Download Offers of classes for Erasmus students from the Faculty of

Offers of classes for Erasmus students from the Faculty of Mathematics and Applied
Physics:
Dept of Mathematics
1) Linear algebra (a course for undergraduate students)
Requirements: Basic knowledge of mathematics at a secondary school level. Ability to use
basic mathematical tools at a high school level.
ECTS: 7
Semester: winter and summer
The syllabus of the module:
1. Complex numbers: the algebraic form and the trigonometric form of a complex number, the
conjugate complex number, the absolute value of a complex number, extraction of roots.
2. Matrices and determinants: operations on matrices, properties of determinants, the inverse
matrix, the rank of a matrix.
3. Systems of linear equations: Cramer's Theorem, Theorem of Kronecker-Capelli, the
elimination method of Gauss.
4. Basic algebraic structures: a group, a ring, a field.
5. Linear spaces: a linear combination of vectors, linear dependence and independance of
vectors, a basis and the dimension of a linear space, the subspace of a space.
2) Mathematical analysis (a course for undergraduate students)
Requirements: Basic knowledge of mathematics at a secondary school level. Ability to use
basic mathematical tools at a high school level.
ECTS: 7
Semester: winter and summer
The syllabus of the module:
1. Functions: theimage and the preimage of a set by a function, the inverse function, the
injection,the surjection and the bijection. Cyclometric functions.The composition of functions.
2. Sequences: the limit of a sequence of real numbers, basic properties of limits of sequences.
Euler’s number.
3. Limits of real functions. Continuous functions.
4. The derivative of a function at a point. Monotonicity and the extremum of functions.
Derivatives of higher orders.Convexity, concavity and inflection points of functions.
Asymptotes of a function.
5. Indefinite integrals. Methods of calculation of indefinite integrals. Calculating of integrals
of fundamental classes of functions.
3) Discrete Mathematics (a course for undergraduate students)
Requirements: Basic knowledge of mathematics.
ECTS: 3
Semester: winter and summer
The syllabus of the module:
1.Basic terms and concepts of combinatorics: counting methods, partitions, decompositions.
2.Recurrence relations and generating functions.
3.Graphs: paths and cycles, trees, graphs products
4.Matchings and independent sets.
5.Planar graphs
6.Coloring in graphs
7. Methods of combinatorial optimization.
4) Decisions and Interaction – Elements of Game Theory (a course for undergraduate
students)
Requirements: familiarity with the mathematical notation and basic calculus, an elementary
understanding of probability.
ECTS: 5
Semester: winter and summer
The syllabus of the module:
1. Simple decision models: optimization, making decisions, rational and optimal behaviour.
2. Simple decision processes: decision trees, strategic behaviour, randomising and optimal
strategies.
3. Static games: interactive decision problems, describing and solving games, dominance,
Nash equilibria, classification of games, games with n-players.
4. Finite dynamic games: game trees, Nash equilibria, information sets, behavioural strategies.
5. Games with infinite (and continuous) strategy sets: the Cournot and the Stackelberg
duopoly models.
6. Infinite dynamic games: repeated games, subgame perfection, folk theorems.
5) Mathematical Analysis III (a course for undergraduate students)
Requirements: Mathematical Analysis I and II
ECTS: 5
Semester: winter and summer
The syllabus of the module:
1.Series of real and complex numbers.
2.Functional sequences and series.
3.Calculus of functions of several variables-selected topics.
6) Probability and Statistics (a course for undergraduate students)
Requirements: Completed the course of mathematics of the first and second semester of the
first degree studies. Ability to use basic mathematical apparatus for the analysis and
probability.
ECTS: 3
Semester: winter
The syllabus of the module:
1. Event and probability. Various types of probability. Independent events. Conditional
probability. Bayes formula. Bernoulli trials.
2. Random variables and their distributions. Distribution function. Discrete and continuous
random varibles. Function of random variables.
3. Parameters of random variables. Expectation. Deviation and standard deviation.
4. Two-dimensional random variable. Marginal distributions. Independence of random
variables. Covariation, correlation. Conditional distributions. Regression.
5. Limit theorems. Sequences of random variables. Types of convergence of sequences of
random variables. Laws of large numbers.
6. Elements of mathematical statistics. Descriptive statistics. Estimation. Types of estimators.
Various types of distribution which appear in statistics. Statistical hypotheses and their
verification.
Dept of Physics
1) Physics (a course of general physics at the undergraduate level)
Requirements: Knowledge of mathematics at the high school level and the grounds of
calculus.
ECTS: 4
Semester: winter and summer
The syllabus of the module:
1. Electrostatics. Electric charge. Coulomb law. Electric field. Gauss’ law. Electric potential.
Electric current. Current density. Resistance. Ohm’s law. Kirchhoff’s laws.
2. Fundamentals of the magnetism theory. Magnetic interactions. Magnetic field and
induction. Lorentz force. Force acting on a wire with electric current. Ampere’s law.
Faraday’s induction law. Lenz principle. Magnetic induction. Maxwell’s equations. Electric
current oscillations and electromagnetic waves.
3. Quantum mechanics. Photoelectric effect. Origin of light. Wave character of matter.
Schroedinger’s equation. Uncertainty principle of Heisenberg. Bohr’s model of the hydrogen
atom. Properties of atoms. X-ray radiation.
4. Nuclear physics. Experiment of Rutherford. Table of nuclides. Nuclear reactions. Nuclear
fusion.
