History of Mathematics and Informatics

Garant předmětu:

Alena Šolcová

Přednášející:

Alena Šolcová

Cvičící:

Alena Šolcová

Předmět zajišťuje:

katedra aplikované matematiky

Anotace:

The course focuses on selected topics from calculus, general algebra, number theory, numerical mathematics and logic - useful for today computer science The topics are selected for finding some relations between computer science and mathematical methods. Some examples of applications of mathematics to computer sciences will be showed.

Požadavky:

Knowledge of high school mathematics and of basic courses at the faculty and an ability to solve concrete basic tasks from mathematics and informatics.

Osnova přednášek:

1. Mathematics in the 17th Century. First steps of Calculus - Newton, Leibniz. Sources in Greek mathematics - introduction to the programme of course.

2. The role of Pierre Fermat in the probability theory.

Mathematics in the celestial mechanics. From J. Keplera and P. Laplace to A. Seydler.

3. Descartes' „Discourse de la Méthode“. Algorithms of arithmetic operations, Leibniz and Pelikán binary arithmetics.

4. The oldest mechanical calculators. Schickard, Pascal, Leibniz.

Combinatorics in „kabbala“. The applications in the number theory.

5. The Pell equation and the development of algebra. Lagrange's results and its applications.

6. Mathematics of the 18th Century: Approximations of functions - L. Euler, Ch. Fourier, FFT (Fast Fourier Transform).

7. Solution of the system of the linear equations.

(Cramer Rule, Gauss Elimination Method, Least Square Method, Jacobi and Seidel Method, Cauchy and unlinear epilogue).

8. Number Theory (Gauss congruence, factorization algorithms, Pépin's test).

Development of the number systems and its applications: Complex numbers, Hamilton's quaternions.

9. General algebra - Symmetries and searching for Lie groups. E. Galois. Eliptic curves from Adam.

Change of dimension - Abbot's Flatland, 100 years of hypercube, Hermann Minkowski.

10. From mathematical linguistic (kvantitative, algebraic, computer linguistic).

The development of the typography. (A. Duerer, D. Knuth, etc.).

11. The 19th Century in Computer Science - Analytical Engine, Charles Babbage, Ada Byron.

From logic of the 20th Century: A. Whitehead, B. Russel - Principia mathematica, K. Gödel, S. C. Kleene - recursive functions.

12. Mathematics, informatics and the development of computer science. Computers in the 20th Century. A. Svoboda and V. Vand, its ideas and applications.

History of the Czech Technical University in Prague.

13. On the character of matematical thinking - H. Poincaré. Hilbert's problems for the 20th Century and opem problems for the 21st Century (Kepler hypothesis, etc.).

Osnova cvičení:

1. Methodological introduction and work with historical sources in exact sciences.

2. Interesting calculus, joy of solving, discussion on individual essays.

3. Descartes questions and problems. An introduction to the Leibniz binary system of numbers. „Arithmeticus perfectus“ of Václav Josef Pelikán (1713).

4..Mathematical Topography of Prague. First computers in Prague. (A lecture in the streets.)

5. Bernoulli numbers, their properties and Ada Lovelace. Approximations of functions.

6. Boolean algebra and Boole's Mathematical Analysis of Logic. Brief development of symbols and description of algorithms. A presentation of student's individual works.

Cíle studia:

To know some important relations between mathematical methods and computer science through history. To get an overview of basic steps in the development of mathematical methods and computer science.

Studijní materiály:

1. Chabert, J.-L. et all: A History of Algorithms. From the Pebble to the Microchip, Springer, Berlin-Heidelberg-New York, 1999

2. Graham, R., Knuth, D., Patashnik, O.: 'Concrete Mathematics: A Foundation for Computer Science', Addison-Wesley, Reading, Mass., 1989.

3. Lovász, L.: 'Combinatorial Problems and Exercises', 2nd Ed., Akademiai Kiadó Budapest and North- Holland, Amsterdam, 1993.

4. Schroeder, R. M.: ''Number Theory in Science and Communication'', Springer, Berlin, 2006.

5. Křížek, M., Luca, F., Somer, L.: ''17 Lectures on Fermat Numbers: From Number Theory to Geometry'', Springer, New York, 2001

6. Bentley, P. J.: The Book of Numbers, Octopus Publishing Group, 2008.

7. Pickover, C. A. Mathematical Book. 2009

8. Crilly, T.: Mathematics 50 Mathematical Ideas You Really to Know, Quercus, 2007.

And some internet sources, etc.

Poznámka:

Information about the course and courseware are available at https://moodle.fit.cvut.cz/courses/MIE-HMI/

Další informace:

https://moodle.fit.cvut.cz/courses/MIE-HMI/

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