To master the fundamentals of the common language and basic technical terms of particular fields of study; with pre-intermediate students emphasis on reading technical texts and presentations of technical articles, application of French Internet websites.
Capelle Guy, Menand Robert: Le Nouveau taxi 1, Hachette FLE Paris, France 2009, ISBN 978-2-01-155548 - 9
test
French language is taught at two levels: beginner and intermediate. Students opt for one of them depending on whether they wish to obtain the fundamentals of the language or wish to maintain and/or improve previous knowledge of French.
100/0
To master the fundamentals of the common language and basic technical terms of particular fields of study; with pre-intermediate students emphasis on reading technical texts and presentations of technical articles, application of French Internet websites.
Capelle Guy, Menand Robert: Le Nouveau taxi 1, Hachette FLE Paris, France 2009, ISBN 978-2-01-155548 - 9
test
The subject continues the program of French language (1) and provides courses of essential and intermediate French language.
100/0
To master the fundamentals of the common language and basic technical terms of particular fields of study (depending on the student's level of German proficiency )
Appropriate study material is supplied by teacher based on the participants’level of German proficiency.
test
German language is taught at three levels: beginner, intermediate and advanced. Students opt for one of them depending on whether they need to learn the fundamentals or maintain and/or improve their previous knowledge.
This course’s focus is to master the fundamentals of the common language and basic technical terms of particular fields of study (depending on the student's level of German proficiency )
100/0
To master the fundamentals of the common language and basic technical terms of particular fields of study (depending on the student's level of German proficiency )
To master the fundamentals of the common language and basic technical terms of particular fields of study (depending on the student's level of German proficiency )
Appropriate study material is supplied by teacher based on the participants’level of German proficiency
test
German language is taught at two levels: beginner and intermediate. Students opt for one of them depending on whether they wish to obtain the fundamentals of the language or wish to maintain and/or improve previous knowledge of German.
This course’s focus is to to master the fundamentals of the common language and basic technical terms of particular fields of study (depending on the student's level of German proficiency )
100/0
To master the fundamentals of the common language and basic technical terms of particular fields of study (depending on the student's level of German proficiency )
To master the fundamentals of general Russian. The language level is A1.
Learning the Cyrillic (Russian) alphabet, gaining basic language competence, building up skills and confidence in dealing with unfamiliar authentic and semi-authentic texts.
The textbook: : Точка Ру A1 (Ольга Долматова, Екатерина Новачац), рracovné karty Падежи 1 (Л.С. Безкоровайная, В.Е. Штыленко).
test
To master the fundamentals of general Russian. The language level is A1.
Learning the Cyrillic (Russian) alphabet, gaining basic language competence, building up skills and confidence in dealing with unfamiliar authentic and semi-authentic texts.
The subject provides a course in Russian language for beginners.
100/0
Basic communication in Russian, developing other Russian language skills - listening comprehension, reading and writing.
To master the fundamentals of general Russian.
Learning the Cyrillic (Russian) alphabet, gaining basic language competence, building up skills and confidence in dealing with unfamiliar authentic and semi-authentic texts.
Textbook: Точка Ру A1 (Ольга Долматова, Екатерина Новачац), рracovné karty Падежи 1 (Л.С. Безкоровайная, В.Е. Штыленко).
test
To master the fundamentals of general Russian.
Learning the Cyrillic (Russian) alphabet, gaining basic language competence, building up skills and confidence in dealing with unfamiliar authentic and semi-authentic texts.
The subject continues the program of Russian language (1) and provides a course of Russian for beginners.
100/0
Basic communication in Russian, developing other Russian language skills - listening comprehension, reading and writing.
Krížom- Krážom Slovenčina 1, additional material to further support the covered topics.
tests
Course prerequisites:
https://fmph.uniba.sk/microsites/kjp/katedra-jazykovej-pripravy/poziadavky-na-udelenie-priebezneho-hodnotenia-aj1aj2aj3-ostatne-kurzy/
The sylabus is targeted at the comprehension of the basics of the Slovak language for the absolute beginners (A1).
100/0
This course is aimed for foreign students to learn the fundamentals of the Slovak language with the focus on basic communication as well as all other language skills- listening comprehension,reading and writing.
Krížom- Krážom Slovenčina 1, additional material to further support the covered topics
tests
Course prerequisites:
https://fmph.uniba.sk/microsites/kjp/katedra-jazykovej-pripravy/poziadavky-na-udelenie-priebezneho-hodnotenia-aj1aj2aj3-ostatne-kurzy/
The sylabus is targeted at the comprehension of the basics of the Slovak language for the absolute beginners (A1) and this course is a follow up course to the Slovak language course 1.
100/0
This course is aimed for foreign students to learn the fundamentals of the Slovak language with the focus on basic communication as well as all other language skills- listening comprehension,reading and writing.
Continual improvement of all language skills focused on communication/speaking, listening comprehension and writing. The emphasis is on discourse, lexicology and morphology, word-bank broadening of communicational English as well as English for specific purposes appropriate for university students. This course is a follow up of the previously taught ESP course.
Appropriate study material is supplied based on the participants’level of English by the lecturer.(Sources- The Guardian, The Herald Morning Sun. The Nine News, The West Australian, BBC News and podcasts, CNN podcasts).
tests, oral presentations, essays
tests, presentations, essays
Course prerequisites:
https://fmph.uniba.sk/microsites/kjp/katedra-jazykovej-pripravy/poziadavky-na-udelenie-priebezneho-hodnotenia-aj1aj2aj3-ostatne-kurzy/
This course’s focus is to broaden spoken/communicational English for students with B2/C1 level of English knowledge.
100/0
Continual improvement of all language skills focused on communication/speaking, listening comprehension and writing. The emphasis is on discourse, lexicology and morphology, word-bank broadening of communicational English as well as English for specific purposes appropriate for university students. This course is a follow up of the previously taught ESP course.
tests
Continual improvement of all language skills focused on communication/speaking, listening comprehension and writing. The emphasis is on discourse, lexicology and morphology, word-bank broadening of communicational/spoken English as well as English for specific purpose appropriate for university students. This course is a follow up of the Conversational English course 1.
Appropriate study material is supplied based on the participants’level of English by the lecturer.(Sources- The Guardian, The Herald Morning Sun. The Nine News, The West Australian, BBC News and podcasts, CNN podcasts).
tests, oral presentations, essays
tests, oral presentations, essays
Course prerequisites:
https://fmph.uniba.sk/microsites/kjp/katedra-jazykovej-pripravy/poziadavky-na-udelenie-priebezneho-hodnotenia-aj1aj2aj3-ostatne-kurzy/
This course’s focus is to broaden spoken/communicational English for students with B2/C1 level of English knowledge( Upper-Intermediate/Lower Advanced).
100/0
Continual improvement of all language skills focused on communication/speaking, listening comprehension and writing. The emphasis is on discourse, lexicology and morphology, word-bank broadening of communicational/spoken English as well as English for specific purpose appropriate for university students. This course is a follow up of the Conversational English course 1.
tests
Improvement of common language, deepening the knowledge of technical terms, reading technical texts, presentations of popular scientific articles.
Capelle Guy, Menand Robert: Le Nouveau taxi 1, Hachette FLE Paris, France 2009, ISBN 978-2-01-155548 - 9
test
The subject provides a course of intermediate French language, covering not only general, but also technical language.
100/0
Improvement of common language, deepening the knowledge of technical terms, reading technical texts, presentations of popular scientific articles.
Menand Robert: Le Nouveau taxi 2, Hachette FLE, Paris, France 2009, ISBN 978-2-01-155551 - 9
test
The subject provides a course of intermediate French covering not only general, but also technical French language.
100/0
Deepening the knowledge of the common language and basic technical terms of particular fields of study (depending on the student's level of German proficiency).
Appropriate study material is supplied by teacher based on the participants’level of German proficiency.
test
The course is a follow-up to the German language (1,2). The subject provides a course of intermediate or advanced German language.
This course’s focus is to deepen the knowledge of the common language and basic technical terms of particular fields of study (depending on the student's level of German proficiency).
100/0
Master the basics of general language and basic professional terminology of individual fields of study (depending on the advanced level of students)
Deepening the knowledge of the common language and basic technical terms of particular fields of study (depending on the student's level of German proficiency).
Appropriate study material is supplied by teacher based on the participants’level of German proficiency.
test
The course is a follow-up to the German language (1-3). It provides a course of intermediate and advanced German language.
This course’s focus is to deepen the knowledge of the common language and basic technical terms of particular fields of study (depending on the student's level of German proficiency).
100/0
Master the basics of general language and basic professional terminology of individual fields of study (depending on the advanced level of students)
Learning the handwritten Russian (Russian Cursive Cyrillic), developing further language skills, gaining knowledge of Russian culture, history and way of life, pre-intermediate to intermediate grammar and vocabulary.
Точка Ру A2 (Ольга Долматова, Екатерина Новачац) a Short Stories in Russian (Olly Richards, Alex Rowlings)
test
Learning the handwritten Russian (Russian Cursive Cyrillic), developing further language skills, gaining knowledge of Russian culture, history and way of life, pre-intermediate to intermediate grammar and vocabulary.
The course "Russian for Intermediate Students" is a follow-up to "Russian for Beginners". The subject of the course is general Russian in the range appropriate to the given level.
100/0
Basic communication in Russian, developing other Russian language skills - listening comprehension, reading and writing.
