistic and the UMVU (uniformlyminimum variance unbiased)estimator for θ=p 1−p 2.(b)Give the Cramer-Rao bound for the variance of the unbiasedestimators for v (p 1)=p 1(1−p 1)or the UMVU estimator for v (p 1).(c)Compute the asymptotic power of the test with critical region |√n (ˆp 1−ˆp 2)/ 2ˆp ˆq |≥z 1−αwhen p 1=p and p 2=p +n −1/2∆,where ˆp =0.5ˆp 1+0.5ˆp 2.6.Suppose that an experiment is conducted to measure a constant θ.Independent unbiased measurements y of θcan be made with either of two instruments,both of which measure with normal errors:fori =1,2,instrument i produces independent errors with a N (0,σ2i )distribution.The two error variances σ21and σ22are known.When ameasurement y is made,a record is kept of the instrument used so that after n measurements the data is (a 1,y 1),...,(a n ,y n ),where a m =i if y m is obtained using instrument i .The choice between instruments is made independently for each observation in such a way thatP (a m =1)=P (a m =2)=0.5,1≤m ≤n.Applied Math.Prob.Stat.,2011-Individual 3Let x denote the entire set of data available to the statistician,in this case (a 1,y 1),...,(a n ,y n ),and let l θ(x )denote the corresponding log likelihood function for θ.Let a =n m =1(2−a m ).(a)Show that the maximum likelihood estimate of θis given by ˆθ= n m =11/σ2a m −1 n m =1y m /σ2a m.(b)Express the expected Fisher information I θand the observedFisher information I x in terms of n ,σ21,σ22,and a .What hap-pens to the quantity I θ/I x as n →∞?(c)Show that a is an ancillary statistic,and that the conditional variance of ˆθgiven a equals 1/I x .Of the two approximations ˆθ·∼N (θ,1/I θ)and ˆθ·∼N (θ,1/I x ),which (if either)would you use for the purposes of inference,and why?。
丘成桐大学生数学竞赛试卷
S.-T.Yau College Student Mathematics Contests 2010Analysis and Differential EquationsTeam(Please select 5problems to solve)1.a)Let f (z )be holomorphic in D :|z |<1and |f (z )|≤1(z ∈D ).Prove that|f (0)|−|z |1+|f (0)||z |≤|f (z )|≤|f (0)|+|z |1−|f (0)||z |.(z ∈D )b)For any finite complex value a ,prove that 12π 2π0log |a −e iθ|dθ=max {log |a |,0}.2.Let f ∈C 1(R ),f (x +1)=f (x ),for all x ,then we have ||f ||∞≤ 10|f (t )|dt + 10|f (t )|dt.3.Consider the equation¨x +(1+f (t ))x =0.We assume that ∞|f (t )|dt <∞.Study the Lyapunov stability of the solution (x,˙x )=(0,0).4.Suppose f :[a,b ]→R be a L 1-integrable function.Extend f to be 0outside the interval [a,b ].Letφ(x )=12h x +h x −hf Show thatb a |φ|≤ b a |f |.5.Suppose f ∈L 1[0,2π],ˆf (n )=12π 2π0f (x )e −inx dx ,prove that 1)∞ |n |=0|ˆf(n )|2<∞implies f ∈L 2[0,2π],2)n |n ˆf (n )|<∞implies that f =f 0,a.e.,f 0∈C 1[0,2π],where C 1[0,2π]is the space of functions f over [0,1]such that both f and its derivative f are continuous functions.126.SupposeΩ⊂R3to be a simply connected domain andΩ1⊂Ωwith boundaryΓ.Let u be a harmonic function inΩand M0=(x0,y0,z0)∈Ω1.Calculate the integral:II=−Γu∂∂n(1r)−1r∂u∂ndS,where 1r=1(x−x0)2+(y−x0)2+(z−x0)2and∂∂ndenotes theout normal derivative with respect to boundaryΓof the domainΩ1.(Hint:use the formula∂v∂n dS=∂v∂xdy∧dz+∂v∂ydz∧dx+∂v∂zdx∧dy.)S.-T.Yau College Student Mathematics Contests 2010Applied Math.,Computational Math.,Probability and StatisticsTeam(Please select 5problems to solve)1.Let X 1,···,X n be independent and identically distributed random variables with continuous distribution functions F (x 1),···,F (x n ),re-spectively.Let Y 1<···<Y n be the order statistics of X 1,···,X n .Prove that Z j =F (Y j )has the beta (j,n −j +1)distribution (j =1,···,n ).2.Let X 1,···,X n be i.i.d.random variable with a continuous density f at point 0.LetY n,i =34b n (1−X 2i /b 2n )I (|X i |≤b n ).Show that n i =1(Y n,i −EY n,i )(b n n i =1Y n,i )1/2L −→N (0,3/5),provided b n →0and nb n →∞.3.Let X 1,···,X n be independently and indentically distributed ran-dom variables with X i ∼N (θ,1).Suppose that it is known that |θ|≤τ,where τis given.Showmin a 1,···,a n +1sup |θ|≤τE (n i =1a i X i +a n +1−θ)2=τ2n −1τ2+n −1.Hint:Carefully use the sufficiency principle.4.The rules for “1and 1”foul shooting in basketball are as follows.The shooter gets to try to make a basket from the foul line.If he succeeds,he gets another try.More precisely,he make 0baskets by missing the first time,1basket by making the first shot and xsmissing the second one,or 2baskets by making both shots.Let n be a fixed integer,and suppose a player gets n tries at “1and 1”shooting.Let N 0,N 1,and N 2be the random variables recording the number of times he makes 0,1,or 2baskets,respectively.Note that N 0+N 1+N 2=n .Suppose that shots are independent Bernoulli trails with probability p for making a basket.