• 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.

1. Skřivánková V.-Skřivánek J.: Kvantitatívne metódy finančných operácií, IURA Edition, Bratislava, 2006.

2. Elliott R.J.-Kopp P.E.: Mathematics of Financial Markets, Springer, New York, 2005.

3. Janssen at al.: Mathematical Finance, ISTE / Wiley, 2009.

4. Ross S.M.: Mathematical Finance, Cambridge University Press, 2011.

5. Sharpe W.F.- Alexander G.J.: Investments, Prentice-Hall, New Jersey,1994.

6. Shreve S.E.: Stochastic Calculus for Finance, Springer, 2004.

Written tests during the semester. Final evaluation is based on written tests and oral exam.

Slovak

Financial markets, institutions and instruments. Stochastic methods of valuation of financial products. Risk and return, analysis of portfolio of securities. Characteristics of portfolio, mean and variance, measures of dependencies. Admissible, efficient and optimal portfolio. Indiference curves, utility functions. Financial market models. Markowitz´s mean-variance model and its modifications, model of capital market line (CML). Sharpe´s model and its modifications. Capital assets pricing model (CAPM), security market line model (SML). Decomposition of total risk, market risk and specific risk. Diversification of portfolio. Measurement of performance. Investment and financial decisions. Financial derivatives, their classification and pricing. Financial time series and their decomposition. Analytical and adaptive methods of smoothing. Financial forecasting. Hypothesis of randomness.

To provide stochastic methods for investments, financial market analysis and financial forecasting.

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.

1. E. A. Robinson, D. H. Ullmann: A mathematical look at politics, CRC Press, 2010

2. J. Rothe (ed.): Economics and Computation, Springer, 2016

3. F. Brandt, V. Conitzer, U. Endriss, J. Lang, A.D. Procaccia: Handbook of computational social choice, Cambridge University Press, 2016

4. F. Pukelsheim, Proportional Representation, Springer, 2013

5. Donald G. Saari, Geometry of Voting, 1994

6. Sherif El-Helaly: TheMathematics of Voting and Apportionment, Birkhäuser, 2019

7. G. G. Szpiro, Numbers rule, Princeton University Press, 2010

Final evaluation is given by continuous assessment, project and oral exam.

Slovak

1. Historical examples, examples of paradoxes: Condorcet, Borda. Basic formal definitions.

2. Elections with two candidates. May's theorem.

3. Voting systems: plurality, approval, Borda, scoring systems. Properties of scoring systems. Saari' s theorems and Fishburn's theorem.

4.Other properties of voting systems: monotonicity, Paroto optimality, independence. Impossibility theorems.

5. Condorcet, Copeland, Llul, Copelandov-alfa, Dodgso systems. Computational complexity of winner determination.

6. Election manipulation. Gibbard-Sattertwaite theorem.

7. Complexity of manipulation.

8. Other models: incomplete information, possible and necessary winner.

9. Proportional reprezentation. Paradoxes. Divisor and quota methods. Balinski-Young theorem.

10. Extension of proportional representation: combining proportionality with personal elections, voting districts.

11. Electoral college

12. Students' presentations.

To know various voting systems, their properties, modelling and analysis using quantitative methods. Students are able to construct counterexamples, paradoxes, verify properties, design algorithms and understand principles of computational complexity analysis. Historical and political context is stressed.

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.