Learning the handwritten Russian (Russian Cursive Cyrillic), developing further language skills, gaining knowledge of Russian culture, history and way of life, pre-intermediate to intermediate grammar and vocabulary.
Точка Ру A2 (Ольга Долматова, Екатерина Новачац) a Short Stories in Russian (Olly Richards, Alex Rowlings)
test
Learning the handwritten Russian (Russian Cursive Cyrillic), developing further language skills, gaining knowledge of Russian culture, history and way of life, pre-intermediate to intermediate grammar and vocabulary.
The course "Russian for Intermediate Students" is a follow-up to "Russian for Beginners". The subject of the course is general Russian in the range appropriate to the given level.
100/0
Learning the handwritten Russian (Russian Cursive Cyrillic), developing further language skills, gaining knowledge of Russian culture, history and way of life, pre-intermediate to intermediate grammar and vocabulary.
Krížom-Krážom Slovenčina 2, additional material to further support the covered topics.
tests
Course prerequisites:
https://fmph.uniba.sk/microsites/kjp/katedra-jazykovej-pripravy/poziadavky-na-udelenie-priebezneho-hodnotenia-aj1aj2aj3-ostatne-kurzy/
The sylabus is targeted at the comprehension of all the language skills of the Slovak language , and it is a follow up course to the Slovak language course 2.
100/0
This course is aimed for foreign students to better comprehend all the language skills important to enable correct usage of the Slovak language – listening comprehension, reading, writing and speaking.
Krížom-Krážom Slovenčina 2, additional material to further support the covered topics.
tests
Course prerequisites:
https://fmph.uniba.sk/microsites/kjp/katedra-jazykovej-pripravy/poziadavky-na-udelenie-priebezneho-hodnotenia-aj1aj2aj3-ostatne-kurzy/
The sylabus is targeted at the comprehension of all the language skills of the Slovak language , and it is a follow up course to the Slovak language course 3.
100/0
This course is aimed for foreign students to better comprehend all the language skills important to enable correct usage of the Slovak language – listening comprehension, reading, writing and speaking.
Slovník demografických pojmov / Danuša Jurčová. Bratislava : INFOSTAT – Inštitút informatiky a štatistiky, Výskumné demografické centrum, Edícia: Akty, 2005, ISBN 80-85659-40-9 [available on the Internet];
Methods of Demographic Analysis / Farhat Yusuf, Jo. M. Martins, David A. Swanson. First Edition, Dordrecht : Springer Netherlands, 2014, ISBN 978-94-007-6784-3;
Fundamentals of Demographic Analysis: Concepts, Measures and Methods / Gordon A. Carmichael. First Edition, Cham : Springer International Publishing, Switzerland, 2016, ISBN 978-3-319-23255-3;
study materials of the lecturer.
The course assessment consists of three parts. The semester evaluation has a weight of 33% and consists of practical test. In order to meet the minimal condition to sit the final written exam, the student must obtain at least half of the possible points from the semester evaluation. During the exam period, an exam will be written on which an additional 34% of points can be obtained. The student must obtain at least half of the possible points on the written exam in order to be admitted to the oral exam. The last part of the evaluation - the oral exam - has a weight of 33%.
Grade thresholds: A: at least 90.00%; B: 89.99% – 80.00%; C: 79.99% – 70.00%; D: 69.99% – 60.00%; E: 59.99% – 50.00%; Fx: 49.99% – 0.00%.
Slovak, English
Basics of demography. Demographic data, sources of demographic data. Demographic indicators. Standardization and decomposition. Probability models for the number of demographic events. Construction of life tables from statistical data. Force of mortality, mathematical modeling of the force of mortality. Multiple decrement tables. Actuarial demography.
Practical test during semester 33% / final exam 67% (written and oral exam).
After completing the course students will control the basics of demographics for the purpose of actuarial practice. They will be able to construct life tables and assess the statistical properties of demographic indicators and apply statistical methods in demographic analyzes.
Numerická matematika II : Numerické řešení diferenciálních rovnic / Emil Vitásek. Praha : Univerzita Karlova, 1981
Numerické metody / Emil Vitásek. Praha : Státní nakladatelství technické literatury, 1987
Numerické metódy / Jela Babušíková, Marián Slodička, Juraj Weisz. Bratislava : Univerzita Komenského, 2000
Preliminary assessment: 2 written tests 10 points each, project 20 points
Final examination: written exam for 50 points and oral exam for 10 points
Grading scheme: A 90%, B 80%, C 70%, D 60%, E 50%
Slovak, English
Numerical methods for ordinary differential equations.
Initial value problem, one-step and multistep methods of Runge-Kutta type.
Methods for Solving Ordinary Differential Equations.
Applications of ODR numerical methods in physics and biology.
Numerical solution of boundary value problems for ordinary differential equations, difference method, shooting method.
Partial differential equation: Finite difference method for parabolic, hyperbolic and elliptic problems in 2D, explicit and implicit methods, stability, alternating direction method.
Applications of PDR numerical methods in physics and biology.
Implementation of numerical algorithms in Matlab and Python.
40/60
Student will be able to solve ordinary and partial differential equations by modern numerical methods.
Introduction to modern time series analysis / Gebhard Kirchgässner, Jürgen Wolters. Berlin : Springer, 2008
Introductory time series with R / Paul S. P. Cowpertwait, Andrew V. Metcalfe. Dordrecht : Springer, 2009
Forecasting With Univariate Box-Jenkins Models : Concepts and Cases / Alan Pankratz. New York : John Wiley, 1983
Applied econometric time series / Walter Enders. New York : John Wiley, 2004
Time series analysis and its applications : with R examples / Robert H. Shumway, David S. Stoffer. New York : Springer, 2011
Basics of statistics (linear regression, analysis of residuals, testing hypotheses) and use of R language, complex numbers
Assessment during the term: homework (50%).
Exam: Written exam (50%).
Grading: A: 90 and more, B: [80, 90), C: [70, 80), D: [60, 70), E: [50, 60), FX: less than 50
Slovak, English
Introduction. Time series and their moments. Stationarity and ergodicity. White noise. Wold representation. Correlations between values of the process, autocorrelation function. Testing white noise, Ljung-Box Q-statistic.
Autoregressive models (AR), moving average models (MA), ARMA models. Stationarity and invertibility conditions. Computation of expected value, variance and covariances. Autocorrelation and partial autocorrelation function and their use in identification of the model. Predictions.
Differencing time series, integrated processes. Testing unit root. ADF test.
Seasonality, SARIMA models.
Modelling volatility, ARCH and GARCH models, their generalizations.
Modelling trend - exponential smoothing, Holt-Winters method, Hodrick-Prescott filter.
Spectral analysis of time series.
50/50
Student wll be able to model univariate time series using Box-Jenkins methodology and will know its theoretical background.
Introduce the basic properties of Markov chains and their application in Queueing theory and Inventory theory.
K. Janková, S. Kilianová, P. Brunovský, P. Bokes: Markovove reťazce a ich aplikácie. Epos, 2015
D. Gross, J. F. Shortle, J. M. Thompson C. M. Harris: Fundamentals of Queueing Theory, Fourth Edition. Wiley, 2008.
individual work, homeworks
Old syllabus.
Random chains, continuous and discrete Markov chains, Poisson process, Queueing theory (M/M/1, M/M/n, M/M/n/m, M/D/1), Inventory theory (basic deterministic and stochastic models). Newsvendor problem.
0/100
exam
Financial Calculus : An Introduction to derivate Pricing / Martin Baxter, Andrew Rennie. Cambridge : Cambridge University Press, 1996
Mathematical techniques in finance : Tools for incomplete markets / Aleš Černý. Princeton : Princeton University Press, 2009
Written test during the semester;
Written and oral exam (50/50)
Indicative assessment scale: A 90%, B 80%, C 70%, D 60%, E 50%
Slovak/english
1. Theory of Lebesgue integral
2. Wiener process, Ito’s integral, Ito’s lemma
3. Black-Scholes model of pricing asset and currency derivatives
4. Models of interest rate development and pricing interest rate derivatives
40/60
Mastering the priciples of the Lebesgue integral theory, stochastic calculus and valuation of derivatives based on the Wiener process. The aim of the course is also to acquire key knowledge of classical methods of valuation of financial derivatives.
To introduce non-cooperative game theory with focus on games with incomplete information.
Microeconomic Theory / Andreu Mas-Colell, Michael D. Whinston, Jerry R. Green. New York : Oxford University Press, 4.Ed., 2022
Game theory / Drew Fudenberg, Jean Tirole. Cambridge, Mass. : MIT Press, 1998
individual work, homework
Continuous Assessment: Separate Work, Homework (40 %), Midterm Test (30 %)
Test: Written exam
Indicative assessment scale: A 100-90%, B 89-80%, C 79-70%, D 69-60%, E 59-50%
Slovak, English
1. Static Bayesian Games
1.1 Static Games of Incomplete Information
Introductory examples: Battle of Sexes and Cournot duopoly with incomplete information. Definition of static game of incomplete information (Bayesian Game). Players, actions, types, strategies, believes and payoffs.
1.2 Bayesian Equilibrium
Associated game to a static Bayesian game. Definition of Bayesian Equilibrium. Separating equilibrium, pooling equilibrium.
1.3 Computing Bayesian Equilibria
Battle of Sexes with incomplete information about types of players. Cournot duopoly with incomplete information about costs.