(a)Write down the likelihood for (N 0,N 1,N 2).12(b)Show that the maximum likelihood estimator of p is ˆp =N 1+2N 2N 0+2N 1+2N 2.(c)Is ˆp an unbiased estimator for p ?Prove or disprove.(Hint:E ˆp is a polynomial in p ,whose order is higher than 1for p ∈(0,1).)(d)Find the asymptotic distribution of ˆp as n tends to ∞.5.When considering finite difference schemes approximating partial differential equations (PDEs),for example,the scheme(1)u n +1j =u n j −λ(u n j −u n j −1)where λ=∆t ∆x ,approximating the PDE (2)u t +u x =0,we are often interested in stability,namely(3)||u n ||≤C ||u 0||,n ∆t ≤T for a constant C =C (T )independent of the time step ∆t and the spa-tial mesh size ∆x .Here ||·||is a given norm,for example the L 2norm orthe L ∞norm,of the numerical solution vector u n =(u n 1,u n 2,···,u n N ).The mesh points are x j =j ∆x ,t n =n ∆t ,and the numerical solutionu n j approximates the exact solution u (x j ,t n )of the PDE (2)with aperiodic boundary condition.(i)Prove that the scheme (1)is stable in the sense of (3)for boththe L 2norm and the L ∞norm under the time step restriction λ≤1.(ii)Since the numerical solution u n is in a finite dimensional space,Student A argues that the stability (3),once proved for a spe-cific norm ||·||a ,would also automatically hold for any other norm ||·||b .His argument is based on the equivalency of all norms in a finite dimensional space,namely for any two norms ||·||a and ||·||b on a finite dimensional space W ,there exists a constant δ>0such thatδ||u ||b ≤||u ||a ≤1δ||u ||b .Do you agree with his argument?If yes,please give a detailed proof of the following theorem:If a scheme is stable,namely (3)holds for one particular norm (e.g.the L 2norm),then it is also stable for any other norm.If not,please explain the mistake made by Student A.6.We have the following 3PDEs(4)u t +Au x =0,(5)u t +Bu x =0,3 (6)u t+Cu x=0,C=A+B.Here u is a vector of size m and A and B are m×m real matrices. We assume m≥2and both A and B are diagonalizable with only real eigenvalues.We also assume periodic initial condition for these PDEs.(i)Prove that(4)and(5)are both well-posed in the L2-norm.Recall that a PDE is well-posed if its solution satisfies||u(·,t)||≤C(T)||u(·,0)||,0≤t≤Tfor a constant C(T)which depends only on T.(ii)Is(6)guaranteed to be well-posed as well?If yes,give a proof;if not,give a counter example.(iii)Suppose we have afinite difference schemeu n+1=A h u nfor approximating(4)and another schemeu n+1=B h u nfor approximating(5).Suppose both schemes are stable in theL2-norm,namely(3)holds for both schemes.If we now formthe splitting schemeu n+1=B h A h u nwhich is a consistent scheme for solving(6),is this scheme guar-anteed to be L2stable as well?If yes,give a proof;if not,givea counter example.S.-T.Yau College Student Mathematics Contests2010Geometry and TopologyTeam(Please select5problems to solve)1.Let S n⊂R n+1be the unit sphere,and R n⊂R n+1the equator n-plane through the center of S n.Let N be the north pole of S n.Define a mappingπ:S n\{N}→R n called the stereographic projection that takes A∈S n\{N}into the intersection A ∈R n of the equator n-plane R n with the line which passes through A and N.Prove that the stereographic projection is a conformal change,and derive the standard metric of S n by the stereographic projection.2.Let M be a(connected)Riemannian manifold of dimension2.Let f be a smooth non-constant function on M such that f is bounded from above and∆f≥0everywhere on M.Show that there does not exist any point p∈M such that f(p)=sup{f(x):x∈M}.3.Let M be a compact smooth manifold of dimension d.Prove that there exists some n∈Z+such that M can be regularly embedded in the Euclidean space R n.4.Show that any C∞function f on a compact smooth manifold M (without boundary)must have at least two critical points.When M is the2-torus,show that f must have more than two critical points.5.Construct a space X with H0(X)=Z,H1(X)=Z2×Z3,H2(X)= Z,and all other homology groups of X vanishing.6.