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CURRENT AFFILIATION: BEIJING INSITUTE OF MATHEMATICAL SCIENCES AND APPLICATIONS (BIMSA) & YAU MATHEMATICAL SCEINCES CENTER (YMSC), TSINGHUA UNIVERSITY
MATH 4000-Problem Solving for Putnam, Fall 2020, Texas Tech University
Information about the competition and seminar (course description)
The William Lowell Putnam Mathematical Competition is the premiere competition for undergraduate students in North America. More than 500 universities compete in this contest organized by the Mathematical Association of America (MAA).
Putnam Examination (2020)
The Seventy Eight Putnam Examination will be held on Saturday, February 20th, 2021.
It will consist of two sessions of three hours each:
- Morning Session: 9:00am-12:00pm, location: online.
- Afternoon Session: 2:00pm-5:00pm, location: online.
The test is supervised by faculty members of each participating school. Every problem is graded on a scale of 0-10. The problems are usually listed in increasing order of difficulty, with A1 and B1 the easiest, and A6 and B6 the hardest. Top 5 scoring students on the Putnam exam are named Putnam Fellows. A student can take this exam maximum four times and the Putnam official team of the university consists of 3 members. The purpose of this class is to provide a comprehensive introductioninto the world of problem solving in different branches of mathematics such as: real & complex analysis, linear algebra, abstract algebra, combinatorics, probability, geometry and trigonometry and number theory.
This course (MATH 4000-Problem Solving for Putnam) teaches important skills in problem solving that are not taught in a systematic way in any other course. These skills are extremely valuable in preparing students for jobs and for graduate-level research. The teaching style will be a mixture of a lecture and a problem-solving session. By the end of this course, students should develop fundamental problem solving skills, and become accustomed to concentrating on a problem for an extended period of time. Indeed, this seminar concentrates on the raw creative problem-solving skills which can serve as an essential ingredient in almost every field of activity.
Course number, office hours and webpage
- MATH 4000-Problem Solving for Putnam, Main Campus (Online via Zoom), Fall 2020
- Lecturer: Cezar Lupu (Postdoctoral scholar)
- Topics covered: Elementary, Abstract & Linear Algebra, Real Analysis, Combinatorics, Geometry & Trigonometry, Number Theory, Probability
- Office hours: by appointment via email
- Webpage: https://lupucezar.wordpress.com/competitions/ & https://lupucezar.wordpress.com/teaching/
Schedule and locations
- Tuesday, 12.30-1.50 PM online via Zoom (Zoom link will be sent via email)
This is a lecture given by the instructor on a certain topic. The students will learn different concepts and techniques. Moreover, the lecturer will also present solutions of some problems and will assign other problems as homework for the students.
- Thursday, 1-1.50 PM online via Zoom (Zoom link will be sent via email)
This is more like a recitation rather than a lecture. The 80 minutes will be divided into a problem solving time (30 minutes) to think of problems assigned prior which will be followed by a discussion of the problems (50 minutes). You don’t need to be logged in on Zoom during the problem solving time.
Syllabus, grading criteria and references
The grade will be determined by the following three factors:
- Seminar attendance: 50%
- Homework and seminar activity: 50%
Any student must attend at least 10/15 of the lectures to get full credit. Homework will be assigned biweekly and posted on the teaching section (https://lupucezar.wordpress.com/teaching/) of my webpage at the end of each lecture on Monday. It will consist mostly of problems of A1-B1 or A2-B2 from previous Putnam exams. The homework will be submitted as a PDF file via my email. The deadline for each homework will appear on my teaching page.
The main references include the following:
- R. Gelca, T. Andreescu, Putnam and Beyond, Springer Verlag, 2007.
- K. Kedlaya, B. Poonen, R. Vakil- The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions and Commentary, The Mathematical Association of America, Washington, D.C., 2002.
- L. Larson, Problem-Solving Through Problems, Springer Verlag, 1983.
Detailed schedule (lectures & recitations)
Week 0. An invitation to the Putnam competition
Abstract: In this lecture, we will discuss what is Putnam competition and how to train for it.
Date: August 27 (Presentation)
Week 1. (Elementary) Algebra I
Abstract: This lecture will cover problems on topics such as real algebraic identities and inequalities, and complex numbers.
Date: September 1 (Lecture), September 3 (Recitation)
Week 2. (Elementary) Algebra II
Abstract: This seminar will focus more on mathematical induction, functional equations and polynomials (integer polynomials, roots of polynomials).
Date: September 8 (Lecture), September 10 (Recitation)
Week 3. Geometry and Trigonometry
Abstract: This will cover problems on topics such as vectors, conics, quadratics, and other curves in the plane as well as trigonometric formulas.
Date: September 15 (Lecture), September 17 (Recitation)
Week 4. Combinatorics
Abstract: This will cover problems on topics combinatorial geometry, pigeonhole principle, generating functions graph theory, binomial identities and counting strategies.
Date: September 22 (Lecture) & September 24 (Recitation)
Week 5. Number Theory I
Abstract: This will cover problems on topics such as integer-valued sequences and functions, congruences, divisibility and arithmetic functions.
Date: September 29 (Lecture) & October 1 (Recitation)
Week 6. Number Theory II
Abstract: This will cover problems on topics such as quadratic residues, diophantine equations, and analytic methods in number theory.
