Cezar Lupu

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OFFICE: ROOM 205 (2nd FLOOR),
E-MAIL: lupucezar@gmail.com, Cezar.Lupu@ttu.edu, cel47@pitt.edu

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The 79th William Lowell Putnam Mathematical Competition, December 1st, 2018, Texas Tech University

The 79th annual William Lowell Putnam Mathematical Competition took place on December 1st in rooms 014 & 015, Math Building. Twelve TTU students had the mission of solving the challenging Putnam problem sets. As usual, there were two sessions of 6 problems each. The official TTU Team members were Alexander Morgan-Fleming, Jackson Kulik and Abubakarr Yillah. Other participating students were: Sergio Baez, Nathan Dortch, Jose Duran, Abdulla Hilmy, Claudia Munoz, Yu Jung Nam, Ezequiel Tovar, Jack Weiland, and Xueting Xia. Below one can find this year’s Putnam problems. Congratulations to all participants!

SESSION A:

Problem A1.  Find all ordered pairs (a, b) of positive integers for which

\displaystyle\frac{1}{a}+\frac{1}{b}=\frac{3}{2018}.

Problem A2. Let S_1, S_2, \dots, S_{2^n - 1} be the nonempty subsets of {1, 2, \dots, n} in some order, and let M be the (2^n - 1) \times (2^n - 1) matrix whose (i, j) entry is \displaystyle m_{ij} =0 if S_{i}\cap S_{j}=\emptyset and 1 otherwise.

Calculate the determinant of M.

 

Problem A3. Determine the greatest possible value of \displaystyle\sum_{i = 1}^{10} \cos(3x_i) for real numbers \displaystyle x_1, x_2, \dots, x_{10} satisfying \displaystyle\sum_{i = 1}^{10} \cos(x_i) = 0.

Problem A4. Let m and n be positive integers with \gcd(m, n) = 1, and let \displaystyle a_k = \left\lfloor \frac{mk}{n} \right\rfloor - \left\lfloor \frac{m(k-1)}{n} \right\rfloor.

for k = 1, 2, \dots, n. Suppose that g and h are elements in a group G and that

gh^{a_1} gh^{a_2} \cdots gh^{a_n} = e,

where e is the identity element. Show that gh = hg. (As usual, \lfloor x \rfloor denotes the greatest integer less than or equal to x.)

 

Problem A5.  Let f: \mathbb{R} \to \mathbb{R} be an infinitely differentiable function satisfying f(0) = 0, f(1) = 1, and f(x) \ge 0 for all x \in \mathbb{R}. Show that there exist a positive integer n and a real number x such that f^{(n)}(x) < 0.

Problem A6. Suppose that A, B, C, and D are distinct points, no three of which lie on a line, in the Euclidean plane. Show that if the squares of the lengths of the line segments AB, AC, AD, BC, BD, and CD are rational numbers, then the quotient


\displaystyle\frac{\mathrm{area}(\triangle ABC)}{\mathrm{area}(\triangle ABD)}


is a rational number.

 

SESSION B:

Problem B1. Let \mathcal{P} be the set of vectors defined by

\displaystyle\mathcal{P} =\left\{\binom{a}{b}, 0\leq a\leq 2, 0\leq b\leq 100, a, b\in\mathbb{Z}\right\} .

Find all \mathbf{v} \in \mathcal{P} such that the set \mathcal{P}\setminus{\mathbf{v}} obtained by omitting vector \mathbf{v} from \mathcal{P} can be partitioned into two sets of equal size and equal sum.

Problem B2. Let n be a positive integer, and let \displaystyle f_n(z) = n + (n-1)z + (n-2)z^2 + \dots + z^{n-1}. Prove that f_n has no roots in the closed unit disk \{z \in \mathbb{C}: |z| \le 1\}

 

Problem B3. Find all positive integers n < 10^{100} for which simultaneously n divides 2^n, n-1 divides 2^n - 1, and n-2 divides 2^n - 2.

 

Problem B4. Given a real number a, we define a sequence by x_0 = 1, x_1 = x_2 = a, and \displaystyle x_{n+1} = 2x_nx_{n-1} - x_{n-2} for n \ge 2. Prove that if x_n = 0 for some n, then the sequence is periodic.

