Curricula and Syllabi of FNSPE CTU in Prague

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Aktualizace dat: 15.10.2017

česky

Bachelor degree programmeMathematical Engineering
Mathematical Physics
Year 3
course code teacher ws ss ws cr. ss cr.

Compulsory courses

Quantum Mechanics02KVAN Hlavatý, Štefaňák 4+2 z,zk - - 6 -
Course:Quantum Mechanics02KVANprof. RNDr. Hlavatý Ladislav DrSc. / Ing. Štefaňák Martin Ph.D.4+2 Z,ZK-6-
Abstract:The lecture describes the birth of quantum mechanics and description of one particle and more particles by elements of the Hilbert space as well as its time evolution. Besides that it includes description of observable quantities by operators in the Hilbert space and calculation of their spectra.
Outline:1. Experiments leading to the birth of QM
2. De Broglie's conjecture, Schroedinger's equation
3. Description of states in QM
4. Elements of Hilbert space theory and operators
5. Harmonic oscilator
6. Quantization of angular momentum
7. Particle in the Coulomb field
8. Mean values of observables and transition probabilities
9. Time evolution of states
10. Particle in the electromagnetic field. Spin
11. Perturbation methods
12. Many particle systems
13. Potential scattering, tunnel phenomenon

Outline (exercises):Free particle
Harmonic oscilator
Coulomb potential
Goals:knowledge:
The goal of the lecture is to explain fundamentals and mathematical methods of the quantum mechanics.

abilities:
apply mathematical methods to problems of quantum mechanics
Requirements:Absolutely necessary is good knowledge of hamiltonian formulation of classical mechanics, linear algebra including operation on infinitely dimensional spaces, calculus in several variables and Fourier analysis. Contact lecturer before inscription.
Key words:quantum mechanics, Hilbert space, wave function, probability prediction
ReferencesKey references:
[1] P.A.M. Dirac, Principles of Quantum Mechanics, Oxford University Press,
Oxford 1958.
Recommended references:
[2] L. D. Faddeev and O. A. Yakubovskii: Lectures on Quantum Mechanics for Mathematics Students (Student Mathematical Library), AMS 2009.

Quantum Mechanics 202KVAN2 Potoček, Šnobl - - 2+2 z,zk - 4
Course:Quantum Mechanics 202KVAN2Ing. Potoček Václav Ph.D. / doc. Ing. Šnobl Libor Ph.D.-2+2 Z,ZK-4
Abstract:Introduction to more advanced topics in quantum mechanics. General formalism of quantum theory, approximate methods and path integral.
Outline:1) Addition of angular momenta, tensor operators
2) Various representations of quantum theory
3) Density matrix
4) JWKB approximation
5) Variational method
6) Time-dependent perturbation theory
7) Propagator, Green function
8) Path integral in quantum mechanics
9) Perturbative expansion of path integral, Feynman diagrams
10) Path integral description of scattering
11) Occupation numbers, annihilation and creation operators, Fock space
12) Brief review of quantum field theory
Outline (exercises):Solution of topical problems in
1) Addition of angular momenta, tensor operators
2) Various representations of quantum theory
3) Density matrix
4) JWKB approximation
5) Variational method
6) Time-dependent perturbation theory
7) Propagator, Green function
8) Path integral in quantum mechanics
9) Perturbative expansion of path integral, Feynman diagrams
10) Path integral description of scattering
11) Occupation numbers, annihilation and creation operators, Fock space
Goals:Knowledge:
Introduction to more advanced topics in quantum mechanics.

Abilities:
Application of general formalism of quantum theory, approximation methods and path integral
Requirements:02 KVAN Quantum Mechanics
Key words:quantum mechanics, approximate methods, path integral
ReferencesKey references:
[1] P.A.M. Dirac, Principles of Quantum Mechanics, Oxford University Press, Oxford 1958.

