Ngô Quốc Anh

October 12, 2012

Đề thi Cao học ĐHKHTN Hà Nội

Filed under: Đề Thi — Ngô Quốc Anh @ 10:20

Đề thi Cao học Đại học Khoa học Tự Nhiên Hà Nội

__________Môn Đại số__________ __________Môn Giải tích__________







Vì lý do khách quan nên đề thi trong 1 hoặc 2 năm gần nhất sẽ chưa được cập nhật ở trang web này. Mọi đóng góp về đề thi cũng như những chỉnh sửa về sai sót luôn được chào đón.

Happy reading :).

December 21, 2009

The QE – Department of Mathematics, Rutgers University

Filed under: Đề Thi — Ngô Quốc Anh @ 12:58

The Mathematics Ph.D. program at Rutgers includes two qualifying examinations, a written exam and an oral exam . The written exam is taken first and covers advanced calculus, elementary topology (metric spaces, compactness, and related topics), and the material of 501 (real analysis), 503 (complex analysis), and 551 (algebra). It is offered twice a year, near the beginning of each semester.

The syllabus represents a common core of material required of all Rutgers Ph.D.’s. In particular, the exam is designed with the goal that a pass on this exam shows a level of mathematical knowledge and ability appropriate for teaching the central undergraduate classes in mathematics.

Each student is required to take the exam by the beginning of the student’s second year; the program director may allow a student who has entered with less preparation than the norm to take the exam a specified number of semesters later.

Students who fail this exam may take it again during the semester following the one in which the exam was failed. Students who fail on the second attempt or who do not take the exams on schedule (as determined by the program director) will not be allowed to continue in the Ph.D. program.


December 7, 2009

The QE – UCLA Department of Mathematics

Filed under: Đề Thi — Ngô Quốc Anh @ 11:36

There are two types of qualifying exam: the Basic exam and the Area exams. The Basic exam is designed to be passed by well-trained students before they commence study at UCLA. It examines fundamental topics of the undergraduate mathematics curriculum. The Area exams are graduate level exams. For each Area exam there is a preparatory course sequence. There are Area exams in Algebra, Analysis, Applied Differential Equations, Numerical Analysis, Geometry /Topology, and Logic. Students may attempt any number of examinations in each examination period.

MA students must pass the Basic Exam only. PhD students must pass the Basic exam and two Area exams. MA students must pass the Basic by the beginning of the sixth quarter of study. PhD students must pass the Basic by the fourth quarter of graduate study. A PhD student must pass the one Area examination by the sixth quarter. A PhD student must pass the second Area examination by the seventh quarter of graduate study.

The exams are offered in the Fall and in the Spring, usually just before the beginning of those quarters. Precise dates and times are posted well in advance of the exams. Students must sign-up for the exams in the Graduate Office. Each exam lasts 4 hours. Copies of past exams may be downloaded from our website by clicking here: Download Exams. However, it is strongly recommended that students prepare for exams by studying their syllabi theme by theme, and by doing numerous exercises other than those on old exams. Experience shows that study organized around working old exams is not as efficacious as thematically organized study.

Each exam is written and graded by a committee created for that purpose. The Graduate Studies Committee approves exam results (passing or failing), taking into account recommendations of the examination committee. Shortly after the Graduate Studies Committee’s decision, students are notified of their exam results. Students are reminded that the grading of exams is a complex matter, and that final result (Pass or Fail) is not usually determined by the total score of all work on all problems. Students should read and follow carefully the instructions of an exam.

Graded exams are kept in the Graduate Office for six months and then destroyed. They may be examined in the Graduate Office during this time. After the results of the exams are announced, there is a one week appeal period during which students may petition, in writing, to a Qual Committee for regrading of problems. Appeals must be submitted via the Graduate Office. The Qual Committee will respond, usually in writing, to any appeal within one week.


September 13, 2009

QE in Department of Mathematics, National University of Singapore, August 2009

I have just passed QE held in August 2009 for my first attendance, I hereby show you the analysis paper

Question 1 [10 marks]. Suppose f and g are both measurable functions on the interval (0,1) such that for all t \in \mathbb R^1

| \{ x \in(0,1):f(x)\geq t\}| = |\{ x\in (0,1):g(x)\geq t\}|

Assume that f and g both are monotone decreasing and continuous from left. Can you conclude that f(x)=g(x) for all x \in (0,1)? Give the reason to support your answer.