Problem B5. Let f = (f_1, f_2) be a function from \mathbb{R}^2 to \mathbb{R}^2 with continuous partial derivatives \tfrac{\partial f_i}{\partial x_j} that are positive everywhere. Suppose that


\displaystyle \frac{\partial f_1}{\partial x_1} \frac{\partial f_2}{\partial x_2} - \frac{1}{4} \left(\frac{\partial f_1}{\partial x_2} + \frac{\partial f_2}{\partial x_1} \right)^2 > 0


everywhere. Prove that f is one-to-one.

Problem B6. Let S be the set of sequences of length 2018 whose terms are in the set \{1, 2, 3, 4, 5, 6, 10\} and sum to 3860. Prove that the cardinality of S is at most


\displaystyle 2^{3860} \cdot \left(\frac{2018}{2048}\right)^{2018}.

 


OFFICIAL SOLUTIONS


MATH 4360: Foundations of Algebra II, Fall 2018, Texas Tech University

Course description and purpose

This course is a continuation of MATH 3360 and is intended to continue the student’s introduction to abstract algebra, with proofs.

Students learn how to think and reason abstractly in the context of algebraic structures, and learn how to write correct and clear mathematical arguments in this context. This second part of the sequence will provide sufficient background for the students to start learning Galois theory.
Course number, office hours and webpage
Schedule and locations
Grading criteria, homework, exams and references                                             

The grade will be determined by the following three factors:

  • 2 midterm exams: 40% (each worth 20%)
  • Homework assignments: 20%
  • Final exam: 40%
Homework will be assigned throughout the semester as written assignments, usually on a bi-weekly basis. There will be around 5 homework assignments this semester and they will be posted on my teaching page: https://lupucezar.wordpress.com/teaching/. Late homework will not be accepted.
Midterm exams will be announced in advance in class and on my website. The final exam will take place on Friday, 7.30-10 AM, December 7 in our regular classroom.  The final exam is comprehensive, i.e., it will examine material from the entire course. Final exam dates are set by the university-please confirm the date of our exam on the university website.

 Letter grades will then be assigned in accordance with the following correspondence:

Letter grade A = a percentile grade of 90% of higher
Letter grade B = a percentile grade of 80% or higher, that is lower than 90%
Letter grade C = a percentile grade of 70% or higher, that is lower than 80%
Letter grade D = a percentile grade of 60% or higher, that is lower than 70%
Letter grade F = a percentile grade lower than 60%

 

The main references include the following:

  1. A. Papantonopoulou, Algebra, Pure and Applied, Pearson Publishers, 2002.
  2. Thomas W. Judson, Abstract Algebra. Theory and Applications, Online edition, 2018.
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, Fall 2018, 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 (2018)

The Seventy Eight Putnam Examination will be held on Saturday, December 1st, 2018.

It will consist of two sessions of three hours each:

  • Morning Session: 10:00am-1:00pm, location to be determined.
  • Afternoon Session: 3:00pm-6:00pm, location to be determined.

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) 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
Schedule and locations

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.

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 of 3-4 problems A1-B1 from previous Putnam exams. The homework will be discussed in the recitation on Wednesday and will be returned in class the following Monday. The participation in the Putnam exam is mandatory for any student who wants to get full credit.

The main references include the following:

  1. R. Gelca, T. Andreescu, Putnam and Beyond, Springer Verlag, 2007.
  2. 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.
  3. 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 5 (Lecture)


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 formulae.

Date: September 10 (Lecture), September 12 (Recitation)


Week 4. Abstract Algebra

Abstract: This will cover problems on topics such as groups, rings, and finite fields.

Date: September 17 (Lecture) & September 19 (Recitation)


Week 5. Linear Algebra I

Abstract: This will cover topics on 2\times 2 and 3\times 3 matrices and determinants.

Date: September 24 (Lecture) & September 26 (Recitation)


Week 6. Linear Algebra II

Abstract: This will cover problems on topics such as vectors spaces, linear transformations, characteristic and minimal polynomials, eigenvalues, eigenvectors.

Date: October 1 (Lecture) & October 3 (Recitation)


Week 7 Number Theory I

Abstract: This will cover problems on topics such as integer-valued sequences and functions, congruences, divisibility and arithmetic functions.

Date: October 8 (Lecture) & October 10 (Recitation)


Week 8. Number Theory II

Abstract: This will cover problems on topics such as quadratic residues and diophantine eqations.

