course 
code 
teacher 
ws 
ss 
ws cr. 
ss cr. 
Algebra  01ALGE 
Šťovíček 
4+1 z,zk 
  
6 
 
Course:  Algebra  01ALGE  prof. Ing. Šťovíček Pavel DrSc.          Abstract:  Firstly, the Peano axioms are treated in detail. Elements of the set theory cover only: equivalence and subvalence, the CantorovBernstein 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
 References  Key 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. 

Algebraic structures in theoretical informatics  01ALTI 
Pošta, Svobodová 
1+1 zk 
  
3 
 
Course:  Algebraic structures in theoretical informatics  01ALTI  prof. Ing. Pelantová Edita CSc. / doc. Ing. Pošta Severin Ph.D.          Abstract:  The course is devoted to the applications of some special algebraic structures. The first part of the course is devoted to the Gröbner bases of ideals of polynomial rings and their use for solving of systems of algebraic equations and other applications. The second part of the course is devoted to the ring of integers of algebraic number fields, used to constructions of various representations of numbers utilized in fast effective algorithms for arithmetic operations and evaluations of elementary functions.  Outline:  1) Gröbner bases in rings of polynomials of several variables
2) Ideals of polynomial rings, basis of an ideal
3) Monomial orderings, rewrite rules
4) Buchberger algorithm
5) The solution of systems of algebraic equations, coloring of graphs, automatic theorem proving
6) Unique position representations with integer bases in the ring of integers Z and in the ring of Gaussian integers, NAF representations, arithmetic operations in these systems and finite automata.
7) Redundant representations in real and complex cases, parallel algorithms for addition and substraction and the relation to the redundancy level, online algorithms for multiplication and division.
8) Representations of numbers for shift and add algorithms for the evaluations of elementary functions.
 Outline (exercises):  The first part of the exercises is devoted to the constructions of Gröbner bases of some concretely specified ideals.
The second part of the exercises is devoted to the construction of representations of numbers in various systems, algorithms for arithmetic operations in redundant systems, optimalization of parameters for given basis. Students individually present their solutions of theoretical and practical exercises.  Goals:  The main goal of the course is to learn to manipulate with ideals of rings of polynomials of several variables and to construct their Gröbner bases. The second goal is to learn to manipulate various representations of numbers and to construct the algorithms for effective performing of arithmetic operations and evaluations of elementary functions.  Requirements:  The knowledge of basics of general algebra (01ALGE) and algebraic number theory (01TC).  Key words:  Gröbner basis, numeral system, online algorithm, shiftandadd algorithm  References  [1] JeanMichel Muller, Elementary Functions: Algorithms and Implementation, 3rd ed., Birkhauser 2016
[2] Michel Rigo, Formal languages, Automata and Numeration systems, Wiley, 2014
[3] William Adams, Phillippe Loustaunau, An Introduction to Grobner Bases, AMS, 1994


Aperiodic Structures 1  01APST12 
Masáková 
2+0 z 
2+0 z 
2 
2 
Course:  Aperiodic Structures 1  01APST1  prof. Ing. Masáková Zuzana Ph.D.          Abstract:  The seminar is devoted to combinatorics on infinite words, nonstandard numeration systems and aperiodic tilings of the space.
The seminar often hosts foreign researcher. Students participate actively in solution of open problems in the field.  Outline:  1. Combinatorics on words over a finite alphabet, infinite aperiodic words with low complexity. Morphisms and substitutions, their incidence matrix.
2. Aperiodic tilings of the space, selfsimilarity, aperiodic Delone sets, cutandproject sets, mathematical models of quasicrystals.
3. Number representation in systems with irrational base, arithmetics in such systems.  Outline (exercises):   Goals:  The seminar is an iniciation to research.
Acquired knowledge: sources of relevant mathematical literature
Acquired skills: independent research  Requirements:  Knowledge of mathematics in the extent of the
FNSPE bachelor's specialization Mathematical Informatics or Mathematical modeling.  Key words:  combinatorics on words, nonstandard numeration systems, mathematical models of quasicrystals  References  P. Fogg, Substitutions in Dynamics, Arithmetics, and Combinatorics (Lecture Notes in Mathematics, Vol. 1794).
M. Lothaire, Algebraic Combinatorics on Words Cambridge University Press, 2002. 
Course:  Aperiodic Structures 2  01APST2  prof. Ing. Masáková Zuzana Ph.D.          Abstract:  The seminar is a continuation of 01APST1. It is devoted to advanced issues of combinatorics on infinite words, nonstandard numeration systems and aperiodic tilings of the space.
The seminar often hosts foreign researcher. Students participate actively in solution of open problems in the field.  Outline:  1. Properties of infinite words constructed as fixed points of morphisms, palindromic and pseudopalindromic closures, codings of interval exchanges.
2. Aperiodic tilings of the space, selfsimilarity, aperiodic Delone sets, cutandproject sets, mathematical models of quasicrystals.
3. Number systems with complex alphabets or complex digits. Algorithms in nonstandard numeration systems.  Outline (exercises):   Goals:  The seminar is an iniciation to research.
Acquired knowledge: sources of relevant mathematical literature
Acquired skills: independent research  Requirements:  Knowledge of mathematics in the extent of the
FNSPE bachelor's specialization Mathematical Informatics or Mathematical modeling.  Key words:  combinatorics on words, nonstandard numeration systems, mathematical models of quasicrystals  References  P. Fogg, Substitutions in Dynamics, Arithmetics, and Combinatorics (Lecture Notes in Mathematics, Vol. 1794).
M. Lothaire, Algebraic Combinatorics on Words Cambridge University Press, 2002. 

Application of Statistical Methods  01ASM 
Hobza 
  
2+0 kz 
 
2 
Course:  Application of Statistical Methods  01ASM  doc. Ing. Hobza Tomáš Ph.D.          Abstract:  The course focuses on applications of selected methods of statistical data analysis to concrete problems including their solutions using statistical software. Namely we will deal with: hypotheses tests about parameters of normal distribution, nonparametric methods, contingency tables, linear regression and correlation, analysis of variance.  Outline:  1. Hypothesis tests about parameters of normal distribution.
2. Goodnesoffit tests.
3. Nonparametric tests  sign and rank tests, Wilcoxon test, KruskalWallis test.
4. Contingency tables  tests of independence and homogeneity, McNemar's test.
5. Linear regression and correlation.
6. Oneway and twoway analysis of variance.
 Outline (exercises):   Goals:  Knowledge:
Basic statistical procedures for data analysis and data visualization.
Skills:
Application of theoretically studied statistical procedures to practical problems of data analysis including use of these methods on computer in the MATLAB environment.  Requirements:  Basic course of Calculus and Probability (in the extent of the courses 01MAB3, 01MAB4 and 01PRST held at the FNSPE CTU in Prague).  Key words:  Hypothesis testing, goodnessoffit tests, linear regression, ANOVA, nonparametric tests, contingency tables.  References  Key references:
[1] Shao, Jun: Mathematical Statistics, Springer, New York 1999
Recommended references:
[2] J.P. Marques de Sá: Applied statistics using SPSS, STATISTICA, MATLAB and R, Springer, 2007. 

Assistive Technology  01ASTE 
Seifert 
0+1 z 
  
2 
 
Course:  Assistive Technology  01ASTE  Seifert Radek  0+1 Z    2    Abstract:  The aim of the course is to introduce the field of Assistive Technology for people with a visual impairment. Besides the technological background of the tools in use, emphasis is put on the general principles of usage and special demands of the target group of users.  Outline:  1.Term "Assistive Technology?
2.Computer hardware and software
3.Webpage accessibility and the most important guidelines
4.Digitisation and archiving of textual documents
5.Basic formats accessibility
6.Braille code and tactile graphics
7.Navigation systems
 Outline (exercises):   Goals:   Requirements:   Key words:  Assistive Technology, Braille display, screenreader, accessibility, document digitalising and archiving, Braille code, tactile graphics, orientation systems  References  [1] Assistive technology. In Wikipedia : the free encyclopedia [online]. St. Petersburg (Florida) : Wikipedia Foundation, [cit. 20101222]. [2] KONECNY, Josef. Blind Friendly [online]. 2007 [cit. 20101222]. Malé nahlédnutí do historie hlasovych syntéz [3] PAVLICEK, Radek. Blind Friendly [online]. 2.3. 20050331 [cit. 20101222]. Metodika Blind Friendly Web. 