(a).Define the degree deg f of a C∞map f:S2−→S2and prove that deg f as you present it is well-defined and independent of any choices you need to make in your definition.(b).Prove in detail that for each integer k(possibly negative),there is a C∞map f:S2−→S2of degree k.1S.-T.Yau College Student Mathematics Contests 2010Algebra,Number Theory andCombinatoricsTeam(Please select 5problems to solve)1.For a real number r ,let [r ]denote the maximal integer less or equal than r .Let a and b be two positive irrational numbers such that 1a +1b = 1.Show that the two sequences of integers [ax ],[bx ]for x =1,2,3,···contain all natural numbers without repetition.2.Let n ≥2be an integer and consider the Fermat equationX n +Y n =Z n ,X,Y,Z ∈C [t ].Find all nontrivial solution (X,Y,Z )of the above equation in the sense that X,Y,Z have no common zero and are not all constant.3.Let p ≥7be an odd prime number.(a)Evaluate the rational number cos(π/7)·cos(2π/7)·cos(3π/7).(b)Show that (p −1)/2n =1cos(nπ/p )is a rational number and deter-mine its value.4.For a positive integer a ,consider the polynomialf a =x 6+3ax 4+3x 3+3ax 2+1.Show that it is irreducible.Let F be the splitting field of f a .Show that its Galois group is solvable.5.Prove that a group of order 150is not simple.6.Let V ∼=C 2be the standard representation of SL 2(C ).(a)Show that the n -th symmetric power V n =Sym n V is irre-ducible.(b)Which V n appear in the decomposition of the tensor productV 2⊗V 3into irreducible representations?1。
丘成桐大学生数学竞赛2021年笔试真题probability_and_statistics_21s
S.-T.Yau College Student Mathematics Contests2021Probability and StatisticsSolve every problem.Part I:ProbabilityProblem1.Suppose that a sequence{X n}of real-valued random variables converges to X in distribution and there are positive constants r and C such that E|X n|r≤C for all n.Show thatlimn→∞E|X n|s=E|X|sfor all0<s<r.Problem2.Let p(x,y)be the(one-step)transition function of a Markov chain on a discrete state space S and p n(x,y)be the n-step transition function.Show that for any positive integers L and N and any two states x and y we haveN+Ln=L p n(x,y)≤Nn=0p n(y,y).Part II:StatisticsProblem5.You have been asked to help design a randomized trial of a new drug,call it drug A,to be used in place of the current drug,call it drug B,for a particular medical condition.The budget isfixed to have 1000patients treated with A and1000treated with drug B.The issue is how to do the allocation of patients, because we have many pre-randomization measurements on each patient,roughly200,such as blood pressurerecordings,age,sex,and a large collection of genetics measurements.Obviously it is desirable to have the A group similar to the B group with respect to all pre-treatment covariates and non-linear functions of them that are expected to influence the effectiveness of the drugs with respect to the outcome variables.Complete(or simple)randomization does this in expectation,but with many covariates,some covariates will not be balanced between the A and B groups in any one single randomized allocation.Standard blocking used in traditional experimental design can force balance on a few covariates,but the designer of drug A wants to have an experimental design that creates balance on many covariates,and feels that you,as a modern applied mathematician/statistician,should be able to do this.Describe a class of methods that achieves this goal where each patient has a positive probability of receiving drug A and a positive probability of receiving drug B.Provide enough detail that you are describing an explicit algorithm.Problem6.You are given the results of a randomized experiment of two drugs,A and B.The experiment was not conducted in the usual way,however,but rather by allocating patients by a machine-learning algorithm under which each patient has a positive probability of receiving A and of receiving B;moreover the algorithm is completely specified and is built to create better than random balance on the covariates.