### CONTACT INFO

OFFICE 711 (7TH FLOOR), THACKERAY HALL
E-MAIL: lupucezar@gmail.com, cel47@pitt.edu

• 82,341 hits

## Job Application Materials

I am currently a 6th year graduate student at the University of Pittsburgh, Department of Mathematics and I shall be graduating in June, 2018. I am currently on the job market for Fall 2018. My advisors are professors Piotr Hajlasz and William C. Troy. Most of my research is centered around special values of L-functions and multiple zeta functions which play an important role at the interface of analysis, number theory, geometry and physics with applications ranging from periods of mixed Tate motives to evaluating Feynman integrals in quantum field theory. I employ methods from real analysis and special functions. Broadly I am interested in analytic number theory and real, geometric, functional and harmonic analysis even with some PDEs flavour in it. Feel free to contact me at cel47@pitt.edu or lupucezar@gmail.com.  For more details, you can find below some of my application materials:

1. The Arithmetic Seminar, SUNY-University at Binghamton, March ??, 2018
2. Algebra Seminar, University of Connecticut, February 21, 2018
3. Analysis Seminar, Cornell University, January 29, 2018
4. Algorithms, Combinatorics and Optimization Seminar, Carnegie Mellon University, January 25, 2018
5. Contributed Talk at Number Theory I: Joint Mathematics Meeting of AMS & MAA, San Diego, January 10-13, 2018
6. Analysis Seminar, University of South Florida, December 1, 2017
7. Colloquim, University of South Florida, December 1, 2017
8. Algebra, Geometry and Topology Seminar, University of Pittsburgh, November 28, 2017
9. Algebra & Number Theory Seminar, Texas Tech University, November 15, 2017
10. Number Theory Seminar, University of California Irvine, November 9, 2017
11. Special Session in Preparing Students for AMC: AMS Sectional Meeting, University of California Riverside, November 5, 2017
12. Algebra, Geometry and Topology Seminar, University of Pittsburgh, October 24, 2017
13. Undergraduate Mathematics Seminar, University of Pittsburgh, October 17, 2017
14. Talk at the Parallel Session I: Northeastern Analysis Meeting (NEAM 2), SUNY-University at Albany, October 13-15, 2017
15. “Nicolae Popescu” Number Theory Seminar, Simion Stoilow Institute of Mathematics of the Romanian Academy, September 27, 2017
16. Undergraduate Mathematics Seminar, University of Pittsburgh, September 12, 2017
17. Algebra, Geometry and Combinatorics Graduate Student Research Seminar, University of Pittsburgh, September 7, 2017

Advertisements

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

The 77th 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, Ercis 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 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

## The Putnam Seminar (MATH-1010), Fall 2017, University of Pittsburgh

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 (2017)

The Seventy Eight Putnam Examination will be held on Saturday, December 2nd, 2017.

It will consist of two sessions of three hours each:

• Morning Session: 10:00am-1:00pm, in Thackeray 705.
• Afternoon Session: 3:00pm-6:00pm, in Thackeray 705.

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.

The 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 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.  On the other hand, starting this Fall, the Putnam seminar has honors designation.

Last but not least, since the department wants to revamp the Putnam tradition at Pitt, Thomas Hales will work with the undergraduate committee to establish a special prize for performers.

In 2015, Pitt official Team ranked 24th in the nation and this marks the best performance since 2002. More details about this can be found here and here.

Course number, lecturers 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 coordinators the problems assigned by the lecturer 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 10/14 seminars to get full credit. Homework will be assigned weekly and posted on the teaching section of my webpage (https://lupucezar.wordpress.com/teaching/) at the end of each lecture on Tuesdays. It will consist of 3-4 problems A1-B1 from previous Putnam exams. The homework will be discussed in the recitation on Thursday and will be placed in Cezar’s mailbox by Friday. It will be returned graded the following week. 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 seminar will cover problems on topics such as algebraic identities and inequalities.

Lecturer: Cezar Lupu
Date: August 29

Recitation instructors: Cezar Lupu & George Sparling
Date: August 31

Week 2. (Elementary) Algebra II

Abstract: This seminar will focus more on mathematical induction, functional equations and polynomials (integer polynomials, roots of polynomials).