Question 2 [10 marks]. Compute the volume of the region bounded by

{\left( {{a_{11}}x + {a_{12}}y + {a_{13}}z} \right)^2} + {\left( {{a_{21}}x + {a_{22}}y + {a_{23}}z} \right)^2} + {\left( {{a_{31}}x + {a_{32}}y + {a_{33}}z} \right)^2} = 1

where the determinant of the 3 \times 3 matrix (a_{ij}) is NOT equal to zero.

Question 3 [10 marks]. Let D be a measureable set in \mathbb R^n with finite measure. Suppose \phi(x,t) is a real valued continuous function on D \times \mathbb R^1 such that for almost every x \in D, \phi(x,t) is a continuous function of t and for every real number t, \phi(x,t) is measurable function of x. If \{f_n\} is a sequence of measurable functions on D that converges to f in measure, show that \{\phi(x,f_n(x))\} converges to \phi(x,f(x)) in measure.

Question 4 [10 marks]. Find the function

I\left( y \right) =\displaystyle\int\limits_0^\infty {{e^{ - a{x^2}}}\cos \left( {yx} \right)dx}

if a>0 is a constant. Justify your answer.

Question 5 [10 marks]. Compute the intergal

\displaystyle\int\limits_0^\pi {\frac{{x\sin x}}{{1 + {a^2} - 2a\cos x}}dx}

where a>0 is a constant.

Question 6 [10 marks]. Supposet f(z) is a holomorphic function on the complex plane \mathbb C. If f locally keeps the area invariant, what will the function f be?

Question 7 [10 marks]. Is there an analytic function f on \Delta (unit disk in the complex plane with center 0) such that |f(z)|<1 for |z|<1 with f(0)=\frac{1}{2} and f'(0)=\frac{3}{4}? If so, find such an f. Is it unique?

Question 8 [10 marks]. Let m<n be two positive integers and \Omega and G be open subsets in \mathbb R^n and \mathbb R^m respectively. Does there exist a map f :\Omega \to G such that f and the inverse of f are both C^1? Justify your answer.

Question 9 [10 marks]. Is there a square integrable function f on [0,\pi] such that both inequalities

\displaystyle\int\limits_0^\pi{{{\left( {f\left( x \right)-\sin x} \right)}^2}dx}\leq\frac{4}{9}


\displaystyle\int\limits_0^\pi{{{\left( {f\left( x \right)-\cos x} \right)}^2}dx}\leq\frac{1}{9}

hold? Justify your answer.

Question 10 [10 marks]. Let \alpha_k for k=1,2,...,n be n real numbers such that 0<\alpha_k<\pi for any k. Define


Show that

\displaystyle{\left( {\prod\limits_{k = 1}^n {\frac{{\sin {\alpha _k}}}{{{\alpha _k}}}} } \right)^{\frac{1}{n}}} \leq \frac{{\sin \alpha }}{\alpha }.

May 19, 2009

The QE – Purdue University, Department of Mathematics

Filed under: Đề Thi — Ngô Quốc Anh @ 14:06

The student must pass four written examinations chosen as described below. The exams are based on material that is covered in the courses listed and on material from undergraduate prerequisites. Credit for passing a similar examination at another university cannot be transferred.

The Qualifying Examinations are written examinations offered twice a year during week long Qualifier Exam Sessions the week before classes start in August and January. Each examination is written and graded by a faculty member or a committee of faculty members chosen by the Graduate Committee.

The following four subject areas are called the Core 4 Areas:.

  • Complex Analysis (MA 530)
  • Real Analysis (MA 544)
  • Abstract Algebra (MA 553)
  • Linear Algebra (MA 554)

The qualifier exam subject areas are the Core 4 Areas plus the following Area Exams:

  • Numerical Analysis (MA 514)
  • Probability (MA 519)
  • Partial Differential Equations (MA 523)
  • Differential Geometry (MA 562)
  • Topology (MA 571)
  • Mathematical Logic (MA 585)

The student must pass at least two exams from the Core 4 Areas, including at least one of 544 or 553. They must also pass two more exams from the Area Exams and the unused two exams from the Core 4.

