2016年北京大学数学科学学院直博生摸底考试
2015年北京大学数学科学学院应用数学考博专业介绍,考博真题,真题解析
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【全国百所名校定向保录】 【才思教育由命题组领专业化辅导】
comprehension of the formation and development as well as **************. If possible, I will go on with my study for doctorate degree. In a word, I am looking forward to making a solid foundation for future profession after two years study here.
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【全国百所名校定向保录】
12. 图像重建与图像处 理
13. 微分方程在图像处 理和信 号分析中的应 用
14. 信息物理融合系统
【才思教育由命题组领专业化辅导】
1. 自我介绍(self-introduce) Good morning. I am glad to be here for this interview. First let me introduce myself. My name is ***, 24. I come from ******,the capital of *******Province. I graduated from the ******* department of *****University in July ,2001.In the past two years I have been prepareing for the postgraduate examination while I have been teaching *****in NO.****middle School and I was a head-teacher of a class in junior grade two.Now all my hard work has got a result since I have a chance to be interview by you . I am open-minded ,quick in thought and very fond of history.In my spare time,I have broad interests like many other youngers.I like reading books, especially those about *******.Frequently I exchange with other people by making comments in the forum on line.In addition ,during my college years,I was once a Net-bar technician.So, I have a comparative good command of network application.I am able to operate the computer well.I am skillful in searching for information in Internet.I am a football fan for years.Italian team is my favorite.Anyway,I feel great pity for our country’s team. I always believe that one will easily lag behind unless he keeps on learning .Of course, if I am given a chance to study ****** in this famous University,I will stare no effort to master a good command of advance ******.
北京大学2016年博士研究生招生简章(校本部)
校按有关规定给予处分，直至开除学籍；对在职考生，将通知考生所在单位，由 考生所在单位视情节给予党纪或政纪处分。构成犯罪的，由司法机关依法追究刑 事责任。
十四、信息公开与监督 1、按照“公开、公平、公正”的原则，研究生院或各学院（系、所、中心） 将通过网站公布其博士生招生的相关说明、审核程序、进入复审或复试的名单和 拟录取名单。 、对招生录取过程中出现的争议，报名者可向各学院（系、所、中心）博 士研究生招生委员会提出申诉；若仍有争议，可向北京大学研究生招生办公室提 出申诉。 十五、其他事项 1、2016 年我校继续在多学科交叉领域招收博士研究生，交叉学科情况介绍 详见《2016 年重点交叉学科招生简介》（网址： /zsxx/bszs/bozsjzjml/index.htm）。 2、北京大学医学部、深圳研究生院博士研究生招生及奖助学金等有关信息， 详见其招生简章。 3、若国家在 2016 招生年度出台新的博士研究生招生政策，我校将做相应调 整，并将及时予以公布。
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3、健康状况符合北京大学研究生入学体检标准。 4、报考专业学位博士者，除满足申请基本条件外，还须符合院系对申请者 提出的相关领域工作年限等条件的要求。 四、报名时间 报名时间从 2015 年 10 月中旬开始，届时详见博士研究生报名公告（网址： /zsxx/wsbm/）。 五、提交报名材料 符合报名条件的人员须按照招生简章、报名公告和院系招生说明的要求在规 定期限内向报考院系提交下列材料（交、寄报名材料地址：考生所报北京大学各 个招生院系研究生教务办公室，具体地点查询网址： /zsxx/dtxx/35835.htm）： 1、北京大学博士研究生入学报名表：请在规定的报名时间内登陆北京大学 研究生招生网（网址：/bm）进行网上报名，上传 相关材料，并打印《报名表》； 2、最高学历、学位证书复印件（应届硕士毕业生提交所在单位研究生院或 研究生部的证明信，在录取后须补交学历学位证明文件，经确认后方可报到注册）； 3、硕士学位论文（应届硕士毕业生可提供论文摘要和论文目录等）； 4、毕业院校的正式成绩单原件； 5、个人陈述（含对报考学科专业的认识、拟定研究计划，3000 字左右，下 载网址：/zsxx/bszs/bobmxx/）； 6、两封所报考学科专业领域内的副教授（含）以上或具有相当专业技术职 称专家的推荐信（下载网址：/zsxx/bszs/bobmxx/）； 7、报考采用“申请-考核制”招生院系的，还需要提交三年内的外语考试成 绩证明复印件，复审时携带原件进行确认（具体要求见各院系招生说明）。2015 年 12 月底，我校将首次举行“北京大学博士研究生英语水平考试”，预计于 2016 年 1 月发布成绩（若有时间变化，另行通知）。该项考试成绩将作为我校采用“申 请-考核制”招生院系认可的有效外语成绩。报考者可根据实际情况在网上报名 时自主选择是否参加该项考试（具体考试时间和地点另行通知）；
北师大历年博士入学考试题
北师大历年专业博士入学考试试题教育学原理2008（A）1. 从现代教育的基本价值和功能分析教育对个人发展的意义（25 分）2. 试论教学过程的特殊性及对教学工作的意义（25 分）3. 互联网对青少年交往行为的影响并就学校教育由此面临的问题等谈谈你的看法。
（20 分）剩下的一题为各个方向规定选择的试题。
（30 分）今年实行的是3+1 模式。
2008 （B）（其中第2,3两题与教育学A卷一样，希望后来者有借鉴）1. 论述义务教育的基本特征以及政府应该承担的责任.（35 分）0 a O9 H\&d9L#F C'q"e（d4R2. 试论教学过程的特殊性及对教学工作的指导意义（35分）2……3. 论述互联网对青少年学生交往行为的影响并就学校教育由此面临的问题等谈谈你的对策。
（30 分）2007（A）1、试论现代教育的基本价值和功能及其对个人发展的意义。
（35 分）2、谈谈近年来教育研究方法论方面的若干变化及其意义。
（30 分）3、要求根据材料写篇小文章。
材料太长，我的准考证实在抄不下来哦，但是出自《发展教育学年鉴2 00 3 》，首都师大出版社2003 版。
具体内容是关于现代德育方面的。
（35 分）2007（B）1. 现代教育的价值取向以及与人美好生活的关系2. 评析基础教育课程改革的价值取向3. 有关网络道德教育插入：2007 年教育政治学与教育法真题1、怎样理解法律上的权利与义务的概念及分类。
2、试述教育与法律的关系。
（ ?（h8u-u4o7U'n8|4B3、高校与学生的法律关系辨析。
2007 经管学院的博士考试1. 教育财政公平与效率关系；2. 大学生就业难成因与对策分析；3. 各级教育中“市场化”问题分析；4. 择校问题成因分析；5. 分析人力资本理论与筛选理论关于教育作用的观点。
2006（B）1、从教育的本质属性角度，评析“教育产业化”。
2、现在德育概念有泛化的趋势，广义的德育包括（思想政治教育、道德品质教育、法制教育、心理健康教育等）以及狭义的德育（道德品质教育）概念的各自内涵及相互关系。
北京大学2016年数学分析试题及解答
10. 充分性: ∀ε > 0, ∃N > 0, 当 n > m > N 时,
令 x → R− 得
∑n akxk < ε, ∀x ∈ [0, R).
k=m
∑n
∑ ∞
akRk < ε =⇒ anRn 收敛.
k=m
n=1
必要性: 首先注意到
∑ ∞
anxn
=
∑ ∞
anRn
( x )n R
,
n=1
n=1
又因为
中的开集映为开集.