Recommended references:
[2] L. D. Faddeev and O. A. Yakubovskii: Lectures on Quantum Mechanics for Mathematics Students (Student Mathematical Library), AMS 2009.
[3] A.Messiah, Quantum Mechanics, Two Volumes Bound as One, (Dover Publications, New York, 1999).
[4] L. H. Ryder, Quantum Field Theory, Cambridge University Press, Cambridge 1996.

Nuclear Physics02ZJF Wagner 3+2 z,zk - - 6 -
Course:Nuclear Physics02ZJFRNDr. Wagner Vladimír CSc.3+2 Z,ZK-6-
Abstract:This scientific field presents formidable challenges both experimentally and theoretically, simply because we are dealing with the submicroscopic domain, where much of our classical intuition regarding the behaviour of objects fails us. The lecture is a basic introduction to very interesting regions of subatomic physics.
Outline:1.Introduction - basic definitions and historical review
2.Collision kinematics
3.Cross section properties
4.Basic features of atomic nuclei and nuclear forces
5.Nuclear models
6.Radioactive decay of nuclei
7.Experimental techniques of nuclear and particle physics
8.Nuclear reactions
9.Nuclear matter, its study and features
10.Particles and their interactions
11.Unified theories of matter and interactions
12.Applications of nuclear and particle physics, nuclear astrophysics
Outline (exercises):Testing knowledge on particular problems from chapters:
1.Introduction - basic definitions and historical review
2.Collision kinematics
3.Cross section properties
4.Basic features of atomic nuclei and nuclear forces
5.Nuclear models
6.Radioactive decay of nuclei
7.Experimental techniques of nuclear and particle physics
8.Nuclear reactions
9.Nuclear matter, its study and features
10.Particles and their interactions
11.Unified theories of matter and interactions
12.Applications of nuclear and particle physics, nuclear astrophysics
Goals:Knowledge:
Fundamentals of nuclear and subnuclear physics, laws of microworld, understanding of experimental methods at subatomic physics.

Skills:
Basic understanding and calculations in nuclear and subnuclear physics
Requirements:Basic course of physics. Knowledge from classical mechanics, theory of relativity, electricity and magnetisms as well as thermodynamics.
Key words:Radioactivity, nuclear decay, nuclear reactions, elementary particles, quarks
ReferencesKey references:
[1] W.S.C. Williams : Nuclear and Particle Physics, Oxford Science Publications, 2001
[2] B. Povh, K. Rith, Ch. Scholz, F. Zetsche: Particles and Nuclei. An Introduction to the Physical Concepts, Springer 2004

Recommended references:
[3] Ashok Das, Thomas Ferbel: Introduction to Nuclear and Particle Physics, John Wiley and Sons, 1994
[4] P.E.Hodgson, E Gadioli and E.Gadioli Erba: Introductory Nuclear Physics, Oxford Science Publications, 1997
[5] A. Beiser: Concepts of Modern Physics, McGraw-Hill Companies Date Published, 1995

Media and tools:
Lecture room with dataprojector

Functional Analysis 101FAN1 Šťovíček 2+2 z,zk - - 4 -
Course:Functional Analysis 101FAN1prof. Ing. Šťovíček Pavel DrSc.----
Abstract:Basic notions and results are addressed concerning successively topological spaces, metric spaces, topological vector spaces, normed and Banach spaces, Hilbert spaces.
Outline:1. Topological spaces
2. Metric spaces, compactness criteria, completion of a metric space
3. Topological vector spaces
4. Minkowski functional, the Hahn-Banach theorem
6. Metric vector spaces, Fréchet spaces
6. Normed vector spaces, bounded linear mappings, the operator norm
7. Banach spaces, extension of a bounded operator
8. Banach spaces of integrable functions
9. Hilbert spaces, orthogonal projection, orthogonal basis
10. The Riesz representation theorem, adjoint operator
Outline (exercises):Exercise is closely linked to the lecture, which is illustrated by appropriate examples. Accent is placed on the correctness of the calculation.
1. Basics of topology, repetition.
2. Basics of metric spaces and of banach spaces.
3. Banach spaces and linear bounded mappings
4. Resolvent formula, Fourier transform
5. Scalar product, isomorphism of Hilbert spaces orthogonality
7. Norms, continuity, linear extension, projectors, types of convergence
8. Spectral properties of normal and compact operators, ideals of compact operators
Goals:Knowledge: basics of Banach and Hilbert spaces and linear operators in these spaces, and as a background sufficiently profound knowledge of topological and metric spaces