Date: October 6 (Lecture) & October 8 (Recitation)
Week 7. Abstract Algebra
Abstract: This will cover problems on topics such as groups, rings, and finite fields.
Date: October 13 (Lecture) & October 15 (Recitation)
Week 8. Linear Algebra I
Abstract: This will cover topics on 
Date: October 20 (Lecture) & October 22 (Recitation)
Week 9. Linear Algebra II
Abstract: This will cover problems on topics such as vectors spaces, linear transformations, characteristic and minimal polynomials, eigenvalues, eigenvectors.
Date: October 27 (Lecture) & October 29 (Recitation)
Week 10. Linear Algebra III
Abstract: This will cover some special topics in linear algebra (Jordan canonical form) and beyond.
Date: November 3 (Lecture) & November 5 (Recitation)
Week 11. Real Analysis I
Abstract: This will cover problems on topics such as sequences of real numbers.
Date: November 10 (Lecture) & November 12 (Recitation)
Week 12. Real Analysis II
Abstract: This will cover problems on topics such as series of real numbers.
Date: November 17 (Lecture) & November 19 (Recitation)
Week 13. Real Analysis III
Abstract: This will cover problems on topics such as limits of functions and continuity.
Date: November 24 (Lecture)& November 26 (Recitation)
Week 14. Real Analysis IV
Abstract: This will cover problems on topics such as differentiability of functions.
Date: December 1 (Lecture) & December 3 (Recitation)
Week 15. Real Analysis V
Abstract: This will cover problems on integrability (Riemann integrals, continuity of integrals).
Date: December 8 (Lecture) & December 10 (Recitation)
Week 16. Real Analysis VI
Abstract: This will cover problems on multivariable differential and integral calculus.
Date: December 15 (Lecture) & December 17 (Recitation)
THE 2020 MATH 4000-PROBLEM SOLVING FOR PUTNAM
The 2020 Math Olympiad Summer Program (MOP), July 8-28, online
The 2020 Mathematical Olympiad Summer Program will take place online via Google Meet between July 8-28. The camp is organized by the Mathematical Association of America and it is run by the CMU faculty Po-Shen Loh (director) with the help of other instructors (faculty, postdocs and Ph.D. students) from schools such as Massachusetts Institute of Technology (MIT), Stanford University, Harvard University, North Carolina State University, Carnegie Mellon University (CMU), Princeton University, Ohio State University, and Texas Tech University.
- Students will be separated into four groups.
- Black (~15 students): approx IMO gold.
- Blue (~15 students): approx IMO silver
- Green (~15 students): approx IMO bronze
- Red (~15 students): approx IMO honorable mention.
- The timetable will be (all times Eastern):
- Noon – 1:30pm (Class)
- 2:00pm – 4:00pm (Test/Quiz/Other)
- 4:00pm – 6:30pm (Test/Quiz/Other)
- 7:00pm – 8:00pm (Optional non-Olympiad seminar)
- 8:30pm – 9:00pm (Panel)
My schedule consists of 6 lectures and two seminars. More details are given below:
Lecture 1. Geometric inequalities .
Group: Black, July 9
Location: online via Google Meet
Lecture 2. Algebraic integers.
Group: Green, July 16
Location: online via Google Meet
Lecture 3. Algebraic integers. .
Group: Red, July 17
Location: online via Google Meet
Seminar. Euler’s single and double zeta values.
Group: Black, Blue, Green, Red, July 19
Location: online via Google Meet
Lecture 4. Sequences and series of real numbers and their inequalities.
Group: Blue, July 20
Location: online via Google Meet
Lecture 5. Sequences and series of real numbers and their inequalities.
Group: Black, July 23
Location: online via Google Meet
Lecture 6. Sequences and series of real numbers and their inequalities .
Group: Green, July 24
Location: online via Google Meet
Seminar. An elementary problem equivalent to the Riemann hypothesis.
Group: Black, Blue, Green, Red , July 25
Location: online via Google Meet
MATH 2360-Linear Algebra D-01, Summer 2020, Texas Tech University
Course description and purpose
Students learn how to perform basic vector algebra, and compute their bases, express a linear transformation as a matrix, perform basic matrix manipulations, and compute the determinant of a matrix, compute eigenvalues and eigenvectors, and use the Gram-Schmidt process. Mainly, we shall cover the following topics:
- Systems of Linear Equations.
1.1 Introduction to systems of linear equations.
1.2 Gaussian elimination and Gauss-Jordan elimination. - Matrices
2.1 Operations with matrices.
2.2 Properties of matrix operations.
2.3 The inverse of a matrix.
2.4 Elementary matrices. - Determinants
3.1 The determinant of a matrix.
3.2 Determinants and elementary operations.
3.3 Properties of determinants. - Vector Spaces
4.1 Vectors in.
4.2 Vector spaces.
4.3 Subspaces of a vector space.
4.4 Spanning sets and linear independence.
4.5 Basis and dimension.
4.6 Rank of a matrix and systems of linear equations.
4.7 Coordinates and change of basis. - Linear Transformations
6.1 Introduction to linear transformations.
6.2 The kernel and the range of a linear transformation.
6.3 Matrices for linear transformations.
6.4 Transition matrices and similarity. - Eigenvalues and Eigenvectors
7.1 Eigenvalues and eigenvectors.
7.2 Diagonalization.
Course number, office hours and webpage
- MATH 2360-Linear Algebra, Distance-Online
- Lecturer: Cezar Lupu (Postdoctoral scholar)
- Topics covered: systems of linear equations, matrices, determinants, finite-dimensional vector spaces, linear transformation, eigenvalues and eigenvectors.