Asymptotical Methods  01ASY 
Mikyška 
  
2+1 z,zk 
 
3 
Course:  Asymptotical Methods  01ASY  doc. Ing. Mikyška Jiří Ph.D.  2+1 Z,ZK    3    Abstract:  Examples. Addition parts of mathematical analysis (generalized Lebesgue integral, parametric integrals.) Asymptotic relations a expansions  properties; algebraical and analytical operations. Applied asymptotics of sequences and sums; integrals of Laplace and Fourier type.  Outline:  1. Landau symbols
2. Asymptotic sequences and Asymptotic expansions of functions.
3. Basic properties of asymptotic expansions and algebraic operations with them
4. Differentiation and integration of the asymptotic relations
5. Asymptotics of sequences
6. Asymptotics of series
7. Asymptotics of the roots of algebraic equations
8. Supplement to mathematical analysis  generalized Lebesgue integral
9. Asymptotics of the Laplace integrals, Laplace theorem, Watson's lemma.
10. Examples, applications of the asymptotic methods.  Outline (exercises):  1. Examples of the asymptotic expansions of functions and their properties
2. Basic properties of asymptotic expansions and algebraic operations with them
3. Asymptotics of sequences, Stirling's formula
4. Asymptotics of series, approximation of pi.
5. Asymptotics of the roots of algebraic equations
6. Asymptotics of the Laplace integrals, applications of the Laplace theorem and Watson's lemma
7. Examples, applications of the asymptotic methods.  Goals:  EulerMaclaurin summation formula, perturbation methods, Laplace method, Watson's lemma.
Skills: Application of the asymptotical methods for investigation of the asymptotics of sequences, series, and integrals of Laplace and Fourier type.  Requirements:  Basic courses of Calculus (in the extent of the courses 01MA1, 01MAA24 held at the FNSPE
CTU in Prague).  Key words:  Asymptotic expansions, asymptotics of sequences, asymptotics of series, asymptotics of roots of algebraic equations, Laplace method, Watson's lemma, generalized Lebesgue integral.  References  Key references:
[1] P. D. Miller: Applied Asymptotic Analysis, Graduate Studies in Applied Mathematics, Vol. 75, American Mathematical Society, 2006.
Recommended references:
[2] E. T. Copson: Asymptotic Expansions, Cambridge University Press, 1965.
[3] N. G. de Bruin: Asymptotic Methods in Analysis, North Holland Publishing Co., 1958.
[4] F. J. Olver: Asymptotics and special functions, Academic press, New York (1974) 

Bayesian principles in statistics  01BAPS 
Kůs 
3+0 zk 
  
3 
 
Course:  Bayesian principles in statistics  01BAPS  Ing. Kůs Václav Ph.D.          Abstract:  The main goal of the subject is to provide decision making mathematical principles with random effects, optimal and robust strategies and their mutual links together with computational aspects for the real applications. The techniques are illustrated within practical examples originating from point and interval estimation and statistical hypothesis testing.  Outline:  1. Sufficient statistics, general principles of classical statistics, conditionality, likelihood, sequential principles and their relations, Bayesian principle, Bayesian complete model and its advantages.
2. Loss and risk functions, utility function and its existence, general decision functions. Optimal decision and complete classes of optimal strategies.
3. Convex loss functions, RaoBlackwell theorem, uniformly best strategy, unbiasness, UMVUE construction, examples.
4. Bayes optimal decision strategy, prior and posterior Bayesian risk. Families of aprior informations, uncertainty principle.
5. Jeffreys densities, conjugated systems, limit aposteriory densities, examples for standard families.
6. Minimax strategies, admissibility principle and its consequences within classical and Bayesian statistics, Stein effect.
7. Score functions and their robust properties, Shannon entropy, fdivergences, maximum entropy principle, new extended families of divergences and its metric and robust properties.
8. Minimum distance point estimators, minimum Kolmogorov, Lévy and discrepancy decision functions and its L1consistency and qualitative robustness, Kolmogorov entropy, VapnikChervonenkis dimension.
9. Numerical procedures, approximative calculations in higher dimensions, MonteCarlo approaches, importance sampling, convergence, Metropolis algorithm.
10. Second order Laplace asymptotic expansion, fully exponential forms, regularity assumptions for stochastic expansion/approximation, the results of KassTierneyKadaneho.
11. Hierarchic Bayes, Empirical Bayes, Variational Bayes  principles and examples.
12. Bayesian hypothesis testing for various loss functions, properties.  Outline (exercises):   Goals:  Knowledge:
Extension of the decision makinng principles with random effects and their application in stochastic optimization tasks, mainly in Bayesian methods.
Skills:
Orientation in various stochastical approaches and their properties. Computational aspects.  Requirements:  Basic course of Calculus and Probability  in the extent of the courses 01MAA34 or 01MAB34, 01MIP or 01PRST.  Key words:  Statistical point estimators, loss functions, prior information, Bayesian risk, robust optimal strategies, minimum distance method, fdivergences, MonteCarlo calculations.  References  Key references:
[1] Berger J.O., Statistical Decision Theory and Bayesian Analysis, Springer, N.Y., 1985.
[2] Maitra A.P., Sudderth W.D., Discrete Gambling and Sochastic Games, Springer, 1996.
Recommended references:
[3] Fishman G.S., Monte Carlo, Springer, 1996.
[4] Bernardo J.M., Smith A.F.M., Bayesian Theory, Wiley, 1994.


Bachelor's Degree Project 1  01BPAI12 
Strachota 
0+5 z 
0+10 z 
5 
10 
Course:  Bachelor's Degree Project 1  01BPAI1  Ing. Strachota Pavel Ph.D.          Abstract:  Bachelor's Degree project on the selected topic under the supervision. Supervision and regular checking of the bachalor project under preparation.  Outline:  Bachelor's Degree project on the selected topic under the supervision.  Outline (exercises):   Goals:   Requirements:  The ability of the independent students work and skills.  Key words:   References  
Course:  Bachelor's Degree Project 2  01BPAI2  Ing. Strachota Pavel Ph.D.          Abstract:  Bachelor's Degree project on the selected topic under the supervision. Supervision and regular checking of the bachalor project under preparation.  Outline:  Bachelor's Degree project on the selected topic under the supervision.  Outline (exercises):   Goals:   Requirements:   Key words:   References  

Bachelor Thesis 1  01BPAM12 
Strachota 
0+5 z 
0+10 z 
5 
10 
Course:  Bachelor Thesis 1  01BPAM1  Ing. Strachota Pavel Ph.D.          Abstract:  Bachelor's Degree project on the selected topic under the supervision. Supervision and regular checking of the bachalor project under preparation.  Outline:  Bachelor's Degree project on the selected topic under the supervision.  Outline (exercises):   Goals:  Knowledge: particular theme depending
on a given project topic.
Abilities: working unaided on a given task, understanding the problem, producing the original specialist text.  Requirements:  The ability of the independent students work and skills.  Key words:  Bachelor's degree project, research, development, mathematical models, applications.  References  Individual 
Course:  Bachelor Thesis 2  01BPAM2  Ing. Strachota Pavel Ph.D.          Abstract:  Bachelor's Degree project on the selected topic under the supervision. Supervision and regular checking of the bachalor project under preparation.  Outline:  Bachelor's Degree project on the selected topic under the supervision.  Outline (exercises):   Goals:  Knowledge: particular theme depending
on a given project topic.
Abilities: working unaided on a given task, understanding the problem, producing the original specialist text.
 Requirements:  The ability of independent students work and skills.  Key words:  Bachelor's degree project, research, development, mathematical models, applications.  References  Individual 