Skills: applications of the apparatus of Banach and Hilbert spaces
Requirements:The complete introductory course in mathematical analysis and linear algebra on level A or B given at the Faculty of Nuclear Sciences and Physical Engineering
Key words:compact topological space, complete metric space, topological vector space, operator norm, the Hahn-Banach theorem, Banach space, Hilbert space, orthogonal projection, orthogonal basis, adjoint operator
ReferencesKey references:
[1] J.Blank.P,Exner,M.Havlíček: Hilbert Space Operators in Quantum Physics, Springer,2008.

Recommended references:
[2] M. Reed, B. Simon : Methods of Modern Mathematical Physics I.. ACADEMIC PRESS, N.Z. 1972
[3] W. Rudin: Real and Complex Analysis, (McGrew-Hill, Inc., New York, 1974)
[4] A. N. Kolmogorov, S. V. Fomin: Elements of the Theory of Functions and Functional Analysis, (Dover Publications, 1999)
[5] A. E. Taylor: Introduction to Functional Analysis, (John Wiley and Sons, Inc., New York, 1976)

Functional Analysis 201FA2 Šťovíček - - 2+2 z,zk - 4
Course:Functional Analysis 201FA2prof. Ing. Šťovíček Pavel DrSc.-2+2 Z,ZK-4
Abstract:The course aims to present selected fundamental results from functional analysis including basic theorems of the theory of Banach spaces, closed operators and their spectrum, Hilbert-Schmidt operators, spectral decomposition of bounded self-adjoint operators.
Outline:1. The Baire theorem, the Banach-Steinhaus theorem (the principle of uniform boundedness), the open mapping theorem, the closed graph theorem.
2. Spectrum of closed operators in Banach spaces, the graph of an operator, analytic properties of a resolvent, the spectral radius.
3. Compact operators, the Arzela-Ascoli theorem, Hilbert-Schmidt operators.
4. The Weyl criterion for normal operators, properties of spectra of bounded self-adjoint operators.
5, The spectral decomposition of bounded self-adjoint operators, functional calculus.
Outline (exercises):1. Exercises devoted to basic properties of Hilbert spaces and to the orthogonal projection theorem.
2. The quotient of a Banach space by a closed subspace.
3. Properties of projection operators in Banach spaces and orthogonal projections in Hilbert spaces.
4. Examples of the application of the principle of uniform boundedness.
5. Exercises focused on integral operators, Hilbert-Schmidt operators.
6. Examples of the spectral decomposition of bounded self-adjoint operators.
Goals:Knowledge: Basics of the theory of Banach spaces, selected results about compact operators and the spectral analysis in Hilbert spaces.

Skills: Application of this knowledge in subsequent studies aimed at partial differential equations, integral equations, and problems of mathematical physics.
Requirements:01FA1
Key words:Banach space, Hilbert space, spectrum, uniform boundedness principle, open mapping theorem, the Arzela-Ascoli theorem, Hilbert-Schmidt operators, spectrum of a bounded operator, self-adjoint operator, spectral decomposition
ReferencesKey references:
[1] J. Blank, P. Exner, M. Havlíček: Hilbert Space Operators in Quantum Physics, (American Institute of Physics, New York, 1994)