6.
(15
分)
x1
=
√ 2, xn+1
√ = 2 + xn.
证明
{xn}
收敛并求极限值.
7. (15 分) 证明 ∫ +∞ sin x dx 收敛并求值. 写出计算过程.
0
x
8. (15 分)
∫b
(1) 证明存在 [a, b] 上的多项式序列 {pn(x)} 使得 pi(x)pj(x) dx = δij 并使得对于 [a, b] 上的连续函数
准则
(不用证明)
并
i=1
用你叙述的 Cauchy 准则证明闭区间上的单调函数可积.
3. (15 分) (a, b) 上的连续函数 f (x) 有反函数. 证明反函数连续.
4.
(15
分)
f (x1, x2, x3)
是
C2
映射,
∂f ∂x1
(x01
,
x02,
x03
)
̸=
0.
证明
f (x1, x2, x3)
对于任意 n > m, x0 ∈ U, 因为 rank (J (f )|x=x0 ) = m, 不妨设 J (f )|x=x0 的前 m 列是线性无关的. 定义
北京大学数学科学学院关于2016年博士生招生的说明
北京大学数学科学学院关于2016年博士生招生的说明数学科学学院2016年招收博士研究生将试行以综合素质能力为基础的“申请-考核制”。
申请人须按照我校博士生招生简章和我院的相关要求进行报名并提交申请材料。
我院招生委员会将对申请人的材料审核评估后确认是否给予考核资格，并对获得考核资格者进行考核，最后确定是否录取。
一、基本条件1、报名者须身体健康，品行端正，遵守法律、法规和遵守学校的规章制度。
2、申请者必须符合下述条件之一：（1）已获得硕士或博士学位；（2）应届硕士毕业生须在博士入学前取得硕士学位；（3）获得学士学位后，在报考学科、专业或相近领域已工作6年或6年以上（从获得学位到录取为博士生当年的9月1日）的人员只能以同等学力资格申请。
3、健康状况符合北京大学研究生入学体检标准。
二、报名申请1、采取网上申报。
具体时间和程序请于10月中下旬登录北大研究生院主页查询。
2、申请者于2016年1月8日前，向我院研究生教务办公室寄（送）达以下申请材料：（1）通过网上报名系统打印的《报名登记表》；（2）学位、学历证书的复印件（应届硕士毕业生提交所在单位研究生主管部门的证明信）；如果是在国外获得学位，还需要提供教育部留学服务中心出具的《国外学历学位认证书》；（3）硕士期间课程成绩单的原件或复印件（提供复印件的同学须在复试时提交原件）；（4）硕士学位论文（应届硕士毕业生可提供论文摘要和目录等）；（5）两位与所申请学科相关的副教授（或相当于副教授）以上职称专家的推荐信；（模板请在/zsxx/wsbm/上下载）（6）攻读博士学位期间本人想致力研究的问题和设想的陈述书；（模板请在/zsxx/wsbm/上下载）（7）公开发表的学术论文、所获专利和其他原创性研究成果的陈述和证明；（8）英语水平证明（至少一项），请提交复印件：1)北京大学博士研究生英语水平考试；2)T OEFL成绩；3)GRE成绩；4) WSK(PETS 5)考试合格；5)国家英语四级、六级考试成绩；6) 国家英语专业四级考试；7) 雅思成绩；8）英语国家或地区的学位证；以上英语水平证明有效期为三年。
全国大学生数学竞赛模拟试卷六套详解(汪宏喜选编)
2ne e
n
1 ln(2 − 1 ) − (1 + ln 2) 1 − ln(1 − 1 ) > 0 ,为此引入辅助函数
n
n
n
n
g(x) = x ln(2 − x) − (1 + ln 2)x − ln(1 − x)(0 < x < 1) ,只需要证明 g(x) > 0 .
g′(x) = ln(2 − x) − x − (1 + ln 2) + 1 ,
适合 2016-2017 年全国大学生数学竞赛（非数学类）
汪宏喜教授设计选编
2016 年全国大学生数学竞赛模拟试卷（一）
考试形式： 闭卷 考试时间： 150 分钟 满分： 100 分.
一、下列各题（共 24 分,每小题各 6 分,要求写出重要步骤）.
（ 1 ） 设 函 数 f (x) 在 点 x = 0 处 有 定 义 , 且 f (0) = 1 , lim ln(1 − x) + sin x ⋅ f (x) = 0 . 求 x→0 ln(1 + x) ⋅ arctan x
.
∫ ∫ | ∫ 解：
π 2
f (x)dx = 2
π
2 f (x)d
π
π
x = 2 f (x) x 2 − 2 2
xdf (x)
0x
0
0
0
∫ ∫ ∫ π
=− 2
1 dx （令 t = tan x ） = −
0 1 + t;
1 t)(1 +
t
2
dx )
=
1 2
+∞ ⎜⎛ − 0⎝
c
线.
（1）求 f (x), g(x) 使得 f (0) = g(0) = 0 .（2）计算任一条曲线从点 (0,0) 到 (1,1) 积分.