Date: October 15 (Lecture) & October 17 (Recitation)


Week 9. Real Analysis I

Abstract: This will cover problems on topics such as sequences and series of real numbers.

Date: October  22 (Lecture) & October 24 (Recitation)


Week 10. Real Anaysis II

Abstract: This will cover problems on topics such as intermediate value property, continuity and differentiability of functions of a single variable.

Date: October 29 (Lecture) & October 31 (Recitation)


Week 11. Combinatorics

Abstract: This will cover problems on topics combinatorial arguments in set theory and geometry, graph theory, binomial identities and counting strategies.

Date: November 5 (Lecture) & November 7 (Recitation)


Week 12. Real Analysis III

Abstract: This will cover problems on topics such as Riemann integral and continuity of integrals.

Date: November 12 (Lecture) & November 14 (Recitation)


Week 13. Linear Algebra III

Abstract: This will cover some special topics in linear algebra and beyond.

Date: November 19 (Lecture)

Thanksgiving break: No recitation this week!


Week 14. Real Analysis IV

Abstract: This will cover problems on topics such as applications of multivariable calculus and other miscellaneous topics related to analysis.

Date: November 26 (Lecture) & November 28 (Recitation)


Week 15. The 2018 Putnam Competition-Problems discussion

Abstract: This week we discuss the problems from the 2018 Putnam exam.

Date: December 3 & 5 (Discussion of the problems)

 


THE 2018 PUTNAM SEMINAR POSTER

putnamban2

My PhD Thesis Defense, June 25th 2018, University of Pittsburgh

Title: Analytic Aspects of the Riemann Zeta and Multiple Zeta Values. (Slides)

PhD candidate: Cezar Lupu
Advisors: Piotr Hajlasz (Pitt), William C. Troy (Pitt)
Committee members: Thomas C. Hales (Pitt), Camil Muscalu (Cornell), George Sparling (Pitt)

Abstract: We study Riemann zeta and multiple zeta functions and their special values,

\displaystyle\zeta(k)=\sum_{n=1}^{k}\frac{1}{n^k}, k> 1,
and
                              \displaystyle\zeta(k_{1}, k_{2}, \ldots, k_{r})=\sum_{1\leq n_{1}<n_{2}<\ldots<n_{r}}\frac{1}{n_{1}^{k_{1}}n_{2}^{k_{2}}\ldots n_{r}^{k_{r}}}, k_{1}, k_{2}, \ldots , k_{r-1}\geq 1, k_{r}\geq 2.

 

This manuscript contains two parts. The first part contains fast converging series representations involving \zeta(2n) for Apery’s constant \zeta(3). These representations are obtained via Clausen acceleration formulae. Moreover, we also find evaluations for more general rational zeta series involving \zeta(2n) and binomial coefficients.

The second part will be devoted to the multiple zeta and special Hurwitz zeta values (multiple t-values). In this part, using a new approach involving integer powers of \arcsin which come from particular values of the Gauss hypergeometric function, we are able to provide new proofs for the evaluations of \zeta(2, 2, \ldots, 2), and t(2, 2, \ldots, 2). Moreover, we are able to evaluate \zeta(2, 2, \ldots, 2, 3), and t(2, 2, \ldots, 2, 3) in terms of rational zeta series involving \zeta(2n). On the other hand, using properties of the Clausen functions we can express these rational zeta series as a finite \mathbb{Q}-linear combinations of powers of \pi and odd zeta values. In particular, we deduce the famous formula of Zagier for the Hoffman elements in a special case.

Zagier’s formula is a remarkable example of both strength and the limits of the motivic formalism used by Brown in proving Hoffman’s conjecture where the motivic argument does not give us a precise value for the special multiple zeta values \displaystyle\zeta(2, 2, \ldots, 2, 3, 2, 2,\ldots, 2) as rational linear combinations of products \zeta(m)\pi^{2n} with m odd.

In Zagier’s paper (http://annals.math.princeton.edu/wp-content/uploads/annals-v175-n2-p11-p.pdf), the formula is proven indirectly by computing the generating functions of both sides in closed form and then showing that both are entire functions of exponential growth and that they agree at sufficiently many points to force their equality.