The Qualifier Deadline for students who enter the program with a master’s degree is the January Qualifier Exam Session of their second year. The Qualifier Deadline for students without a master’s degree is the January Qualifier Exam Session of their third year. Students who have not passed the four exams on or before the session of their Qualifier Deadline will have their privileges to continue in the mathematics PhD program terminated.

Each qualifier exam can be attempted a maximum of three times and students may attempt as many qualifier exams as they wish at any Qualifier Session on or before their Qualifier Deadline.

Once an exam is passed, it cannot be retaken to improve the grade from B to A.


The QE – Texas A&M University, Department of Mathematics

Filed under: Đề Thi — Ngô Quốc Anh @ 13:49

This is a combination of Qualifying Exams and Basic Coursework. We offer 5 distinct qualifying exams, twice a year (January and August), consisting of 4hour long written exams in the following subjects:

  • Algebra
  • Complex Analysis
  • Geometry/Topology
  • Numerical/Applied Analysis
  • Real Analysis

These exams are designed to assess that student’s competence in basic mathematical skill and knowledge in each of these areas at the level of the core courses displayed below. In order to fulfill the requirement, a student must pass at least 2 qualifying exams and take at least 2 full-year core sequences (with at least a B grade)out of the following table of basic level courses:

Group I Group II Group III Group IV
AlgebraMath 653/Math 654 Real AnalysisMath 607/Math 608 Diff. GeometryMath 622/Math623 Applied AnalysisMath 641/Math 642
Discrete Math/Number TheoryMath 613/Math 630/Math 627 Complex AnalysisMath 617/Math 618 TopologyMath 636/Math 637 Numerical AnalysisMath 609/Math 610

Furthermore, the choices must be made so that the combination of qualifying exams and core sequences cover at least three out of the four groups displayed in the columns of the table above. A typical student is expect to have fulfilled the qualifying requirements by the end of the second year of enrollment.

Timetable for the Qualifying Exams

To be considered in good academic standing in the Ph.D. program, a student must have passed two qualifying exams by the end of their second year in the Ph.D. program. For the purpose of the qualifying exam timetable, students will be considered to have begun their Ph.D. program with the Fall semester occurring in the calendar year in which they first enroll in the program.

Except for students being admitted to the Ph.D. program upon completion of a M.S. degree in mathematics at Texas A&M University, the following guidelines apply:

  • Students are expected to have passed at least one qualifying exam by the end of their first year in the Ph.D. program.
  • Students are expected to have passed at least two qualifying exams by the end of their second year in the Ph.D. program.

Since students being admitted to the Ph.D. program upon completion of a M.S. degree in Mathematics at Texas A&M University must have already passed at least one Ph.D. qualifying exam prior to admission to the Ph.D. program, the following amended guideline will apply:

  • Students are expected to have passed their second qualifying exam by the end of their first year in the Ph.D. program.

An archive with previous Qualifying Exams can be found here. Students have the option of passing additional qualifying exams in place of taking the additional two full qualifying exam course sequences.


The QE – Indiana University, Department of Mathematics

Filed under: Đề Thi — Ngô Quốc Anh @ 13:39

The Math Department “qualifying exam” is comprised of a 3-tier system designed to help determine as quickly and efficiently as possible whether students have mastered basic graduate level mathematics, exhibit the necessary abilities and self-discipline, and have prepared themselves to pursue the independent research necessary for the Ph.D. within a 2 to 3 year period.

Tier-1: Comprehensive 400-level written exams

Ph.D. students will take a 2-part written exam on 400-level Analysis and Algebra. These exams will be given the week before classes begin in the fall and in the spring. New students may take either or both of the Tier-1 exams in August when they first arrive. A student is allowed to try each exam each time it is offered, but s/he must pass both exams prior to the end of the second year of study. Students pursuing a Master’s Degree are not required to take the Tier-1 exams.

Tier 1 Exams

Tier-2: Committee Review

Each spring, the Graduate Policy Committee will review the record of every Ph.D. student who has either:

  1. Completed 2 years in the program without previous review, or
  2. Passed the Tier-1 exams on entrance to the program and elects the review at the end of their first year.