北京大学2016数学科学夏令营初赛-评分标准
延长 QO 交 BC 于 R，由 CBO 90 A ABH ， BOR BHP ， BO BH ．
可得△BOR≌△BHP，从而 BR BP c ．
B
2
于是 cos C CM ，即 a2 b2 c2
1b 2
（20 分）．
CR
2ab
a1c
2
再由 ABC 60 可得 b2 a2 c2 ac ．
若
Px
ax
b（
a
1），则
a1
b a
，
a2
b
1 a
1 a2
，……．
一般的，有 an
b
an a
1
1 an
，由于 an
为整数，故只可能 a
1 ．
但此时 a3 a1 b ，不可能．（20 分）
若 P x 的次数不小于 2．且存在数列an 满足要求．
北京大学北20京16大年学数中学学科生学数夏学奖令营初赛 评分标准
个人能力挑战赛——试题评分标准
2016 年 7 月
本试卷共 4 题，每题 30 分，满分 120 分．考试时间 180 分钟．
1. 由 ABC 60 ．计算可得 BH AC cos B OB OQ ．
从而 BOQH 为菱形．（10 分）
综上即可得证．（30 分）
3. 注意到编号为 1 的牌永远在牌堆上方，编号为 2n 的牌永远在牌堆下方． 去掉这两张牌，只考虑中间 2n 2 张牌．此时相当于将所有从上往下数奇数位置的牌抽 出来，保持顺序放到牌堆的下方．（10 分） 假设此时某张牌在 k 次操作后所在的位置为从上往下数第 ak 张．
2016年北京大学博雅计划测试-物理学科试题及答案解析
2016年北京大学博雅计划测试物理学科注意事项：1．本试卷满分100分，和语文、数学、英语、化学共同测试，每个科目100分； 2．试卷为考生回忆版本，本回忆版本，为部分试题。
1．（1）两个一样的乒乓球，一个涂黑，另一个涂白，一盏白织灯放在两个小球的中间照明一段时间，涂____的球表面更烫。
（2）两个一样的白炽灯，同样一个涂黑一个涂白，将两个灯同时打开一段时间，涂____的白炽灯表面更烫。
2．一个带正电的均匀橡皮气球，在不断被吹大的过程中：（ ） A ．球内电场强度不变 B ．球内电势不变C ．球面上一点受其它电荷的作用力不变D ．气球掠过空间中某一点时，该点的电势变化是连续的3．如图，一个半径为R 的圆状物体中过圆心的杆上有一质量为m 的物体（忽略形状）固定在离圆心R /2处，下面正确的是A ．θ＝10°时，有两个稳定平衡位置B ．θ＝20°时，有一个稳定平衡位置，有一个不稳定平衡位置C ．θ＝30°时，有一个稳定平衡位置D ．θ＝40°时，没有稳定平衡位置4．质子的质量为2938Mev c ，电子的质量为20.511Mevc，其中c ＝3×108m /s ，基态氢原子的能量为-13.6eV ，氢原子从第一激发态跃迁回基态，放出光子的频率为f 1，氢原子从基态跃迁到第一激发态，吸收的光子频率为f 2。
求：1）初始静止的处于第一激发态的氢原子，跃迁回基态后的速度v2）相对偏差121||f f f −5．两个质量为m 的斜劈和一个质量为4m ，半径为R 的球在外力作用下保持如图所示的静止状态，忽略所有摩擦，然后某时刻撤去外力，求球掉到平面上所需要的时间。
6．一个通电螺线管，单位长度上的匝数为n ，长度为L 。
在螺线管中有一个等腰梯形线框，且OAB ，OCD 各在一条直线上，OB ＝OC ＝BC ＝l ，12AO OB =，通电螺线管中通有大小为I ＝kt +m ，（k ，t 为正的常数）的电流，某时刻产生的磁场如图。
2016北京大学金秋营数学试题(部分含答案)
2016 年北京大学优秀中学生数学金秋营试题学科专业能力测试一第一天2016 年 10 月 14 日下午 14:00— 17:301、在△ ABC内部有一点 P 满足∠ PAB=∠PCB=，L在AC上且BL平分∠ ABC，延长 PL 交△ APC的外接圆于 Q。
证明： BQ平分∠ AQC.2、对于f(的一个排列)={.}定义函数求所有的排列中，f() 的最小值。
3、求所有正整数a,b,c满足对任意实数u,v,0≤u＜v≤1. 存在正整数n, 使得{} ∈(u,v)成立 .4、设 p 为奇素数， p≡1(mod 4). 正整数 a,b 满足-p =1. 设 q 也为奇素数，(q,bp)=1.考虑同余方程-2a +1≡0(mod q). 证明下述 3 个论述等价 :（1） p 为模 q 的二次剩余；（2）同余方程存在一个解；（3）同余方程存在四个互不相同的解。
学科专业能力测试二第二天2016 年 10 月 15 日上午 09:00-12 ： 005、设函数 f(x)=, 且 x∈[-1,1]时，|f(x)|≤1，求|| 的最大可能值。
6、一个班里有 50 人，相互之间发短信，若在三个人 A,B,C 之间，仅有 A 给 B 发过短信， B 给 C 发过短信， C给 A 发过短信。
则称 A,B,C 三个人构成一个“循环”，试求这 50 个人中“循环”个数的最大可能值。
7、试求所有正整数 a, 使得对任意正整数 k , 都存在正整数 n, 使得 an+2016 是一个正整数的 k 次方。
8、对（ 0，1）中的实数称其中两个为相邻的，如果这两个数的十进制表示中只有一位不同，是否可以将（ 0，1）中的实数 10 染色，使得任意两个相邻的数颜色都不相同？【部分试题参考解答】第 6 题参考解答第 7 题参考解答。
北京大学数学科学学院考博计算数学专业考博笔记考博参考书考博真题
北京大学数学科学学院计算数学专业考博考试复习资料-育明考博一、北京大学数学科学学院计算数学专业考博考试内容分析（育明考博辅导中心)专业招生人数初审复试内容070102计算数学年份计划招生数推荐直博人数申请—考核制我院将根据各专业的具体要求对申请者的专业能力进行考核，方式为面试和笔试相结合。
考核时间一般安排在3月上中旬2016年62人47人育明考博辅导中心王老师解析：①北京大学数学科学学院计算数学专业考博的报录比平均在3:1左右②本专业有33个研究方向：01.有限群及其模表示论02.置换群及代数组合论03.球堆积与密码04.拓扑学05.低维拓扑06.微分几何及其应用07.微分几何08.子流形的整体微分几何09.非线性分析10.几何分析11.微分几何与PDE12.常微分方程与动力系统13.微分遍历论14.微分动力系统15.动力系统和哈密顿系统16.非线性偏微分方程17.代数几何18.偏微分方程及其应用19.密码学与信息安全理论20.数论：算术几何，p-进上同调21.调和分析及其应用22.李群及其作用23.调和分析与偏微方程24.表示理论与数学物理25.偏微分方程与几何分析26.共形几何与微分方程、广义相对论中的微分几何27.辛几何与数学物理28.微分几何与数学物理29.偏微分方程与数学物理30.拓扑量子计算31.组合数学；图论32.调和分析与非线性发展方程33.几何群论③以同等学力资格申请的申请者，还须提交两篇公开发表的学术论文复印件。
④2016年北京大学实行“申请—考核制”，没有提供雅思、托福等英语成绩等级证明的同学，需要参加“北京大学博士研究生英语水平考试”。
育明教育考博分校针对北京大学计算数学专业考博开设的辅导课程有：考博英语课程班·专业课课程班·视频班·复试保过班·高端协议班。
每年专业课课程班的平均通过率都在80%以上。
根植育明学校从2006年开始积累的深厚高校资源，整合利用历届育明优秀学员的成功经验与高分资料，为每一位学员构建考博成功的基础保障。
2014—2016年北京大学博士研究生入学考试考博英语试题及参考答案
2014—2016年北京大学博士研究生入学考试英语真题及参考答案高清版2014真题02年部分II. Reading Comprehension (25 points)Directions: There are five passages in this part. Each of the passages is followed by five questions or unfinished statements. For each of them there are four choices marked A, B, C and D. Choose the best one and mark your answer on the ANSWER SHEET with a single line through the center.1There is a new type of small advertisement becoming increasingly common in newspaper classified columns. It is sometimes placed among “situations vacant”, although it does not offer anyone a job; and sometimes it appears among “situations wanted”, although it is not placed by someone looking for a job either. What it does is to offer help in applying for a job. “Contact us before writing application”, or “Make use of our long experience in preparing your curriculum vitae, or job history”, is how it is usually expressed. The growth and apparent success of such a specialized service is , of course, a reflection on the current high levels of unemployment. It is also, an indication of growing importance of the curriculum vitae.(or job history), with the suggestion that it may now qualify as an art form in its own right.There was a time when job seekers simply wrote letters of application. “JustA. there is lack of jobs available for artistic peopleB. there are so many top-level jobs availableC. there are so many people out of workD. the job history is considered to be a work of art .18. In the past it was expected that first-job hunters would .A. write an initial letter giving their life historyB. pass some exams before applying for a jobC. have no qualifications other than being able to read and writeD. keep any detailed information until they obtained an interview19. Later, as one went on to apply more important jobs, one was advised to include in the letter.something that would attract attention to one’s applicationa personal opinion about the organization one was trying to join something that would offered that person reading ita lie that one could easily get with telling20. The job history has become such an important document because .A. there has been a decrease in the number of jobs advertisedB. there has been an increase in the number of “qualified” job huntersC. jobs are becoming much more complicated nowadaysD. the other processes of applying for jobs are more complicated2Pity those who aspire to put the initials PhD after their names. After 16 years of closely supervised education, prospective doctors of philosophyare left more or less alone to write the equivalent of a large book. Most social-science postgraduates have still not completed their theses by the time their grant runs out after three years. They must then get a job and finish in their spare time, which can often take a further three years. By then , most new doctors are sick to death of the narrowly defined subject which has blighted their holidays and ruined their evenings.The Economic and Social Research Council, which gives grants to postgraduate social scientists, wants to get better value for money by cutting short this agony. It would like to see faster completion rates: until recently, only about 25% of PhD candidates were finishing within four years. The ESRC’s response has been to stop PhD grants to all institutions where the proportion taking less than four years is below 10%; in the first year of this policy the national average shot up to 39%. The ESRC feels vindicated in its toughness, and will progressively raise the threshold to 40% in two years. Unless completion rates improve further, this would exclude 55 out of 73 universities and polytechnics-including Oxford University, the London School of Economics