• Rao: Linear statistical inference and its applications, Wiley, 1973

• Seber: Linear regression analysis, Wiley, 1977

• Searle: Linear models, Wiley, 1997

• Sen, Srivastava: Regression analysis (Theory, Methods, and Applications), Springer, 1990

• Christensen: Plane answers to complex questions (The Theory of Linear Models), Springer, 1987

Given at the basis of statistical processing of real data. Final evaluation is given at the basis of partial examination, computing part, and oral part of the exam.

Slovak

o Matrices and linear spaces, g-inversions, projections

o Normal distribution and related distributions

o Hotelling's test

o Probability foundations of regression and correlation

o General linear model with full rank

o Model with incomplete rank

o Submodels testing

o Regression analysis, basic models

o Assesing the quality of a model

o Analysis of variance

o One-way ANOVA, multiple comparison procedures, problem of heteroskedasticity

o Balanced factorial models, hierarchical models

o Analysis of covariance

o Statistical software for linear modeling

Learning most frequently applied statistical methods.

BILASOVÁ, V. – ANDREANSKÝ, E.: Epistemológia a metodológia vedy. Prešov: FF PU 2007.

FAJKUS, B.: Filosofie a metodologie vědy. Praha: Academia 2005.

BEDNÁRIKOVÁ, M. Úvod do metodológie vied. Trnavská univerzita: Trnava 2013.

DÉMUTH, A. Filozofické aspekty dejín vedy. Trnavská univerzita: Trnava 2013.

FEYERABEND, P.: Proti metodě. Prel. J. Fiala. Praha: Aurora 2001.

KUHN, T. S.: Štruktúra vedeckých revolúcií. Prel. Ľ. Valentová. Bratislava 1982.

Attendance: A student may have one unexcused absence in seminar at the most. Absence in more than one seminar must be reasoned and substituted by consultations. Conditions of continuous and final control: during the semester a student is continuously checked and assessed according to his/her activity. To be awarded the credits, a student must pass a test from knowledge obtained in the lectures and seminars. Results of the test will make up the final grade.

Slovak

• Falsificationism and critical realism by K. R. Popper.

• Development and critique of the Popper’s concept.

• Understanding the science development in the work by T. S. Kuhn.

• Methodology of scientific research programmes of I. Lakatos.

• Methodological anarchism of P. Feyerabend.

• W.V.O. Quine – the issue of relation between theory and empiricism.

The course is aimed at getting familiar with the basic issues of methodology and philosophy of science. Significant part will be devoted to presenting the main concepts of the philosophy of science in the 20th century and this aim will be achieved by reading the source and interpretive texts.

I. Anderson, I. Honkala: A short course in combinatorial designs, http://www.utu.fi/~honkala/cover.html

D.R. Stinson: Combinatorial Designs: Constructions and Analysis, Springer 2004

W.D. Wallis: Combinatorial designs, Marcel Dekker 1988

To complete the course, it is necessary to demonstrate the ability to formulate definitions and theorems from the lectured material, to present the proofs of theorems and to solve selected problems based on theory of combinatorial designs.

The exam itakes written form by elaborating a test containing three questions of a theoretical nature and two questions of a practical / computational nature; the maximum number of points that can be obtained for answering each question is 20. To pass the exam, it is necessary to obtain more than half of the maximum number of 100 points (otherwise the exam is evaluated by FX), while the rating E is given in the case of point range 51-59, D in case of 60-69, C in case of 70-79, B in case of 80-89 and A in case of more than 90 points.

Slovak or English

Basic knowledge of arithmetic, linear algebra and graph theory is required, as well as basic knowledge of working with the Maple computer algebra system.

Week 1: Motivational problems using combinatorial designs, definition and basic properties of BIBDs.

Week 2: Incidence matrix of a design and its properties.

Week 3: Constructions of BIBDs.

Week 4: Symmetrical BIBDs, derived and residual design.

Week 5: Order of symmetric BIBD, its relation to the number of points.

Week 6: Finite projective planes.

Week 7: Hadamard designs and Hadamard matrices.

Week 8: Conditions for the existence of symmetric BIBDs, Lagrange's four-square theorem.

Week 9: Bruck-Ryser-Chowla theorem and its consequences.

Week 10 and 11: Steiner triple systems and their constructions.

Week 12: Orthogonal Latin squares, orthogonal arrays.

Week 13: Strongly regular graphs.

Week 14: Selected applications of combinatorial designs (group testing of samples, sharing of secret information).

After completing the course, the student is acquainted with the basics of the theory of combinatorial designs, its applications in the natural sciences and relations to other parts of mathematics.

