- Code: DT0761
- Unit Coordinator: Ida Germana Minelli
- Programme: RealMaths
- ECTS Credits: 9
- Semester: 2
- Year: 1
- Campus: University of L'Aquila
- Language: English
- Delivery: In-class
- Aims:
The course aims to give an introduction to the theory of stochastic processes with special emphasis on applications and examples. On successful completion of this module the students should become familiar with some of the most known stochastic processes and to acquire both the mathematical tools and intuition for being able to describe systems randomly evolving in time and to analyze their properties.
- Content:
A measure theoretic approach to probability. Conditional expectation, properties, interpretations and computations. Filtrations, Martingales, examples and applications. Stopping times. The optional sampling Theorem. Martingale inequalities. Martingale convergence theorems. Uniformly integrable martingales and convergence in L^1. Continuous time processes, definition, finite dimensional distributions. Poisson process and its properties. Additive processes. The strong Markov property of Poisson process. Brownian motion: definition and main properties.
- Pre-requisites:
Basic notions of probability theory, measure theory and integration. Markov Chains.
- Reading list:
D. Williams “Probability with martingales”,
P. Billingsley “Probability and measure”,
Z. Brzezniak, T. Zastawniak “Basic stochastic processes”
- Unit Coordinator: dr inż. Witold Tomaszewski
- ECTS Credits: 5
- Year: 2
- Campus: Silesian University of Technology
- Language: English
- Aims:
The aim of the course is to familiarize students with various aspects of modern cryptography, using number theory, modular arithmetic, group theory and the theory of rings and fields.
- Content:
1. Number theory algorithms: Euclidean algorithm, Diophantine equations, Chinese remainder theorem.
2. Elements of group theory and commutative ring theory.
3. Modular arithmetic.
4. Classical cyphers: Caesar cypher, affine cypher, Vigenère cypher, matrix cyphers.
5. Public-key cryptography.
6. Selected cryptographic protocols.
7. Finite fields: constructions, properties and arithmetic.
8. Applications of finite fields in cryptography.
9. Elliptic-curve cryptography.
- ECTS Credits: 6
- Semester: 1
- Year: 2
- Campus: Ivan Franko National University of Lviv
- Language: English
- Unit Coordinator: Zdeněk Karpíšek
- ECTS Credits: 4
- Semester: 2
- Year: 2
- Campus: Brno University of Technology
- Language: English
- Aims:
The course objective is to make students majoring in Mathematical Engineering and Physical Engineering acquainted with important selected methods of mathematical statistics used for a technical problems solution.
- Content:
1.One-way analysis of variance.
2.Two-way analysis of variance.
3.Regression model identification.
4.Nonlinear regression analysis.
5.Regression diagnostic.
6.Nonparametric methods.
7.Correlation analysis.
8.Principle components.
9.Factor analysis.
10.Cluster analysis.
11.Continuous probability distributions estimation.
12.Discrete probability distributions estimation.
13.Stochastic modeling of the engineering problems.
- Pre-requisites:
Descriptive statistics, probability, random variable, random vector, random sample, parameters estimation, hypotheses testing, and regression analysis.
- Reading list:
- Ryan, T. P.: Modern Regression Methods. New York : John Wiley, 2004.
- Montgomery, D. C. - Renger, G.: Applied Statistics and Probability for Engineers. New York: John Wiley & Sons, 2003.
- Hahn, G. J. - Shapiro, S. S.: Statistical Models in Engineering. New York: John Wiley & Sons, 1994.
- Additional info:
The course is concerned with the selected parts of mathematical statistics for stochastic modeling of the engineering experiments: analysis of variance (ANOVA), regression models, nonparametric methods, multivariate methods, and probability distributions estimation.
Computations are carried out using the software as follows: Statistica, Minitab, and QCExpert.