Recommended references:
[2] W. Rudin: Real and Complex Analysis, (McGrew-Hill, Inc., New York, 1974)
[3] A. N. Kolmogorov, S. V. Fomin: Elements of the Theory of Functions and Functional Analysis, (Dover Publications, 1999)
[4] A. E. Taylor: Introduction to Functional Analysis, (John Wiley and Sons, Inc., New York, 1976)

The Equations of Mathematical Physics01RMF Klika 4+2 z,zk - - 6 -
Course:The Equations of Mathematical Physics01RMFdoc. Ing. Klika Václav Ph.D. / Mgr. Kozák Michal4+2 Z,ZK-6-
Abstract:The subject of this course is solving integral equations, theory of generalized functions, classification of partial differential equations, theory of integral transformations, and solution of partial differential equations (boundary value problem for eliptic PDE, mixed boundary problem for eliptic PDE).
Outline:1. Introduction to functional analysis - factor space, Hilbert space, scalar product, orthonormal basis, fourier series, orthogonal polynoms, hermite operators, operator spectrum and its properties, bounded operators, continuous operators, eliptic operators
2. Integral equations - integral operator and its properties, separable kernel of operator, sequential approximation method, iterated degenerate kernel method, Fredholm integral equations, Volterra integral equations.
3. Classification of partial differential equations - definitions, types of PDE, transformations of partial differential equations into normal form, classification of PDE, equations of mathematical physics.
4. Theory of generalized functions - test functions, generalized functions, elementary operations in distributions, generalized functions with positive support, tensor product and convolution, temepered distributions.
5. Theory of integral transformations - classical and generalized Fourier transformation, classical and generalized Laplace transform, applications.
6. Solving differential equations - fundamental solution of operators, solutions of problems of mathematical physics.
7. Boundary value problem for eliptic partial differential equation.
8. Mixed boundary problem for eliptic partial differential equation.
Outline (exercises):1. Hilbert space
2. Linear operators on Hilbert spaces
3. Integral equations
4. Partial differential equations
5. Theory of generalized functions
6. Laplace transform
7. Fourier transform
8. Fundamental solution of operators
9. Equations of mathematical physics
10. Eliptic differential equations
11. Mixed boundary problem
Goals:Get acquainted with theory of generalized functions and its application to solving partial differential equations including mixed boundary problem.
Requirements:Basic course of Calculus, Linear Algebra and selected topics in mathematical analysis (in the extent of the courses 01MA1, 01MAA2-4, 01LA1, 01LAA2, 01VYMA held at the FNSPE
CTU in Prague).
Key words:Mathematical methods in physics, distributions, integral transfomations, partial differential equations
ReferencesKey References:
P. Stovícek: Methods of Mathematical Physics : Theory of generalized functions, CVUT, Praha, 2004. (in czech),
P. Stovícek: Methods of Mathematical Physics II: Theory of generalized functions II. Integral equations, elliptic operators, CVUT, Praha, 2017. (in czech),
V.S. Vladimirov : Equations of Mathematical Physics, Marcel Dekker, New York, 1971
Č. Burdík, O. Navrátil : Rovnice matematické fyziky, Česká technika - nakladatelství ČVUT, 2008

Recommended literature:
L. Schwartz - Mathematics for the Physical Sciences, Dover Publication, 2008
I. M. Gel'fand, G. E. Shilov, Generalized Functions. Volume I: Properties and Operations, Birkhäuser Boston, 2004