Bachelor Thesis 1  01BPMM12 
Strachota 
0+5 z 
0+10 z 
5 
10 
Course:  Bachelor Thesis 1  01BPMM1  Ing. Strachota Pavel Ph.D.  0+5 Z    5    Abstract:  Bachelor's Degree project on the selected topic under the supervision. Supervision and regular checking of the bachalor project under preparation.  Outline:  Bachelor's Degree project on the selected topic under the supervision.  Outline (exercises):   Goals:  Knowledge: particular theme depending
on a given project topic.
Abilities: working unaided on a given task, understanding the problem, producing the original specialist text.  Requirements:  The ability of the independent students work and skills.  Key words:  Bachelor's degree project, research, development, mathematical models, applications.  References  Individual 
Course:  Bachelor Thesis 2  01BPMM2  Ing. Strachota Pavel Ph.D.    0+10 Z    10  Abstract:  Bachelor's Degree project on the selected topic under the supervision. Supervision and regular checking of the bachalor project under preparation.  Outline:  Bachelor's Degree project on the selected topic under the supervision.  Outline (exercises):   Goals:  Knowledge: particular theme depending
on a given project topic.
Abilities: working unaided on a given task, understanding the problem, producing the original specialist text.
 Requirements:  The ability of independent students work and skills.  Key words:  Bachelor's degree project, research, development, mathematical models, applications.  References  Individual 

Bachelor Thesis 1  01BPSI12 
Strachota 
0+5 z 
0+10 z 
5 
10 
Course:  Bachelor Thesis 1  01BPSI1  Ing. Strachota Pavel Ph.D.  0+5 Z    5    Abstract:  Bachelor's Degree project on the selected topic under the supervision. Supervision and regular checking of the bachalor project under preparation.  Outline:  Bachelor's Degree project on the selected topic under the supervision.  Outline (exercises):   Goals:  Knowledge: particular theme depending
on a given project topic.
Abilities: working unaided on a given task, understanding the problem, producing the original specialist text.
 Requirements:  The ability of the independent students work and skills.  Key words:  Bachelor's degree project, research, development, mathematical models, applications.  References  Individual 
Course:  Bachelor Thesis 2  01BPSI2  Ing. Strachota Pavel Ph.D.    0+10 Z    10  Abstract:  Bachelor's Degree project on the selected topic under the supervision. Supervision and regular checking of the bachalor project under preparation.  Outline:  Bachelor's Degree project on the selected topic under the supervision.  Outline (exercises):   Goals:  Knowledge: particular theme depending
on a given project topic.
Abilities: working unaided on a given task, understanding the problem, producing the original specialist text.  Requirements:  The ability of the independent students work and skills.  Key words:  Bachelor's degree project, research, development, mathematical models, applications.  References  Individual 

Bachelor Seminar  01BSEM 
Strachota 
  
0+2 z 
 
2 
Course:  Bachelor Seminar  01BSEM  Ing. Strachota Pavel Ph.D.    0+2 Z    2  Abstract:  Bachelor seminar  technical details of bachelor thesis, format and processing, prerequisities, individual student presentations of their research results.  Outline:  Bachelor seminar  technical details of bachelor thesis, format and processing, prerequisities, individual student presentations of their research results.  Outline (exercises):   Goals:  Verify the quality of bachelor thesis and the presentation skills of individual student.  Requirements:  The ability of creating own professional presentation on the topic of bacholor thesis.  Key words:  Bachelor thesis, its defence, form of presentation, presentation itself.  References  Individual 

Differential Equations  01DIFR 
Beneš 
  
2+2 z,zk 
 
4 
Course:  Differential Equations  01DIFR  prof. Dr. Ing. Beneš Michal    3+1 Z,ZK    4  Abstract:  The course contains introduction in the solution of ordinary differential equations. It contains a survey of equation types solvable analytically, basics of the existence theory, solution of linear types of equations and introduction in the theory of boundaryvalue problems.  Outline:  1. Introduction  motivation in applications
2. Basics  theory of ordinary differential equations
3. Particular types of 1storder ODEs.
 separated and separable equations
 homogeneous equations
 equations with the rational argument of the righthand side
 linear equations
 Bernoulli equations
 Riccati equations
 Equations x=f(y') a y=f(y')
4. Existence theory for equations y'=f(x,y)
 Peano theorem
 Osgood theorem
5. Sensitivity on the righthand side and on the initial conditions
6. Linear nth order differential equations
7. Systems of 1st order linear differential equations
8. Boundaryvalue problems  Outline (exercises):  1. Equations with separated variables
2. Separable equations
3. Homogeneous differential equations
4. Generalized (quasihomogeneous) differential equations
5. Equations with rational righthandside argument s racionálním argumentem
6. Linear 1storder differential equations
7. Bernoulli equations
8. Riccati equations
9. Differential equations x=f(y') a y=f(y')
10. Linear nth order differential equations
with constant coefficients
11. Fundamental system for linear nth order differential equations
12. Systems of linear 1st order differential equations with constant coefficients  Goals:  Knowledge:
analytical solution of selected types of equations, the basics of the existence theory, solution of linear types of equations
Skills:
Analytical solution of the known types of ordinary differential equations, mathematical analysis of the initialvalue problems, solution of linear nth order differential equations and of the system of 1storder linear ordinary differential equations.  Requirements:  Basic course of Calculus, Linear Algebra (in the extent of the courses 01MA1, 01MAA24, 01LA1, 01LAA2 held at the FNSPE CTU in Prague).
 Key words:  Initialvalue problems for differential equations, Euler approximation, Peano theorem, fundamental system, wronskian, method of variation of constants.
 References  Key references:
[1] D. Schaeffer and J. Cain, Ordinary Differential Equations: Basics and Beyond, SpringerVerlag New York Inc., 2016
[2] F.Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer, Berlin 1990
[3] L.S.Pontrjagin, Obyknovennyje differencialnyje uravnenija, Nauka, Moskva 1965
[4] M.W.Hirsch and S.Smale, Differential Equations, Dynamical systems, and Linear Algebra, Academic Press, Boston, 1974
Recommended references:
[5] A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, Chapman and Hall/CRC Press, Boca Raton, 2003
[6] W. Walter, Gewöhnliche Differenzialgleichungen, Springer, Berlin 1990
[7] J. Kluvánek, L. Mišík a M. Švec. Mathematics II, SVTL Bratislava 1961 (in Slovak)
[8] K. Rektorys a kol. Survey of Applied Mathematics, Prometheus, Praha 1995 (in Czech) 

Discrete Mathematics 1  01DIM12 
Masáková 
2+0 z 
2+0 z 
2 
2 
Course:  Discrete Mathematics 1  01DIM1  prof. Ing. Masáková Zuzana Ph.D.  2+0 Z    2    Abstract:  The seminar is devoted to elementary number theory and applications. It includes individual problem solving.  Outline:  1. Divisibility, congruences, Femat's little theorem.
2. Linear diofantic equations, linear congruences, Chinese remainder theorem.
3. Euler's function, Euler´s theorem, Moebius function, inclusion exclusion principle.
4. Perfect numbers, Mersenne's primes, Fermat's numbers.
5. Primality testing. Public key cryptographic systems: RSA, knapsack problem.  Outline (exercises):   Goals:  Acquired knowledge: Students learn to solve some types of elementary number theoretical problems.
Acquired skills: The emphasis is put on correct formulation of mathematical ideas and logic process.  Requirements:  Knowledge of grammer school mathematics is assumed.  Key words:  Modular arithmetics, Euler's function, primes, RSA  References  Ronald L. Graham, Donald E. Knuth, Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, Reading, Massachusetts: AddisonWesley, 1994
J. Herman, R. Kučera, J. Šimša,
Equations and Inequalities: Elementary Problems and Theorems in Algebra and Number Theory. 1. vyd. New York : SpringerVerlag,
2000. 355 s. Canadian Mathematical Society Books in Math.
P. Erdös, J. Surányi, Topics in the Theory of Numbers,
SpringerVerlag, 2001.
M. Křížek, F. Luca, L. Somer,
17 Lectures on Fermat Numbers: From Number Theory to Geometry, CMS Books in Mathematics, vol. 9, SpringerVerlag, New
York, 2001.