Lecturer: Cezar Lupu
Date: September 5

Recitation instructors: Cezar Lupu & George Sparling
Date: September 7

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.

Lecturer: Derek Orr
Date: September 12

Recitation instructors: Derek Orr and George Sparling
Date: September 14

Week 4. Abstract Algebra

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

Lecturer: George Sparling
Date: September 19

Special lecture! Generating Functions and Applications

Lecturer: Cody Johnson
Date: September 21

Week 5. Linear Algebra I

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

Lecturer: Bogdan Ion
Date: September 26

Recitation instructors: Bogdan Ion & George Sparling
Date: September 28

Week 6. Linear Algebra II

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

Lecturer: Cezar Lupu
Date: October 3

Recitation instructors: Cezar Lupu & George Sparling
Date: October 5

Week 7 Number Theory I

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

Lecturer: Thomas Hales
Date: October 10

Lecturer: Thomas Hales
Date: October 12

Week 8. Number Theory II

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

Lecturer: Roman Fedorov
Date: October 17

Recitation instructors: Roman Fedorov & George Sparling
Date: October 19

Week 9. Real Analysis I

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

Lecturer: Cezar Lupu
Date: October 24

Recitation instructors: Cezar Lupu & George Sparling
Date: October 26

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.

Lecturer: Cezar Lupu
Date: October 31

Recitation instructors: Cezar Lupu & George Sparling
Date: November 2

Week 11. Combinatorics

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

Lecturer: Bogdan Ion
Date: November 7

Recitation instructors: Bogdan Ion & George Sparling
Date: November 9

Week 12. Real Analysis III

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

Lecturer: Cezar Lupu
Date: November 14

Recitation instructors: Cezar Lupu & George Sparling
Date: Novermber 16

Week 13. Problems and Theorems in Linear Algebra

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

Lecturer: Cezar Lupu
Date: November 21

Thanksgiving break: No recitation this week!

Week 14. Real Analysis IV

Abstract: This will cover problems on topics such as applications of multivariable calculus.

Lecturer: Cezar Lupu
Date: November 28

Special lecture! Problems and Theorems in Real Analysis

Lecturer: Piotr Hajlasz
Date: November 30

Week 15. The 2017 Putnam Competition-Problems discussion

Abstract: This week we discuss the the 2017 Putnam exam.

Lecturer: Cezar Lupu
Date: December 5

Recitation instructors: Cezar Lupu & George Sparling
Date: December 7

THE 2017 PUTNAM SEMINAR POSTER

## The 2017 Mathematical Olympiad Summer Program, Carnegie Mellon University, Pittsburgh, PA, June 7-July 1

The 2017 Mathematical Olympiad Summer Program will take place at Carnegie Mellon University between June 7-July 1. 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 its associate director Razvan Gelca of Texas Tech University. They will be accompanied by 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), and University of Pittsburgh.  Moreover, this year is for the second time when MOP welcomes 16 international students from countries such as China, Hong Kong, India, Romania, Russia, and Singapore. The two amazing Romanian students are Ciprian Bonciocat and Mihnea Ocian from Tudor Vianu High School of Bucharest.  Another news is the fact that four Romanians will serve as academic instructors. Besides the associate director, Razvan Gelca and myself, Bogdan Ion and Irina-Roxana Popescu of University of Pittsburgh are in the staff as well. This is for the third time in the history of MOP when this happens. The last two times when MOP had four Romanian instructors was back in 2002 and 2016.

Classrooms are Wean 5403, Wean 5421, Margaret Morrison A14, Wean 8220, Gates 5222.

• Students will be separated into four groups.
• Black (24): USAMO winners and IMO team and many IMO-level students from other countries. The group will split dynamically into two parts for each class.
• Blue (14): next top few from USAMO
• Green (13) / Red (24): students in grades 9 and 10, plus girls, split into two groups. Green includes all returning students, and will be faster.
• Red students will come with no prior MOP experience. Black level is quite impressive.
• The timetable will be:
• 8:30am – 10:00am (Lecture 1)
• 10:15am – 11:45am (Lecture 2)
• 1:15pm – 2:45pm (Lecture 3), or 1:15pm – 5:45pm
• 7:30pm: optional-attendance evening research seminar

My schedule consists of 19 lectures and one seminar. More details are given below:

Lecture 1. Real analysis techniques in solving elementary problems I.

