At the end of the course, the successful students:
Silvio Capobianco, contact: firstname.familyname(at)taltech.ee
Ronald Graham, Donald Knuth, and Oren Patashnik. Concrete Mathematics: a Foundation for Computer Science. Second edition. Addison-Wesley 1994.
These are the topics covered in the 2021 edition of the course.
For the first five weeks, lecture slides and classroom exercises
(without solutions) are available.
Week | Subject | Lecture | Exercises | Talks |
Week 1 | Introduction to the course. | Lecture 1 -- updated to 2023 edition | Exercise session 1 -- updated to 2023 edition | |
Week 2 |
Recurrent problems. Tower of Hanoi. Lines in the plane. The
Josephus problem. Binary representation. The repertoire
method.
Chapter 1, pages 1--15. |
Lecture 2 | Exercise session 2 | |
Week 3 |
Recurrent problems. Generalization of the Josephus
problem. The repertoire method.
Sums. Notation. Sums and recurrences. The perturbation method. Summation factors. Chapter 1, pages 15--16; Chapter 2, pages 21--26, 32--33. |
Lecture 3 | Exercise session 3 | |
Week 4 |
Sums. Summation factors. Efficiency of
quicksort. Manipulation of sums. Multiple sums. General
methods.
Chapter 2, pages 28--46. |
Lecture 4 | Exercise session 4 | |
Week 5 |
Sums. Finite and infinite calculus. Infinite
sums. Cesàro and Abel summations.
Chapter 2, pages 47--62. |
Lecture 5 | Exercise session 5 | |
Week 6 |
Integer functions. Floors and ceilings. Floor/Ceiling
applications. Floor/Ceiling recurrences. "mod":
the binary operation. Floor/Ceiling sums.
Chapter 3, pages 67--78, 81--85. |
|||
Week 7 |
Number Theory. Divisibility. Primes. Prime
examples. Factorial factors. Relative primality.
Chapter 4, pages 102--114, 115--118. |
|||
Week 8 |
Number theory. Relative primality. Modular
arithmetics. Additional applications. Primality
tests. Fermat's test. Rabin-Miller test. Euler and
Möbius functions.
Chapter 4, pages 119--122, 123--126, 129--136. |
|||
Week 9 |
Binomial coefficients. Basic identities. Basic
practice. Generating functions. Intermezzo: analytic
functions.
Chapter 5, pages 153--175, 196--197. Midterm test. |
|||
Week 10 |
Binomial coefficients. Generating functions. Operations on
generating functions. Building generating functions that
count. Identities in Pascal's triangle.
Chapter 5, pages 196--204. |
|||
Week 11 |
Special numbers. Stirling numbers of the first and second
kind. Stirling's inversion formula. Generating function of
falling and rising factorials. Fibonacci numbers. Generating
function of Fibonacci numbers.
Chapter 6, pages 257--267, 290--292, 297--299. |
|||
Week 12 |
Special numbers. Fibonacci numbers. Cassini's
identity. Harmonic numbers. Harmonic summations. Bernoulli
numbers.
Chapter 6, pages 272--284, 292--297. |
|||
Week 13 |
Generating functions. Solving recurrences with generating
functions. Partial fraction expansion. The Rational
Expansion Theorem.
Chapter 7, pages 331--341. |
|||
Week 14 |
Generating functions. Use of
derivatives. Convolutions. Catalan numbers. Exponential
generating functions.
Chapter 7, pages 343--368. |
|||
Week 15 |
Asymptotics. A hierarchy. Big-O notation. Big-O manipulation.
Chapter 9, pages 439--457. |
|||
Week 16 |
Asymptotics. Big-O manipulation. Two asymptotic
tricks. Euler's summation formula. Stirling's approximation
for the logarithm of the factorial.
Chapter 9, pages 463--475, 481--489. |
Last updated: 03.02.2023 Silvio Capobianco