(a)Can unbiased estimates of the causal effect of drug A versus B be found,and if so,show why.(b)Can exact small sample,non-parametric inferences for the causal effect in part(a)be derived,basedsolely on the randomization distribution of some statistic?For example,can wefind exact significance levels under a sharp null hypothesis?If so,outline how to accomplish this goal.。
丘成桐大学生数学竞赛参考书
丘成桐大学生数学竞赛参考书Geometry and Topology (the second draft)Space curves and surfacesCurves and Parametrization, Regular Surfaces; Inverse Images of Regular Values.Gauss Map and Fundamental Properties; Isometries; Conformal Maps; Rigidity of the Sphere.Topological spaceSpace, maps, compactness and connectedness, quotients; Paths and Homotopy. The Fundamental Group of the Circle. Induced Homomorphisms. Free Products of Groups. The van Kampen Theorem. Covering Spaces and Lifting Properties; Simplex and complexes. Triangulations. Surfaces and its classification.Differential ManifoldsDifferentiable Manifolds and Submanifolds, Differentiable Functions and Mappings; The Tangent Space, Vector Field and Covector Fields. Tensors and Tensor Fields and differential forms. The Riemannian Metrics as examples, Orientation and Volume Element; Exterior Differentiation and Frobenius's Theorem; Integration on manifolds, Manifolds with Boundary and Stokes' Theorem.Homology and cohomologySimplicial and Singular Homology. Homotopy Invariance. Exact Sequences and Excision. Degree. Cellular Homology. Mayer-Vietoris Sequences. Homology with Coefficients. The Universal Coefficient Theorem. Cohomology of Spaces. The Cohomology Ring. A Kunneth Formula. Spaces with Polynomial Cohomology. Orientations and Homology. Cup Product and Duality.Riemannian ManifoldsDifferentiation and connection, Constant Vector Fields and Parallel DisplacementRiemann Curvatures and the Equations of Structure Manifolds of Constant Curvature,Spaces of Positive Curvature, Spaces of Zero Curvature, Spaces of Constant Negative CurvatureReferences:M. do Carmo , Differentia geometry of curves and surfaces.Prentice- Hall, 1976 (25th printing)Chen Qing and Chia Kuai Peng, Differential GeometryM. Armstrong, Basic Topology Undergraduate texts in mathematicsW.M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry Academic Press, Inc., Orlando, FL, 1986M. Spivak, A comprehensive introduction to differential geometryN. Hicks, Notes on differential geometry, Van Nostrand.T. Frenkel, Geometry of PhysicsJ. Milnor, Morse TheoryA Hatcher, Algebraic Topology(/~hatcher/AT/ATpage.html)J. Milnor, Topology from the differentiable viewpointR. Bott and L. Tu, Differential forms in algebraic topologyV. Guillemin, A. Pollack, Differential topologyAlgebra, Number Theory and Combinatorics (second draft)Linear AlgebraAbstract vector spaces; subspaces; dimension; matrices and linear transformations; matrix algebras and groups; determinants and traces; eigenvectors and eigenvalues, characteristic and minimal polynomials; diagonalization and triangularization of operators; invariant subspaces and canonical forms; inner products and orthogonalbases; reduction of quadratic forms; hermitian and unitary operators, bilinear forms; dual spaces; adjoints. tensor products and tensor algebras;Integers and polynomialsIntegers, Euclidean algorithm, unique decomposition; congruence and the Chinese Remainder theorem; Quadratic reciprocity ; Indeterminate Equations. Polynomials, Euclidean algorithm, uniqueness decomposition, zeros; The fundamental theorem of algebra; Polynomials of integer coefficients, the Gauss lemma and the Eisenstein criterion; Polynomials of several variables, homogenous and symmetric polynomials, the fundamental theorem of symmetric polynomials.GroupGroups and homomorphisms, Sylow