- Office hours: by appointment via e-mail
- Webpage: https://lupucezar.wordpress.com/teaching/
Schedule for the Zoom online lectures
- Tuesday, 2.-3.20 PM on Zoom link provided in the email.
- Lecture 1 (June 2nd). Syllabus of the course and the big picture of linear algebra.
- Lecture 2 (June 9th). Systems of linear equations.
- Lecture 3 (June 16th). Matrices.
- Lecture 4 (June 23rd). Determinants.
- Lecture 5 (June 30th). Vector spaces I.
- Lecture 6. (July 7th). Vector spaces II.
- Lecture 7 (July 14th). Linear transformations I.
- Lecture 8 (July 21st). Linear transformations II.
- Lecture 9 (July 28th). Eigenvalues and eigenvectors I.
- Lecture 10 (August 1st) Eigenvalues and eigenvectors II.
- Review session for the final exam (August 4th, 6-8 PM).
Grading criteria, homework, exams and references
The grade will be determined by the following three factors:
- 1 midterm exam: 30%
- Homework assignments (WebWork): 40%
- Final exam: 30%
- Attendance: 10% (extra credit!)
Grade Inquiry Policy: Any student who has questions about any grade received on an assessment should meet with the instructor in person, if possible, within a one-week period of receiving the graded assessment. At this meeting, the instructor will provide an explanation of the grading and computation of the score. Inquires received after the one-week period will not be considered.
Reading: Reading the material from the textbook (paper or electronic form) is mandatory.
WeBWorK: WeBWorK is an internet-based method for delivering homework problems to
students. Visit the course webpage for more information on how to access WeBWorK and how to enter your solutions; see Helpful Documents. You will need your eRaider username and student ID number with the R to log into WeBWorK. The WeBWorK system responds by telling you whether an answer (or set of answers) is correct or incorrect and also records whether you answered the question correctly or incorrectly. You are free to try a problem as many times as you wish until the due date. It is your responsibility to check WeBWorK for new assignments.
Please do not wait until the day the assignment is due to begin and/or send questions. I will not answer questions e-mailed to me within 24 hours from the HW deadline.
Regarding the Comprehensive Final Exam: The final represents a course requirement. A student who did not complete the final exam, but otherwise completed all the other
requirements successfully, cannot be assigned a passing letter grade. Each designated
instructor has to keep his/her copy of partial scores and grades for each student for 2 calendar years from the date of recording the grade. The final exam will be given in a proctored environment. The date for the final exam was scheduled prior to start of the semester; the location will be announced at the end of the semester. The date and time of the exam cannot be changed. Final exam conflicts may arise only between two or more math courses, in which case the course with lowest number has priority in maintaining the exam date and time.
For the final exam, students should not purchase a blue book. The exam will be distributed with on paper, with spaces to fill in/fill out. No other printed materials and electronic devices are allowed during the final. Scratch papers may be checked by the instructors.
There are no make-up WeBWorK homework sets assignments except for absence due to religious observance or absence to due officially approved trips (see Class Attendance below). If a student misses a WeBWorK homework sets for one of the above reasons, the homework set will be reopened.
Attendance will be taken regularly. Students who miss no more than 2 weeks of classroom time during the whole semester will receive a bonus (extra credit) of 10% to their overall grade.
Letter grades will then be assigned in accordance with the following correspondence:
The main references include the following:
- Ron Larson, Elementary Linear Algebra (8th edition), published by Cengage.
- Sergei Treil, Linear Algebra Done Wrong, published by Brown University.
- Tom Denton, Andrew Waldron, Linear Algebra in Twenty Five Lectures, UC Davis course.
- Gilbert Strang, Introduction to Linear Algebra, MIT course.
Accommodations and schedule conflicts
If you have a disability for which you are or may be requesting an accommodation, you are encouraged to contact both me and Student Disability Services in West Hall or call 806-742-2405, before the end of the second week of classes. Please see Operating Policy and Procedure 34.22 for a complete description of Texas Tech’s policy on accommodations or students with disabilities. Similarly, students who have any conflicts (including religious holy days) with the scheduled examination dates should notify me before the end of the second week of classes. A student who is absent from classes for the observance of a religious holy day shall be allowed to take an examination or complete an assignment scheduled for that day within a reasonable time after the absence.
Syllabus Updates for the Spring Semester Courses, 2020 (due to Texas Tech University response to COVID-19)
Following Texas Tech University’s general guidance, starting the week of March 30th, our classes will resume via online delivery.
TTU is open with faculty and staff working remotely. Most facilities will continue to operate, although many will post reduced hours of operation. Faculty and staff should be on campus only to complete functions that are essential and cannot be completed remotely.