Group: Black 2

Time and date: 8.30-10 AM,June 7

Location: Gates Hall 5222

Lecture 2. Real analysis techniques in solving elementary problems II.

Group: Black 2
Time and date: 10.15-11.45 AM, June 9

Location: Gates Hall 5222

Lecture 3. Real Analysis techniques in solving elementary problems.

Group: Blue

Time and date: 1.15-2.45 PM, June 9

Location: Margaret Morrison A 14 Hall

Lecture 4. Algebraic integers and applications.

Group: Green

Time and date: 8.30-10 AM, June 12

Location: Wean Hall 5421

Lecture 5. Algebraic integers and applications.

Group: Red

Time and date: 10.15-11.45 AM, June 12

Location: Wean Hall 5403

Lecture 6. Real Analysis techniques in solving elementary problems.

Group: Black 1

Time and date: 8.30-10 AM, June 13

Location: Wean Hall 8220

Lecture 7. Sequences, series of real numbers and inequalities.

Group: Black 1

Time and date: 8.30-11, June 15

Location: Wean Hall 8220

Lecture 8. Sequences, series of real numbers and inequalities.

Group: Black 2

Time and date: 1.15-2.45 PM, June 15

Location: Gates Hall 5222

Lecture 9Sequences, series of real numbers and inequalities.

Group: Blue

Time and date: 8.30-10 AM, June 16

Location: Margaret Morrison A 14 Hall

Seminar. Euler’s formula(s) for Apery’s constant $\zeta(3)$.

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

Time and Date: 8-9 PM, June 17

Location: Stever dorm

Lecture 10. Romanian Olympiad gems.

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

Location: Margaret Morrison A 14 Hall

Lecture 11. Romanian Olympiad gems.

Group: Black 2
Time and date: 1.15-2.45 PM, June 21

Location: Gates Hall 5222

Lecture 12. Maxima and minima in Euclidian geometry and beyond. Geometric and trigonometric inequalities.

Group: Blue

Time and date: 8.30-10 AM, June 26

Location: Margaret Morrison A 14 Hall

Lecture 13. Maxima and minima in Euclidian geometry and beyond. Geometric and trigonometric inequalities.

Group: Black 2
Time and date: 8.30-10 AM,June 27

Location: Gates Hall 5222

Lecture 14. Maxima and minima in Euclidian geometry and beyond. Geometric and trigonometric inequalities.

Group: Green
Time and date: 1.15-2.45 PM, June 27

Location: Wean Hall 5421

Lecture 15. Advanced analytic methods in number theory I.
Group: Black 2
Time and date: 8.30-11 AM, June 28

Location: Gates Hall 5222

Lecture 16. Advanced analytic methods in number theory II.

Group: Black 2
Time and date: 8.30-11 AM, June 29

Location: Gates Hall 5222

Lecture 17. Maxima and minima in Euclidian geometry and beyond. Geometric and trigonometric inequalities.

Group: Red
Time and date: 1.15-2.45 PM, June 29

Location: Wean Hall 5403

Lecture 18. Romanian Olympiad gems.

Group: Green
Time and date: 8.30-10 AM, June 30

Location: Wean Hall 5421

Lecture 19. Romanian Olympiad gems.

Group: Red
Time and date: 10.15-11.45 AM, June 30

Location: Wean Hall 5403

This slideshow requires JavaScript.