Geometric Methods in Physics 102GMF1 Šnobl - - 2+2 z,zk - 4
Course:Geometric Methods in Physics 102GMF1doc. Ing. Šnobl Libor Ph.D.-2+2 Z,ZK-4
Abstract:Foundations of analysis on manifolds. Differential forms. Integration, Stokes theorem.
Outline:1. Manifolds.
2. Tangent vectors, tangent spaces.
3. Vector fields, integral curves.
4. Covectors, p-forms.
5. Differential forms, wedge product, outer derivation.
6. Induced maps of tensorial objects.
7. Lie derivative.
8. Geometric formulation of Hamilton´s mechanics.
9. Integration of forms, Stokes´ theorem
10. Metrics and curvature.
Outline (exercises):Solving problems on the following topics:
1. Manifolds.
2. Tangent vectors, tangent spaces.
3. Vector fields.
4. Covectors, forms.
5. Differential forms, wedge product, outer derivation.
6. Induced maps of tensorial objects.
7. Lie derivative.
8. Geometric formulation of Hamilton´s mechanics.
9. Integration of forms, Stokes´ theorem
10. Metrics and curvature.
Goals:Knowledge:
Foundations of analysis on manifolds

Skills:
Application of geometrical methods in theoretical physics
Requirements:The course of theoretical physics (02TEF1, 02TEF2)
Key words:differentiable manifold, vector fields, p-form, integration of forms, Stokes theorem
ReferencesKey references:
[1]L. Krump, V. Souček, J.A. Těšínský: Mathematical Analysis on Manifolds (in Czech), Karolinum Praha 1998

Recommended references:
[2] M. Nakahara: Geometry, Topology and Physics, IOP Publishing, Bristol 1998

General Relativity02OR Semerák - - 3+0 zk - 3
Course:General Relativity02ORdoc.RNDr. Semerák Oldřich Dr., DSc.-3+0 ZK-3
Abstract:Introduction to general theory of relativity: principle of equivalence and principle of general covariance, parallel transport and geodesic equation, gravitational redshift; curvature and Einstein's gravitational law. Schwarzschild solution of Einstein equations and black holes. General relativity in astrophysics and cosmology: relativistic stellar models, final stages of stellar evolution; Friedmann's cosmological models.
Outline:1. Theory of gravitation and physical picture of the world.
2. Equivalence principle. General covariance (general relativity) principle.
3. Parallel transport and equation of geodesic; affine connection.
4. Covariant derivative.
5. Riemann curvature tensor. Ricci tensor and scalar curvature.
6. Einstein´s gravitational law.
7. Schwarzschild´s solution of Einstein equations.
8. Schwarzschild black hole - horizon, singularity.
9. Black holes. Kerr´s solution of Einstein equations. Gravitational collapse.
10. Astrophysical aspects of black holes.
11. Relativistic stellar models.
12. Fundamentals of relativistic cosmology.
Outline (exercises):
Goals:Knowledge:
Learn the fundamentals of contemporary gravitation theory.

Skills:
Solving simple examples from gravitaion theory.
Requirements:02TEF2, 02GMF1
Key words:Equivalence principle, general covariance principle, affine connection, Einstein´s gravitational law, black holes
ReferencesKey references:
[1] Dvořák L.: General Relativity Theory and Modern Physical View of the Universe (in Czech), SPN, Praha 1984

Recommended references:
[2] P. Hájíček: An introduction to the relativistic theory of gravitation, Springer, Berlin 2008Recommended references:

Bachelor Thesis 102BPMF12 Hlavatý, Tolar 0+5 z 0+10 z 5 10
Course:Bachelor Thesis 102BPMF1prof. RNDr. Hlavatý Ladislav DrSc. / prof. Ing. Tolar Jiří DrSc.0+5 Z-5-
Abstract:Bachelor's thesis on a chosen subject supervised by an adviser.
Outline:Bachelor's thesis on a chosen subject supervised by an adviser.
Outline (exercises):
Goals:Knowledge:
a particular field depending on a given project topic.

Skills:
working unaided on a given task, understanding the problem, producing an original specialist text.
Requirements:
Key words:
ReferencesReferences are done according to the subject.