 

 

The 2018 Mathematical Olympiad Summer Program (June 3-27), Carnegie Mellon University, Pittsburgh, PA

The 2018 Mathematical Olympiad Summer Program will take place at Carnegie Mellon University between June 3-27. 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, Columbia University, Carnegie Mellon University (CMU), Princeton University, Ohio State University and University of Pittsburgh. Moreover, this year is for the second time when MOP welcomes 20 international students from countries such as Bulgaria, China, Czech Republic, Hong Kong, India, Romania, Ukraine and Singapore. The two amazing Romanian students are Denis Chirita and George Picu from International High School of Bucharest.  Another news is the fact that two Romanians will serve as academic instructors. Also, Irina-Roxana Popescu of University of Pittsburgh is in the staff as well.

  • Classrooms are Gates 5222, Margaret Morrison A14, Wean 8201, Wean 8220, Gates 4101, Gates 4102.
  • Students will be separated into four groups.
    • Black (22 students): approx IMO gold.
    • Blue (21 students): approx IMO silver
    • Green (24 students): approx IMO bronze
    • Red (13 students): approx IMO honorable mention.
  • The timetable will be:
    • 8:30am – 10:00am (Lecture 1)
    • 10:15am – 11:45am (Lecture 2)
    • 7:30pm: optional-attendance evening research seminar

My schedule consists of 7 lectures and one seminar. More details are given below:


Lecture 1. Maxima and minima in Euclidian geometry and beyond. Geometric and trigonometric inequalities.

Group: Black 2

Time and date: 8.30-10 AM, June 8

Location: Gates Hall 4102


Lecture 2. Maxima and minima in Euclidian geometry and beyond. Geometric and trigonometric inequalities.

Group: Blue 2

Time and date: 8.30-10 AM, June 13

Location: Wean Hall 8220


Lecture 3. Romanian Olympiad (Algebra) gems.

Group: Green
Time and date: 8.30-10 AM, June 20

Location: Margaret Morison A14


Lecture 4. Romanian Olympiad (Algebra) gems.

Group: Red
Time and date: 10.15-11.45 AM, June 20

Location: Gates Hall 5222


Lecture 5. Romanian Olympiad (Algebra) gems.

Group: Blue 2
Time and date: 10.15-11.45 AM, June 21

Location: Wean Hall 8220


Seminar. Zeta regularization and the golden nugget: “1+2+3+\ldots =-\frac{1}{12}.

Group: Black (1 & 2), Blue (1 & 2), Red & Green

Time and Date: 7.30-8.30 PM, June 22

Location: Stever dorm


Lecture 6. Algebraic integers and applications.

Group: Red

Time and date: 8.30-10 AM, June 26

Location: Gates Hall 5222


Lecture 7. Algebraic integers and applications.

Group: Blue 2

Time and date: 8.30-10 AM, June 26

Location: Wean Hall 8220


 


TEAM USA RETAKES 1ST PLACE AT THE IMO 2018

The 2018 Joint Mathematics Meeting of the AMS & MAA, San Diego, January 10-13, 2018

The 2018 Joint Mathematics Meeting of the American Mathematical Society and Mathematical Association of America is the largest mathematics meeting in the world. More than 6000 mathematicians and math enthusiasts met in the beautiful city of San Diego, CA between January 10-13. The location is San Diego Convention Center. This is the 101th annual winter meeting of MAA and the 124rd annual meeting of AMS. My talk (slides) is part of the AMS Contributed Paper Session on Number Theory I , Wednesday, January 10, 2017, 10.15 a.m.-10.30 a.m at the Mezzanine, room 13A.

Abstract book                                     Program                                      Presenters

 

jmm_2017maa-ams-logos

The 2017 William Lowell Putnam Competition Exam at the University of Pittsburgh

The 78th annual William Lowell Putnam Mathematical Competition took place on December 2nd in 705 Thackeray Hall. Twelve Pitt students had the mission of solving the challenging Putnam problem sets. As usual, there were two sessions of 6 problems each. The official Pitt Team members were Andrew Tindall, Tianke Li and Haihui Zhu. Other participating students were: Matthew Gerstbrein, Abraham Harris, Andrew Klang, Eric Peterson, Tanmoy Sarker, Anthony Sicillia, Ed Terrell, Carlos Vazquez Gomez, and Lu Zhao. Below one can find this year’s Putnam problems. Congratulations to all participants!