Course:  Discrete Mathematics 2  01DIM2  prof. Ing. Masáková Zuzana Ph.D.    2+0 Z    2  Abstract:  The seminar is devoted to recurrence relations. It includes individual problem solving.  Outline:  1. Recurrence relations: Linear difference equations, some types of nonlinear recurrences, inverting formula.
2. Josephus problem.
3. Fibonacci numbers and Wythoff's game.
4. Integer coefficient polynomials and their rational roots, Viete relations.
5. Finite groups.  Outline (exercises):   Goals:  Acquired knowledge: Students learn to solve linear recurrence relations with constant coefficients and some other types of difference equations.
Acquired skills: The emphasis is put on correct formulation of mathematical ideas and logic process.
 Requirements:  Knowledge of grammer school mathematics is assumed. Also knowledge of FNSPE courses 01MA1, 01LA1 is needed.  Key words:  recurrence relations, difference equations,
Josephus problem, Fibonacci numbers  References  Ronald L. Graham, Donald E. Knuth, Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, Reading, Massachusetts: AddisonWesley, 1994
P. Cull, M. Flahive, R. Robson, Difference Equations, Springer, 2005.
J. Herman, R. Kučera, J. Šimša,
Equations and Inequalities: Elementary Problems and Theorems in Algebra and Number Theory. 1. vyd. New York : SpringerVerlag,
2000. 355 s. Canadian Mathematical Society Books in Math.


Discrete Mathematics 3  01DIM3 
Masáková 
2+0 z 
  
2 
 
Course:  Discrete Mathematics 3  01DIM3  prof. Ing. Masáková Zuzana Ph.D.  2+0 Z    2    Abstract:  The subject is devoted to elementary proofs of nontrivial combinatoriwal identities and to generating functions and their applications. In the seminar students present a problem with solution chosen from the given literature.  Outline:  1. Methods of combinatorial proof.
2. Stirling, Bernoulli, Catalan and Bell numbers.
3. Ordinary, exponential and Dirichlet generating functions. 4. Evaluation of sums, solution of linear and nonlinear difference equations.
5. Combinatorial interpretation of product and composition of generating functions.
6. Applications in number theory and graph theory.  Outline (exercises):   Goals:  Students learn methods of combinatorial proof, use of generating functions for solution of difference equations and for proving combinatorial identities. Students also learn comprehension of English written mathematical text and learn to present it to others.  Requirements:  Knowledge of FNSPE courses 01MA1, 01MAA2, 01LA1, 01LAA2 is required.  Key words:  generating functions, combinatorial identities, difference equations  References  M. Aigner, G. M. Ziegler, Proofs from the Book, SpringerVerlag 2004
A. T. Benjamin, J. J. Quinn, Proofs that Really Count, The Art of Combinatorial Proof, The Mathematical Association of America, 2003.
A. M. Yaglom, I. M. Yaglom, Challenging Mathematical Problems with Elementary Solutions, Dover Publications, 1987.
H. Dörrie, 100 Great Problems of Elementary Mathematics, Dover Publications, 1965.
Kombinatorické počítání 1999 , KAMDIMATIA Series preprint no. 451 (1999), 59 p 

Master Thesis 1  01DPAM12 
Burdík 
0+10 z 
0+20 z 
10 
20 
Course:  Master Thesis 1  01DPAM1  Ing. Ambrož Petr Ph.D. / prof. RNDr. Burdík Čestmír DrSc.          Abstract:  Master's thesis preparation.  Outline:   Outline (exercises):   Goals:  Knowledge: a particular field depending on the given project topic.
Abilities: working unaided on a given task, understanding the problem, producing an original specialist text.  Requirements:   Key words:  Master's thesis.  References  
Course:  Master Thesis 2  01DPAM2  Ing. Ambrož Petr Ph.D. / prof. RNDr. Burdík Čestmír DrSc.          Abstract:  Master's thesis preparation.  Outline:   Outline (exercises):   Goals:  Knowledge: a particular field depending on the given project topic.
Abilities: working unaided on a given task, understanding the problem, producing an original specialist text.  Requirements:   Key words:  Master's thesis.  References  

Master Thesis 1  01DPMM12 
Burdík 
0+10 z 
0+20 z 
10 
20 
Course:  Master Thesis 1  01DPMM1  Ing. Ambrož Petr Ph.D. / prof. RNDr. Burdík Čestmír DrSc.  0+10 Z    10    Abstract:  Master's thesis preparation.  Outline:   Outline (exercises):   Goals:  Knowledge: a particular field depending on the given project topic.
Abilities: working unaided on a given task, understanding the problem, producing an original specialist text.  Requirements:   Key words:  master's thesis  References  
Course:  Master Thesis 2  01DPMM2  Ing. Ambrož Petr Ph.D. / prof. RNDr. Burdík Čestmír DrSc.    0+20 Z    20  Abstract:  Master's thesis preparation.  Outline:   Outline (exercises):   Goals:  Knowledge: a particular field depending on the given project topic.
Abilities: working unaided on a given task, understanding the problem, producing an original specialist text.  Requirements:   Key words:  Master's thesis.  References  

Master Thesis 1  01DPSI12 
Burdík 
0+10 z 
0+20 z 
10 
20 
Course:  Master Thesis 1  01DPSI1  Ing. Ambrož Petr Ph.D. / prof. RNDr. Burdík Čestmír DrSc.  0+10 Z    10    Abstract:  Master's thesis preparation.  Outline:   Outline (exercises):   Goals:  Knowledge: a particular field depending on the given project topic.
Abilities: working unaided on a given task, understanding the problem, producing an original specialist text.  Requirements:   Key words:  master's thesis  References  
Course:  Master Thesis 2  01DPSI2  Ing. Ambrož Petr Ph.D. / prof. RNDr. Burdík Čestmír DrSc.    0+20 Z    20  Abstract:  Master's thesis preparation.  Outline:   Outline (exercises):   Goals:  Knowledge: a particular field depending on the given project topic.
Abilities: working unaided on a given task, understanding the problem, producing an original specialist text.  Requirements:   Key words:  master's thesis  References  

Differential Calculus on Manifolds  01DPV 
Tušek 
  
2+0 zk 
 
2 
Course:  Differential Calculus on Manifolds  01DPV  Ing. Tušek Matěj Ph.D.    2+0 ZK    2  Abstract:  Smooth manifold, tangent space differential forms, tensors, Riemannian metrics and manifold, covariant derivative, parallel transport, orientation of manifold, itegration on manifold and Stokes theorem.
 Outline:  1. Smooth manifolds 2. Tangent and cotangent space 3. Tensors, differential forms 4. Orientation of manifold, integration on manifold 5. Stokes theorem 6. Riemannian manifold.  Outline (exercises):   Goals:  Knowledge: To get acquainted with basic notions of differential geometry with emphasis on mathematical details.
Abilities: Consequently, to be able to selfstudy advanced physical (not only) literature.  Requirements:  Basic course in Calculus and Linear Algebra and topology (in the extent of the courses 01MA1, 01MAA24, 01LA1, 01LAA2, 01TOP held at the FNSPE CTU in Prague).  Key words:  Differential geometry, Riemannian manifold, Stokes theorem.  References  key references:
[1] J.M. Lee: Introduction to Smooth Manifolds, Springer, 2003.
recommended references:
[2] J. M Lee: Riemannian Manifolds: An Introduction to Curvature, Springer, 1997.
[3] M. Spivak: Calculus on Manifolds, AddisonWesley Publishing Company, 1965.
[4] F. Morgan: Riemannian Geometry: A Begginer's Guide, Jones and Bartlett Publishers, 1993. 