Course:Bachelor Thesis 202BPMF2prof. RNDr. Hlavatý Ladislav DrSc. / prof. Ing. Tolar Jiří DrSc.----
Abstract:Bachelor's thesis on a chosen subject supervised by an adviser.
Outline:Bachelor's thesis on a chosen subject supervised by an adviser.
Outline (exercises):
Goals:Knowledge:
a particular field depending on a given project topic.

Skills:
working unaided on a given task, understanding the problem, producing an original specialist text.
Requirements:
Key words:
ReferencesReferences are done according to the subject.

Výuka jazyků04... KJ - - - - - -

Optional courses

Differential Equations, Symmetries and Groups02DRG Šnobl 2+2 z - - 4 -
Course:Differential Equations, Symmetries and Groups02DRGdoc. Ing. Šnobl Libor Ph.D.2+2 Z-4-
Abstract:The purpose of the lecture is to teach students computation of symmetries of the differential equations.
Outline:1. Symmetries in physics and mathematics
2. Groups.
3. One-parameter subgroups, generators.
4. Group actions.
5. Local and infinitesimal group actions.
6. Point transformations.
7. Symmetries of equations.
8. Determination of infinitesimal symmetries.
9. Symmetry-based reduction of order of ODEs
10. Selfsimilar solutions of PDEs
Outline (exercises):1. Calculation of symmetries of a given ODE
2. Solution of ODE via oredr reduction
3. Calculation of symmetries of a given PDE (Heat equation, KdV equation, ...)
4. Interpretation of the symmetries
5. Determination of the Lie algebra of the symmetries
6. Construction of selfsimilar solutions.
Goals:Knowledge:
Lie symmetries of differential equations

Abilities:
Computation of point symmetries of differential equations and their application in solution of given ODEs and PDEs.
Requirements:01MAA34, 01DIFR, 02TEF12
Key words:Lie groups, Lie algebras, symmetries of differential equations
ReferencesKey references:
[1] P.J.Olver, Applications of Lie Groups to Differential Equations, Springer 2000
[2] P.E. Hydon: Symmetry Methods for Differential Equations: A Beginner's Guide (Cambridge Texts in Applied Mathematics), CUP 2000

Recommended references:
[3] N.Kh. Ibragimov: Group analysis of ordinary differential equations an the invariance principle in mathematical physics, Uspekhi Mat Nauk 47:4 (1992) 83-144 Russian Math. Surveys 47:4 (1992) 89-156

Simulations and Data Analysis Tools02NSAD Hubáček 2+0 z - - 2 -
Course:Simulations and Data Analysis Tools02NSADIng. Hubáček Zdeněk Ph.D.----
Abstract:Data analysis and simulations of high energy elementary particle collisions. ROOT and Pythia programs.
Outline:1.ROOT program.
2.Data storing in the ROOT program.
3.Working with histograms.
4.Working with trees.
5.Monte - Carlo collision generators.
6.Pythia program.
7.Pythia generator parameters.
8.Generating high-energy collision.
9.Storing of the generated collision in the ROOT program.
10.Making histogram from generated collision.
11.Selection priciples - cuts
12. I/O operations in ROOT.
13.Fitting in ROOT
Outline (exercises):
Goals:Knowledge:
Data analysis in high energy particle physics - methods and programs, simulation of particle collisions

Abilities:
Individual simulation of particle collision and consequent analysis using proper tools
Requirements:Secondary school knowledges.
Key words:ROOT, pythia, C++
ReferencesKey references:
[1] ROOT User's guide - root.cern.ch/drupal/
[2] ROOT Reference guide - root.cern.ch/drupal/
[3] Pythia manual pythia6.hepforge.org

Recommended references:
[4] M.Virius: Programování v C++, ČVUT Praha 2009(in czech)
[5] M. Virius: Metoda Monte Carlo, ČVUT Praha 2010(in czech)