and the London Business School.Predictably, howls of protest have come from the universities, who view the blacklisting of whole institutions as arbitrary and negative. They point out that many of the best students go quickly into jobs where they can apply their research skills, but consequently take longer to finis their theses. Polytechnics with as few as two PhD candidates complain that they are penalized by random fluctuations in student performance. The colleges say there is no hard evidence to prove that faster completion rates result fromgreater efficiency rather than lower standards or less ambitious doctoral topics.The ESRC thinks it might not be a bad thing if PhD students were more modest in their aims. It would prefer to see more systematic teaching of research skills and fewer unrealistic expectations placed on young men and women who are undertaking their first piece of serious research. So in future its grants will be given only where it is convinced that students are being trained as researchers, rather than carrying out purely knowledge-based studies.The ESRC can not dictate the standard of thesis required by external examiners, or force departments to give graduates more teaching time. The most it can do is to try to persuade universities to change their ways. Recalcitrant professors should note that students want more research training and a less elaborate style of thesis, too.21. By time new doctors get a job and try to finish their theses in spare time, .A. most of them died of some sicknessB. their holidays and evenings have been ruined by their jobsC. most of them are completely tired of the narrowly defined subjectD. most of their grants run out22. Oxford University would be excluded out of those universities that receive PhD grants from ESRC, because the completion rate of its PhD students’ theses within four years is lower than.A. 25%B. 40%C. 39%D. 10%23. All the following statements are the arguments against ESRC’s policy except .A. all the institutions on the blacklist are arbitrary and negativeB. there is no hard evidence to prove that faster completion rates result from greater efficiency rather than lower standards or less ambitious doctoral topics.C. many of the best students go quickly into jobs where they can apply their research skills, but consequently take longer to finish their theses.D. some polytechnics are penalized by random fluctuations in student performance24. The ESRC would prefer .A. that the students were carrying out purely knowledge-based studies rather than being trained as researchers.B. to see higher standards of PhD students’ theses and more ambitious doctoral topicsC. more systematic teaching of research skills to fewer unrealistic expectations placed on inexperienced young PhD students.D. that PhD students were less modest in their aims25. what the ESRC can do is to .A. force departments to give graduates more teaching timeB. try to persuade universities to change their waysC. dictate the standard of thesis required by external examinersD. note that students want more research training and less elaborate styleof thesis3Influenza should not be dismissed as a trivial disease. It kills thousands of people every year at a very high cost to the economy, hits hardest the young and the elderly, and is most dangerous for people over the age of 65. influenza is mainly a seasonal illness of the winter months, though in tropical and subtropical areas of Asia and the Pacific it can occur all the year round.The damaging effects of influenza can be prevented by immunization, but constant changes of antigenic specificity of the virus necessitate a different composition of the vaccine from one year to another. The network of WHO Collaborating Centers for Influenza and national institutes carries out influenza surveillance activities to monitor the evaluation of influenza virus strains, and WHO hold an annual consultation at the end of February to recommend the composition of the vaccine for the forthcoming epidemiological season. These recommendations are published immediately in the Weekly epidemiological record.Vaccination each year against influenza is recommended for certain high-risk populations. In closed or semi-closed settings, maximum benefit from immunization is likely to be achieved when more than three-quarters of the population are vaccinated so that the benefit of “herd immunity” can be exploited. Special care should be taken of the following groups:--adults and children with chronic disorders of the pulmonary or cardiovascular systems requiring regular medical follow-up or who had beenhospitalized during the previous year, including children with asthma; --residents of nursing homes and other establishments for patients of any age with chronic medical conditions;--all people over the age of 65.Physicians, nurses, and other personal in primary and intensive care units, who are potentially capable of transmitting influenza to high risk persons, should be immunized; visiting nurses and volunteer workers providing home care to high-risk persons should also be included.26. This passage .A. concerns the damaging effects of influenzaB. mentions the steps of fighting against the harmful effects of influenzaC. emphasizes the worry expressed by all age groupsD. both A and B27. That a different component part of the vaccine is necessary is principally due to the variable change of .A. virusB. strainC. antigenD. immunization28. Which has been done by World Health Organization in combating the bad effects of influenza?A. supervising the assessment of influenza virus strains.B. Holding meetings twice a year to provide the latest data concerning the composition of the vaccines.C. Publishing the related information in a WHO almanac.D. Stressing the importance of preventing influenza for people living in tropical areas of Asia.29. According to the passage, high-risk persons exclude which of thefollowing kinds of people ?A. Children suffering from asthma.B. The elderly with chronic pulmonary diseases.C. Middle aged people with chronic heart diseases.D. Nurses taking special care of the sick.30. In which of the following publications would this passage most likely be printed?A. A surgery book.B. A psychology bookC. An epidemiology book.D. An obstetrics book4In science the meaning of the word “explain”suffers with civilization’s every step in search of reality. Science can not really explain electricity, magnetism, and gravitation; their effects can be measured and predicted, but of their nature no more is known to the modern scientist than to Thales who first speculated on the electrification of amber. Most contemporary physicists reject the notion that man can ever discover what these mysterious forces “really”are. Electricity, Bertrand Russell says, “is not a thing, like St. Paul’s Cathedral; it is a way in which things behave. When we have told how