Dynamic Decision Making 1  01DRO1 
Guy, Kárný 
  
2+0 zk 
 
2 
Course:  Dynamic Decision Making 1  01DRO1  Ing. Guy Tatiana Ph.D. / Ing. Kárný Miroslav          Abstract:  1. An abstraction of real decisionmaking problems
2. Decisionmaking elements (decision maker, its environment, behaviour of the closed decision loop, strategy, constraints)
3. Quantification of the decisionmaking task (harmonised quantitative modelling of preferences among behaviours and strategies)
4. A final formalised decisionmaking task and its elements (probabilistic models and performance index)
5. Fully probabilistic design as the optimisation of a universal expected performance index.
6. Tools for solving of dynamic decisionmaking task (dynamic programming, its additive and datadriven versions)
7. General tools for constructing decsionmaking elements
 Outline:  1. An abstraction of real decisionmaking problems
2. Decisionmaking elements (decision maker, its environment, behaviour of the closed decision loop, strategy, constraints)
3. Quantification of the decisionmaking task (harmonised quantitative modelling of preferences among behaviours and strategies)
4. A final formalised decisionmaking task and its elements (probabilistic models and performance index)
5. Fully probabilistic design as the optimisation of a universal expected performance index.
6. Tools for solving of dynamic decisionmaking task (dynamic programming, its additive and datadriven versions)
7. General tools for constructing decsionmaking elements
 Outline (exercises):   Goals:  Acquired knowledge: Abstraction of dynamic decision making under uncertainty, incomplete knowledge and constraints (technological, informational and computational); a general methodology of the formalisation and solution of the decisionmaking task
Abilities: to understand the formalisation way of a general decsionmaking problem, to grasp what are its elements and methods for their construction as well as methods for solving the optimal decsionmaking problems
 Requirements:   Key words:   References  Recommended literature: selected parts from
[1] M. Kárný, J. Bohm, T.V. Guy, L. Jirsa, I. Nagy, P. Nedoma, and L. Tesař. Optimized Bayesian Dynamic Advising: Theory and Algorithms. Springer, London, 2006.
[2] M. Kárný, T.V. Guy. Fully probabilistic control design. Systems & Control Letters, 55(4), 2006.
[3] M. Kárný and T. Kroupa. Axiomatisation of fully probabilistic design. Information Sciences, 186(1), 2012.
Needed support: lecture room with projector


Dynamic Decision Making 2  01DRO2 
Guy, Kárný 
2+0 zk 
  
2 
 
Course:  Dynamic Decision Making 2  01DRO2  Ing. Guy Tatiana Valentine Ph.D. / Ing. Kárný Miroslav          Abstract:  1.Overview of the formalised decisionmaking task and tools for its solution
2.Application of the general fully probabilistic design of decisionmaking strategies for Markov chains and linearGaussian models
3.Aproximation and completion of probabilities serving to processing databased as well as probabilistic knowledge and preferences for Markov chains
4.Introduction into multiparticipants decision making and its formalisation
5.Usability of general tools for knowledge sharing and cooperation within multipleparticipants decision making
6.Ilustrative case studies of solving decisionmaking tasks
7.Open decisionmaking problems
 Outline:  1.Overview of the formalised decisionmaking task and tools for its solution
2.Application of the general fully probabilistic design of decisionmaking strategies for Markov chains and linearGaussian models
3.Aproximation and completion of probabilities serving to processing databased as well as probabilistic knowledge and preferences for Markov chains
4.Introduction into multiparticipants decision making and its formalisation
5.Usability of general tools for knowledge sharing and cooperation within multipleparticipants decision making
6.Ilustrative case studies of solving decisionmaking tasks
7.Open decisionmaking problems
 Outline (exercises):   Goals:  Acquired knowledge: Deepening of the insight into the general formalisation and the solution methodology of reallife decisionmaking tasks addressed under uncertainty and incomplete knowledge: all this is gained during the lecture 01DRO1
Abilities: To formalise a specific reallife decisionmaking problem, to fill its elements with appropriately chosen methods for their constructing as well as for solving the resulting formalised decisionmaking problem
 Requirements:   Key words:   References  Recommended literature: selected parts from
[1] M. Kárný, J. Bohm, T.V. Guy, L. Jirsa, I. Nagy, P. Nedoma, and L. Tesař. Optimized Bayesian Dynamic Advising: Theory and Algorithms. Springer, London, 2006.
[2] M. Kárný, T.V. Guy. Fully probabilistic control design. Systems & Control Letters, 55(4), 2006.
[3] M. Kárný, T.V. Guy Tatiana Valentine: On the Origins of Imperfection and Apparent NonRationality, 5792, in T.V. Guy, M. Kárný, D.H. Wolpert, Decision Making: Uncertainty, Im perfection, Deliberation and Scalability, Springer, Studies in Computational Intelligence 538, 2014
The needed support: lecture room with projector


Seminar Course on Dynamic Decision Making  01DROS 
Guy, Kárný 
  
2+0 z 
 
2 
Course:  Seminar Course on Dynamic Decision Making  01DROS  Ing. Guy Tatiana Ph.D. / Ing. Kárný Miroslav          Abstract:  The seminar is devoted to the actual topics and trends in decision making, machine learning (ML) and artificial intelligence (AI). It will extend the topics learned in the lecture course 01DRO1, in particular formalisation of DM problem and its solution incl. techniques to tackle the problem; multiagent DM and related tasks incl. possible ways of agents? interaction.
A subselection of relevant articles presented at the main DM, ML and AI conferences will be discussed.  Outline:  A part of the seminal will be devoted to practical tasks, in particular:
 Practical introduction into ML and business intelligence will be demonstrated on a real project. The seminar will connect various courses studied within MI or AMSM with practical applications (taxis, insurance, ecommerce,...), visualization in Qlik (Tableau), programming in R/Python.
 Description of workflow needed for using linear models and logistic regression (data preparation, preprocessing, scaling, etc.) will be illustrated on unfinished calls in an insurance company call centre.
 Practical use of decision trees, random forests, and gradient boosting technique will be demonstrated. Preparing a customer?s action for an insurance company will serve as a practical example.
 Processing of natural language will be demonstrated on searching potential employees using methods TFIDF and Word2vec serving for evaluation texts in CV databases.
 Introduction into futures trading incl. main principles, common strategies used and open problems.
 Concept of lazy learning for DM will be illustrated on a nontrivial problem of helicopter flight stabilisation.  Outline (exercises):   Goals:  Acquired knowledge:
The elective seminar course focuses on using methods and algorithms of dynamical decision making under uncertainty (DM). Theoretical knowledge of DM will be complemented by examples of their practical use. This will enhance the students? ability to solve practical decision tasks and inspire further theoretic and algorithmic research.
Acquired skills:
The seminar will support understanding how to design elements and methods inevitable for optimised decision making.
 Requirements:  The seminar is intended for all students who attend the lecture course 01DRO1 and are interested in theoretic solutions and/or their subsequent practical use.
Individual work of students will consist of active participation to seminars, which is necessary condition for getting the allocated credits.  Key words:  dynamic decision making; uncertainty and incomplete knowledge; environment, its dynamics, models and their learning; aims and preferences  References  Compulsory literature:
[1] M. L. Puterman: Markov Decision Processes, Wiley, 1994. (selected chapters).
[2] R. S. Sutton, A. G. Barto: Reinforcement Learning: An Introduction MIT Press, Cambridge, 1998 (selected chapters).
Optional literature:
[3] S. French. Decision Theory. Halsted Press, 1986.
[4] L. Savage. The Foundations of Statistics. Wiley, 1954.
[5] D. P. Bertsekas. Dynamic Programming & Optimal Control, 1,2. Athena Scientific Press, 2005 

Diploma Seminar  01DSEMI 
Burdík 
  
0+2 z 
 
3 
Course:  Diploma Seminar  01DSEMI  Ing. Ambrož Petr Ph.D. / prof. RNDr. Burdík Čestmír DrSc.          Abstract:  Preparation of the thesis defense.  Outline:   Outline (exercises):   Goals:  Knowledge:
Essentials of the diploma thesis defense, process of the state exams.
Abilities:
Preparation of slides and presentation of result of the master's thesis to a scientific audience.  Requirements:  To be enrolled exclusively in the last term.  Key words:  Master's thesis, defence.  References  