1.4 Auctions
Auction formats (open bid auctions – ascending and descending bids, sealed bid auctions – first price and second price auctions, private value and common value auctions). Auctions with independent private value: English auction, Dutch auction, First price auction, Second price auction. Equivalence of auction formats. Common Value auction and Winner’s course.
1.5 Correlated Equilibria
Correlated equilibrium with observable signals. Correlated equilibrium with unobservable signals. Correlated random device. Examples.
1.6 Harsanyi Purification of Mixed Action Equilibria
Other approaches to interpretation of mixed action equilibrium. Bayesian game with infinite many types of players. Examples.
2. Dynamic Bayesian Games
2.1 Dynamic Games of Incomplete Information
Definition. Players, actions, game trees, strategies, information sets, believes and payoffs. Pure strategy, mixed strategy, behavioral strategy. Perfect recall. Strategies and outcomes. Expected payoffs. Equivalence of strategies. Behavior strategy equivalent to a mixed strategy. Mixed strategies equivalent to a behavior strategy.
2.2 Solution Concepts of Dynamic Bayesian Games
Associated game to a dynamic Bayesian game. Nash Equilibrium. Example of an implausible Nash equilibrium.
2.3 Subgames and continuations of a game
Subgame. Subgame perfect Nash equilibrium. Example of an implausible subgame perfect Nash equilibrium. The source of implausibility. Continuation of a game.
2.3 Bayesian assumptions, Weak Bayesian Perfect Equilibrium
Beliefs and assessments. First Bayesian assumption (existence of an assessment). Sequential rationality. Second Bayesian assumption (on sequential rationality). Weak consistency. Third Bayesian assumption (on weak consistency on equilibrium path). Fourth Bayesian assumption (on weak consistency everywhere it goes). Weak Bayesian perfect equilibrium.
Bayes games, Bayes-Nash equilibrium. Dynamic games with incomplete information. Four Bayes assumptions. Sequential equilibrium. Weak perfect Bayes equilibrium. Perfect Bayes equilibrium.
70/30
Students will be able to built models of strategic decision making in the presence of uncertainty. They will be able to decide in such situations as well as to determine optimal approaches.
exam
Analytické a numerické metódy oceňovania finančných derivátov / Daniel Ševčovič, Beáta Stehlíková, Karol Mikula. Bratislava : Slovenská technická univerzita, 2009
Analytical and numerical methods for pricing financial derivatives / Daniel Sevcovic, Beáta Stehlíková, Karol Mikula. New York : Nova Science, 2011
Mathematical Models of Financial Derivatives / Yue Kuen Kwok. Singapore : Springer, 1998
Knowledge of stochastic processes and partial differential equations.
Assessment during the term: Homework (25%), test (15%).
Exam: Written exam (40%), oral exam (20%).
Grading: A: 90 and more, B: [80, 90), C: [70, 80), D: [60, 70), E: [50, 60), FX: less than 50
Slovak, English
Financial derivatives. Call and put option on a stock. Put-call parity. Bounds of option proces. Combined option strategies.
Stochastic processes. Wiener process, Brownian motion, geometric Brownian motion. Probability distribution, expected value and variance of a GBM. GBM as a model for a stock price, estimation of its parameters from stock prices. One-dimensional and multi-dimensional Ito lemma.
Black-Scholes and Merton model. Derivation of the partial differential equation of the derivative price - two approaches (Black-Scholes and Merton). Terminal conditions.
Pricing of call and put options - solving the Black-Scholes PDE. Pricing put options using put-call parity. Pricing option strategies. Comparison of theoretical pricing results wit real market data.
Volatility. Historical volatility of stocks. Implied volatility. Volatility smile.
Greeks - sensitivity to parameters. Delta of an option, using delta to secuting portfolio - delta hedging. Gamma of an option. Other factors: theta (sensitivity to time), vega (sensitivity to a change of volatility), rho (sensitivity to interest rate).
Modelling transaction costs - Leland model. Nonlinear PDE for derivatives prices, its explicit solution in the case of call and put options. Modelling bid-ask spreads. Computation of implied parameters (volatility, time between two adjustments of a portfolio) from data. Infomatively other nonlinear models.
Numerical methods of pricing European derivatives. Explicit scheme for solving Black-Scholes equation. Binomial and trinomial tree. Implicit scheme and solving the corresponding system of linear equations, SOR method.
Americal type of derivatives and their pricing. Pricing American options via free boundary problems Pricing American options via linear complementarity problems.
Numerical methods for pricing American derivatives. Projected SOR method. Identification of early exercise boundary for an American option. Infomatively - research in the area of early exercise boundary analysis.
Modelling instantaneous interest rate. One-factor models, Vasicek, CIR and CKLS models. Density of a stochastic process, Fokker-Planck equation. Calibration of the Vasicek model by the maximum likelihood method and Nowman estimates of the CKLS model - basic ideas. Infomatively various types of two-factor models.
Pricing interest rate derivatives. PDE for the price of an interest rate derivative. Bonds and time structure of interest rates. Pricing bonds in concrete models.
Modelling exotic financial derivatives. Asian options, barrier options. Basket options and options on indices. Overview of other types of exotic options.
40/60
Students learn basics of modelling financial derivatives based on stochastic processes and partial differential equations. They gain key competences in the field of financial mathematics, analytical and numerical methods of pricing financial derivatives.
Parciálne diferenciálne rovnice a ich aplikácie / Daniel Ševčovič. Bratislava : Iris, 2008
Linear partial differential equations for scientists and engineers / Tyn Myint-U, Lokenath Debnath. Boston : Birkhäuser, 2007
Sbornik zadač po differenciaľnym uravnenijam / Aleksej Fedorovič Filippov. Moskva : Nauka, 1979
Assessment during the term: two tests (2x20%).
Exam: written and oral exam (60%).
Grading: A 90%, B 80%, C 70%, D 60%, E 50%
Slovak, English
Partial differential equations and mathematical modeling. Derivation of partial differential equations of continuity, heat equation and equation of vibrating string. Derivation of the Black-Scholes model for valuation of derivatives. First order linear PDEs. The first integral and method of characteristics. Second order linear PDEs. Parabolic equations. Green function method on an unbounded interval. Explicit solution of European call and put options. Fourier method of separation of variables for problems on a bounded interval. Comparison principle and smoothing of solutions of parabolic equations. Hyperbolic equations. D'Almbert's formula for the oscillations of an unbounded string. Elliptic equations and harmonic functions. Green's representation of the solution to the Poisson equation. The maximum principle and its use in the proof of the fundamental theorem of algebra.
40/60
The student will master the basics of classical methods of solving partial differential equations with an emphasis on applications used in financial mathematics. The output of the class is obtaining key knowledge and competences in the field of partial differential equations.
The course builds on the lecture of Optimal control (1), explains the basic techniques that allow quality analysis and discusses the solution of continuous optimal control problems that leads to Pontryagin’s maximum principle.
M. Halická, P. Jurča: Optimálne riadenie 2 (učebný text http://pc2.iam.fmph.uniba.sk/institute/halicka/)
M.I. Kamien, N.L. Schwartz: Dynamic Optimization. The Calculus of Variations and Optimal Control in Economics and Management, ELSEVIER 1995
individual work, test
Interim evaluation: The course does not have an interim evaluation.
Examination: written exam (the obtained evaluation can be adjusted by one degree by oral examination)
Indicative assessment scale: A 90%, B 80%, C 70%, D 60%, E 50%
I. Introduction to continuous optimal control problems
1. Formulation of continuous optimal control problems: Motivating examples (optimal consumption problem, time optimal control). General scheme of the continuous optimal control problem. Concepts of control and its response, admissible control and optimal control. Classification of optimal control problems. Relation between optimal control and calculus of variations.
2. Calculus of variations: Formulation of the fundamental calculus of variations problem. Derivation of the Euler equation. Necessary conditions of higher order. Sufficient conditions of optimality. The optimal consumption problem as a calculus of variations problem and its solution. Problem solving.
3. Special calculus of variations problems: Modification of the derivation of necessary conditions for free endpoint and free time problems. Higher dimensional problems. Higher-order problems. Bolza problem. The isometric problem. Derivation of necessary conditions and problem solving.
II. Finite time continuous optimal control problems
4. Pontryagin’s maximum principle for the standard problem: Formulation of the standard optimal control problem (autonomous problem with free time, with control and endpoint constraints, without state variable constraints, Lagrange form of objective function). Formulation of Potryagin’s maximum principle, idea of the proof, its importance for solving problems. Solving various problems.
5. Non-autonomous problems. Fixed-time problems: Derivation of the necessary conditions for non-autonomous and fixed-time problems from the conditions for the standard problem. Solving various problems.
6. Extensions of Pontryagin’s maximum principle: Pontryagin’s maximum principle for the discounted problem. Relation between Potryagin’s maximum principle and Euler’s equations. Notes about the zero-index multiplicator of the objective function.
7. Pontryagin’s maximum principle and qualitative analysis: Importance of the maximum principle for qualitative analysis. Illustration by the solution of the optimal consumption problem. Solving the production planning problem.
8. Pontryagin’s maximum principle as a tool for numerical solution of problems: Necessary optimality conditions in the form of boundary-value problems. Numerical solution of the boundary-value problem. Numerical solution of various problems.
9. The Ramsey model and phase portraits analysis: Formulation of the finite horizon Ramsey model and its relation to the optimal consumption problem. The boundary-value problem for the Ramsey model. Phase portrait analysis for the Ramsey model. Numerical solution of the Ramsey model. General view on phase portrait analysis. Problem solving.