5. Conductivity of solids. Electric properties of solids. Energy levels in crystals.
Semiconductors.
2) Solid State Physics (a general course for 3rd year students specializing in experimental or
theoretical physics)
Requirements: Knowledge of physics at the level of the course textbook Fundamentals of
Physics Vol 1 - 5 by Halliday, Resnick and Walker, differential and integral calculus.
Understanding the material tensors.
ECTS: 7
Semester: winter and summer
The syllabus of the module:
1. Crystal lattice. Types of lattices. Simple crystalline structures.
2. X-ray diffraction in crystals. Bragg’s law. Laue diffraction equations. Inverse lattice.
Geometrical structure factor. Atomic scattering factor.
3. Crystal binding and the lattice energy. Ionic, covalent and metallic crystals.
4. Lattice oscillations and phonons. Quantization of lattice oscillations. Crystal lattice with
two atoms per cell.
5. Thermal properties of insulators. Specific heat of lattice. Anharmonic oscillations. Thermal
conductivity.
6. Fermi gas of electrons. Energy levels and density of states in a one-dimensional model.
Electron gas in three-dimensional model.
7. Theory of conductivity in metals. Electric resistance of metals. Hall effect.
Magnetoresistance.
8. Electronic band structure. Particle in a periodic potential. Stationary states. Bloch function.
Kroenig-Penny model. Fermi surface. Effective mass of electron.
9. Semiconductor crystals. Intrinsic conductivity. Energy gap. Conduction of doped
semiconductors. Relaxation time and recombination. Thermoelectric effect.
10. Basic properties of superconductors.
11. Main properties of insulators.
12. Ferroelectric crystals. Phonon soft modes.
13. Diamagnetism and paramagnetism. Theory of Langevin. Curie’s law.
14. Ferromagnetism and antiferromagnetism. Magnetic ordering. Magnetic domains.
15. Optical properties of insulators. Excitons. Photoconductivity. Luminescence.
16. Dislocations in crystals.
3) Condensed Matter Physics (a course for graduate students specializing in experimental or
theoretical physics)
Requirements: Knowledge of basics of vector analysis, structure of atoms and molecules,
basics of quantum mechanics, electromagnetism.
ECTS: 7
Semester: winter and summer
The syllabus of the module:
1. Phases and states of matter. Basics of crystallography. Chemical bonds in solids. Elements
of the theory of group representations. Diffraction in periodic structures. Dynamics of atoms
in crystals. Phonons. Symmetry and thermal properties of the crystal lattice. Phase
transitions. Ehrenfest classification of phase transitions. Order parameter. Theory of Landau.
Critical indices.
2. Free electrons in metals. Electronic band structure of solids. Insulators. Metals. Electron
motion and transport properties. Electric conduction in metals. Thermoelectric effects.
3. Magnetism. Diamagnetism and paramagnetism. Band theory of magnetism. Spin waves.
4. Semiconductors. Density of carriers in intrinsic semiconductors. Doping of
semiconductors. Conductivity of semiconductors.
5. Superconductivity. Basic properties of a superconducting state. Superfluidity.
6. Liquid crystals. Nematic, chiral and smectic phases. Order parameters of nematics and
smectics. Textures.
7. Physics of surfaces and interfaces. Experimental methods in condensed matter physics.
4) Advanced Condensed Matter Physics (a course for graduate students specializing in
experimental or theoretical physics.
Requirements: Knowledge of basics of vector analysis, structure of atoms and molecules,
basics of quantum mechanics, statistical physics, electromagnetism.
ECTS: 7
Semester: winter and summer
The syllabus of the module:
1. Tight-binding approximation. K-p method
2. Kane & Luttinger models. Semiconductors, graphene, topological insulators
3. Transport phenomena: charge and spin currents
4. Transverse transport. Anomalous and spin Hall effect
5. Thermoelectricity and spin thermoelectric effects
6. Localization physics. Weak localization
7. Q-theory and supersymmetry in localization
8. Quantum Hall Effect: theories of Thouless and Pruisken
9. Anomalous and spin Hall effects: topology and the Berry phase
10. Gauge fields in insulators: theory of Ioffe and Larkin
11. Bosonization method
5) Nuclear Physics (a general course for 3rd year students specializing in experimental or
theoretical physics)
Requirements: Knowledge of physics at the level of the course textbook Fundamentals of
Physics Vol 1 - 5 by Halliday, Resnick and Walker, differential and integral calculus.
ECTS: 7
Semester: winter and summer
The syllabus of the module:
1. Atom. History. Bohr model.
2. Discovery of nucleus. Rutherford experiment and theory.
3. Characteristics of nucleus.
4. Chart of nucleus as a function of Z and A
5. Nuclear energy. Nuclear Bombs. Nuclear reactors and existing technologies.
6) Biophysics (a course for graduate students specializing in Medical Engineering)
Requirements: Knowledge of basics Physics, Chemistry and Biology.
ECTS: 7
Semester: winter and summer
The syllabus of the module:
1. Hierarchy of living organisms.
2. Atom and molecules.
3. Introduction to Bioenergetics and Thermokinetics.
4. Elements of Molecular Biophysics.
5. Introduction to Biophysics of Cell.
6. Biophysics of tissues.
7. Biophysics of organs.
8. Interaction of living organism with environment
9. Modern techniques of imaging in Biology and Medicine
10. Modern methods of cancer treatment