## The 3rd Algebra, Geometry and Topology Graduate Student Conference, June 2-4, Philadelphia, PA

The 3rd Annual Graduate Student Conference in Algebra, Geometry and Topology will take place between June 3-5 at Temple University in Philadelphia, PA. This meeting reunites Ph.D. students and postdoctoral scholars from the most prestigious universities from all around US and Canada. The organizers are graduate students and faculty from the department of mathematics from Temple University. The talks were delivered by Ph.D. students from universities such as Princeton, University of Pennsylvania, Tufts, University of Virginia, Cornell, Caltech, University of Maryland College Park, University of Iowa, University of Chicago, University of New Hampshire, University of Pittsburgh, Dartmouth College, Purdue University, University of Illinois Chicago, CUNY, University of California Davis, North Carolina State University, University of Toronto and University of Michigan.

## Overview of the Calculus III (MATH 0240) course, University of Pittsburgh, 1st 6weeks, Summer 2017

This Summer I shall be teaching Calculus III (MATH 0240) course at the University of Pittsburgh, between May 15-June 25. Mainly, we shall cover vectors, vector functions and space functions, arc length and curvature, functions of several variables, double and triple integrals, line integrals, surface integrals and some other related topics.

About the course

This is the third sequence of three calculus courses for science and engineering students. The goal is to prepare you to make use of calculus as a practical problem-solving tool.

Prerequisite

Math 0230 or equivalent, with grade of C or better.

Lectures and Recitations

• Lectures are from Monday to Thursday, 6-7.45 PM  in 226 Benedum Hall.
• Recitations are from Monday to Thursday, 8-9 PM  in Benedum Hall. The recitation instructor for this class is Mohan Wu. His office hours are Wednesday, 3-5 PM in the MAC.
• Moreover, you can also check the regular teaching page for this class: https://lupucezar.wordpress.com/teaching/ (Calculus 3-Summer 2017).
• Quizzes and exams will appear on the teaching section above.

Office Hours

My office hours are from Monday to Thursday, 5-6 PM in the MAC. If you want to schedule an additional meeting with me in my office (Thackeray hall, room 711) do not hesitate to send me an e-mail to lupucezar@gmail.com.

References (Lecture notes, textbook, and practice exams)

1. P. Hajlasz, Lecture Notes in Calculus III, Part 1, Part 2, Part 3.
2. J. Stewart, Essential Calculus, Early Transcendentals, 2nd edition.
3. A. Athanas, Calculus 3-practice exams.
4. I. Sysoeva, Calculus 3-practice exams.
5. E. Trofimov, Calculus 3-webpage.
6. A. Yarosh, Calculus 3-webpage.

Grades

Your final grade will be determined as follows:

• Homework: 20% (2 homeworks of 10% each)
• Quizzes: 20% (4 quizzes of 5% each)
• Two exams: 60% (2 exams of 30% each)
• LON CAPA homework: 5% (extra credit)

Homework

I shall assign two homeworks which will appear on the teaching section. The first homework is hereEach one will be due before the exams. Additionally, one can do problems from the LON CAPA system. If you find it really annoying, you can also do this.

Final Grade Policy

A: 90-100%, A-: 85-90%, B+: 80-85%, B: 75-80%, B-: 70-75%, C+: 65-70%, C: 60-65%.

Exam Dates

The first exam will be on Thursday, June 1st, 6-8 PM in Benedum 226.

The second exam will be on Thursday, June 22nd, 6-9 PM in Benedum 226.

Disability Resource Services

If you have any disability for which you are or may be requesting an accomodation, you are encouraged to contact both your instructor and the Office of Disability and Services, 216 William Pitt Union (412) 624-7890 as early as possible in the term.

## The 7th Upstate New York Number Theory Conference, Binghamton, NY, 5-7 May, 2017

The 7th Upstate New York Number Theory Conference took place at the SUNY-University of Binghamton, Binghamton, NY between 5-7 May. More than 30 number theorists and related areas affiliated with universities from USA, Canada, and Germany participated.
The conference was organized by Alexander Borisov, Jaiung Jun, Marcin Mazur,  and Adrian Vasiu of SUNY-University of Binghamton and it included plenary speakers such as Michael Filaseta, Thomas Hales, Jeffrey Lagarias, Melvyn Nathanson, Alexandra Shlapentokh, and Joseph Silverman. The 3 contributed sections involved all sorts of number theoretical aspects. More details about the conference including the abstract of the talks are given below:

This slideshow requires JavaScript.