theorem, finitely generated abelian groups. Examples: permutation groups, cyclic groups, dihedral groups, matrix groups, simple groups, Jordan-Holder theorem, linear groups (GL(n, F) and its subgroups), p-groups, solvable and nilpotent groups, group extensions, semi-direct products, free groups, amalgamated products and group presentations.RingBasic properties of rings, units, ideals, homomorphisms, quotient rings, prime and maximal ideals, fields of fractions, Euclidean domains, principal ideal domains and unique factorization domains, polynomial and power series rings, Chinese Remainder Theorem, local rings and localization, Nakayama's lemma, chain conditions and Noetherian rings, Hilbert basis theorem, Artin rings, integral ring extensions, Nullstellensatz, Dedekind domains,algebraic sets, Spec(A).ModuleModules and algebra Free and projective; tensor products; irreducible modules and Schur’s lemma; semisimple, simple and primitive rings; density and Wederburn theorems; the structure of finitely generated modules over principal ideal domains, with application to abelian groups and canonical forms; categories and functors; complexes, injective modues, cohomology; Tor and Ext.FieldField extensions, algebraic extensions, transcendence bases; cyclic and cyclotomic extensions; solvability of polynomial equations; finite fields; separable and inseparable extensions; Galois theory, norms and traces, cyclic extensions, Galois theory of number fields, transcendence degree, function fields.Group representationIrreducible representations, Schur's lemma, characters, Schur orthogonality, character tables, semisimple group rings, induced representations, Frobenius reciprocity, tensor products, symmetric and exterior powers, complex, real, and rational representations.Lie AlgebraBasic concepts, semisimple Lie algebras, root systems, isomorphism and conjugacy theorems, representation theory.Combinatorics (TBA)References:Strang, Linear algebra, Academic Press.I.M. Gelfand, Linear Algebra《整数与多项式》冯克勤余红兵著高等教育出版社Jacobson, Nathan Basic algebra. I. Second edition. W. H. Freeman and Company, New York, 1985. xviii+499 pp.。
2016 丘成桐大学生数学竞赛优胜奖名单
2016丘成桐大学生数学竞赛优胜奖名单分析与微分方程清华大学陈钇帆清华大学董从翰北京大学董子超清华大学干盛文武汉大学郝子墨复旦大学何东辰清华大学贾瑨清华大学贾楸烨台灣大學江泓复旦大学金羽佳清华大学李林骏台灣大學李自然清华大学刘冠华中国科技大学马翘楚中国科技大学马骁山东大学乔一坤复旦大学石佳清华大学涂绪山北京大学王飞骋清华大学王可预浙江大学王慎融北京大学王翔清华大学王怡清华大学熊昊仁清华大学张翔宇北京大学赵洛晨清华大学赵瑞屾复旦大学周易铖清华大学朱永兴复旦大学邹嘉骅几何与拓扑复旦大学査承晗上海交通大学刁亦杰清华大学董从翰中国科技大学何声中国科技大学胡曦煜清华大学贾楸烨台灣大學江泓台灣大學李龍欣清华大学刘冠华清华大学罗鑫涛中国科技大学马骁复旦大学孟筱枫南京大学阮宇平北京大学孙成章北京大学王翔中国科技大学王轶堃山东大学吴保君复旦大学吴海涵南开大学颜俊榕四川大学姚琪厦门大学张良肇清华大学赵瑞屾复旦大学周易铖清华大学朱晶泽复旦大学邹嘉骅代数、数论与组合北京大学韩松奇中国科技大学何厚睿南京大学胡龙玺台灣大學江泓台灣大學李自然清华大学刘冠华中国科技大学毛天乐山东大学毛通复旦大学牛文君清华大学齐仁睿复旦大学钱列清华大学秦翊宸北京大学沈澈湖南大学沈子轩北京大学孙成章中国科技大学孙泽铭中国科技大学汪亦桐北京大学王翔华中师范大学吴晓旭清华大学吴志翔中国科技大学鲜文瀚上海交通大学张驰麟北京大学赵梓文兰州大学周胜铉应用数学与计算数学北京大学陈成中国科技大学陈皓清华大学陈钇帆中国科技大学高英瓒清华大学郭怡辰北京大学韩松奇清华大学贾楸烨台灣大學江泓复旦大学金正中北京大学李大为北京大学李冠淳武汉大学李轩复旦大学任骋力复旦大学任奕复旦大学石佳清华大学王柏然中国科技大学王子丰台灣大學吴博生北京大学辛未清华大学熊昊仁北京大学徐芦泽清华大学杨羽轩北京大学袁宏霖台灣大學趙庭偉清华大学朱晶泽概率统计复旦大学陈品翰中国政法大学储备哈尔滨工业大学（威海）黄元开北京大学金晨子北京大学李兴远北京大学刘驰洲中国科技大学马明辉中国科技大学马翘楚南开大学任少康北京大学孙成章南京大学王瑾旸中南大学王鲸中国科技大学王炜复旦大学吴佩学北京大学肖非依北京大学辛未中国地质大学（武汉）熊东北京大学严煜凌武汉大学喻洋中国科技大学袁望钧北京大学张浩然中国科技大学张羽丰北京大学赵洛晨北京大学郑亦如中国科技大学朱秋雨。
2015年丘成桐大学生数学竞赛团体赛题目
5. Let T : H1 → H2 be a bounded operator of Hilbert spaces H1, H2. Let S : H1 → H2 be a compact operator, that is, for every bounded sequence {vn} ∈ H1, Svn has a con-
is its L1-norm.
S.-T. Yau College Student Mathematics Contests 2015
Geometry and Topology
Team Please solve 5 out of the following 6 problems.
1. Let SO(3) be the set of all 3 × 3 real matrices A with determinant 1 and satisfying tAA = I, where I is the identity matrix and tA is the transpose of A. Show that SO(3)
Problem 5. Consider the following penalized least-squares problem (Lasso):
1 2
∥Y
−
Xβ∥2
+
λ∥β∥1
Let β be a minimizer and ∆ = β − β∗ for any given β∗. If λ > 2∥XT (Y − Xβ∗)∥∞, show that
3. Let F : M → N be a smooth map between two manifolds. Let X1, X2 be smooth vector fields on M and let Y1, Y2 be smooth vector fields on N . Prove that if Y1 = F∗X1 and Y2 = F∗X2, then F∗[X1, X2] = [Y1, Y2], where [ , ] is the Lie bracket.