10. The Bolza problem: Derivation of Pontryagin’s maximum principle for the Bolza problem from the maximum principle for the Lagrange’s problem. Solving the comodity business problem. Solving various problems.
11. Economic interpretation of the maximum principle. The dynamic programming equation: Concept of the value function. Derivation of the dynamic programming eqution. Derivation of the formula allowing the economic interpretation of the adjoint variable. Economic interpretation of the adjoint variable. Interpretation of the maximum principle.
12. Problems with state variable and endpoint constraints in an inequality form: Principles of necessary optimality conditions for problems with target-state constraints in the inequality-form, for problems with mixed state and control constraints and for problems with pure state constraints. Problem solving for the first two types of problems.
13. Optimality of the controls obeying the maximum principle. Existence of the optimal control: Formulation of the optimal control existence theorem. Sufficiency of Pontryagin’s maximum principle (Formulation and proof of Arrow’s theorem and it’s corrollaries).
14. The linear time optimal control problem: Formulation of the linear problem of fastest reocation. Existence of optimal solutions and the form of the necessary conditions. Problem of the optimal control synthesis (solution of the fastest relocation problem). Normality. Bang-bang controls and uniqueness. Solution of problems involving synthesis.
15. Singular optimal controls: Problems of singular control (modified Solow’s model). Definition of singular control. Necessary optimality conditions of higher order. Synthesis of optimal control for Solow’s model. Solving various problems.
III. Infinite horizon continuous optimal control problems
16. Infinite horizon optimal control problems: Formulation of the standard problem. Concept of admissible solution. Pontryagin’s maximum principle and the idea of its derivation from the finite horizon problems. Missing condition of transversality. Solution of the optimal fixed endpoint consumption problem.
17. Autonomous problems with a discount factor: Idea of deriving the limit condition for the Hamiltonian. Sufficient optimality conditions, solving the infinite horizon Ramsey model. Problem solving.
Problems for calculus of variations and Euler’s equation. Formulating the continuous optimal control problems. Pontryagin’s maximum principle for different types of problems. Methods of analytic and numeric solutions (boundary-value problem), analysis of phase portraits. Solving problems. Problem of optimal consumption and Ramsey’s model. Equation of dynamic programming, economic interpretation of the adjoint variable. Existence of optimal control. Sufficient conditions of optimality. Singular control. Problems with infinite time horizon.
0/100
exam
Introduce the theory, methods and applications of convex optimization.
M. Trnovská: Konvexná optimalizácia, elektronický text.
Boyd, Vandenberghe: Convex Optimization, Cambridge Univ.Press 2004
CVX: Matlab Software for Disciplined Convex Programming www.stanford.edu/~boyd/cvxbook
Ben-Tal, Nemirovski: Lectures on Modern Convex Optimization, SIAM 2001
Nonlinear programming, Linear programming
team homework, test
Ongoing assessment: homework assignments (20%), in-class tests (30%)
Final assessment: semester project (20%), final exam (30%)
Grading A 91%, B 81%, C 71%, D 61%, E 51%
Slovak, English
Convex optimization problems in standard form
(Convex functions, problems, duality, KKT optimality conditions)
Generalization of standard convex problems
(Convex cones, conic programs, transformation of standard convex problems into a conic form)
Conic convex problems
(Basic conic linear programming classes, LP, SDP, SOCP,.. and relations between them, conic representable functions and problems)
Geometry of convex cones
(Proper cones, dual and self dual cones, properties)
Duality theory for conic linear programs
(Dual problem derivation, Generalized Farkas Lemma, Duality theory for conic linear problems, optimality conditions)
Applications of convex conic problems
(Portfolio optimization, Optimal experiment design, Minimal surface problem, Linear and quadratic discrimination)
Conic relaxations
(Non-convex quadratic problems relaxation, MAX-CUT)
Interior point methods
Convex optimization problems in standard form
Generalization of standard convex problems
Conic convex problems (SDP, SOCP,..)
Geometry of convex cones
Duality theory for conic linear programs
Applications of convex conic problems
Conic relaxations
Interior point methods
50% /50%
Student learn the basic theory of convex analysis and convex (conic) optimization,
basic classes of convex conic programming, and methods for solving them, they learn the duality theory of conic linear programming. They are able to use Matlab and CVX (cvxopt) modeling system for solving convex problems, they are able to solve various practical problems and applications.
Project
Literature:
M. Halická, P. Brunovský, P. Jurča: Optimálne riadenie. Viacetapové rozhodovacie procesy v
ekonómii a financiách, EPOS, Bratislava 2009 (English translation)
A total of 100 points can be obtained from the subject Optimal Control I. It is possible to have a maximum of 40 points from the exercises. Points can be obtained by project 40p, by solving problems at the board - 0 to 2 points per exercise. A necessary condition for the evaluation of E and better is at least 15 points from the exercises. It is possible to obtain 60 points from the written exam.
If the necessary conditions (15C) are met, the classification is as follows:
Classification: A 90 and more points B 80-89 points C 70-79 points D 60-69 points E 50-59 points
The obtained evaluation can be improved or worsened by one degree by a voluntary oral exam, or improved or worsened by an oral exam at the initiative of the teacher.
Weight of midterm / final assessment: Mid-term assessment 40% / 60% final exam.
Slovak, English
Brief outline:
Formulation of deterministic discrete problems of optimal control. Equation of dynamic
programming for problems with finite and infinite time horizon. Methods of its solutions.
Stochastic problems. Control and feedback control. Solving problems. Necessary conditions of
optimality of variation type for discrete problems. Principle of maximum, economic
interpretation of the adjoint variable, discrete Euler equation. Quality analysis.
30/70
The course provides an overview of the optimal control theory for solving optimization
problems and reviews its main applications. The course introduces the basic methods of solving
discrete problems that lead to the problems of dynamic programming. Moreover, methods
allowing quantitative analysis for discrete problems are discussed.
To introduce games in which players have asymetric information.
E. Rasmusen: Games and Information, An Introduction to Game Theory, 4th Edition. Blackwell Publishers, 2006
Mas-Collel, Whinston, Green: Microeconomic Analysis. Oxford University Press, 1995
Fudenberg , Tirole: Game Theory. MIT Press,1998
individual work, homework
Continuous assessment four equivalent home-works.
Approximate rating scale: A 100-90%, B 89-80%, C 79-70%, D 69-60%, E 59-50%
Slovak, English
Classes of models with asymmetric information. Subjective risk with hidden action or hidden information. Adverse selection. Mechanism design and post-contractual hidden knowledge. Signalling and detection.
100/0
The student will be able to decide in an asymmetric information environment. It will be able to analyze the relevant models and compose contracts in asymmetric information.
exam
To introduce the theoretical and game models of markets and industries.
Industrial organization : Theory and applications / Oz Shy. Massachusetts : The MIT Press, 1995
The Theory of Industrial Organization. Cambridge, Mass. : MIT Press, 1997.
individual work
Continuous assessment four equivalent home-works.
Approximate rating scale: A 100-90%, B 89-80%, C 79-70%, D 69-60%, E 59-50%
Slovak, English
Marker structure and competition: perfect competition; monopoly; oligopoly; costs.
Obstacles in entering the market, fusions. Vertical obstacles and vertical connecting.
Firms' entering to market, leaving market, obstacles by entering.
Strategies of price determination and product differentiation.
Fusions, cartels, agreements.
Research and development: investments to research and development, licences.
Advertisement. Marketing strategies. Quality and endurance of products.
100/0
Students will be able to recognize the individual types of markets to build their models and look for equilibrium outputs.
exam
Teach the basics of interior-point methods in linear programming.
R. J. Vanderbei: Linear Programming: Foundations and Exensions, Kluwer, Boston 1977, available on http://www.princeton.edu/irvdb/LPbook
M. Kabát: Metódy vnútorného bodu v lineárnom programovaní a ich aplikácie vo financiách, Diploma thesis (supervisor: M.Halická), FMFI UK, Bratislava, 2013
individual work of solving assignments, test
I. Introduction to interior-point methods
1. Fundamental idea of interior-point methods and the history of interior-point methods in mathematic programming
2. Recapitulation of the fundamentals of duality in linear programming. Expression of the necessary and sufficient optimality conditions. Proof of the theorem that connects the existence of the interior-point in the primal problem and the boundedness of optimal solutions in the corresponding dual problem. Concept of the optimal decomposition and the expression of the optimal solution sets via optimal decomposition. Relative interiors of optimal solution sets.
II. Theory of the interior-point methods in linear programming.
3. Two fundamental assumptions and their importance. Logarithmic barrier function for the primal and dual problems. Transformed problems. Existence and uniqueness of the transformed problem solutions. Necessary and sufficient optimality conditions for the transformed problems (centering system). The relation between the centering system and the optimality conditions for the primal and dual problems. Concept of the central path.
4. Central path and its properties. Central path as a smooth curve in the interior of the admissible solution set. Boundedness of the central path. The limit points of the central path are strictly complementary optimal solutions. Definition of the analytic center of the optimal solution set. Its existence and uniqueness. Convergence of the central path to the analytic centre of the optimal solution set.
III. Algorithms of interior-point methods in linear programming
5. Overview of the interior-point algorithms (admissible vs. non-admissible, primal, dual, primal-dual, algorithm of potential reduction, affine scaling algorithms, algorithms of following the central path). Algorithms with implicit and explicit change of the barrier parameter. Predictor-corrector algorithms.