A decision on the need to cancel or move online face-to-face Maymester or Summer courses will be made in mid-April. Therefore, our classes will be delivered as follows:
MATH 4351-ADVANCED CALCULUS II
- 11-11.30 AM -reading time
- 11.30 AM-12.20 PM-Zoom lecture/discussion
MATH 2360-LINEAR ALGEBRA
- 12.30-1 PM -reading time
- 1-1.50 PM-Zoom lecture/discussion
MATH 4351-Advanced Calculus 2, Spring 2020, Texas Tech University
Course description and purpose
Students learn how to operate with differentiable functions, Riemann integrals, and study the convergence of sequences and series of functions and real numbers. Mainly, we shall cover the following topics:
- Differentiability.
1.1 The derivative.
1.2 The mean value theorem.
1.3 L’Hospital rules.
1.4 Taylor’s theorem. - The Riemann Integral.
2.1 Riemann integral.
2.2 Riemann Integrable Functions.
2.3 The Fundamental Theorem.
2.4 The Darboux Integral.
2.5 Approximate Integration. - Sequences of Functions.
3.1 Pointwise and Uniform Convergence.
3.2 Interchange of Limits.
3.3 The Exponential and Logarithmic Functions.
3.4 The Trigonometric Functions. - Infinite Series.
4.1 Absolute Convergence.
4.2 Tests for Absolute Convergence.
4.3 Tests for Nonabsolute Convergence.
4.4 Series of Functions.
Course number, office hours and webpage
- MATH 2360-Advanced Calculus 2, Main Campus (TTU Mathematics Building)
- Lecturer: Cezar Lupu (Postdoctoral scholar)
- Topics covered: differentiability, Riemann integrals, sequences and series of functions, infinite series of real numbers.
- Office hours: Tuesday & Thursday: 2-3.30 PM, and by appointment via e-mail
- Webpage: https://lupucezar.wordpress.com/teaching/
Schedule and locations
- Tuesday, 11 AM-12.20 PM in TTU Mathematics Building, room 109
- Thursday, 11 AM-12.20 PM in TTU Mathematics Building, room 109
Grading criteria, homework, exams and references
The grade will be determined by the following three factors:
- 1 midterm exam: 30%
- Homework assignments: 40% (4 assignments of 10% each!+ 10% extra credit for typing them in LaTeX)
- Final exam: 40%
- Final Project: 10%
- Attendance: 5% (extra credit!)
Grade Inquiry Policy: Any student who has questions about any grade received on an assessment should meet with the instructor in person, if possible, within a one-week period of receiving the graded assessment. At this meeting, the instructor will provide an explanation of the grading and computation of the score. Inquires received after the one-week period will not be considered.
Calculator: A graphing calculator is a useful tool for this course. However, calculators or other electronic devices will NOT be permitted on in-class exams, and the final exam.
Reading: Reading the material from the textbook (paper or electronic form) is mandatory.
Regarding the Comprehensive Final Exam: The final represents a course requirement.
A student who did not complete the final exam, but otherwise completed all the other
requirements successfully, cannot be assigned a passing letter grade. Each designated
instructor has to keep his/her copy of partial scores and grades for each student for 2 calendar years from the date of recording the grade. The final exam will be given in a proctored environment. The date for the final exam was scheduled prior to start of the semester; the location will be announced at the end of the semester. The date and time of the exam cannot be changed. Final exam conflicts may arise only between two or more math courses, in which case the course with lowest number has priority in maintaining the exam date and time.
For the final exam, students should not purchase a blue book. The exam will be distributed with on paper, with spaces to fill in/fill out. No other printed materials and electronic devices are allowed during the final. Scratch papers may be checked by the instructors.
Attendance will be taken regularly. Students who miss no more than 2 weeks of classroom time during the whole semester will receive a bonus (extra credit) of 5% to their overall grade.
Letter grades will then be assigned in accordance with the following correspondence:
The main references include the following:
- R Bartle, D. R. Sherbert, Introduction to Real Analysis (4th edition), published by John Wiley & Sons Inc., 2011.
- J. Lebl, Introduction to Analysis (Volume 1), online.
Accommodations and schedule conflicts
If you have a disability for which you are or may be requesting an accommodation, you are encouraged to contact both me and Student Disability Services in West Hall or call 806-742-2405, before the end of the second week of classes. Please see Operating Policy and Procedure 34.22 for a complete description of Texas Tech’s policy on accommodations or students with disabilities. Similarly, students who have any conflicts (including religious holy days) with the scheduled examination dates should notify me before the end of the second week of classes. A student who is absent from classes for the observance of a religious holy day shall be allowed to take an examination or complete an assignment scheduled for that day within a reasonable time after the absence.
MATH 2360-Linear Algebra, Spring 2020, Texas Tech University
Course description and purpose
Students learn how to perform basic vector algebra, and compute their bases, express a linear transformation as a matrix, perform basic matrix manipulations, and compute the determinant of a matrix, compute eigenvalues and eigenvectors, and use the Gram-Schmidt process. Mainly, we shall cover the following topics:
- Systems of Linear Equations.