Introduction to Continuum Dynamics  01DYK 
Fučík, Strachota 
  
0+2 z 
 
2 
Course:  Introduction to Continuum Dynamics  01DYK  Ing. Fučík Radek Ph.D. / Ing. Strachota Pavel Ph.D.          Abstract:  This course is an introduction to the mathematical description of continuum dynamics. It summarizes the necessary mathematical apparatus with emphasis on vector and tensor calculus, differential forms, and integration on manifolds. It includes the basic concepts of continuum mechanics such as strain and stress tensors or substantial derivative, by means of which it is possible to derive the fundamental laws of conservation of mass, momentum, angular momentum, and energy in integral and differential form. In the last part of the course, these conservation laws are adapted to the case of viscous and inviscid fluid and linear and nonlinear elastic body.  Outline:  1. Mathematical background
a) vector and tensor calculus
b) differential forms
c) integration on manifolds
2. Basic concepts of continuum mechanics
a) movement and deformation of continuum
b) the strain tensor and small strain tensor
c) decomposition of deformation, rotation
d) substantial derivative of scalar, vector and volume quantities
3. Conservation laws
a) conservation of mass
b) conservation of momentum
c) conservation of angular momentum
d) conservation of mechanical energy
e) conservation of total energy
4. Constitutive relations
a) inviscid fluid
b) viscous fluid
c) nonlinear elastic body
d) linear elastic body
5. Selected applications
 Outline (exercises):   Goals:  Knowledge:
The basic principles of continuum mechanics description. Conservation laws for mass, momentum, angular momentum, and energy. Constitutive equations for viscous and inviscid fluid. Constitutive relations for linear and nonlinear elastic body.
Abilities:
Derivation of basic conservation laws. Derivation of the constitutive relations for the case of fluid or elastic body.  Requirements:  Basic courses in calculus, linear algebra, theoretical physics and differential equations (according lectures at CTU in Prague 01DIFR, 01LA1, 01LAA2, 01MA1, 01MAA2, 01MAA3, 02TEF1).
 Key words:  strain rate tensor, stress tensor, Stokesian fluid, ideal fluid, Newtonian fluid, continuity equation, Euler equations, NavierStokes equations. Conservation laws.  References  Mandatory reading:
[1] Gurtin, Morton E. An introduction to continuum mechanics. Vol. 158. Academic Pr, 1981.
[2] Anderson, John D. Computational Fluid Dynamics: The Basics with Applications. McGrawHill, 1995.
Recommended reading:
[2] Chorin, Alexandre Joel, and Jerrold E. Marsden. A mathematical introduction to fluid mechanics. Springer, 1990.
[3] Maršík, F. Termodynamika kontinua. Academia, 1999.


Theory of Dynamic Systems  01DYSY 
Rehák 
  
3+0 zk 
 
3 
Course:  Theory of Dynamic Systems  01DYSY  Mgr. RNDr. Augustová Petra Ph.D.    3+0 ZK    3  Abstract:  The course provides an introduction to system theory with emphasis on control theory and understanding of the fundamental concepts of systems and control theory. First, we build up the understanding of the dynamical behavior of systems as well as provide the necessary mathematical background. Internal and external system descriptions are described in detail, including state variable, impulse response and transfer function, polynomial matrix, and fractional representations. Stability, controllability, observability, and realizations are explained with the emphasis always being on fundamental results. State feedback, state estimation, and eigenvalue assignment are discussed in detail. All stabilizing feedback controllers are also parameterized using polynomial and fractional system representations. The emphasis in this primer is on linear timeinvariant systems, both continuous and discrete time.  Outline:  1. Introduction to the general theory of systems (decision, control, control structures, object, model, system).
2. Description of the systems (inputoutput and state space description of the system, stochastic processes and systems, system coupling).
3. Innner dynamics, input output constraints (solution of state space equations, modes of the system, response of continuous and discrete time systems, stability, reachability and observability).
4. Modification of dynamic properties of the system (state feedback, state reconstruction, separation principle, decomposition and system realization, system sensitivity analysis).
5. Control (state feedback, feedback control systems).
 Outline (exercises):   Goals:  Knowledges: Students will emerge with a clear picture of the dynamical behavior of linear systems and their advantages and limitations.
Skills: They will be able to describe the system, analyze its properties (stability, controllability, observability) and apply the system theory to particular examples in physics and engineering.
 Requirements:  Undergraduatelevel differential equations and linear algebra (in the extent of the courses 01DIFR, 01LA1, 01LAA2 held at the FNSPE CTU in Prague).  Key words:  Dynamic systems, linear systems, stability, controllability, observability, linearization, control theory.  References  Key references
[1] P. J. Antsaklis, A. N. Michel: A Linear Systems Primer. Birkhäuser, 2007. ISBN13: 9780817644604
Recommended references
[1] Mikleš, J. a Fikar, M., Process Modelling, Identification, and Control, Springer Verlag, Berlin, 2007. ISBN13: 9783540719694
[2] P. J. Antsaklis and A. N. Michel, Linear Systems, Birkhäuser, Boston, MA, 2006. ISBN13: 9780817644345
[3] T. Kailath: Linear systems. PrenticeHall, Englewood Cliffs, NJ, 1980. ISBN13: 9780135369616


Elementary Introduction to Graph Theory  01EIGR 
Ambrož, Masáková 
2+0 kz 
  
2 
 
Course:  Elementary Introduction to Graph Theory  01EIGR  Ing. Ambrož Petr Ph.D. / prof. Ing. Masáková Zuzana Ph.D.          Abstract:  The course provides an explanation of basic graph theory followed by a survey of common graph algorithms.  Outline:  1. Basic and enumerative combinatorics.
2. The Notion of a graph.
3. Trees and spanning trees.
4. Eulerian trails and Hamiltonian cycles.
5. Flows in networks.
6. Colouring and matching.
7. Planar graphs.
 Outline (exercises):   Goals:  Knowledge:
Basic notions of graph theory.
Skills:
Orienation in elementary graph algorithms and their use for solving practical problems.  Requirements:   Key words:   References  Povinná literatura:
[1] J.A. Bondy, U.S.R. Murty: Graph theory. Graduate Texts in Mathematics 244. Springer, New York, (2008).
Doporučená literatura:
[2] M. Bóna. A Walk Through Combinatorics. World Scientific, Singapoore (2006)
[3] Ján Plesník. Grafové algoritmy. Veda, Bratislava, (1983).


Functional Analysis 2  01FA2 
Šťovíček 
  
2+2 z,zk 
 
4 
Course:  Functional Analysis 2  01FA2  prof. 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, HilbertSchmidt operators, spectral decomposition of bounded selfadjoint operators.
 Outline:  1. The Baire theorem, the BanachSteinhaus 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 ArzelaAscoli theorem, HilbertSchmidt operators.
4. The Weyl criterion for normal operators, properties of spectra of bounded selfadjoint operators.
5, The spectral decomposition of bounded selfadjoint 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, HilbertSchmidt operators.
6. Examples of the spectral decomposition of bounded selfadjoint 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 ArzelaAscoli theorem, HilbertSchmidt operators, spectrum of a bounded operator, selfadjoint operator, spectral decomposition  References  Key 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, (McGrewHill, 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) 

Functional Analysis 3  01FA3 
Šťovíček 
2+1 z,zk 
  
3 
 
Course:  Functional Analysis 3  01FA3  prof. Ing. Šťovíček Pavel DrSc.  2+1 Z,ZK    3    Abstract:  Advanced parts of functional analysis needed for modern quantum theory.  Outline:  1. Bounded operators in Hilbert spaces, summarization.
2. Topological vector spaces. Schwartz space.
3. Fourier transform.
4. Compact operators.
5. Closed operators, unbounded operators.
6. Symmetric operators, selfadjoint operators.
7. Symmetric extension of symmetric operators. Problem of selfadjoint extensions.
8. Cayley transform.
9. First and Second John von Neumann formula.
 Outline (exercises):  1. Summarization of properties of bounded operators on Hilbert spaces.
2. Compact operators.
3. Symmetric, selfadjoint, closed operators. Essential spectrum.  Goals:  The goal of study is to finish the basic functional analysis course oriented mainly on modern quantum theory and solving of problems which arise in physical and technical applications.  Requirements:  Basic calculus and linear algebra courses (01MANA, 01MAA24, 01LALA, 01LAA2), the first two parts of Functional analysis course (01FAN1, 01FA2).  Key words:  Linear operators, spectrum of linear operator, selfadjoint operator, Fourier transform.  References  Key references:
[1] Blank, Exner, Havlíček: Hilbert Space Operators in Quantum Physics, Springer 2008


Functional Analysis 1  01FAN1 
Šťovíček 
2+2 z,zk 
  
4 
 
Course:  Functional Analysis 1  01FAN1  prof. 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 HahnBanach 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 HahnBanach theorem, Banach space, Hilbert space, orthogonal projection, orthogonal basis, adjoint operator  References  Key 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, (McGrewHill, 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) 