things behave when they are electrified, we have told all thee is to tell.”Until recently scientists would have disapproved of such an idea. Aristotle, for example, whose natural science dominated Western thought for two thousand years, believed that man could arrive at an understanding of reality by reasoning from self-evident principles. He felt, for example, that it is a self-evident principle that everything in the universe has itsproper place, hence one can conclude that objects fall to the ground because that is where they belong, and smoking goes up because that is where it belongs. The goal of Aristotelian science was to explain why things happen. Modern science was born when Galileo began trying to explain how things happen and thus originated the method of controlled experiment which now forms the basis of scientific investigation.31. The aim of controlled scientific experiments is .A. to explain why things happenB. to explain how things happenC. to describe self-evident principlesD. to support Aristotelian science32. what principles most influenced scientific thought for two thousand years?A. The speculations of ThalesB. The forces of electricity, magnetism, and gravityC. Aristotle’s natural scienceD. Galileo’s discoveries33. Bertrand Russell’s notion about electricity is .A. disapproved of by most modern scientistsB. in agreement with Aristotle’s theory of self-evident principlesC. in agreement with scientific investigation directed toward “how”things happenD. in agreement with scientific investigation directed toward “why ”things happen34. The passage says that until recently scientists disagreed with the idea .A. that there are mysterious forces in the universeB. that man can not discover what forces “really” areC. that there are self-evident principlesD. that we can discover why things behave as they do35. Which of the following is the topic most likely to be discussed right after the passage?A. The most recent definition of “explain”B. The relationship between science and religionC. The limitations of scienceD. Galileo and the birth of modern science. 5Some weeks ago, riding in a cab from Boston to Cambridge, my driver turned and asked me what I did for a living . “Teach English”, I said. “Is that so? ”The young man continued. “I was an English major”But then, instead of chatting idly about Joyce or dropping the subject altogether, this driver caught me short. “You guys,”he said, turning back so that his furry face pressed into the glass partition, “ought to be shot”I think he meant it . The guilty party in this present state of affairs is not really the academic discipline. It is not the fault of English and philosophy and biology that engineering and accounting and computer science afford students better job opportunities and increased flexibility in career choice. Literature and an understanding of, say, man’s evolutionary past are as important as ever. They simply are no longer perceived in today’s market as salable. That is a harsh economic fact. And it is not only true in the United States. Employment prospects for liberal arts graduates in Canada, for example, aresaid to be the worst since the 1930s.What to do? I think it would be shortsighted for colleges and universities to advise students against majoring in certain subjects that do not appear linked (at least directly) to careers. Where our energies should be directed instead is toward the development of educational programs that combine course sequences in the liberal arts with course in the viable professions. Double majors---one for enrichment, one for earning one’s bread---have never been promoted very seriously in our institutions of higher learning, mainly because liberal arts and professional-vocational faculties have long been suspicious or contemptuous of one another. Thus students have been directed to one path or the other, to the disadvantage of both students and faculty.A hopeful cue could be taken, it seems to me, from new attempts in the health profession(nursing and pharmacy, for example), where jobs are still plentiful, to give the humanities and social sciences a greater share of the curriculum. Why could not the traditional history major in the college of arts and sciences be pointed toward additional courses in the business school, or to engineering, or to physical therapy? This strategy requires a new commitment from both the institution and the student and demands a much harder look at the allocation of time and resources. But in an age of adversity, double majors are one way liberal arts students can more effectively prepare for the world outside.36. What is the chief purpose of double majors?A. To help graduates of history major become successful businessmen.B. To provide liberal arts graduates with a method of meeting effectively the challenge in employment.C. To extend their knowledge learnt in the college.D. To moderate the tension between liberal arts and vocational faculties.37. In paragraph 1, the sentence “You guys ought to be shot” shows that at heart the driver .A. felt greatly regretted about the major he had chosenB. felt a deep hatred for all the English teachers in his former collegeC. complained that his teachers hadn’t taught him how to survive in this competitive society.D. held a deep contempt in the author because of his scholastic manner38. It can be inferred from the passage that the blame for the present state of affairs lies in the fact that .A. the course sequences themselves are unreliable.B. more and more students start to select science majorsC. almost none of the specialties the students major in might be salable in today’s marketD. the opportunities of employment are scarce for graduates of non-science majors39. The obstacles in course sequences in academic schooling are indicated in all of the following EXCEPT .A. the misguidance of major-selection in some of the institutions of higher learningB. the current curriculum couldn’t keep up with the development of thesocietyC. the inharmonious relation among the teaching facultiesD. the authorities of higher learning attach only little importance to course sequences40. This passage can best be titled as .A. Harsh Economic FactB. Double Majors, a Way OutC. Careers, Schooling fro BetterD. Market for Graduates6Does an unborn baby know his mother’s voice? psychology professor Anthony DeCasper advised an ingenious experiment to find out. He placed padded earphones over a newborn’s ears and gave him a bottle nipple attached to a closed rubber tube. Changes in pressure in the tube switched channels on a tape recorder. If the baby paused extra long between bursts of sucking, he heard on channel; if he paused shorter than average, he heard the other. The baby now had the ability, in effect to change channels.DeCasper found that newborns choose the recording of their mother’s voice over that of another woman’s. The