1.1 Introduction to systems of linear equations.
1.2 Gaussian elimination and Gauss-Jordan elimination. - Matrices
2.1 Operations with matrices.
2.2 Properties of matrix operations.
2.3 The inverse of a matrix.
2.4 Elementary matrices. - Determinants
3.1 The determinant of a matrix.
3.2 Determinants and elementary operations.
3.3 Properties of determinants. - Vector Spaces
4.1 Vectors in
4.2 Vector spaces.
4.3 Subspaces of a vector space.
4.4 Spanning sets and linear independence.
4.5 Basis and dimension.
4.6 Rank of a matrix and systems of linear equations.
4.7 Coordinates and change of basis. - Linear Transformations
6.1 Introduction to linear transformations.
6.2 The kernel and the range of a linear transformation.
6.3 Matrices for linear transformations.
6.4 Transition matrices and similarity. - Eigenvalues and Eigenvectors
7.1 Eigenvalues and eigenvectors.
7.2 Diagonalization.
7.3 Symmetric matrices and orthogonal diagonalization. - Inner Product Spaces
5.1 Length and dot product in
5.2 Inner product spaces.
5.3 Orthonormal bases: Gram-Schmidt process.
Course number, office hours and webpage
- MATH 2360-Linear Algebra, Main Campus (TTU Electrical Engineering Building)
- Lecturer: Cezar Lupu (Postdoctoral scholar)
- Topics covered: systems of linear equations, matrices, determinants, finite-dimensional vector spaces, inner product spaces, linear transformation, eigenvalues and eigenvectors.
- Office hours: Tuesday & Thursday: 2-3.30 PM, and by appointment via e-mail
- Webpage: https://lupucezar.wordpress.com/teaching/
Schedule and locations
- Tuesday, 12.30-1.50 PM in TTU Mathematics Building, room 111
- Thursday, 12.30-1.50 PM in TTU Mathematics Building, room 111
Grading criteria, homework, exams and references
The grade will be determined by the following three factors:
- 1 midterm exam: 30% (there will be 10% extra bonus questions!)
- Homework assignments (WebWork): 30%
- Final exam: 40% (there will be 10% extra bonus questions!)
- Attendance: 5% (extra credit!)
Grade Inquiry Policy: Any student who has questions about any grade received on an assessment should meet with the instructor in person, if possible, within a one-week period of receiving the graded assessment. At this meeting, the instructor will provide an explanation of the grading and computation of the score. Inquires received after the one-week period will not be considered.
Calculator: A graphing calculator is a useful tool for this course. However, calculators or other electronic devices will NOT be permitted on quizzes, in-class exams, and the final exam.
Reading: Reading the material from the textbook (paper or electronic form) is mandatory.
WeBWorK: WeBWorK is an internet-based method for delivering homework problems to
students. Visit the course webpage for more information on how to access WeBWorK and how to enter your solutions; see Helpful Documents. You will need your eRaider username and student ID number with the R to log into WeBWorK. The WeBWorK system responds by telling you whether an answer (or set of answers) is correct or incorrect and also records whether you answered the question correctly or incorrectly. You are free to try a problem as many times as you wish until the due date. It is your responsibility to check WeBWorK for new assignments.
Please do not wait until the day the assignment is due to begin and/or send questions. I will not answer questions e-mailed to me within 24 hours from the HW deadline.
Regarding the Comprehensive Final Exam: The final represents a course requirement. A student who did not complete the final exam, but otherwise completed all the other
requirements successfully, cannot be assigned a passing letter grade. Each designated
instructor has to keep his/her copy of partial scores and grades for each student for 2 calendar years from the date of recording the grade. The final exam will be given in a proctored environment. The date for the final exam was scheduled prior to start of the semester; the location will be announced at the end of the semester. The date and time of the exam cannot be changed. Final exam conflicts may arise only between two or more math courses, in which case the course with lowest number has priority in maintaining the exam date and time.
For the final exam, students should not purchase a blue book. The exam will be distributed with on paper, with spaces to fill in/fill out. No other printed materials and electronic devices are allowed during the final. Scratch papers may be checked by the instructors.
There are no make-up WeBWorK homework sets assignments except for absence due to religious observance or absence to due officially approved trips (see Class Attendance below). If a student misses a WeBWorK homework sets for one of the above reasons, the homework set will be reopened.
Attendance will be taken regularly. Students who miss no more than 2 weeks of classroom time during the whole semester will receive a bonus (extra credit) of 5% to their overall grade.
Letter grades will then be assigned in accordance with the following correspondence:
The main references include the following:
- Ron Larson, Elementary Linear Algebra (8th edition), published by Cengage.
- Sergei Treil, Linear Algebra Done Wrong, published by Brown University.
- Tom Denton, Andrew Waldron, Linear Algebra in Twenty Five Lectures, UC Davis course.
- Gilbert Strang, Introduction to Linear Algebra, MIT course.
Accommodations and schedule conflicts
If you have a disability for which you are or may be requesting an accommodation, you are encouraged to contact both me and Student Disability Services in West Hall or call 806-742-2405, before the end of the second week of classes. Please see Operating Policy and Procedure 34.22 for a complete description of Texas Tech’s policy on accommodations or students with disabilities. Similarly, students who have any conflicts (including religious holy days) with the scheduled examination dates should notify me before the end of the second week of classes. A student who is absent from classes for the observance of a religious holy day shall be allowed to take an examination or complete an assignment scheduled for that day within a reasonable time after the absence.
MATH 4000-Problem Solving for Putnam, Spring 2020, Texas Tech University
Course description and purpose
The MATH 4000-Problem Solving for Putnam is a continuation of the previous semester’s problem solving course. This semester this course is organized such as an independent study session. We shall focus more on solving problems on certain topics, and I shall emphasize on real analysis, combinatorics, number theory, and abstract & linear algebra.