6. Newton directions for the centering system. Overview of the directions used in interior-point methods. Overview of the distance measures from the points of the central path. Formulation of the primal-dual algorithm of following the central path. Estimation of the polynomial complexity of the primal-dual algorithm of following the central path. Implementation of algorithms.
The fundamentals and history of interior-point methods in mathematic programming. Basic theories of interior-point methods in linear programming. Central path and its properties. Overview of interior-point algorithms. Estimation of the polynomial complexity of the primal-dual algorithms of following the central path. Implementation of algorithms.
100/0
Daniel Schneller, Udo Schwedt, MySQL Admin Cookbook, Packt Publishing Ltd. 2010
https://goalkicker.com/MySQLBook/MySQLNotesForProfessionals.pdf
Graeme Simsion, Graham Witt, Data Modeling Essentials, Elsevier 2004, ISBN: 9780080488677
Preliminary semester evaluation: active participation
Final examination: oral, semestral project
Approximate grade thresholds: A 90%, B 80%, C 70%, D 60%, E 50%
Slovak, English
Introduction to databases, database technologies, database models
Relational database model
SQL language
Relational databases
User rights in SQL
Database API
Interfaces for working with databases (web interface, CLI, GUI)
SQL programming (implementation in R, Matlab, Python, C ++)
Scale of assessment (preliminary/final): 20/80
By completing the course, the student will gain the ability to work with databases, including database model design, creation in common SQL databases, and user interface programming in python, R, C ++, Matlab
Modern Actuarial Risk Theory Using R / Rob Kaas, Marc Goovaerts, Jan Dhaene, Michel Denuit. Heidelberg : Springer, 2008, ISBN: 978-3-540-70998-5;
Jazyk R v aktuárskych analýzach / Michal Páleš. Bratislava : Vydavateľstvo EKONÓM, 2017, ISBN 978-80-225-4331-6;
Jazyk R pre aktuárov / Michal Páleš. Bratislava : Vydavateľstvo EKONÓM, 2019, ISBN 978-80-225-4331-6;
Zurich Insurance Company Ltd internal training materials.
During the semester, the student can get 100% of points, for solving individual assignments (50%) and elaboration and submission of the semester project (50%).
Grade thresholds: A: at least 90.00%; B: 89.99% – 80.00%; C: 79.99% – 70.00%; D: 69.99% – 60.00%; E: 59.99% – 50.00%; Fx: 49.99% – 0.00%.
Slovak, English
Limit: maximum 18 students.
It is recommended to have basic R-software skills. At the seminars, we present quantitative techniques used by actuaries and use real data sets from the area of life and non-life insurance.
Actuarial models in life insurance. Modelling the development of insurance contracts from the perspective of the client. Development of the number of insurance contracts with respect to the expected probability of death and lapse. Development of the mathematical reserves in the entire portfolio. Modelling of other variables affecting the profit of the insurance company. Modelling the future profits using the direct method. Modelling the future profits using the indirect method. Modelling of the present value of financial indicators. Impact of changes in assumptions on the profit of an insurance company.
Data preparation for modelling using the software R. Application of Generalized Linear Models (GLM) using the software R – selecting of appropriate parameters, presentation of results using R Shiny package.
Portfolio management in the insurance company using actuarial and financial indicators and relationships between them.
Practical work during semester 100% / final exam 0%.
After completing the course, the student will master basic methods of work in life and non-life insurance in an international insurance company, focusing on methods of cash-flow projection. The student will be able to calculate actuarial and financial indicators.
Applied multivariate statistical analysis / Wolfgang Karl Härdle, Léopold Simar. Heidelberg : Springer, 2012
Multivariate statistics: : Exercises and solutions / Wolfgang Härdle, Zdeněk Hlávka. New York : Springer, 2007
Preliminary assessment: test (60%)
Final examination: oral examination (40%)
Approximate final assessment: A 90%, B 80%, C 70%, D 60%, E 50%
Slovak, English
Additional information: http://www.iam.fmph.uniba.sk/ospm/Filova/teaching.htm
- Random vectors. (Random vectors, multivariate distributions, their densities, moments, and characteristic functions, transformations.)
- Multivariate normal distribution. (Properties of the multivariate normal distribution, marginal and conditional distributions.)
- Distribution of quadratic forms. (Wishart distribution, Hotelling distribution.)
- Estimating the parameters of multivariate linear models. (Likelihood function, maximum likelihood estimation, Cramer-Rao inequality.)
- Testing the hypotheses in multivariate linear models. (Likelihood ratio test, tests on parameters of the normal distribution, linear hypotheses.)
- Multivariate linear regression. (Linear regression model, least squares method.)
- Multivariate analysis of variance. (Analysis of variance model, one- and two-factor models, repeated measures model, profile analysis, growth curve models, multivariate analysis of variance.)
- Analysis of covariance.
http://www.iam.fmph.uniba.sk/ospm/Filova/teaching.htm
60/40
After completing the course, students will master multivariate regression analysis, analysis of variance and covariance; further, they will be able to formulate and use multivariate statistical analyses in practical situations.
A. C. Fowler, Mathematical Models in the Applied Sciences, Cambridge University Press, 1997
A. Quarteroni, P. Gervasio, A Primer on Mathematical Modelling, Springer, 2020
Interim assessment during the semester has a weight of 40% (homeworks 30%, bonus exercises 10%). The two semester exam papers have a total weight of 60% (the first paper taken in the middle of the semester, the second paper taken at the end of the semester). The student must obtain at least half of the points from each semester exam paper. The final evaluation can be adjusted by an oral exam (theoretical questions, written preparation).
Grading: A (100-91), B (90-81), C (80-71), D (70-61), E (60-51), FX (50-0)
Slovak, English
Basic principles of modeling.
Principle of nondimensionalisation. Buckingham Pi-theorem. Dimensionless parameters.
Asymptotic expansion, convergence vs. divergence, uniformity. Matched asymptotic approximations.
Application of asymptotic methods: Van der Pol oscillator.
Heat transfer model. Degenerate diffusion.
Material derivative. Vorticity. Viscous flow.
Flow instability and transition to turbulence.
Weight of the intermediate / final evaluation: 40/60
By completing this course, the student will gain knowledge of the principles of mathematical modeling of phenomena in the natural and technical sciences.
Poisťovníctvo / Anna Majtánová a kolektív. Bratislava : Wolters Kluwer (Iura Edition), 2009;
Lecturer's notes and handouts.
During the semester the student can obtain 100% of points, for active participation in seminars (80%), solving tasks during seminars (20%), and possibly for additional individual practical work.
Grade thresholds: A: 100.00% – 90.00%; B: 89.99% – 80.00%; C: 79.99% – 70.00%; D: 69.99% – 60.00%; E: 59.99% – 50.00%; Fx: 49.99% – 0.00%.
Slovak, English
Basic financial and insurance segments. Banks and insurance companies - joint-stock companies and their legislative background.
The role of financial analysts, risk managers and actuaries in banks, insurance companies and in other financial institutions. International professional qualification of actuaries, stages in actuarial qualification, the career of an actuary. Profession of Actuary in Slovakia and globally.
Professional software used in financial and insurance practice.
Practical work during semester 100% / final exam 0%.
After completing the course, the student will master the basics of work in the financial and insurance business. Student should also learn about the roles of the actuary, risk manager and financial analyst.
Poisťovníctvo / Anna Majtánová a kolektív. Bratislava : Wolters Kluwer (Iura Edition), 2009;
Lecturer's notes and handouts.
During the semester the student can obtain 100% of points, for active participation in seminars (80%), solving tasks during seminars (20%), and possibly for additional individual practical work.
Grade thresholds: A: 100.00% – 90.00%; B: 89.99% – 80.00%; C: 79.99% – 70.00%; D: 69.99% – 60.00%; E: 59.99% – 50.00%; Fx: 49.99% – 0.00%.
Slovak, English
Basic financial and insurance segments. Actuarial model development in life and non-life insurance. Financial modelling.
The role of financial analysts, risk managers and actuaries in banks, insurance companies and in other financial institutions. Professional requirements for financial analysts, risk managers and actuaries.
Professional software used in financial and insurance practice.
Practical work during semester 100% / final exam 0%.
After completing the course, the student will master the basics of work in the financial and insurance business. Student should also learn about the roles of the actuary, risk manager and financial analyst.
KOLACZYK, Eric D.; CSÁRDI, Gábor. Statistical analysis of network data with R. New York: Springer, 2014.
LUKE, Douglas A. A user's guide to network analysis in R. London, England: Springer, 2015.
Journal papers.
Basics of R language
Assessment during the term: homework (20%), project (80%).
Grading: A: 90 and more, B: [80, 90), C: [70, 80), D: [60, 70), E: [50, 60), FX: less than 50
Slovak, English
Basic notions of graph theory, examples of graphs/networks, their visualization. Node centralities - classical centralities (degree, closeness, betweenness, eigenvector centrality), selected new centrality measures, applications. Detection of communities in networks - concept of modularity, modularity optimization, selected other methods, applications. Networks based on correlations between time series. Random networks and their basic properties.
100/0
Students gain an overview of different methods used in analysis of social networks. They will be able to apply them to real data using R software.