SESSION A:

Problem A1.  Let S be the smallest set of positive integers such that

a) 2 is in $latex  S$,
b) n is in S whenever n^2 is in S, and
c) (n+5)^2 is in S whenever n is in S.

Which positive integers are not in S?

(The set S is “smallest” in the sense that S is contained in any other such set.)

Problem A2. Let Q_0(x)=1, Q_1(x)=x, and

\displaystyle Q_n(x)=\frac{(Q_{n-1}(x))^2-1}{Q_{n-2}(x)}

for all n\ge 2. Show that, whenever n is a positive integer, Q_n(x) is equal to a polynomial with integer coefficients.

Problem A3. Let a and b be real numbers with a<b, and let f and g be continuous functions from [a,b] to (0,\infty) such that \int_a^b f(x)\,dx=\int_a^b g(x)\,dx but f\ne g. For every positive integer n, define

\displaystyle I_n=\int_a^b\frac{(f(x))^{n+1}}{(g(x))^n}\,dx.

Show that I_1,I_2,I_3,\dots is an increasing sequence with \displaystyle\lim_{n\to\infty}I_n=\infty.

Problem A4. A class with 2N students took a quiz, on which the possible scores were 0,1,\dots,10. Each of these scores occurred at least once, and the average score was exactly 7.4. Show that the class can be divided into two groups of N students in such a way that the average score for each group was exactly 7.4.

Problem A5.  Each of the integers from 1 to n is written on a separate card, and then the cards are combined into a deck and shuffled. Three players, A,B, and C, take turns in the order A,B,C,A, \dots choosing one card at random from the deck. (Each card in the deck is equally likely to be chosen.) After a card is chosen, that card and all higher-numbered cards are removed from the deck, and the remaining cards are reshuffled before the next turn. Play continues until one of the three players wins the game by drawing the card numbered 1.

Show that for each of the three players, there are arbitrarily large values of n for which that player has the highest probability among the three players of winning the game.

Problem A6. The 30 edges of a regular icosahedron are distinguished by labeling them 1,2, \dots, 30. How many different ways are there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color?

SESSION B:

Problem B1. Let L_1 and L_2 be distinct lines in the plane. Prove that L_1 and L_2 intersect if and only if, for every real number \lambda\ne 0 and every point P not on L_1 or L_2, there exist points A_1 on L_1 and A_2 on L_2 such that \overrightarrow{PA_2}=\lambda\overrightarrow{PA_1}.

Problem B2. Suppose that a positive integer N can be expressed as the sum of k consecutive positive integers

\displaystyle N=a+(a+1)+(a+2)+\cdots+(a+k-1)

for \displaystyle k=2017 but for no other values of k>1. Considering all positive integers N with this property, what is the smallest positive integer a that occurs in any of these expressions?

Problem B3. Suppose that

\displaystyle f(x) = \sum_{i=0}^\infty c_ix^i

is a power series for which each coefficient c_i is 0 or 1. Show that if f(2/3) = 3/2, then f(1/2) must be irrational.

Problem B4. Evaluate the sum

\displaystyle\sum_{k=0}^{\infty}\left(3\cdot\frac{\ln(4k+2)}{4k+2}-\frac{\ln(4k+3)}{4k+3}-\frac{\ln(4k+4)}{4k+4}-\frac{\ln(4k+5)}{4k+5}\right)
\displaystyle=3\cdot\frac{\ln 2}2-\frac{\ln 3}3-\frac{\ln 4}4-\frac{\ln 5}5+3\cdot\frac{\ln 6}6-\frac{\ln 7}7-\frac{\ln 8}8-\frac{\ln 9}9+3\cdot\frac{\ln 10}{10}-\cdots.

(As usual, \ln x denotes the natural logarithm of x.)

Problem B5. A line in the plane of a triangle T is called an equalizer if it divides T into two regions having equal area and equal perimeter. Find positive integers a>b>c, with a as small as possible, such that there exists a triangle with side lengths a,b,c that has exactly two distinct equalizers.

Problem B6. Find the number of ordered 64-tuples \{x_0,x_1,\dots,x_{63}\} such that x_0,x_1,\dots,x_{63} are distinct elements of \{1,2,\dots,2017\} and

\displaystyle x_0+x_1+2x_2+3x_3+\cdots+63x_{63}

is divisible by 2017.


OFFICIAL SOLUTIONS