This course teaches important skills in problem solving that are not taught in a systematic way in any other course. These skills are extremely valuable in preparing students for jobs and for graduate-level research. The teaching style will be a problem-solving session. By the end of this course, students should develop fundamental problem solving skills, and become accustomed to concentrating on a problem for an extended period of time. Indeed, this seminar concentrates on the raw creative problem-solving skills which can serve as an essential ingredient in almost every field of activity.
The ultimate goal of this course is for students to acquire necessary skills to be able to solve the difficult problems from the Putnam Mathematical Competition. For more details, about the competition and how the class was organized in Fall see here. Mainly, we shall solve the homework problems from Fall 2019.
Homework 1 (Elementary Algebra & Geometry), Homework 2 (Combinatorics & Number Theory), Homework 3 (Abstract & Linear Algebra), Homework 4 (Real Analysis)
Course number, office hours and webpage
- MATH 4000-Problem Solving for Putnam, Main Campus (TTU Math Building)
- Lecturer: Cezar Lupu (Postdoctoral scholar)
- Topics covered: Elementary Algebra (identities and inequalities for real variables, complex numbers, polynomials, functional equations), Geometry (Euclidian geometry, vectors, complex numbers in geometry, conics), Combinatorics (pigeonhole principle, combinatorial geometry, generating functions, graph theory), Number Theory (divisibility, arithmetic functions, quadratic residues), Abstract Algebra (monoids, groups, rings, fields), Real Analysis (sequences and series of real numbers, continuity and differentiability of functions of a single variable, Riemann integrals and continuity), and Linear Algebra (matrices, determinants, eigenvalues and eigenvectors, linear transformations, etc)
- Office hours: Tuesday & Thursday: 2-3.30 PM, and by appointment via e-mail.
Schedule and locations
- Tuesday & Thursday, 3.30-5 PM in TTU Math Building, room 12
Grading criteria, homework, exams and references
The grade will be determined by the following factors:
- attendance and activity (discussions and solving problems) during regular classes. Roughly, we will solve problems from the homework assigned in the Fall 2019.
Letter grades will then be assigned in accordance with the activity during classes.
The main references include the following:
- R. Gelca, T. Andreescu, Putnam and Beyond, Springer Verlag, 2007.
- K. Kedlaya, B. Poonen, R. Vakil- The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions and Commentary, The Mathematical Association of America, Washington, D.C., 2002.
- L. Larson, Problem-Solving Through Problems, Springer Verlag, 1983.
The 2020 Joint Mathematics Meeting, 15-18 January, 2020, Denver, Colorado
The 2020 Joint Mathematics Meeting of the American Mathematical Society is the largest mathematics meeting in the world. More than 4000 mathematicians and math enthusiasts met in the beautiful city of Denver, CO between January 15-18. The location is Colorado Convention Center. This is the 103th annual winter meeting of MAA and the 124rd annual meeting of AMS. My talk (slides) is part of the AMS Special Session on How to Discover and Train Gifted Students , Wednesday, January 15, 2017, 2.15 PM. Also, I am one of the organizers of this special session.
Full Program Program Presenters
MATH 1451-Calculus I with Applications, Fall 2019, Texas Tech University
Course description and purpose
Students learn how to perform . Mainly, we shall cover the following topics:
- Functions and Graphs.
1.1 What is Calculus?
1.2 Preliminaries.
1.3 Lines in the plane; parametric equations.
1.4 Functions and graphs.
1.5. Inverse functions; inverse trigonometric functions. - Limits and Continuity
2.1 The limit of a function.
2.2 Algebraic computation of limits.
2.3 Continuity.
2.4 Exponential and logarithmic functions. - Differentiation.
3.1 An introduction to the derivative: tangents.
3.2 Techniques of differentiation.
3.3 Derivatives of trigonometric, exponential, and logarithmic functions.
3.4 Rates of change: modeling rectilinear motion.
3.5 The chain rule.
3.6 Implicit differentiation.
3.7 Related rates and applications.
3.8 Linear approximation and differentials. - Additional applications and derivatives and integrals.
5.1 Extreme values of a continuous function.
5.2 The mean values theorem.
5.3 Using derivatives to sketch the graph of a function.
5.4 Curve sketching with asymptotics: limits involving infinity.
5.5 L’Hospital rule.
5.6 Optimization in the physical sciences and engineering. - Integration.
4.1 Antidifferentiation.
4.2 Area as the limit of a sum.
4.3 Riemann sums and definite integrals.
4.4 The fundamental theorem of calculus.
4.5 Integration by substitution.
4.6 The mean value theorem for integrals; average value.
4.7 Numerical integration: the trapeizodal rule and Simpson’s rule.
Course number, office hours and webpage
- MATH 1351-CALCULUS I WITH APPLICATIONS, Main Campus (TTU Mechanical Engineering South Building)
- Lecturer: Cezar Lupu (Postdoctoral scholar)
- Topics covered: functions and graphs, limits of functions, continuity, differentiability and integration.