Financial and Insurance Mathematics  01FIMA 
Hora 
2+0 zk 
  
2 
 
Course:  Financial and Insurance Mathematics  01FIMA  Hora Jan Mgr.  2 ZK    2    Abstract:  This course is an introduction to the problems of life and nonlife insurance and financial mathematics.  Outline:  1. Fundamentals of Financial Mathematics (interest, etc.) 2. Fundamentals of Demographics (especially mortality) 3. Life Insurance (premium, reserves, reinsurance) 4. Nonlife insurance (premium, reserves, reinsurance) 5. Financial Mathematics (Securities)  Outline (exercises):   Goals:  Principles of calculation and the calculation of insurance premiums, life insurance reserves, reserves for insurance benefits (especialy IBNR reserves), prices of selected securities  Requirements:  Basic course of probability and mathematical statistics  Key words:  Mortality, interest, net premium, gross premium, reserves, reinsurance, bonds, equities, options  References  Key references:
Actuarial Mathematics, Newton L. Bowers, Hans U. Gerber, James C. Hickman, Donald A. Jones, Cecil J. Nesbitt, Society of Actuaries; 2nd edition (May 1997)
Recommended references:
A Course in Financial Calculus, Alison Etheridge, Cambridge University Press, 2002


Functions of Complex Variable  01FKO 
Šťovíček 
  
2+1 z,zk 
 
3 
Course:  Functions of Complex Variable  01FKO  prof. Ing. Šťovíček Pavel DrSc.          Abstract:  The course starts from outlining the Jordan curve theorem and the RiemannStieltjes integral. Then basic results of complex analysis in one variable are explained in detail: the derivative of a complex function and the CauchyRiemann 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, pathconnected, simply connected spaces, the Jordan curve theorem
2. Variation of a function, length of a curve, the RiemannStieltjes integral (survey)
3. Derivative of a complex function, the CauchyRiemann 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 RiemannStieltjes 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, RiemannStieltjes integral, CauchyRiemann equations, Morera's theorem, isolated singularity, maximum modulus principle, Liouville's theorem, Cauchy estimates, Laurent series, residue theorem  References  Key references:
[1] W. Rudin: Real and Complex Analysis, (McGrewHill, Inc., New York, 1974)
Recommended references:
[2] J. B. Conway: Functions of One Complex Variable I, SpringerVerlag, New York, 1978


Geometric Theory of Ordinary Differential Equations  01GTDR 
Beneš 
0+2 z 
  
2 
 
Course:  Geometric Theory of Ordinary Differential Equations  01GTDR  prof. Dr. Ing. Beneš Michal  0+2 Z    2    Abstract:  The seminar consists of the qualitative theory of ODEs dealing with the geometric and topological properties of the solution. In this context, we mention suitably formulated basic results of the existence and uniqueness, continuous dependence on parameters and initial conditions. Main part is devoted to the autonomous systems.  Outline:  1. Basic theorem on the local existence and uniqueness of the solution
2. Theorem of continuous dependence on parameters
3. Differentiability with respect to parameters
4. Continuous dependence on initial conditions, and dfferentiability with respect to initial conditions
5. Basics of the theory of autonomous systems
6. Analysis of solution of autonomous systems (special solutions, phase space)
7. Exponentials of operators
8. Systems 2 x 2
9. Lyapunov stability
10. Limit cycles
11. Poincaré map
12. First integrals and integral manifolds
 Outline (exercises):   Goals:  Knowledge:
geometric theory of ordinary differential equations, autonomous systems, Lyapunov stability, limit cycles, Poincaré map
Skills:
Formulation of initial value problems for ordinary differential equation. Proving basic mathematical properties of given problems, geometric analysis of the solution.  Requirements:  Basic course of Calculus, Linear Algebra and Ordinary Differential Equations (in the extent of the courses 01MA1, 01MAA24, 01LA1, 01LAA2, 01DIFR held at the FNSPE CTU in Prague).
 Key words:  Ordinary differential equations, qualitative theory, parmeter dependence, autonomous systems, limit cycles, Poincaré map.  References  Key references:
[1] M.W.Hirsch, S.Smale, Differential Equations, Dynamical systems, and Linear Algebra, Academic Press, Boston, 1974
[2] F.Verhulst, Nonlinear Differential Equations and Dynamical Systems, SpringerVerlag, Berlin 1990
Recommended references:
[3] L. S. Pontryagin, Ordinary Differential Equations. AddisonWesley, London 1962
[4] D. Schaeffer and J. Cain, Ordinary Differential Equations: Basics and Beyond, SpringerVerlag New York Inc., 2016 

Hierarchical Bayesian Models  01HBM 
Šmídl 
  
2+0 kz 
 
2 
Course:  Hierarchical Bayesian Models  01HBM  doc. Ing. Šmídl Václav Ph.D.          Abstract:  Keywords:
Bayesian theory, linear regression, signal separation, mixture models, Bayesian filtering  Outline:  1. Fundamentals of Bayesian theory
2. Methods of approximate evaluation of Bayesian calculus (Variational Bayes, Importance Sampling, Gibbs Sampling)
3. Linear regression and structure selection algorithms (spike and slab, horseshoe prior, lasso, fused lasso, automatic relevance determination)
4. Signal separation and its variants as different prior models
5. Mixture models for clustering (using Gaussian and Beta components)
6. Estimation of relevant number of components in a mixture
7. Density representation in high dimensions (mixtures of factor analyzers, deep neural networks)
8. Bayesian filtering (Kalman and particle filter)
 Outline (exercises):   Goals:  Acquired knowledge:
Computational methods suitable for evaluation of hierarchical Bayesian models. Selected hierarchical models for common practical tasks. Relation of these models to classical approaches.
Acquired skills:
Ability to modify standard models to nonstandard problem formulations, incorporation of additional assumption into the model, development of computational method for the modified model
 Requirements:   Key words:   References  Compulsory literature:
[1] Bishop, C., Pattern Recognition and Machine Learning" Springer, New York, 2007.
Optional literature:
[2] Šmídl, Václav, and Anthony Quinn. The Variational Bayes Method in Signal Processing, Springer 2005.
Working environment:
Matlab


Languages, Automata and Computability  01JAVY 
Ambrož, Pelantová 
  
3+1 z,zk 
 
5 
Course:  Languages, Automata and Computability  01JAVY  Ing. Ambrož Petr Ph.D.          Abstract:  Finite automata and regular languages. Context free languages and pushdown automata. Unrestricted languages and Turing machines. Algorithms a algorithmically enumerable functions. Recursive functions, recursive sets and recursively enumerable sets. Algorithmically unsolvable problems.  Outline:  1. Finite automata, regular languages and operations, star lemmas. (3 lectures)
2. Kleene theorem (2 lectures)
3. Determinisation a minimisation (2 lectures)
4. Confextfree grammas and their reductions (2 lectures)
5. Pushdown automata and contextfree languages (2 lectures)
6. Star lemma for CFL, closure properties of CFL (2 lectures)
7. Turing machine, recursive and recursively enumerable languages, methods of design of turing machines (2 lectures)
8. Undecidability (1 lecture)
9. Rice theorem, Post correspondence problem, undecidable properties of CFL (2 lectures)  Outline (exercises):  1. Constructions of finite automata, use of star lemmas
2. Normalized and standard automata, regular operations
3. Conversion automata>regular expression: MNY algorithm, BMC algorithm, Arden lemma
4. Determinisation a minimisation of finite automata
5. Construction of pushdown automata, convesion pushdown automata>contextfree grammar and vice versa
6. Construction of Turing machines, reduction of undecidable problems  Goals:  Acquired knowledge:
Classical results of theory of formal languages, grammars and automata. Recursion theory as a mathematically rigorous definition of intuitive notion of the algorithm, used finite and constructive methods.
Acquired skills:
Orientation in the field of formal languages, finite descriptions of functions and set. Practical applications.
 Requirements:   Key words:  Language, grammar, automata, algorithm  References  Compulsory literature:
[1] J. Mareš: Jazyky, gramatiky a automaty. Vydavatelství ČVUT, Praha 2004.
[2] J. Mareš: Teorie vyčíslitelnosti. Vydavatelství ČVUT, Praha 2008.
Optional literature:
[1] J. Sakarovitch: Elements of Automata Theory, Cambridge University Press 2009.
[2] J.E. Hopcroft, J.D. Ullman: Introduction to Automata Theory, Languages, and Computation
(1st ed.). AddisonWesley 1979.
[3] J.E. Hopcroft, R. Motwani, J.D. Ullman: Introduction to Automata Theory, Languages, and Computation (3rd ed.). Pearson 2013.