baby, however, has no innate interest in his father’s voice, which is heard in the womb only from time to time, while the mother’s voice is ever present. Within two weeks after birth, however, the baby can recognize Dad’s voice too.A newborn is even attuned to the cadence and rhythm of his native language. In a French study using a setup similar to DeCasper’s, French babies given the choice between French and Russian words responded more to the sound of French.Brian Satt, a research specialist in clinical psychology, has parents sing a lullaby-like “womb song”to their babies. The unborn baby often develops a specific, consistent movement pattern when its song is sung. According to Satt, most parents can calm a fussy newborn with the song most of the time, which is a prize worth more than rubies to a new parent.He is roused by a heavy jolt. His mother has tripped and fallen heavily on one hip. He is much too well cushioned to experience any injury, but her pain and the fear that she may have hurt him floods both their bodies with adrenaline and other stress-related hormones. He cries and kicks vigorously, a cry never heard because there is no air to make sound. As she recovers the stress hormones ebb away, and he calms down too.41. Which of the following is NOT mentioned about the unborn baby in the passage?A. An unborn baby can occasionally hear his father’s voice.B. Dc. Casper’s approach proved absolutely effective in a French experiment.C. An unborn baby is able to identify the tone and rhythm of his native language.D. Parents are able to soothe a fussy newly-born baby.42. According to the author, an unborn baby .A. is unable to identify his mother’s lullaby after birthB. is able to identify his mother’s voice rather than that of others’C. is able to help release adrenaline and other stress-related hormonesD. is able to distinguish French accent from Russian accent43. It is known from the passage that .A. mother’s stress, anger, shock or grief might not hurt the unborn baby in the wombB. an unborn baby’s cry might never be heard because of the particular condition of the womb.C. lullabies are the most precious means to young parentsD. an unborn baby has to move at intervals in the womb44. The author believes that the reaction of an unborn baby to his mother’s voice .A. belongs to one of the natural tendenciesB. is an indication which shows an unborn baby can use all his senses after birthC. is but a physiological circulation of any human beingD. is the most important factor which leads an unborn baby to the survival in the womb45. It can be assumed that the paragraph preceding the passage most probably discussed .A. the development of the baby in his mother’ s wombB. the well-developed taste buds of the babyC. the fact that the baby remains motionless just as what he performs in the first month of his mother’s pregnancyD. the fact that the baby can start to use some of his senses by the last few weeks of pregnancyⅢ. Translation and Writing (55 points)Part A TranslationTranslate the following into Chinese (30 points):Engineering is the professional art of applying science to the optimum conversion of the resources of nature to the uses of humankind. Engineering has been defined as the creative application of “scientific principles to design or develop structures, machines, apparatus, or manufacturing processes, or works utilizing them singly or in combination.” The term engineering is sometimes more loosely defined, especially in Great Britain, as the manufacture or assembly of engines, machine tools, and machine parts. Associated with engineering is a great body of special knowledge; preparation for professional practice involves extensive training in the application of that knowledge. The function of the scientist is to know, while that of the engineer is to do. The scientist adds to the store of verified, systematized knowledge of the physical world; the engineer brings this knowledge to bear on practical problems. Engineering is based principally on physics, chemistry, and mathematics and their extensions into materials science, solid and fluid mechanics, thermodynamics, and systems analysis.2Although for the purpose of this article English literature is treated as being confined to writings in English by natives or inhabitants of the British Isles, it is to a certain extent the case that literature---and this is particularly true of the literature written in English---knows nofrontiers. Thus, English literature can be regarded as a cultural whole of which the mainstream literatures of the United States, Australia, New Zealand, and Canada and important elements in the literatures of other commonwealth countries are parts. It can be argued that no single English novel attains the universality of the Russian writer Leo Tolstoy’s War and Peace. Yet in the Middle ages the Old English literature was influenced and gradually changed by the Latin and French writings, eminently foreign in origin in which the churchmen and the Norman conquerors expressed themselves. From this combination emerged a flexible and subtle linguistic instrument exploited by Geoffrey Chaucer and brought to supreme application by William Shakespeare.Translate the following into English (10 points):从二十世纪中叶起，名国政府对科学技术的重视引起了各级教育机构的响应，理论科学和应用科学的巨大进步也激起了人们学习自然科学的兴趣，科学技术因此有了飞速的发展。
2016年北京大学博雅计划自主招生数学试题及解答
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2016博士《现代数学基础》考试复习题及参考答案(1)
2016级博士生数学复习题1.设(X)=IlA^11是实Hilb e rt空间H上的泛函，证明,当XHo, /U)在点X处沿着“方向的GateaUX微分。
P81证明：Iim /(χ+ 〃?)-/(X) = IinI卜+ 〃?卜何=Iim (|卜+呦-闵1)(卜+呦+制)z→0t /TO t7 r(∣∣x + rΛ∣∣ + ∣∣Λ∙∣∣)Iirn卜 + 呦2-卜If =Iim◎ +心 + 呦_心〉=Iiln 2(儿巾〉-仲M)“ r(∣∣x÷∕A∣∣÷ IA-Il) ^ -O /(∣χ÷rA∣∣÷∣∣χ∣∣)_ “，(卜+ 呦+ 卜||)-(^Z o于是，当XHO时，f在X处沿着Ii方向的GateauX微分为：砒(M) = ¥半IkIl处不是FreChet微分。
P84证明：由于X4 + y1≤^∣x∣,V.r,yX3 yG χ÷y+ J . /Uθ,) = S ' X +y- 2.设泛函〔°(x,y)H(0,0)(XJy)=(O,O),证明/(兀刃在点(°,°)所以f在点(0,0)处连续，令" = (■〃)，则有lim∕(O÷r∕Q-/(O)=Iiln∕→<) f t→0tξ + tη +(fξ)3f∏(/鉀+(S’= ξ + fj因此,f 在点（Oe ）处沿方向h 的GateaUX 微分为Zy （（0、0）,（歹,〃）） = § + 〃，但是,如果令ξ = η∖则有Il 州=(孕+〃2)"2=(孕+0严于是=∣∣⅛⅛≠θ所以，f 在点（0,0）处不是FreChet 可微的。
3.设"，$）为10,l ］χ［0,l ］上的二元连续函数，定义以"⑶为积分核的积分算子^ιΔ2（［0,l ］）→Δ2（［0,l ］）为 (/(/)(∕) = ∫∖(r,5)∕(5)√5,∀f ∈Δ2([0,l])V JL证明：对于任意的/,g∈Δ2（［0,l ］）,有㈣,g> = (伽，s)f($)ds,g(f» =』lw ，s)/(s)d$ g{t}dt = =打(S)^k(TS)g(t)dt ds∕ = (∕,f ",s)g(s)"s 则有(∕√)(r) = ∫7(r,5)∕(5M5,V∕'∈L 2([θ,l])4 求证.(K 了)(/)= [殆可G)ds,WeW[°,ll)∙ blH（孑+缈= ∫j ,∕(0 ^k(JTS) g(S ) d ST(K,∙∙∙) = (討∣χ3,∙∙∙,乎和…)^Il r lLP41证明：对于任意的x = (x∖x2√-∙)∈/2,有II rV Il=Z I r(Xl‘ 疋‘…)iHG 吃,彳兀3,X n‘ ∙∙∙)∣∣「1,2、H ■>"|,/2F产)-+(严)÷ -÷(-χw)-÷- )≤ [(吃)2 + (-Vj)2+••■ + (A H)2+∙∙ )]'2≤ [(^1)2 + (AS)2 + (A3)2 + ∙ +(⅞)2+∙ O]'2 =M于是I∣7'∣∣≤ 1,另外，对于5 = (O,∙∙∙,OJ,O,∙∙-),则有其中ci, C2为任意常数，作变化 1［厂y-广→ 1(〃→ S)所以IlrlI = I5.判断下面方程的类型并把它化成标准型:4%+5g+q+/+匕+ 2 = 0.证明：因为判别式△ = 〃2_4必=9〉0,故方程为双曲型。
北大直博历年试题(2011-2018,缺2014,2017)
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001_北京大学数学科学学院考博参考书和考试要求-博士.