丘成桐中国大学生数学竞赛大纲
2010年中国大学生数学竞赛（丘成桐教授发起）竞赛大纲一．Syllabuses for Geometry and TopologyGeometry:Curves and surfaces1) Plane curves and space curves2) The fundamental theorem of curves3) Concept and examples of surfaces4) The first and second fundamental forms5) Normal curvature, principal curvature and the Gauss curvature6) Orthogonal moving frames and structure equations of surfaces7) Existence and uniqueness of surfaces8) Isometric transformation of surfaces9) Covariant derivatives on surfaces10) Geodesic curvatures and geodesics, Geodesic coordinates11) The Gauss-Bonnet formula12) Laplacian operator on surfacesGeometry on manifolds1) Manifolds2) Vector fields and differentials3) Tensors and differential forms4) Stokes formula5) De Rham theorem6) Lie derivatives7) Lie algebras8) Maurer-Cartan equations9) Vector bundles10)Connection and curvatures11) Structure equations12) Riemannian metrics13) The Hodge star operator and Laplacian operator14) The Hodge theoremReferences:M. Do Carmo, Differential geometry of curves and surfaces.S S Chern and Chen Weihuan, Lectures on differential geometry Q. Chen and CK Peng, Differential geometryT. Frenkel: Geometry from physicsJ. Milnor, Morse theoryTopologyPoint Set Topology1) Open set and closed set2) Continuous maps3) Haudorff space, seperability and countable axioms4) Compactness and Heine-Borel theorem5) Connectivity and path connectivity6) Quotient space and quotient topologyFundamental groups1) Definition of fundamental groups, homotopic maps2) Computation of fundamental groups: Van Kampen theorem3) Covering maps and covering spaces4) Applications: Brouwer fixed point theorem, Lefschetz fixed point theoremComplexes and homology groups1) Simplex, complexes and polyhedron2) Barycentric subdivision and simplex approximation3) Computation of fundamental groups of complexes4) Classification of surfaces5) Simplex homology groups6) Application: Lefschetz fixed point theoremDifferential topology1) Smooth manifolds and smooth maps2) Sard’s theorem3) Transversality and intersection4) Vector fileds and Poincare-Hopf theorem5) Differential forms and de Rham complexes6) Orientation and integration7) Poincare Lemma8) Poincare duality9) Meyer-Vietoris sequences10)de Rham theorem11)Vector bundle and Euler classesReferences:Armstrong, Basic topologyJ. Milnor, Topology from the differentiable viewpointV. Guillemin and A. Pollack, Differential topologyBott and Tu, Differential forms in algebraic topology (first chapter)二．Syllabuses on algebra, combinatorics, number theory and representation theoryAlgebra群论（31）：集合论预备知识；对称和群；子群和陪集分解；生成元集和循环群；正规子群、商群和同态定理；置换群和线性群；群在集合上的作用；Sylow定理和单群；自由群和群的表现；有限生成Abel群的结构；小阶群的结构；幂零群和可解群。
丘成桐数学竞赛2021年决赛笔试真题2021+YCMC+Algebra+problems(S)