The committee will decide which students may continue toward Ph.D. candidacy. The committee’s considerations will include:

  1. Performance on the Tier-1 exams
  2. Performance in 500 level coursework
  3. Student’s academic advisor’s report (see below)
  4. Written personal statement by student
  5. Student’s performance of assistantship duties
  6. Student’s performance on A.I. English exams (if applicable)

As indicated above, students can accelerate their progress in the program by passing the Tier-1 exams on entrance into the program and electing the Tier-2 review at the end of their first year. The review committee will treat this as favorable for their case. Students who do not get a recommendation to continue will be encouraged to complete the M.A. degree. If they have financial support at the time of review, they will be entitled to one additional semester of support.


April 6, 2009

The QE – Harvard University, Department of Mathematics

Filed under: Đề Thi — Ngô Quốc Anh @ 9:45

The qualifying exam in mathematics is designed to measure the breadth of a student’s knowledge in mathematics. The exam may identify those areas in which a student’s knowledge is weak. Passing the exam is an indication that a student is ready to begin more specialized study leading to research work.

The exam is given at the very start of each semester. A student may take the exam as often as (s)he likes. There is absolutely no stigma attached to `failing’ the exam. `Failing’ it may well provide more useful information than `passing’ it. `Passing’ the exam early is mainly an indication that a student has been an undergraduate at a university with a broad undergraduate program in mathematics. It is not a good predictor of the quality of the eventual PhD thesis.

Students are strongly encouraged to first take the exam no later than their second semester. Before passing the qualifying exam, students should take three beginning 200 level (or 100 level) math courses each semester. In a semester in which they are teaching they need only take two such courses. After passing the qualifying exam students are usually excused from grades in any math courses they take. Students are expected to pass the qualifying exam by the end of their second year.

The exam consists of three three hour papers on three consecutive days. Each paper typically has 6 questions covering a broad range of mathematics. The questions aim to test your ability to solve concrete problems by identifying and applying important theorems. They should not require great ingenuity. In any given year the exam may not cover every topic on the syllabus, but it should cover a broadly representative set of quals/topics and over time all quals/topics should be examined.


February 4, 2009

Đề thi cuối kỳ I năm học 2008-2009 môn Giải tích

Filed under: Đề Thi — Ngô Quốc Anh @ 21:58

Đề thi cuối kỳ I năm học 2008-2009 môn Giải tích của Khoa Toán-Cơ-Tin học.

January 11, 2009

The QE – University of Michigan, Department of Mathematics

Filed under: Đề Thi — Ngô Quốc Anh @ 3:20

Each student must pass, or demonstrate a knowledge of, six of the eight core courses. These consist of two-course sequences in

  • algebra (593, 594)
  • analysis (596, 597)
  • applied analysis (556, 572)
  • geometry/topology (591, 592)

Each student must pass the Qualifying Review. This consists of written examinations, based on the same syllabuses as the core courses, a course requirement, and a survey by the Doctoral Committee of the student’s record as a graduate student in the Department. The purpose of the Review is to ensure that students have a good knowledge of core graduate mathematics and to evaluate the chances that a student will be able to complete a Ph.D. degree.

The Qualifying Review is conducted three times a year, and should be taken as soon as the student feels ready. It can be taken as many times as necessary with the only stipulation that a student must pass the exams in one area by the beginning of the fourth term in the program, and must complete the entire Review by the beginning of the sixth.

Students are required to take six courses beyond those needed for the Qualifying Review, distributed among at least three of five areas of mathematics.

To ensure greater intellectual breadth, the Graduate School requires that every student must successfully complete four hours of cognate courses before achieving Candidacy. For students in most departments, these must be taken outside the student’s home department, but mathematics students are allowed to take courses within the Mathematics Department under certain restrictions and with Advisor or Doctoral Committee approval. Cognate courses can be taken at any time.

The course requirements listed above should be regarded as the absolute minimum. The Department expects that most students will take more courses distributed so that they achieve a broad background in their specialty and related areas. Students should also participate actively in the Departmental Seminars offered in their area of interest and attend Colloquia since it is there that they can learn about the latest developments and open problems.


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