报考北京大学数学科学学院的考生请注意，在报考前要与所要报考方向的博士生导师取得联系。
招生人数：48人。
其中拟招收本科起点直博生20人，本校硕博连读生20人。
应试生复试权重为50％。
一、招生研究方向及考试科目研究方向名称博士生导师考试科目070101基础数学1 有限群及其模表示论张继平 1.英语2.抽象代数3.拓扑学2 置换群及代数组合论方新贵、冯荣权1.英语2.抽象代数3.拓扑学、代数组合论任选一门3 球堆积与密码；数的几何宗传明1.英语2.离散数学3.实分析、复分析任选一门4 拓扑学姜伯驹、丁帆 1.英语2.抽象代数3.拓扑学5 低维拓扑王诗宬同上6 微分几何及其应用莫小欢 1.英语2.微分几何3.拓扑学7 小波及其在图像处理中的应用刘和平 1.英语2.泛函分析3.实分析8 调和分析与小波分析刘和平同上9 微分几何王长平 1.英语2.微分几何3.拓扑学10 多复分析谭小江 1.英语2.微分几何3.复分析11 非线性分析张恭庆、蒋美跃 1.英语2.泛函分析3.偏微分方程12 几何分析丁伟岳、史宇光 1.英语2.微分几何3.偏微分方程13 微分几何与PDE 田刚、朱小华1.英语2.微分几何3.偏微分方程、拓扑学任选一门14 常微分方程与动力系统李伟固、柳彬、杨家忠1.英语2.常微分方程3.偏微分方程、实分析、复分析任选一门15 动力系统与复杂性理论郑志明1.英语2.常微分方程3.偏微分方程、高等概率论任选一门16 复分析伍胜健 1.英语2.泛函分析3.复分析17 微分动力系统文兰、甘少波 1.英语2.常微分方程3.拓扑学18 微分遍历论孙文祥同上19 动力系统和哈密顿系统夏志宏同上20 非线性偏微分方程王保祥 1.英语2.泛函分析3.偏微分方程21 代数几何蔡金星同上22 偏微分方程及其应用王冠香、周蜀林 1.英语2.泛函分析3.偏微分方程23 密码学与信息安全理论冯荣权1.英语2.抽象代数3.计算机软件和理论、密码学任选一门24 数论：算术几何，p-进上同调田青春1.英语2.抽象代数、微分几何任选一门3.拓扑学25 调和分析及应用王保祥 1.英语2.泛函分析3.实分析0701021 多尺度模型与计算及自适应方法张平文 1.英语2.计算方法3.偏微分方程2 科学计算与随机PDE 鄂维南1.英语2.随机过程、计算方法任选一门3.偏微分方程3 偏微分方程数值解王鸣、许进超 1.英语2.计算方法3.偏微分方程4 矩阵计算及其应用徐树方 1.英语2.计算方法3.最优化方法5 微分方程数值解与计算流体力学汤华中、李若 1.英语2.计算方法3.偏微分方程6 最优化方法及其应用高立 1.英语2.计算方法3.最优化方法7 随机模型、算法及应用李铁军1.英语2.计算方法3.偏微分方程、高等概率论任选一门8 计算机图形学郭百宁 1.英语2.计算方法3.信号处理9 图像处理与图像重建周铁 1.英语2.计算方法3.偏微分方程0701031 概率论与随机分析，随机图与随机复杂网络马志明 1.英语、2.随机过程、3.高等概率论2 马氏过程与相互作用粒子系统陈大岳同上3 测度值马氏过程与非线性偏微分方程任艳霞同上4 生存分析、时间序列分析何书元 1.英语2.高等统计学3.高等概率论5 生物信息学与统计学刘军同上6 因果推断及生物医学统计耿直同上7 统计推断、机器学习、遥感郁彬同上8 生存分析刘力平同上9 可靠性、质量检测与控制、生物医学数据分析房祥忠同上10 统计机器学习、互联网搜索李航、沈向洋同上11 试验设计与分析、应用统计艾明要同上12 随机过程及其应用、非平衡统计物理、系统生物学蒋达权 1.英语2.随机过程3.高等概率论13 生物信息学邓明华 1.英语2.高等统计学3.高等概率论0701041 图像重建与图像分析张恭庆、姜明 1.英语2.计算方法3.偏微分方程2 李群表示论及其应用王正栋 1.英语2.泛函分析3.拓扑学3 数学物理刘旭峰 1.英语2.抽象代数3.实分析4 辛几何与可积系统刘张炬1.英语2.微分几何3.偏微分方程、拓扑学任选一门5 随机动力系统与光滑遍历论刘培东 1.英语2.常微分方程3.微分流形6 人工智能与智能软件林作铨1.英语2.离散数学3.计算机软件和理论7 程序理论、软件形式化方法裘宗燕同上8 随机数理方程、计算机科学的数学基础李未 1.英语2.抽象代数3.偏微分方程9 统计学习与智能信息处理马尽文 1.英语2.模式识别3.信号处理10 密码学与信息安全理论杨义先、徐茂智 1.英语2.抽象代数3.密码学11 信息安全工程杨义先、徐茂智1.英语2.离散数学3.计算机软件和理论12 符号计算与自动推理夏壁灿1.英语2.抽象代数、离散数学任选一门3.计算机软件和理论13 精算学与信用风险管理扬静平1.英语2.随机过程、高等统计学任选一门3.高等概率论二、初试参考书0 英语或法语1 抽象代数《代数学引论》聂灵沼、丁石孙着第二版高等教育出版社2000《抽象代数》II徐明曜、赵春来，北京大学出版社，2007年第一版2 微分几何《黎曼几何初步》伍鸿熙、沈纯理、虞言林着北京大学出版社1989《黎曼几何引论》陈维桓、李兴校着，北京大学出版社2002年，第1章、2、3、4章《黎曼几何初步》伍鸿熙、沈纯理、虞言林着北京大学出版社19893 拓扑学Munkres,J.R.ElementsofAlgebraicTopology,Addison-Wesley,MenloPark,California,1984Guillemin,V.andPollack,A.,DifferentialTopology,Prentice-Hall,NewJersey,1974Hu,Sze-Tsen,HomotopyTheory(Chapters1-6),AcademicPress,NewYork,19594 实分析《实变函数论》周民强《实分析》陆善镇等5 复分析Ahlfors,ComplexAnalysis《复分析引论》李忠北京大学出版社6 泛函分析《泛函分析讲义》上册张恭庆、林源渠北京大学出版社7 常微分方程《常微分方程》丁同仁，李承治《微分方程定性理论》第6、7章张芷芬等8 偏微分方程《数学物理方程讲义》姜礼尚等《偏微分方程》周蜀林北京大学出版社《二阶椭圆型方程与椭圆型方程组》陈亚浙、吴兰成9 高等概率论K.L.Chung,ACourseinProbabilityTheory,SecondEdition,AcademicPress,NewYork,1974R.T.Durrett,Probability,TheoryandExamples,SecondEdition,DuxburyPress,199610 高等统计学《高等统计学》郑忠国，北京大学出版社，199811 随机过程《随机过程论》第二版，钱敏平、龚光鲁，北京大学出版社《应用随机过程》，钱敏平、龚光鲁，北京大学出版社R.N.BhattacharyaandW.C.Waimire,StochasticProcesseswithA pplications,JohnWiley&Sons,199012 计算方法MichaelT.Heath,(ScientificComputing:AnIntroductorySurvey, SecondEdition,McGraw-HillCompanies,Inc.)