Algebra,Number Theory and Combinatorics(2021)Problem1.(Individual round.)Let p be a prime number and Q p thefield of p-adic numbers.Let n≥1be an integer and L=Q p(ζp n),whereζp n denotes a primitivep n-th roots of unity.Determine the image of the norm map N L/Qp :L×→Q×p.Youmay use the inequality[L:Q p]≤(Q×p :N L/Qp(L×))without proof in the case n≥2.Problem2.(Individual round.)Let k be afield and V a k-vector space of dimension n.Consider the group homomorphism:φ:GL(V)→GL(∧2V),f→∧2f.(1)Determine the kernel ofφ.(2)Show thatφinduces a group homomorphismψ:SL(V)→SL(∧2V).Express det(∧2f)in terms of det(f).Problem3.(Individual round.)Let A be a rank2integer matrix of size5×3. Classify all possible groups of the form Z5/A Z3.Solution to Problem 1.We will show that N L/Q p (L ×)=p Z (1+p n Z p ),where Z p denotes the ring of p -adic integers.Let Φ(X )=(X p n −1)/(X p n −1−1)be the p n -th cyclotomic polynomial.Then Φ(X +1)is an Eisenstein polynomial.Thus Φ(X )is the minimal polynomial of ζp n ,so thatN L/Q p (1−ζp n )=Φ(1)=p.We have [L :Q p ]=φ(p n )=p n −p n −1.For p odd,the φ(p n )-th power map on 1+p Z p is the composition1+p Z p log −→∼p Z p φ(p n )−−−→∼p n Z p exp−−→∼1+p n Z p .ThusN L/Q p (L ×)⊇N L/Q p (1+p Z p )=1+p n Z p .For p =2,we may assume n ≥2.The φ(2n )-th power map on 1+4Z 2is the composition1+4Z 2log −→∼4Z 2φ(2n )−−−→∼2n +1Z 2exp −−→∼1+2n +1Z 2.ThusN L/Q 2(L ×)⊇N L/Q 2(1+4Z 2)=1+2n +1Z 2.It is easy to see that1+2n Z 2=(1+2n +1Z 2) 52n −2(1+2n +1Z 2)and52n −2=N L/Q 2(2+ζ4).This finishes the proof of N L/Q p (L ×)⊇p Z (1+p n Z p )in all cases.Let O L denote the integral closure of Z p in L .The residue field of L is F p ,so that N L/Q p |O ×L is compatible with the φ(p n )-th power map on F ×p ,which carries every element to 1.In other words,N L/Q p (L ×)∩Z ×p =N L/Q p (O ×L )⊆1+p Z p .This finishes the proof in the case n =1.For n ≥2,it suffices to apply the giveninequality (Q ×p :N L/Q p (L ×))≥φ(p n )=(Q ×p :p Z (1+p n Z p )).Solution to Problem 2.(1)If n =2,then ∧2V k and φ(f )∈GL(k )is just the multiplication by det (f ),hence the kernel is just SL(V )=SL 2.Now assume n ≥3.By definition,f ∈Ker(φ)if and only if f (x )∧f (y )=x ∧y for all x,y ∈V .We claim x and f (x )are proportional:otherwise expand to a basis e 1=x,e 2=f (x ),e 3,...,e n ,then we have e 2∧f (e 3)=e 1∧e 3which is not possible as e i ∧e j is a basis of ∧2V .Hence x and f (x )are proportional for all x ∈V .This implies that f (x )=ax for some a ∈k .Then f (x )∧f (y )=a 2x ∧y =x ∧y ,thus a =±1.So the kernel is just ±Id .(2)First we show the case for elementary matrices:Take a basis e 1,···,e n of V and consider the endomorphism f ∈GL(V )defined by f (e i )=e i +bδ1,i e 2for all i ,where b is a constant.We have (∧2f )(e 1∧e j )=e 1∧e j +be 2∧e j for all j ≥2,and for 2≤i <j ,(∧2f )(e i ∧e j )=e i ∧e j .Thus ∧2f is lower triangular in the basis of e i ∧e j with 1on the diagonal,which gives det (∧2f )=1.Recall that any matrix of determinant1is a product of elementary matrices, hence by(ii)φsends SL(V)to SL(∧2V).Take t in an extension of k,such that t n det(f)=1.Then det(tf)=1and1=det(∧2(tf))=det(t2∧2f)=(t2)(n2)det(∧2f),which gives det(∧2f)=t−n(n−1)=det(f)n−1.Solution to Problem3.We can change Z bases of Z5and Z3to turn A in to a "canonical form".Equivalently,we can do the usual row-column reduction on A. Since A rank2means that AZ3is a rank2subgroup of Z5,which