- Office hours: Tuesday & Thursday: 2-3.30 PM, and by appointment via e-mail
- Webpage: https://lupucezar.wordpress.com/teaching/
Schedule and locations
- Tuesday, 11.00-12.20 PM in TTU Mechanical Engineering South Building, room 205
- Thursday, 10.00-12.20 in TTU Mechanical Engineering South Building, room 205
Policies, grading criteria, homework, exams and references
The grade will be determined by the following three factors:
- 2 Gateway exams: 20% (each worth 10%)
- Homework assignments (WebWork): 20%
- Midterm Exam: 30%
- Final exam: 30%
- Attendance: 5% (EXTRA credit!)
- Written Homework: 10 % (EXTRA credit!)
All stated examinations are mandatory in order to pass the class.
Each gateway exam contains a set of 10 short questions on differentiation, respectively integration (derivatives and integrals to compute). Each gateway exam will take 30 minutes and will be scheduled by the instructor. Each gateway is administered at a unique time to the entire class – dates TBA. Procedural details will be provided in class.
Grade Inquiry Policy: Any student who has questions about any grade received on an assessment should meet with the instructor in person, if possible, within a one-week period of receiving the graded assessment. At this meeting, the instructor will provide an explanation of the grading and computation of the score. Inquires received after the one-week period will not be considered.
Calculator: A graphing calculator is a useful tool for this course. However, calculators or other electronic devices will NOT be permitted on quizzes, in-class exams, and the final exam.
Reading: Reading the material from the textbook (paper or electronic form) is mandatory.
WeBWorK: WeBWorK is an internet-based method for delivering homework problems to
students. Visit the course webpage for more information on how to access WeBWorK and how to enter your solutions; see Helpful Documents. You will need your eRaider username and student ID number with the R to log into WeBWorK. The WeBWorK system responds by telling you whether an answer (or set of answers) is correct or incorrect and also records whether you answered the question correctly or incorrectly. You are free to try a problem as many times as you wish until the due date. It is your responsibility to check WeBWorK for new assignments.
Please do not wait until the day the assignment is due to begin and/or send questions. I will not answer questions e-mailed to me within 24 hours from the HW deadline.
Written Homework: The written version of the homework will appear on my teaching page and it will consist of 20 problems (some of them will be challenging!). You will obtain the extra 10% if you solve more than 12 of them.
Regarding the Comprehensive Final Exam: The common final represents a course requirement.
A student who did not complete the final exam, but otherwise completed all the other
requirements successfully, cannot be assigned a passing letter grade. Each designated
instructor has to keep his/her copy of partial scores and grades for each student for 2 calendar years from the date of recording the grade. The final exam will be given in a proctored environment. The date for the final exam was scheduled prior to start of the semester; the location will be announced at the end of the semester. The date and time of the exam cannot be changed. Final exam conflicts may arise only between two or more math courses, in which case the course with lowest number has priority in maintaining the exam date and time.
For the final exam, students should not purchase a blue book. The exam will be distributed with on paper, with spaces to fill in/fill out. No other printed materials and electronic devices are allowed during the final. Scratch papers may be checked by the instructors.
There are no make-up WeBWorK homework sets assignments except for absence due to religious observance or absence to due officially approved trips (see Class Attendance below). If a student misses a WeBWorK homework sets for one of the above reasons, the homework set will be reopened.
Attendance will be taken regularly. Students who miss no more than 1 week of classroom time during the whole semester will receive a bonus (extra credit) of 5% to their overall grade.
Letter grades will then be assigned in accordance with the following correspondence:
The main reference include the following:
Accommodations and schedule conflicts
If you have a disability for which you are or may be requesting an accommodation, you are encouraged to contact both me and Student Disability Services in West Hall or call 806-742-2405, before the end of the second week of classes. Please see Operating Policy and Procedure 34.22 for a complete description of Texas Tech’s policy on accommodations or students with disabilities. Similarly, students who have any conflicts (including religious holy days) with the scheduled examination dates should notify me before the end of the second week of classes. A student who is absent from classes for the observance of a religious holy day shall be allowed to take an examination or complete an assignment scheduled for that day within a reasonable time after the absence.
MATH 4000-Problem Solving for Putnam, Fall 2019, Texas Tech University
Information about the competition and seminar (course description)
The William Lowell Putnam Mathematical Competition is the premiere competition for undergraduate students in North America. More than 500 universities compete in this contest organized by the Mathematical Association of America (MAA).
Putnam Examination (2019)
The Seventy Eight Putnam Examination will be held on Saturday, December 7th, 2019.
It will consist of two sessions of three hours each:
- Morning Session: 9:00am-12:00pm, location: MATH 015.
- Afternoon Session: 2:00pm-5:00pm, location: MATH 015.
The test is supervised by faculty members of each participating school. Every problem is graded on a scale of 0-10. The problems are usually listed in increasing order of difficulty, with A1 and B1 the easiest, and A6 and B6 the hardest. Top 5 scoring students on the Putnam exam are named Putnam Fellows. A student can take this exam maximum four times and the Putnam official team of the university consists of 3 members.
The purpose of this class is to provide a comprehensive introduction into the world of problem solving in different branches of mathematics such as: real & complex analysis, linear algebra, abstract algebra, combinatorics, probability, geometry and trigonometry and number theory.