科学计算导论，清华大学出版社影印13 模式识别《模式识别》，边肇祺，清华大学出版社，1988或200014 离散数学（含数理逻辑，集合论与图论，代数结构）《离散数学教程》屈婉铃等，北京大学出版社，2002 《数理逻辑基础上下册》胡世华，陆钟万科学出版社1983 《DiscreteMathematicsandItsApplications》，K.Rosen，McGraw-Hill，1998；影印版，机械工业出版社，199915 最优化方法《最优化方法》，孙文瑜，徐成贤，朱德通，高等教育出版社16 密码学D.R.Stinson,Cryptography,TheoryandPractice,2ndEdition. 中译本，冯登国译，密码学原理与实践（第二版），电子工业出版社，2003O.Goldreich,FoundationsofCryptography,BasicTools,电子工业出版社，2003（此书也有中译本）17 微分流形陈维桓，微分流形初步，高教出版社，第二版，2001 WilliamM.Boothby,AnintroductiontodifferentialmanifoldsandR iemanniangeometry,SecondEdition,AcademicPress,Inc.,1986.18 计算机软件与理论（含数据结构，算法与程序设计，计算理论）《算法与数据结构—C语言描述》，张乃孝，高等教育出版社，2002 《DataStructureandProgrammingDesigninC》（2e）,R.Kruse 等，PrenticeHall,1997；影印版，清华大学出版社，1998 《IntroductiontoAlgorithmC》(2e)，T.Cormen等；影印版，高等教育出版社，2002《C++ProgrammingLanguageC》,B.Stroustrup，Addison-Wesley,1999；影印版，高教出版社，2000；中译本，机械工业出版社，2002或者2005《CProgrammingLanguageC》(2e),B.W.KernighanandD.M.Ritchie,影印版：清华出版社，中译本：机械工业出版社《IntroductiontoAutomataTheory,Languages,andComputation》（2e），J.Hopcroft,R.Motwani，J.Ullman，Addison-Wesley，2001；影印版，清华大学出版社，2002《可计算性与计算复杂性导引C》，张立昂，北京大学出版社，199619 信号处理《数字信号处理》，程干生，北京大学出版社，200320 代数组合论N.Biggs,AlgebraicGraphTheory,secondedition,CambridgeUniversityPress,1993.C.Godsil,AlgebraicCombinatorics,Chapman&Hall,1993.C.GodsilandG.Royle,AlgebraicGraphTheory,GTM207,Springer,2001.以上考试是初试，初试通过后，有关导师可以再按照自己的要求进行专业课的笔试（这时以硕士生课程的水平为准），参考书见下。
2016年北京大学数学科学学院直博生摸底考试
A=
0
0 ...
1
0 பைடு நூலகம்..
··· ... ...
1 ...
1
B
=
0
1 ...
0
0 ...
··· ... ...
0 ...
0
0 ··· 0 0
1 ··· 1 0
试证明两矩阵是相似的，并求出一个矩阵 T ，使得 A = T −1BT
六.(20 分)R[x] 中有多项式 f (x) = x4 + a1x3 + a2x2 + a3x + a4, 试用系数 a1, a2, a3, a4 的关系式给出 f (x) 能表达成某个不可约二次多项式 g(x) 之平方的充分必要条件
0,Φxx, Φyy
在
O
处为
1，Φxy (O)
=
1 2 , Φz(O)
=
−1, 2
则
Φ(x, y, z)
在
O
的领
域存在隐函数 z = z(x, y)(已知 f (0, 0) = 0), 试讨论 z = z(x, y) 在 (0, 0) 点附近的极值
问题。
三.(40 分) 设 z = z(x, y) 为上题所述的函数，Ωδ 是原点 (0, 0) 的 δ 领域，当 δ 充分 小时, 证明如下极限存在并求之
2
∫∫
lim t
e−tz(x,y)dxdy
t→+∞
Ωδ
四.(20 分) 设 A 是一个 2 阶复方阵，考虑 2 阶复方阵的线性空间 M2(C) 上的线性 变换
ϕA : M2(C) → M2(C); X → AX − XA ,
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七.(30 分) 欧式平面上保向的等距变换的一个子群 G，其中每一个非恒同的变换 g 都没有不动点，而且每一个平面上的点 p 在群 G 的作用下得到的轨道 (即点集 {g(p)|g ∈ G}) 在平面上无聚点。证明 G 可由一个或两个平移变换生成, 即 G = {nα|n ∈ Z} 或 G = {nα + mβ|n, m ∈ Z}，其中 Z 为整数集，n, m 为任意整数,α, β 为线性无关的平移 向量 (也表示其对应的平移变换)。nα + mβ 即对应线性组合所表示的平移
问题。
三.(40 分) 设 z = z(x, y) 为上题所述的函数，Ωδ 是原点 (0, 0) 的 δ 领域，当 δ 充分 小时, 证明如下极限存在并求之
∫∫
lim t
e−tz(x,y)dxdyt→+∞Ωδ四.(20 分) 设 A 是一个 2 阶复方阵，考虑 2 阶复方阵的线性空间 M2(C) 上的线性 变换
2
2016 年北京大学数学科学学院直博生摸底考试
2016.4.9 9:00-11:00
一. 证明题 (30 分，每小题 15 分) (1). 设 f (x) 在 (−∞, +∞) 上可导，且 f ′(x) > f (x), ∀x ∈ (−∞, +∞), 证明：f (x) 至多只有一个零点。
(2). 设 f (x) 在 (−∞, +∞) 上二阶可导，且 f ′′(x) > f (x), ∀x ∈ (−∞, +∞), 证明： f (x) 至多只有两个零点。
二.(30 分) 设 Φ(x, y, z) 是原点 O 某个领域上的 C∞ 函数，且 φ, Φx, ΦyΦxz, Φyz 在
原点
O
处为
0,Φxx, Φyy
在
O
处为
1，Φxy (O)
=
1 2 , Φz(O)
=
−1, 2
则
Φ(x, y, z)
在
O
的领
域存在隐函数 z = z(x, y)(已知 f (0, 0) = 0), 试讨论 z = z(x, y) 在 (0, 0) 点附近的极值
ϕA : M2(C) → M2(C); X → AX − XA ,
试确定 dim(ker(ϕA)) 的所有可能的取值
1
五.(30 分) 证明有理数域上的两个 n 阶方阵
A=
0
0 ...
1
0 ...
··· ... ...
1 ...
1
B
=
0
1 ...
0
0 ...
··· ... ...
0 ...
0
0 ··· 0 0
1 ··· 1 0
试证明两矩阵是相似的，并求出一个矩阵 T ，使得 A = T −1BT
六.(20 分)R[x] 中有多项式 f (x) = x4 + a1x3 + a2x2 + a3x + a4, 试用系数 a1, a2, a3, a4 的关系式给出 f (x) 能表达成某个不可约二次多项式 g(x) 之平方的充分必要条件