Algebra01ALGE Šťovíček 4+1 z,zk - - 6 -
Course:Algebra01ALGEprof. Ing. Šťovíček Pavel DrSc.----
Abstract:Firstly, the Peano axioms are treated in detail. Elements of the set theory cover only: equivalence and subvalence, the Cantorov-Bernstein theorem, the axiom of choice and equivalent statements, definition of ordinals and cardinals. Further standard algebraic structures are addressed: semigroups, monoids, groups, rings, integral domains, principal ideal domains, fields, lattices. Independent chapters are devoted to divisibility in integral domains and to finite fields.
Outline:1. Binary relations, equivalence, ordering
2. The Peano axioms for the natural numbers, principle of recursive definition
3. Equivalence and subvalence of sets, the transfinite induction
4. The axiom of choice and equivalent statements
5. Ordinals and cardinals
6. Semigroups, monoids
7. Groups
8. Rings, integral domains, principal ideal domains, fields
9. Divisibility in integral domains
10. Finite fields
11. Lattices
Outline (exercises):
Goals:Knowledge: elements of the set theory - equivalence and subvalence, the axiom of choice and equivalent statements, ordinals and cardinals; basics of algebra - the Peano axioms, monoids, groups, rings, integral domains, principal ideal domains, fields

Skills: using algebraic structures, applying these structures along with some elements of the set theory in other fields of mathematics
Requirements:01LAA2
Key words:binary relation, ordering, axiom of choice, ordinal, cardinal, semigroup, monoid, group, ring, integral domain, principal ideal domain, field, lattice
ReferencesKey references:
[1] Mareš J.: Algebra. Úvod do obecné algebry, 3. vydání. ČVUT, Praha, 1999.

Recommended references:
[2] Mac Lane S., Birkhoff G.: Algebra. Springer, New York, 2005.
[3] Lang S.: Algebra. Springer, New York, 2005.

Probability and Statistics01PRST Hobza 3+1 z,zk - - 4 -
Course:Probability and Statistics01PRSTdoc. Ing. Hobza Tomáš Ph.D.3+1 Z,ZK-4-
Abstract:It is a basic course of probability theory and mathematical statistics. The probability theory is build gradually beginning with the classical definition and continuing till the Kolmogorov definition. The notions as random variable, distribution function of random variable and characteristics of random variable are treated and basic limit theorems are stated and proved. On the basis of this theory the basic methods of mathematical statistics such as estimation of distribution parameters and hypothesis testing are explained.
Outline:1. Classical definition of probability, statistical definition of probability, conditional probability and Bayes's theorem
2. Random variables, distribution functions, discrete and continuous random variables, independent random variables, characteristics of random variable
3. Law of large numbers, central limit theorem
4. Point estimation, confidence intervals
5. Tests of statistical hypotheses, goodness of fit tests
Outline (exercises):1. Combinatorial rules, classical and geometric probability
2. Conditioned probability and related theorems
3. Distribution function of random variable, discrete and continuous random variables, transformation of random variables
4. Characteristics of random variables, mainly expectation and variance, central limit theorem
5. Point estimation of parameters
6. Hypothesis testing, goodness-of-fit tests
Goals:Knowledge:
Fundamentals of probability theory and overview of simple statistical methods.

Skills:
Application of probability theory to solution of concrete examples, statistical analysis and processing of real data, testing hypothesis about the sets of real data.
Requirements:Basic course of Calculus (in the extent of the courses 01MAB3, 01MAB4 held at the FNSPE CTU in Prague).
Key words:Random variable, distribution function, probability mass function, probability density, independence of random variables, expectation, variance, central limit theorem, point estimation of parameters, hypothesis testing, goodness-of-fit tests.
ReferencesKey references:
[1] H. G. Tucker: An introduction to probability and mathematical statistics. Academic Press, 1963
[2] H. Pishro-Nik: Introduction to Probability, Statistics, and Random Processes, Kappa Research, LLC, 2014