This course (MATH 4000-Problem Solving for Putnam) teaches important skills in problem solving that are not taught in a systematic way in any other course. These skills are extremely valuable in preparing students for jobs and for graduate-level research. The teaching style will be a mixture of a lecture and a problem-solving session. By the end of this course, students should develop fundamental problem solving skills, and become accustomed to concentrating on a problem for an extended period of time. Indeed, this seminar concentrates on the raw creative problem-solving skills which can serve as an essential ingredient in almost every field of activity.
Course number, office hours and webpage
- MATH 4000-Problem Solving for Putnam, Main Campus (TTU Media and Communications), Fall 2019
- Lecturer: Cezar Lupu (Postdoctoral scholar)
- Topics covered: Elementary, Abstract & Linear Algebra, Real Analysis, Combinatorics, Geometry & Trigonometry, Number Theory, Probability
- Office hours: Tuesday & Thursday: 2-3.30 PM, and by appointment via email
- Webpage: https://lupucezar.wordpress.com/competitions/ & https://lupucezar.wordpress.com/teaching/
Schedule and locations
- Tuesday, 2-13.20 PM in TTU Media and Communications Building, room 153
This is a lecture given by the instructor on a certain topic. The students will learn different concepts and techniques. Moreover, the lecturer will also present solutions of some problems and will assign other problems as homework for the students.
- Thursday, 2-3.20 PM in TTU Media and Communications Building, room 153
This is more like a recitation rather than a lecture. The students will meet and discuss with the lecturer the problems assigned as homework.
Syllabus, grading criteria and references
The grade will be determined by the following three factors:
- Seminar attendance: 20%
- Homework and seminar activity: 60%
- Participation in the Putnam exam: 20%
Any student must attend at least 10/15 seminars to get full credit. Homework will be assigned biweekly and posted on the teaching section (https://lupucezar.wordpress.com/teaching/) of my webpage at the end of each lecture on Monday. It will consist mostly of problems of A1-B1 or A2-B2 from previous Putnam exams. The homework will be submitted as a PDF file via my email. The deadline for each homework will appear on my teaching page.
The main references include the following:
- R. Gelca, T. Andreescu, Putnam and Beyond, Springer Verlag, 2007.
- K. Kedlaya, B. Poonen, R. Vakil- The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions and Commentary, The Mathematical Association of America, Washington, D.C., 2002.
- L. Larson, Problem-Solving Through Problems, Springer Verlag, 1983.
Detailed schedule (lectures & recitations)
Week 1. (Elementary) Algebra I
Abstract: This lecture will cover problems on topics such as real algebraic identities and inequalities, and complex numbers.
Date: August 27 (Lecture), August 29 (Recitation)
Week 2. (Elementary) Algebra II
Abstract: This seminar will focus more on mathematical induction, functional equations and polynomials (integer polynomials, roots of polynomials).
Date: September 3 (Lecture), September 5 (Recitation)
Week 3. Geometry and Trigonometry
Abstract: This will cover problems on topics such as vectors, conics, quadratics, and other curves in the plane as well as trigonometric formulas.
Date: September 10 (Lecture), September 12 (Recitation)
Week 4. Combinatorics
Abstract: This will cover problems on topics combinatorial geometry, pigeonhole principle, generating functions graph theory, binomial identities and counting strategies.
Date: September 17 (Lecture) & September 19 (Recitation)
Week 5. Number Theory I
Abstract: This will cover problems on topics such as integer-valued sequences and functions, congruences, divisibility and arithmetic functions.
Date: September 24 (Lecture) & September 26 (Recitation)
Week 6. Number Theory II
Abstract: This will cover problems on topics such as quadratic residues, diophantine equations, and analytic methods in number theory.
Date: October 1 (Lecture) & October 3 (Recitation)
Week 7. Abstract Algebra
Abstract: This will cover problems on topics such as groups, rings, and finite fields.
Date: October 8 (Lecture) & October 10 (Recitation)
Week 8. Linear Algebra I
Abstract: This will cover topics on
Date: October 15 (Lecture) & October 17 (Recitation)
Week 9. Linear Algebra II
Abstract: This will cover problems on topics such as vectors spaces, linear transformations, characteristic and minimal polynomials, eigenvalues, eigenvectors.
Date: October 22 (Lecture) & October 24 (Recitation)
Week 10. Linear Algebra III
Abstract: This will cover some special topics in linear algebra (Jordan canonical form) and beyond.
Date: October 29 (Lecture) & October 31 (Recitation)
Week 11. Real Analysis I
Abstract: This will cover problems on topics such as sequences of real numbers.
Date: November 5 (Lecture) & November 7 (Recitation)
Week 12. Real Analysis II
Abstract: This will cover problems on topics such as series of real numbers.
Date: November 12 (Lecture) & November 14 (Recitation)
Week 13. Real Analysis III
Abstract: This will cover problems on topics such as limits of functions and continuity.
Date: November 19 (Lecture)& November 21 (Recitation)
Week 14. Real Analysis IV
Abstract: This will cover problems on topics such as differentiability of functions.
Date: November 26 (Lecture)
Thanksgiving break: No recitation this week!
Week 15. Real Analysis V
Abstract: This will cover problems on integrability (Riemann integrals, continuity of integrals).
Date: December 3 (Lecture) & December 5 (Recitation)
THE 2019 MATH 4000-PROBLEM SOLVING FOR PUTNAM
Cezar Lupu 