Life Insurance Mathematics / Hans U. Gerber. Heidelberg : Springer, 1997, ISBN 978-3-662-03460-6;
Modely v životnom a neživotnom poistení / Rastislav Potocký. Bratislava : Statis, 2012;
Modern Actuarial Risk Theory Using R / Rob Kaas, Marc Goovaerts, Jan Dhaene, Michel Denuit. Second Edition, Heidelberg : Springer-Verlag, 2008;
Poistná matematika / Viera Sekerová, Mária Bilíková, Bratislava : Ekonóm, 2005;
Aplikovaná poistná štatistika / Viera Pacáková. Bratislava : Iura Edition, 2004;
study materials of the lecturer.
The course assessment is based only on the final written exam (with a weight of 100%), which has a possible (supplementary) oral part.
Grade thresholds: A: at least 90.00%; B: 89.99% – 80.00%; C: 79.99% – 70.00%; D: 69.99% – 60.00%; E: 59.99% – 50.00%; Fx: 49.99% – 0.00%.
Slovak, English
The collective risk model in general insurance. Methods of mathematical modeling in life and non-life insurance, actuarial model construction techniques. Deductible, excess, and franchise. Reinsurance: proportional and non-proportional forms of reinsurance. Estimation of technical provisions in non-life insurance, deterministic and stochastic run-off triangles. The stochastic model of life insurance. Yield curves and their applications in insurance. International Financial Reporting Standard (IFRS) 17 Insurance Contracts and its impact on the actuarial calculations.
Practical work during semester 0% / final exam 100%.
After completing the course, the student will master mathematical modeling techniques in life and non-life insurance. The student will be able to solve typical problems, e.g., risk premium calculation and estimation of the technical provisions. The student will also get acquainted with forms of deductible and reinsurance and actuarial modeling techniques.
Life Insurance Mathematics / Hans U. Gerber. Heidelberg : Springer, 1997, ISBN 978-3-662-03460-6;
Modely v životnom a neživotnom poistení / Rastislav Potocký. Bratislava : Statis, 2012;
Modern Actuarial Risk Theory Using R / Rob Kaas, Marc Goovaerts, Jan Dhaene, Michel Denuit. Second Edition, Heidelberg : Springer-Verlag, 2008;
Aplikovaná poistná štatistika / Viera Pacáková. Bratislava : Iura Edition, 2004;
study materials of the lecturer.
2-EFM-201 Insurance Mathematics
During the semester, the student can get 100% points after writing one test, which is usually written during the last two weeks of the semester.
Grade thresholds: A: at least 90.00%; B: 89.99% – 80.00%; C: 79.99% – 70.00%; D: 69.99% – 60.00%; E: 59.99% – 50.00%; Fx: 49.99% – 0.00%.
Slovak, English
Methods of mathematical modeling in life and non-life insurance, actuarial model construction techniques. Deductible, excess, and franchise. Reinsurance: proportional and non-proportional forms of reinsurance. Estimation of technical provisions in non-life insurance, deterministic run-off triangles: chain-ladder method, separation method and other methods. The stochastic model of the life insurance. Yield curves and their applications in insurance.
Practical work during semester 100% / final exam 0%.
After completing the course, the student will master mathematical modeling techniques in life and non-life insurance. The student will be able to solve typical problems, e.g., risk premium calculation and technical provision estimation. The student will also get acquainted with forms of deductible and reinsurance and actuarial modeling methods.
Course is a combination of explanation of new notions from data modeling in a lecture and of feedback to the student´s work on individual project, where the students subsequently build in new data modeling features from lecture.
Continuous work during the semester is needed - cca 1-2 hours weekly.
Students work with a software tool for data modeling.
Lectures and materials from them.
Basics of set theory: set, subset, element, Cartesian product, relation.
Participation in lectures and individual project
English
Relation to courses about SQL and relational databases:
- Typical courses about relational databases contain usually only few about data modeling and
students are usually not able to create a data model for nontrivial requirements.
- Typical courses about data modeling contain usually only few about relational databases and
students are usually not able to transform a logical data model to a functioning database for nontrivial requirements.
That is why it is necessary for a an efficient use in practice separate courses about data modeling and about relational databases.
This first course "Practical data modeling 1" is devoted primarily to the conceptual data modeling.
The second course "Practical data modeling 2" is devoted primarily to the creation of logical data model, to basic principles of relational databases and
to the transformation of the logical data model to the relational database.
Levels of data models (conceptual/business, logical, physical).
Basic principles of entity-relationship data modeling (conceptual level).
Core data modeling elements and their categorizations: entities, relationships, attributes, value domains, entity identifiers and references.
Instance data diagrams and Baker´s notation for data diagrams.
Introduction to data modeling patterns. Some particular data modeling patterns for conceptual data models.
Basics of data model normalization.
Weight of semester evaluation / final exam: 0/100.
Homework enhanced after feedback from teacher shall be collected to final project presentation.
Final grade is based on quality of homework data model, its presentation, its understanding and answers to additional questions about course topics.
A data model is a base for any IT system. Many system functions share the same data so that
their structure and description is the key for consistent services for the system users. Without a data model
you cannot create nontrivial relational database.
Students gain the basic ability to work with conceptual data models, namely:
- creation of new data models,
- reading and understanding of data models,
- application of some data model patterns during conceptual data modeling.
Mathematical biology : 1. : An introduction / J. D. Murray. New York : Springer, 2002
Mathematical biology : 2. : Spatial models and biomedical applications / J. D. Murray. New York : Springer, 2003
Keener, J., Sneyd, J., Mathematical physiology: I. Cellular physiology, 2nd. ed., Springer, New York, 2008
Wilkinson, D., Stochastic modelling for systems biology, 2nd ed., Chapman & Hall/CRC, Boca Raton, 2012.
Continuous assessment: homework, exam during the semester
Exam: written and oral
Approximate grading scale: A 90%, B 80%, C 70%, D 60%, E 50%
Slovak, English
Biological modeling with ordinary differential equations: the principle of mass balance, mass action rule, scaling and nondimensionalisation, one-component models (Michaelis-Menten kinetics, gene autoregulation), multi-component models (biological switches, oscillators, epidemiology). Modeling with differential equations with delay. Models with spatial component: the reaction-diffusion systems, the spread of epidemics, pattern formation. Stochastic models: probability balance equation, Gillespie simulation algorithm, stochastic models of gene expression.
Passing this subject, students will gain a basic understanding and overview of methods in biological modeling.
Course is a combination of explanation of new notions from data modeling in a lecture and of feedback to the student´s work on individual project,
where the students subsequently build into their project new data modeling features from lecture.
Continuous work during the semester is needed - cca 1-2 hours weekly.
Students work with a software tool for data modeling.
Lecture materials.
Course attendance and individual project
English
Relation to courses about SQL and relational databases:
- Typical courses about relational databases contain usually only few about data modeling and
students are usually not able to create a data model for nontrivial requirements.
- Typical courses about data modeling contain usually only few about relational databases and
students are usually not able to transform a logical data model to a functioning database for nontrivial requirements.
That is why it is necessary for a an efficient use in practice separate courses about data modeling and about relational databases.
First course "Practical data modeling 1" is devoted primarily to the conceptual data modeling.
This second course "Practical data modeling 2" is devoted primarily to the creation of logical data model. Moreover, we explain also basic principles of relational databases needed for the understanding of the transformation of the logical data model to the relational database.
Levels of data models (conceptual/business, logical, physical).
Basic principles of relational databases.
Entity design patterns and their application to cover functional requirements.
Completeness of functional requirements on logical data model.
Transformation of logical data model to relational database and preparation for this transformation in logical data model.
Higher levels of data model normalization.
0/100
Homework enhanced after feedback from teacher shall be collected to final project presentation.
Final grade is based on quality of homework data model, its presentation, its understanding and answers to additional questions about course topics.
A detailed logical data model is necessary to create nontrivial relational database of IT system.
Students gain the basic ability to work with conceptual data models, namely:
- to enhance conceptual data model into a logical data model,
- to correctly transform model to structures in relational database,
- ability to apply explained data model patterns logical data models.
To teach the students to present their work to a professional audience and take part in a professional discussion as speaker as well as participant.
By choice of the thesis supervisor
individual work
Presentation (50%), project elaboration (50%).
Indicative assessment scale: A 90%, B 80%, C 70%, D 60%, E 50%
Students present a 15 minute talk on their thesis in the corresponding state of completion and respond to questions and comments of their classmates. In addition to the presentation, the student has to submit several pages of his/her thesis electronically or in a printed form at its end.
100/0
Students will get acquainted with the formal requirements of writing professional literature and with the methods of presenting results on professional forums.
Mathematical physics: Basic equations and special functions / Vasilij Jakovlevič Arsenin ; in Slovak, translated by Jozef Kačur. Bratislava : Alfa, 1977,
Fourier series and integral transforms / Allan Pinkus, Samy Zafrany. Cambridge : Cambridge University Press, 1997
midterm test,
final exam
Slovak, English
Fourier transform, Laplace transform, Bessel functions, Legendre polynomials and their applications.
33/67
Students will learn methods of integral transforms and basic properties of some special functions from the point of view of applications in solving differential equations.
Intermittent assessment: homework (30%), project (15%), project presentation (5%)
Exam: written (50%)
To successfully complete the course, student has to obtain at least 50% of points on the final exam
Final grade: A 90%, B 80%, C 70%, D 60%, E 50%
Slovak, English
Introduction to networks. Computational complexity. Measures and metrics on networks. Labyrinth exploration. Search for strong digraph components. Optimal paths. Time analysis of projects. The cheapest connection networks. Optimal flows. Project cost analysis. Random network models and their characteristics. Network dynamics.