Recommended references:
[3] J. Shao: Mathematical statistics, Springer, 2003

Functions of Complex Variable01FKO Šťovíček - - 2+1 z,zk - 3
Course:Functions of Complex Variable01FKOprof. Ing. Šťovíček Pavel DrSc.----
Abstract:The course starts from outlining the Jordan curve theorem and the Riemann-Stieltjes integral. Then basic results of complex analysis in one variable are explained in detail: the derivative of a complex function and the Cauchy-Riemann equations, holomorphic and analytic functions, the index of a point with respect to a closed curve, Cauchy's integral theorem, Morera's theorem, roots of a holomorphic function, analytic continuation, isolated singularities, the maximum modulus principle, Liouville's theorem, the Cauchy estimates, Laurent series, residue theorem.
Outline:1. Connected, path-connected, simply connected spaces, the Jordan curve theorem
2. Variation of a function, length of a curve, the Riemann-Stieltjes integral (survey)
3. Derivative of a complex function, the Cauchy-Riemann equations
4. Holomorphic functions, power series, analytic functions
5. Regular curves, integration of a function along a curve (contour integral), the index of a point with respect to a closed curve
6. Cauchy's integral theorem for triangles
7. Cauchy's integral formula for convex sets, relation between holomorphic and analytic functions, Morera's theorem
8. Roots of a analytic function, analytic continuation
9. Isolated singularities
10. The maximum modulus principle, Liouville's theorem
11. The Cauchy estimates, uniform convergence of analytic functions
12. Cauchy's integral theorem (general version)
13. The residue theorem
Outline (exercises):
Goals:Knowledge: the Jordan curve theorem, construction of the Riemann-Stieltjes integral, basic results of complex analysis in one variable.

Skills: practical usage of complex analysis, applications in evaluation of integrals.
Requirements:The complete introductory course in mathematical analysis on level A or B given at the Faculty of Nuclear Sciences and Physical Engineering
Key words:Jordan curve theorem, Riemann-Stieltjes integral, Cauchy-Riemann equations, Morera's theorem, isolated singularity, maximum modulus principle, Liouville's theorem, Cauchy estimates, Laurent series, residue theorem
ReferencesKey references:
[1] W. Rudin: Real and Complex Analysis, (McGrew-Hill, Inc., New York, 1974)

Recommended references:
[2] J. B. Conway: Functions of One Complex Variable I, Springer-Verlag, New York, 1978

Topology01TOP Burdík 2+0 zk - - 2 -
Course:Topology01TOPprof. RNDr. Burdík Čestmír DrSc.2+0 ZK-2-
Abstract:The aim of lecture is the systematization and deepening the knowledge of general topology.
Outline:1.Structure on the set. 2. Real number and plane. 3.Sets, products and sums. 4. Graphs. 5. Mathematical structures. 6. Abstract spaces. 7. Structure of topological spases. 8. Separation axioms. 9. Hausdorff spaces. 10. Normal spaces. 11. Compact spaces. 12. Topology of metric. 13. Metric spaces.
Outline (exercises):
Goals:Knowledge : mathematical basis for a general topology. Skills: able to think in the schema definition, theorem and proof, and the use of general topology.
Requirements:Basic course of Calculus and Linear Algebra (in the extent of the courses 01MA, 01MAA2-4, 01LAP, 01LAA2 held at the FNSPE CTU in Prague).
Key words:Topological space, product topology, subspace topology, contiunous function, connected spaces, compact spaces, separation axioms.
ReferencesKey references: [1] John L. Kelly, (Springer, 1975, 315 pp., ISBN10:0-387-90125-6), [2] Bourbaki; Elements of Mathematics: General Topology, Addison-Wesley(1966).
Recommended references: [3] Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6. [4] Basener, William (2006). Topology and Its Applications (1st ed.).Wiley. ISBN 0-471-68755-3.

Physical Training 300TV34 ČVUT - z - z 1 1
Course:Physical Training 300TV3----
Abstract:
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Course:Physical Training 400TV4----
Abstract:
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