means the free part of G is Z3.So,in the reduced form A has3zero rows,and2positive diagonal entries of all possibilities.We can arrange so thatG Z/a⊕Z/b⊕Z3with a≥b>0.Finally list all possible non-isomorphic torsion part Z/a⊕Z/b,by factorizing a,b.Possible follow up:Generalize this as follows:A is rank k of size m×n with m>n>k.Classify all possible abelian groups of the form G=Z m/A Z n.The same method would likewise yield G Z/a1⊕···⊕Z/a n−k⊕Z m−k with a1≥a2≥···≥a>0.。
丘成桐小学试题及答案
丘成桐小学试题及答案一、选择题（每题2分，共10分）1. 下列哪个选项是数学家丘成桐的国籍？A. 中国B. 美国C. 加拿大D. 英国答案：A2. 丘成桐教授在哪个领域做出了杰出贡献？A. 物理学B. 数学C. 化学D. 生物学答案：B3. 丘成桐教授获得的数学界最高荣誉是什么？A. 诺贝尔奖B. 菲尔兹奖C. 沃尔夫奖D. 图灵奖答案：B4. 丘成桐教授在哪一年获得了菲尔兹奖？A. 1982年B. 1986年C. 1990年D. 1994年答案：A5. 丘成桐教授的哪项工作对数学界产生了深远影响？A. 量子力学B. 微分几何C. 拓扑学D. 代数几何答案：D二、填空题（每题2分，共10分）6. 丘成桐教授是_________年获得菲尔兹奖的。
答案：19827. 丘成桐教授的研究成果主要涉及_________和_________等领域。
答案：微分几何；代数几何8. 丘成桐教授在_________年获得了沃尔夫奖。
答案：20109. 丘成桐教授的国籍是_________。
答案：中国10. 丘成桐教授的主要研究领域是_________。
答案：数学三、简答题（每题5分，共20分）11. 请简述丘成桐教授的主要学术成就。
答案：丘成桐教授的主要学术成就包括在微分几何和代数几何领域的开创性工作，特别是对卡拉比-丘猜想的证明，以及在数学物理和弦理论中的应用。
12. 丘成桐教授对数学教育有哪些贡献？答案：丘成桐教授在数学教育方面的贡献包括创办了丘成桐数学奖，推动了数学竞赛的发展，以及在多个国家和地区推广数学教育和研究。
13. 丘成桐教授的哪些工作对现代数学产生了重要影响？答案：丘成桐教授的卡拉比-丘猜想的证明，以及在几何分析和弦理论方面的工作，对现代数学产生了重要影响。
14. 丘成桐教授在数学研究中有哪些创新方法？答案：丘成桐教授在数学研究中创新性地将微分几何和代数几何相结合，发展了几何分析，并且将数学理论应用于物理学，特别是在弦理论中。
2024年陕西省西安市碑林区西北工业大学附属中学丘成桐少年班选拔复试数学试题
2024年陕西省西安市碑林区西北工业大学附属中学丘成桐少年班选拔复试真题第一部分:现学现考2、若有一个直角三角形的两条边长度分别为3、4，求第三条边的长度。
3.∠B=90°，AC=20，AB:BC=3:4，求AB。
4、如图，任意四边形ABCD中，AC=4，BD=6.求的最大值.6、在下图中的正方形格点图中，每个小正方形边长均为1，求sin∠BAC。
7、已知菱形ABCD，BO=DO=4，AO=CO=3，求8、已知，绳长28cm，求物块下降多少cm?9.两个大小正方形的面积分别为49与9，求S大长方形10.[发现探究]则sin(a+β)=__________________(用含a、β的三角函数表示) [实践应用](1)sin18°xcos42°+cos18°xsin42°=_________________；(2)已知sina+sin(a+60°)=1，求sin(a+30°)的值第二部分:知识储备1.=_________________；2.以下五个碎片拼在一起可以组成完整的拼图，拼图中含有一个算式，则算式结果为_______。
3.下图式正八面体的立体图形以及它的展开图，则“?”处是A~H八个面的面________面4.四边形沿对角线切开是2个三角形，则四边形内角和为180°x2=360°，由此可得到一个求多边形内角和的公式，利用你得到公式解决问题，在图2得圆周上铺满图1，已知∠B=∠C=80°，则铺满圆周需要用_________个这样的四边形。
5.将下图①的两个儿何体放入图(②)的水箱中，水面上升____________cm .7.任意连续的两位数字都是相邻的自然数，叫做快乐数.比如说1232是个快乐数，各个数位数字之和为2024的最小的快乐数是个________位数。
8.一名射击运动员打靶训练，已知他共射击了10组，且每组打靶10次，已知第七、八、九组的成绩分别为94，87，95环(每一次环数均为整数)，且前九组平均环数比前六组平均环数高，那么若想让这10组的平均环数不低于92环，则第10组最少打________环。
丘成桐数学8道测试题
丘成桐数学8道测试题丘成桐是一位著名的数学家，他曾经获得过世界最高学术奖项之一的菲尔兹奖，因此他的数学水平是非常高超的。
下面将介绍他出的八道数学测试题，大家可以挑战一下。
题目1：10个箱子，其中9个装有相同的质量不确定的铁球，而另一个箱子装有质量稍重的铁球。
使用偏重计（只能用一次）并且不使用电子秤，请问如何找到装有较重铁球的那个箱子？题目2：用有限数量的直线在平面上切割一个圆形，问能够切成多少块？题目3：在一个死胡同里，有两间房间在拐角处相遇，其中一个门通向金库另一个门通向天台，目前只知道金库的门是被锁上的，而天台的门是开着的。
一个小偷想进入金库拿走里面的财宝，但他只有梯子和夹子。
请问他怎样才能进入金库？题目4：把一个小球放置在一个倒置的圆锥上，当小球滚动到圆锥一侧的底部时，小球会反弹回去并再次滚回圆锥的另一侧。
请问小球从圆锥的一侧滚到另一侧经历多少次反弹？假设小球与圆锥的接触点上下不断反弹。
题目5：在一个10 x 10的网格中，从左下角出发，通过只能向上或者向右走的方式到达右上角的位置，问有多少种不同的走法？题目6：井的深度为100米，第一次跳跃高度为1米，第二次跳跃高度为0.5米，第三次跳跃高度为0.25米，以此类推。
问需要跳跃几次才能从井底跳出去？题目7：有一个半径为100厘米的圆形花坛，在其中随机撒了1000颗种子，每颗种子的落点都是随机的。
请问最多需要选出多少颗种子才能确保所有的种子都被覆盖到了？题目8：3个人到一个小岛上，他们需要对一些动物进行分类。
其中一人只能对猴子进行分类，另一个人只能对长颈鹿进行分类，而另一个人则只能对斑马进行分类。
请问这三个人可以使用什么方法才能确保将所有的动物分好类？以上就是丘成桐出的八道数学测试题，涉及的领域十分丰富，需要有一定的数学功底才能够解决。
不过，挑战一下也是可以的，毕竟数学题目的解决过程也是一种锻炼思维能力的方式。