M. Hamala, M. Trnovská: Nelineárne programovanie, Epos, 2012

M.S. Bazaraa, H.D. Sherali, C.M. Shetty: Nonlinear Programming: Theory and Algorithms, 3rd edition, Wiley-Interscience, 2006

To complete the course, it is necessary to demonstrate the ability to formulate definitions and theorems from the lectured material, to demonstrate the proofs of theorems and to solve selected problems of convex resp. nonlinear programming.

The overall evaluation of the course is awarded on the basis of semester evaluation (which includes the completion of two semester tests focusing on problem solving; for each of them, a maximum of 25 points can be obtained) and the results of an oral exam (consisting of three questions of a theoretical nature, with a total of 50 points). To pass the exam, it is necessary to obtain

more than half of the maximum number of 100 points (otherwise the test is evaluated by FX), whereby

evaluation E is given in case of point gain 51-59, D in case of 60-69, C in case of 70-79, B

in the case of 80-89 and A in the case of more than 90 points.

Slovak or English

Knowledge of the basics of differential calculus of functions of one and more variables, linear algebra and linear programming (simplex method) is required.

Week 1: Practical problems leading to nonlinear programs.

Week 2 - 3: Convex sets and their properties.

Week 4 - 6: Convex functions – properties and criteria of convexity.

Week 7 - 8: Necessary and sufficient conditions of optimality. Karush-Kuhn-Tucker conditions.

Week 9 - 10: Quadratic programming. Duality in nonlinear programming.

After completing the course, the student is acquainted with the basics of nonlinear programming from both theoretical point of view (the topics include properties of convex sets, properties of convex functions, optimality conditions for nonlinear problems, Karush-Kuhn-Tucker theory, quadratic programming), as well as from practical one (illustrations of real problems with underlying models that use nonlinear programming, and methods of their solution using computer algebra systems and computer technology).

Familiarize students with the structure, properties and functions of information macromolecules and their work, focusing primarily on the molecular mechanisms of regulation of DNA replication, gene expression and cell cycle.

E. Mišúrová:Molekulárna biológia. Učebné texty, PF UPJŠ Košice, 1999

E. Mišúrová, P. Solár: Molekulová biológia. Učebné texty, PF UPJŠ, 2007

S.Rosypal:Úvod do molekulární biologie. Grafex Blansko, Brno,1999

D.P. Clark: Molecular Biology, Elsevier Academic Press, London, 2005

D.P. Clark, N.Pazdernik, M. McGehee: Molecular Biology, 3rd Edition, Elsevier 2018

1. Structure and properties of information biomacromolecules.

2. Chromatine molecular structure and dynamics and oragnization of chromosome.

3. Replication of chromosomal and extrachromosomal DNAs.

4. Mutations and DNA reapir.

5. Prokaryotic and eukaryotic genome. Human genome.

6. Mobile gene elements.

7. Transcription and postranscription processing of RNA.

8. Translation and posttraslational modification of proteins. Protein degradation.

9. Interaction of proteins with DNA. Regulation of gene expression in prokaryots.

10. Regulation of gene expression in eukaryots.

11. Cell signaling.

12. Cell cycle and cell cycle control.

Familiarize students with the structure, properties and functions of information macromolecules and their work, focusing primarily on the molecular mechanisms of regulation of DNA replication, gene expression and cell cycle.

Grabisch, M.: Set Functions, Games and Capacities in Decision Making. Theory and Decision Library C~46, Springer International Publishing Switzerland, 2016.

Greco, S., Ehrgott, M., Figueira, J.-R.: Multiple Criteria Decision Analysis: State of the Art Surveys, International Series in Operations Research & Management Science, 233, Springer; 2nd ed. 2016.

The evaluation is given on the basis of elaboration of individual projects related to the topics covered within the subject.

Slovak

Weeks 1 -2: Basics of decision theory, games against nature: examples of practical problems, decision criteria.

Week 3: Decision trees and their applications.

Weeks 4 - 8: Multicriterial optimization: examples of practical problems, methods for evaluating the importance of variant assessment criteria, methods for determining compromise and optimal variants.

Week 9: Utility function, decision-making under risk and uncertainty.

Weeks 10 - 11: Non-additive measures, their interpretation and the Choquet integral in the decision-making process, other selected non-additive methods.

Week 12: CEU, RDEU method and the Choquet-Stieltjes integral, paradoxes (Allais and Ellsberg paradox).