60/40
Complex systems can often be represented as a network of a number of interacting components. The aim of the course is to get explore examples of complex networks in applications in physics, informatics, biology and social sciences, as well as with algorithmic, but also computational and statistical methods for the analysis of their behavior.
Basic practicing and improvement of locomotion skills in various selected sports: aerobics, basketball, volleyball, football, floorball, hockey, badminton, tennis, table tennis, swimming, body building, trampoline jumping, paddling. Development of a harmonic personality in the sense of “kalokagatia” – to cultivate the students´ mental and physical values.
Slovak, English
Practicing of the students´ game skills in collective sports: basketball, volleyball, football, floorball and hockey. Mastering of the basic technique of a particular sport discipline in other sports. In paddling, basic training on still and slightly flowing water. Development of coordination skills, improvement of articular mobility and cardiovascular system.
100/0
Grades: A 90%, B 80%, C 70%, D 60%, E 50%
The condition for the award of 1 or 2 credits is the completion of a multi-day course in its full scope, or the completion of one-day courses in the scope of 4 days. Candidates can apply to the leaders of individual courses. From the presented offer of courses, you can choose the one that suits your interests, abilities and deadlines.
Slovak
KTVŠ does not rent ski equipment.
The student can sign up for the outdoor sports courses offered by the department: skiing, snowboarding. The lessons in the courses are focused on the development of basic and special movement skills and mastering the techniques needed for the sports.
Acquisition and development of basic motor skills and abilities in selected sports: skiing and snowboarding. Mastering the correct technique of performing individual movements, which are necessary for skiing and snowboarding.
Grades: A 90%, B 80%, C 70%, D 60%, E 50%.
The condition for the award of 1 or 2 credits is the completion of a multi-day course in its full scope, or the completion of one-day courses in the scope of 4 days. Candidates can apply to the leaders of individual courses. From the presented offer of courses, you can choose the one that suits your interests, abilities and deadlines.
Slovak
KTVŠ will provide sports equipment.
The student can sign up for the outdoor sports courses offered by the department: water tourism - river rafting, windsurfing, beach volleyball, hiking and other hobby sports. The lessons in the courses are focused on the development of basic and special movement skills and, mastering the techniques needed for the sports.
Creating a positive and lasting relationship with physical activity. Acquisition and mastery of basic motor skills and abilities in outdoor sports: windsurfing, beach volleyball, water tourism - river rafting, hiking and other sports according to interest. Training and improving the technique needed for the sports.
Improvement training in a selected sport aimed at enhancing locomotion skills in the selected sport. To improve the students´ resistance against various vices, stresses and risk factors.
Slovak, English
Practicing of offensive and defensive game combinations and playing with modified rules in collective sports such as basketball, volleyball, football, floorball, hockey. Command of elements of higher difficulty in locomotion skills (swimming - crawl stroke, breast stroke, butterfly stroke, trampoline jumping and aerobics – practicing of areobics compositions, bodybuilding – development of the main muscle groups, paddling on running water. Testing of the level of physical fitness and coordination skills.
100/0
Russell S., Norwig P. (2010). Artificial Intelligence: A Modern Approach, (3rd ed.), Prentice Hall. Available in faculty library.
Marsland S. (2015). Machine Learning: An Algorithmic Perspective, (2nd ed.), CRC Press.
Passing the online course https://course.elementsofai.com/ (in Enlish or Slovak version).
Slovak or English
The course consists of 20 numerical and 5 text-based tasks. Numerical tasks are checked automatically, text-based tasks are evaluated anonymously by students.
1. What is artificial intelligence: related areas, AI philosophy.
2. Troubleshooting and UI: Browsing and troubleshooting, browsing and games
3. Probability and chance, Bayes' theorem, naive Bayesian classification.
4. Machine learning: nearest neighbor classifier, regression.
5. Neural networks: basics, creation, modern techniques.
6. Consequences: on predicting the future, the effects of AI on society, summary.
The student will get acquainted with selected basic concepts of artificial intelligence and their use in solving various practical tasks.
Russell S., Norwig P. (2010). Artificial Intelligence: A Modern Approach, (3rd ed.), Prentice Hall. Available in faculty library.
Marsland S. (2015). Machine Learning: An Algorithmic Perspective, (2nd ed.), CRC Press.
Passing the online course https://course.elementsofai.com/ (in Enlish or Slovak version).
Slovak or English
The course consists of 20 numerical and 5 text-based tasks. Numerical tasks are checked automatically, text-based tasks are evaluated anonymously by students.
1. What is artificial intelligence: related areas, AI philosophy.
2. Troubleshooting and UI: Browsing and troubleshooting, browsing and games
3. Probability and chance, Bayes' theorem, naive Bayesian classification.
4. Machine learning: nearest neighbor classifier, regression.
5. Neural networks: basics, creation, modern techniques.
6. Consequences: on predicting the future, the effects of AI on society, summary.
The student will get acquainted with selected basic concepts of artificial intelligence and their use in solving various practical tasks.
- Teachers' own electronic study materials published on the course website or in the Moodle system
- Gavora, Peter a kol. 2010. Elektronická učebnica pedagogického výskumu. [online]. Bratislava : Univerzita Komenského, 2010. Dostupné na: http://www.e-metodologia.fedu.uniba.sk/ ISBN 978–80–223–2951–4.
- Tharenou, P., Donohue, R. and Cooper, B., 2007. Management research methods. Cambridge University Press.
- Topping, A., 2015: The Quantitative-Qualitative Continium. In: Gerrish, K. and Lathlean, J., The Research Process in Nursing, p. 159-172
- Williamson, K. and Johanson, G. eds., 2017. Research methods: Information, systems, and contexts. Chandos Publishing.
Continuous assessment: active participation in research in an international student team (25%), presentation of work in a workshop (25%), scientific article (50%)
Indicative evaluation scale: A 90 %, B 80 %, C 70 %, D 60 %, E 50 %
English
- Research methodology
- Design and implementation of a research project in an international group (preferably interdisciplinary)
- Methods and tools for collaboration in virtual space, collaboration in science and practice
- Academic writing, presentation of research results through scientific articles; objectives, content and structure of scientific articles; forms of academic publication, publication forums and evaluation of their quality
- Quality assurance and feedback - peer review
- Communication of results through posters or conference presentations
100/0
Students will learn in the team to agree on a common research topic, formulate research questions, determine research methods for the problem, collect and evaluate data, discuss their findings, present research results to the professional public, analyze and evaluate the scientific work of their colleagues, prepare a scientific article suitable for publication
2/1 (lecture / individual work)
Clark, Timothy R., et al. Business Model Generation. Wiley, 2010
The condition for admission to the exam is active participation in at least 80% of the lessons.
The final assessment consists of a presentation of the semester project. To successfully complete the course, it is necessary to achieve at least 50% of the overall score.
Slovak
1. Commercialization of scientific research.
2. Fundamentals of entrepreneurship and startup terminology.
3. Identification of problems and customer needs analysis (design thinking).
4. Technology transfer. Technology Readiness Levels (TRL).
5. Intellectual property and its protection.
6. Market, customer, and market potential of a technological solution.
7. Business Model Canvas. Revenue models.
8. Sources of financing for technological projects.
9. Pitching and communication of the solution.
10. Fundamentals of management and leadership.
11. Innovation support and incubation structures at national and international levels.
0/100
After completing the course, students can describe the possibilities for commercialization of scientific and technological research. They can identify market needs, assess the market potential of a technological solution, and are familiar with the terminology of entrepreneurship, technology transfer, and intellectual property protection. They understand the overall structure of a business plan and the main forms of financing for technological projects. They are familiar with the principles of communication, teamwork, and team leadership and can apply them appropriately in project work and its presentation.
To improve the students´ physical fitness, health and sport performance through sport.
Slovak, English
To improve offensive and defensive game combinations in collective sports. Practicing of tactical and technical elements in individual sports. Compensatory exercises to correct wrong body posture. Stretching. Competition rules in sport disciplines.
100/0
To strengthen the procedural aspect of university sport, stressing its health and psychological importance. To improve the students´ physical fitness.
Slovak, English
Sport training for Faculty Championships in a selected sport with modified rules. Selection of sport-talented students into teams of the Faculty Sport League, University League of Bratislava Faculties, and participation in sport events of the Faculty and University.
100/0
Dean A, Voss D, Draguljic D: Design and Analysis of Experiments (Springer Texts in Statistics) 2nd ed., Springer 2017; Pázman A, Lacko V: Prednášky z regresných modelov - odhadovanie parametrov strednej hodnoty a štatistická optimalizácia experimentu, Univerzita Komenského 2012; A. Atkinson et al.: Optimum Experimental Design, with SAS. Oxford University Press 2007
Regression models 2-PMS-107
Evaluation based on: project (teaching period), oral exam
Approximate grade thresholds: A 90%, B 80%, C 70%, D 60%, E 50%
Slovak, English
Enrollment limit: 20 students
Basic principles of designing experiments, block designs, factorial designs, response surface designs, optimal experimental designs for linear models (exact and approximate design, optimality criteria, algorithmic computing of optimal designs), optimal design of experiments for nonlinear models.
40/60
The students will learn about basic principles of designing statistical experiments and the optimization of the quality of experiments.
Agresti A: Categorical Data Analysis 3rd ed. W Agresti A: Categorical Data Analysis 3rd ed. Wiley 2012;
Anděl J: Statistické metody. Matfyzpress 2007.
project (teaching period), oral exam
Approximate grade thresholds: A 90%, B 80%, C 70%, D 60%, E 50%
Slovak, English
1. Contingency tables, graphical representation, test of independence, homogeneity test, odds ratio, McNemar’s test, Simpson’s paradox and the Cochran-Mantel-Haenszel test, Bowker’s test, Fisher’s exact test.
2. Logistic regression: interpretation of parameters, probability vs. odds, deviance, tests of submodels, Wald tests and confidence intervals, graphical representation, pseudo coefficients of determination, logistic regression as a classifier.
3. Permutation versions of the t-test and ANOVA.
4. Bootstrap: estimation of variation, confidence intervals, application to regression, a demonstration of misuse.
http://www.iam.fmph.uniba.sk/ospm/Somorcik/teaching.htm
70/30
Students will be able to perform some rather advanced statistical analyses, using the software R.
Izenman A: Modern Multivariate Statistical Techniques: Regression, Classification, and Manifold Learning (Springer Texts in Statistics) 1st ed., 2nd printing 2013; James G, Witten D, Hastie T, Tibshirani R: An Introduction to Statistical Learning: with Applications in R (Springer Texts in Statistics) 2nd ed., Springer 2021; Harman R: Multivariate Statistical Analysis (Selected Lecture Notes), study materials of the lecturer, 2021.
Evaluation based on: project (teaching period), oral exam
Approximate grade thresholds: A 90%, B 80%, C 70%, D 60%, E 50%
Slovak, English
The knowledge of the software R is recommended. Enrollment limit: 40 students
Selected multivariate data visualization methods, partitional clustering (k-means, k-medoids, DBSCAN, OPTICS, clustering based on the mixture of Gaussian distributions, spectral clustering), hierarchical clustering, general introduction to the statistical classification methods, Bayes classifier, k nearest neighbors, linear and quadratic discrimination, classification trees and forests, bagging and boosting, support vector machines, multinomial regression as a classification method
50/50
The students understand the principles and practical realization of selected methods of cluster analysis and statistical data classification.
To give an introduction to discrete time Markov chains
Kalas, J: Markovove reťazce, skriptá MFF UK
Norris, J.R.: Markov chains (1998)
Ross, S.M.: Introduction to probability models (2006)
test
Preliminary semester evaluation: test(20%) and homeworks(50%)
Examination: written examination(30%)
Approximate grade thresholds: A 90%, B 80%, C 70%, D 60%, E 50%
Markov property, transition probabilities, transition matrix, Chapman Kolmogorov equation, irreducibility of a chain. Classification of states, recurrent states, transient states, null recurrent states and positive recurrent states, periodicity. Existence of stationary distribution, ergodic distribution, necessary and sufficient conditions for ergodicity. Random walks, branching processes, absorbtion probabilities, mean time to absorbtion. Markov reward chains algorithms and Markov Chain Monte Carlo.
After completing the course the student will master elementary discrete time Markov chains models. He will be able to classify states of a Markov chain and calculate stationary probability distributions.
exam
Ross S: Simulation, Elsevier Academic Press 2006; Study materials of the lecturer.
Evaluation based on: project (teaching period), oral exam
Approximate grade thresholds: A 90%, B 80%, C 70%, D 60%, E 50%
Slovak, English
Enrollment limit: 40 students
Generating random numbers, testing of random number generators, Sampling discrete random variables and vectors, Sampling continuous random variables and vectors, Statistical analysis of simulated data, Classical Monte Carlo methods.
50/50
The students understand the basic methods of computer sampling of random variables and vectors, apply them to the computation of Monte Carlo estimates and use them for the simulation of complex stochastic systems.
Luke S: Essentials of Metaheuristics, Lulu, 2013; Study materials of the lecturer.
Evaluation based on: project (teaching period)
Approximate grade thresholds: A 90%, B 80%, C 70%, D 60%, E 50%
Slovak, English
A brief general overview of optimization. Various types of optimization problems and optimization algorithms. Construction of the initial solution. The basic heuristic optimization algorithms. Covariance matrix adaptation. Simulated annealing. Algorithm Nelder-Mead. Genetic algorithms. Differential evolution. Particle swarm optimization. Basic principles of constrained optimization.
100/0
The students understand the algorithmic principles of a variety of optimization methods, mostly heuristics with stochastic elements, applicable to the problems of discrete optimization as well as non-convex continuous optimization.
Penze: kvantitativní přístup / Tomáš Cipra, Praha : Ekopress, 2012;
Pensions at a Glance 2019 / OECD and G20 Indicators, Paris : OECD Publishing, 2019; https://doi.org/10.1787/b6d3dcfc-en;
An Introduction to Pension Schemes / E. M. Lee, London : Institute and Faculty of Actuaries, 1986;
study materials of lecturers.
During the semester, the student can get 50% of the assessment for one presentation. The final oral exam has a weight of 50%.
Grade thresholds: A: at least 90.00%; B: 89.99% – 80.00%; C: 79.99% – 70.00%; D: 69.99% – 60.00%; E: 59.99% – 50.00%; Fx: 49.99% – 0.00%.
Slovak, English
Three pillars of pension system: compulsory, supplementary, personal. Pension system in Slovakia. The pay-as-you-go system, funded pension scheme. Defined benefit (DB) plans and defined contribution (DC) plans. Benefits not depending on previous earnings, depending on average salary or last salaries. One-time lump sum death benefit. Transfers. Funding plans. Pension funds investment.
Practical work during semester 50% / final exam 50%.
After completing the course, the student will know the basic principles and functions of pension schemes. The student will get acquainted with the pension system of the Slovak Republic and old-age pension schemes of some other countries of the world. The student will be able to valuate assets and liabilities of various types of pension systems, such as defined benefit (DB) schemes and defined contribution (DC) schemes.
Izenman A: Modern Multivariate Statistical Techniques: Regression, Classification, and Manifold Learning (Springer Texts in Statistics) 1st ed., 2nd printing 2013;
James G, Witten D, Hastie T, Tibshirani R: An Introduction to Statistical Learning: with Applications in R (Springer Texts in Statistics) 2nd ed., Springer, 2021;
Hastie T, Tibshirani R, Friedman J: The Elements of Statistical Learning, 2nd ed., Springer, 2016
Evaluation based on: project (teaching period), oral exam
Approximate grade thresholds: A 90%, B 80%, C 70%, D 60%, E 50%
Slovak, English
An electronic version of "An Introduction to Statistical Learning" by James et al. is available at: https://www.statlearning.com/
1. Linear methods of feature extraction – principal component analysis, factor analysis, projection pursuit.
2. Nonlinear methods of feature extraction – nonlinear principal component analysis, metric and nonmetric multidimensional scaling, isomap, t-SNE.
3. Feature selection methods – forward and backward selection, lasso, ridge regression.
4. Subsampling methods – random subsample, leveraging methods.
http://www.iam.fmph.uniba.sk/ospm/Rosa/teaching.htm
50/50
The students will familiarize themselves with the methods of feature extraction and selection and with subsample selection. They will also acquire the ability to apply these methods in practical data analysis.
Rublík F: Neparametrické metódy. Veda 2011;
Lehmann E: Nonparametrics: Statistical Methods Based on Ranks (revised edition). Springer 2006;
Hollander M, Wolfe D A, Chicken E: Nonparametric statistical methods 3rd ed. Wiley 2013;
written exam
Approximate grade thresholds: A 90%, B 80%, C 70%, D 60%, E 50%
Slovak, English
Sign test and a confidence interval for the true median. Wilcoxon signed rank test, Hodges-Lehmann estimator of location and the corresponding confidence interval. Sign test and Wilcoxon test for paired data. Wilcoxon rank sum test and Mann-Whitney test. Hodges-Lehmann estimator of shift and the corresponding confidence interval. Problem of ties. Kruskal-Wallis test and some post-hoc tests. Spearmans’s rho, Kendall’s tau. Theil’s tests, estimators and confidence intervals concerning parameters of simple linear regression. Estimator's robustness to outliers (breakdown point). Kolmogorov-Smirnov tests. Cramér-von Mises test. Multimariate tests: component-wise sign test, Rayleigh test, Randles test by means of interdirections. Availability of the discussed methods in the software R.
http://www.iam.fmph.uniba.sk/ospm/Somorcik/teaching.htm
0/100
Students are able to apply basic nonparametric methods to real data. They also understand the principles and the mathematical background of these methods.
O'Quigley J., Iasonos A., Bornkamp B. (Eds.): Handbook of methods for designing, monitoring, and analyzing dose-finding trials. CRC Press (2017); Filová L: Statistical methods in clinical trials, lecture notes (2021)
Evaluation based on: project (teaching period)
Approximate grade thresholds: A 90%, B 80%, C 70%, D 60%, E 50%
Slovak, English
Enrollment limit: 20 students
Phases and aims of clinical trials, blinding, randomization, maximum tolerated dose, minimum effective dose. Designs in phase 1 of clinical trials: 3+3 design, group design for determining toxicity, continual reassessment method. Designs for phase 2 of clinical trials: two-stage designs, dose-response modeling, estimating target doses, multiple comparisons method. Designs for phase 3 of clinical trials: group sequential methods, adaptive designs. Determining the optimal sample size. Late onset toxicity trials.
http://www.iam.fmph.uniba.sk/ospm/Filova/teaching.htm
100/0
The students will gain knowledge about statistical methods used in the design of clinical trials.