After completing the course, the student has the knowledge on basics of decision theory (games against nature), selected methods of solving multicriterial optimization problems together with advanced topics in decision-making theory (use of non-additive measures and generalized integrals).

Become acquainted with basic knowledge about the structure and evolution of the universe.

1. Carroll, B. W., Ostlie, D. A., An Introduction to Modern Astrophysics, Addison-Wesley Publishing Company, Reading, Massachusetts, 1996;

2. Contopoulos, D. Kotsakis, Cosmology, the structure and evolution of the Universe, Springer, 1984;

3. Pasachoff, J.M., Filippenko, A., The Cosmos: Astronomy in the New Millennium, Cambridge University Press, 2013;

4. Vanýsek, V., Základy astronomie a astrofyziky, Academia, Praha,1980;

5. Čeman, R., Pittich, E., Vesmír 1 - Slnečná sústava, MAPA Slovakia, Bratislava, 2002;

6. Čeman, R., Pittich, E., Vesmír 2 - Hviezdy - Galaxie, MAPA Slovakia, Bratislava, 2003;

Basic knowledge of physics.

1. Seminar essay. Send the title of the selected topic to the lecturer no later than the end of the semester.

To successfully complete the course, the student must demonstrate sufficient understanding of the basic knowledge of the structure and evolution of the universe. Knowledge of the basic properties of stars and methods of their determination, the structure, evolution and energy sources of stars, the structure of matter in the universe and its evolution is required. The condition for obtaining credits is passing a written or oral exam, preparation, and presentation of a semester essay. The credit evaluation of the course considers the following student workload: direct teaching (1 credit), self-study (2 credit) and assessment (1 credits). The minimum threshold for completing the course is to obtain at least 50% of the total score, using the following rating scale: A (90-100%), B (80-89%), C (70-79%), D (60- 69%), E (50-59%), Fx (0-49%).

Slovak, English

Structure of matter in the universe

Basic properties of stars and methods of their determination: Masses, dimensions and distances of stars. Radiation flux, apparent and absolute magnitude. The temperature of stars, black body radiation, spectra of atoms and molecules, non-thermal radiation. Spectral classification, luminosity classes, HR diagram. (1st - 3rd week)

The structure of stars: Basic equations of stellar structure, production and transfer of energy in stars, fusion reactions. (4th week)

The evolution of stars: The origin of stars, interstellar matter, Jeans’ criterion, protostars, main sequence stars, giants and final stages of stellar evolution - white dwarfs, neutron stars and black holes. Planetary systems. (4th - 6th week)

Structure of matter in the universe: Galaxy, its structure, dynamics and evolution. Types of galaxies, quasars, intergalactic matter. Local Group of galaxies, clusters and super-clusters of galaxies. The large-scale structure of the universe. Dark matter and dark energy. (7th week)

Evolution of the universe

Introduction to Cosmology: Historical development of views on the universe, Olbers' paradox, gravitational paradox. Cosmological principle. Isotropy and homogeneity of the universe, cosmic background radiation, expansion of the universe. Steady-state theory. (8th - 9th week)

Relativistic Cosmology: Cosmological solutions of Einstein's equations, models of the universe and their properties. Theory of the expanding universe, the Big Bang, the age of the universe. (10th - 11th week)

The origin of the universe: The initial stages of the expansion of the universe, inflationary expansion and nucleogenesis. The formation of galaxies and galaxy clusters. The arrow of time, the future of the universe. Anthropic principle. (12th - 13th week)

1. Basic properties of stars and methods of their determination: radiation flux, apparent and absolute magnitude, distances of stars, colors of stars.

2. Temperature of stars, black body radiation, spectra of atoms and molecules, non-thermal radiation.

3. Spectral classifications, luminosity classes, HR diagram, masses of stars.

4. Structure of stars: basic equations of stellar structure, transfer of energy by radiation and convection, production of energy in stars, fusion reactions.

5. Evolution of stars: interstellar matter and formation of stars and stellar systems, Jeans’ criterion, protostars.

6. Evolution of stars: main sequence stars, giants, final stages of star evolution - white dwarfs, neutron stars and black holes.

7. Distribution of matter in the universe: Milky Way, its structure, dynamics, and evolution, types of galaxies, quasars, intergalactic matter, local group of galaxies.

8. Clusters and super-clusters of galaxies, large-scale structure of the universe, dark matter, and dark energy.

9. Evolution of the universe: historical development of views on the universe, Olberson's paradox, gravitational paradox, Cosmological principle.

10. Isotropicity and homogeneity of the universe, relic radiation, expansion of the universe. Steady state theory.

11. Relativistic cosmology: cosmological solutions of Einstein's equations, models of the universe and their properties, theory of the expanding universe, the Big Bang, the age of the universe.

12. Origin of the universe: the initial stages of the expansion of the universe, inflationary expansion and nucleogenesis, the formation of galaxies and galaxy clusters.

After completing the lectures, the student will master the basic knowledge about the properties of stars and methods of their determination, structure, evolution and energy sources of stars, the structure of matter in the universe and its evolution. It will also have sufficient physical knowledge and mathematical apparatus to enable independent solving of a wide range of tasks related to space research.

Written or oral exam within the curriculum of the course.

Berlinghoff, W.P., Gouvea, F.Q.: Math through the Ages, MAA Press, 2015.

Čižmár, J . Dejiny matematiky (Od najstarších čias po takmer súčasnosť) Perfekt, 2017.

Hairer, E., Wanner, G.: Analysis by its History, Springer, 2008.

Mareš , M . Příběhy matematiky. Pistorius, 2011.

Conditions for continuous evaluation:

1. Participation in teaching in accordance with the study rules and instructions of the teacher.

2. Activity.

3. Homeworks.

4. Seminar work on the selected topic and its presentation at the seminar

Conditions for successful completion of the course:

1. Participation in teaching in accordance with the study regulations and according to the instructions of the teacher;

2. Credits will be awarded to students who score at least 50% on homework assignments and tests. Additional points can be achieved for the presentation of a seminar paper.

Slovak

1. Algebra and geometry of 16th and 17th century - Tartaglia, Vieta, Descartes

2. Beginning of modern number theory - Mersenne, Fermat

3. Development of infinitesimals -- Newton, Leibniz, Bernoulliovci

4. Complex and hypercomplex numbers -- Hamilton, Cayley, Clifford

5. Combinatory and probability - Pascal, Fermat

6. Algebra in the 18th and 19th century - Gauss, Abel, Galois

7. Non-Euclidean geometries - Gauss, Lobačevskij, Bolyai

8. Mathematical analysis in the 19th century - Cauchy, Bolzano, Weierstrass

9. Set theory - Bolzano, Cantor, Zermelo, Franklin

10. Mathematics in the beginning of 20th century - Peano, Hilbert, Gödel

Students will demonstrate an understanding of the history of the development of some mathematical disciplines and selected concepts. They will demonstrate this understanding by scoring at least 50% on previous topics and homework assignments.

J. Hromkovič (2011). Algorithmics for hard problems : Introduction to combinatorial optimization, randomization, approximation, and heuristics. Berlin - Springer.

J. C. Spall (2003). Introduction to stochastic search and optimization: estimation, simulation, and control. Wiley.

M. J. Kochenderfer and T. A. Wheeler (2019). Algorithms for Optimization. The MIT Press.

P. D. Janos (2010). Global Optimization: Scientific And Engineering Case Studies. Springer Nature.

J. J. Schneider and S. Kirkpatrick (2006). Stochastic Optimization. Springer-Verlag.

J. A. Momoh (2016). Adaptive Stochastic Optimization Techniques with Applications. CRC Press.

Josef Tvrdík: Stochastické algoritmy pro globální optimalizaci, 2010, Ostravská univerzita v Ostravě.

Interim assessment (50% of the total assessment): Solving assigned tasks e.g. in the form of implementation of algorithms or their parts, modification of existing codes or use of available packages in solving real problems. Final examination (50% of the total assessment): It consists of verifying the understanding of the theory taken over and demonstrating the practical skills acquired.

Slovak

Stochastic versions of gradient descent and hill climbing algorithm. Method of differential evolution, parallel tempering and simulated annealing, quantum annealing, cross entropy. Nelder-Mead algorithm, Luus-Jaakola algorithm, stochastic tunneling method, random search algorithm – Scenario optimization. EM algorithm. Bayesian optimization. Optimization based on collective behavior (e.g. particle swarm optimization algorithms – PSO, ACO, GSO, GBA). Methods of stochastic approximation - principle of simultaneous pertubation (disorders), Robbins-Monro and Kiefer-Wolfowitz algrithms. Pattern search (optimization).

After completing the course, student will gain a comprehensive overview of various types of optimization problems, gain theoretical knowledge and practical skills regarding the principles and implementation of algorithms for solving optimization problems with emphasis on heuristic methods of global optimization using random elements. The student should be able to identify specific types of optimization problems in a real context, then he should be able to choose the appropriate method for solving the problem, implement the necessary algorithm in the programming language and / or use the available efficient computational packages.