Simple Compilers  01JEPR 
Čulík 
  
2 z 
 
2 
Course:  Simple Compilers  01JEPR  Ing. Čulík Zdeněk    2 Z    2  Abstract:  Lexical and syntax analysis, code generation, simple optimizations, development environments, reflection.  Outline:  1. Lexical and syntax analysis of programming languages (Pascal, C++, Java)
2. Data structures for representations of expressions, statements, types and declarations 3. Programs for automatic compiler generations (Lex, Yacc, ANTLR) Simple optimizations
4. Code generation
5. Principles of integrated development environments, influence of run time type identification.
6. Overview of syntax analysis and code generation in GNU Compiler Collection.
7. Other GNU programs for software development.  Outline (exercises):  1. Simple lexical analysis written in C programming language
2. Semantic analysis
3. Type and declaration processing, navigation trees in used in development environments
4. Semantic analyzers generated by ANTLR program
5. Simple code generation, register allocation
6. Extension modules for GCC and LLVM/CLang compilers
 Goals:  Knowledge:
Structure of programming language compilers, machine code generation, translation to another programming language
Skills:
To develop syntactic and semantic analysis of simple programming language using modern grammar generators.  Requirements:   Key words:  Programming languages, compilers.  References  [1] N. Wirth: Compiler Construction, Addison Wesley, 1996
[2] S. Pemberton, M. Daniels: Pascal Implementation: The P4 Compiler, Prentice Hall, 1983
[š] D. Grune, C. Jacobs: Parsing Techniques  A Practical Guide, Ellis Horwood, 1990
[4] http://www.antlr.org 

Combinatorics and Probability  01KAP 
Hobza 
2+0 zk 
  
2 
 
Course:  Combinatorics and Probability  01KAP  Ing. Kůs Václav Ph.D.  2+0 ZK    2    Abstract:  The course is devoted to combinatorial rules, definition of the probability, explication of random variable and its characteristics. It explains term of distribution function and examples of discrete and continuous random variables are mentioned. Emphasis is placed on using of these terms and rules.  Outline:  1.Combinatorial rules, variation, combination, permutation (with repetition and without repetition), properties of binomial coefficient, the binomial theorem
2. The classical definition of probability, the geometric probability definition, the mathematical model of probability (events, calculus of events, axioms of probability, dependence and independence of events)
3. Random variables (probability distribution function, discrete and continuous random variables and examples of these variables)
4. Expected value, variance, moments of random variables, the law of large numbers, the central limit theorem  Outline (exercises):   Goals:  Knowledge:
Basic combinatorial rules, fundamentals of probability theory.
Skills:
Application of theoretical knowledge to solution of concrete problems. Ability of calculation of probability (conditional and unconditional), computation of moments of random variables and application of the central limit theorem.  Requirements:  Basic course of Calculus
(in the exten of thecourses 01MAT1, 01MAT2 held at the FNSPE CTU in Prague).
 Key words:  Variation, combination, permutation, probability, event, random variable, distribution function, probability density function, discrete random variable, absolutely continuos variable, expected value, variance, the law of large numbers, the central limit theorem.  References  Keyreferences:
[1] D.C. Montgomery, G.C. Runger: Applied statistics and probability for engineers, Wiley, 2003
Recommended references:
[2] H. G. Tucker: An introduction to probability and mathematical statistics, Academic Press, 1963


Compressed Sensing  01KOS 
Vybíral 
2+0 zk 
  
2 
 
Course:  Compressed Sensing  01KOS  doc. RNDr. Vybíral Jan Ph.D.          Abstract:  The goal of this series of lectures is to get form the very basics of game theory and machine learning all the way to solid understanding of the algorithm used in DeepStack. We will explain how games can be formally modeled, what are meaningful definitions of optimal strategies in games and what is Nash equilibrium in particular. Afterwards, we will focus on simple learning mechanisms in repeated decisionmaking problems called multiarmed bandit problems. We will show basic properties of learning in these models and then investigate what happens if these algorithms are run against each other in a game. This will form the bases of an algorithm for computing Nash equilibria in simple zerosum games, which can be extended to Counterfactual Regret Minimization (CFR) in extensive form games. Next, we will explain why it is complicated to decompose extensive form game to independent parts and how CFRD can solve this problem under certain conditions. Finally, we will briefly introduce deep neural networks and combine all the introduced components to the first algorithm that was able to beat professional poker players.  Outline:   Outline (exercises):   Goals:  Acquired knowledge: Students will learn the basic concepts of game theory. They will acquire the theoretical foundation for understanding the stateofart algorithms for solving the games with incomplete information, like card games.
Acquired skills: Modeling problems from game theory and proposing algorithms of artificial intelligence for their solution.
 Requirements:   Key words:  Sparsity and solution of underdetermined systems of linear equations, basis pursuit, null space property, coherence a restricted isometry property, l1minimization, Gaussian random matrices and JohnsonLindenstrauss imbedding  References  Compulsory literature:
M. Moravčík, M. Schmid, N. Burch, V. Lisý, D. Morrill, N. Bard, T. Davis, K. Waugh, M. Johanson, M. Bowling, DeepStack: Expertlevel artificial intelligence in headsup nolimit poker, Science, Vol. 356, Issue 6337, pp. 508513 (2017)
Optional literature:
J. Nash, Equilibrium points in nperson games, Proceedings of the national academy of sciences 36.1, pp. 4849 (1950)
N. Nisan, T. Roughgarden, E. Tardos, V. Vazirani: Algorithmic Game Theory, Cambridge University Press, 2007


Quantum Groups 1  01KVGR1 
Burdík 
2+0 z 
  
2 
 
Course:  Quantum Groups 1  01KVGR1  prof. RNDr. Burdík Čestmír DrSc.  2+0 Z    2    Abstract:  Quantum Algebra was originated in the 80s in the works of professor L. D. Faddeev and the Leningrad school on the inverse scattering method in order to solve integrable models. They have many applications in mathematics and mathematical physics such as the classification of nodes, in the theory of integrable systems and the string theory.  Outline:  1. Motivation, coalgebras, bialgebras and Hopf algebras. 2. Qcalculus. 3 The quantum algebra U_q(sl(2) and its representations. 4. The quantum group SL_q(2) and its representations. 5. The qOscillator algebras and their representations. 6. DrinfeldJimbo algebras, 7. FiniteDimensional representations of DrinfeldJimbo Algebras. 8. Quasitriangularity and universal R matrix.  Outline (exercises):   Goals:  Knowledge: to acquire the mathematical basis of the quantum group theory. Abilities: able to use the quantum group theory in studying integrable systems.  Requirements:  Basic course of Calculus and Linear Algebra (in particular, the courses 01MA1, 01MAA24, 01LAP, 01LAA2, TRLA held at the FNSPE CTU in Prague).
 Key words:  Hopf algebra, qcalculus, Drinfeld double, quasitriangularity, universal Rmatrix.  References  Key references: [1] Anatoli Klimyk, Konrad Schmudgen , Quantum groups and their representations.SpringerVerlagBerlin 1997
Recommended references: [2] Podles, P.; Muller,E., Introduction to quantum groups, arXiv:qalg/9704002. [3] Kassel, Christian (1995), Quantum groups, Graduate Texts in Mathematics,155, Berlin, New York: SpringerVerlag, MR1321145, ISBN 9780387943701 [3] Majid, Shahn (2002), A quantum groups primer,London Mathematical Society Lecture Note Series, 292, Cambridge University Press, MR1904789, ISBN 9780521010412, [4] Street, Ross (2007), Quantum groups, Australian Mathematical Society Lecture, Series, 19, Cambridge University Press, MR2294803, ISBN9780521695244; 9780521695244. 

Linear Algebra A2  01LAA2 
Dvořáková 
  
2+2 z,zk 
 
6 
Course:  Linear Algebra A2  01LAA2  doc. Ing. Dvořáková Lubomíra Ph.D.    2+2 Z,ZK    6  Abstract:  The subject is devoted to the theory of linear operators on vector spaces (mainly equipped with scalar product). In the same time we introduce the corresponding matrix theory.  Outline:  Inverse matrix and operator. Permutation and determinant. Spectral theory (eigenvalue, eigenvector, diagonalization). Hermitian and quadratic forms. Scalar product and orthogonality. Metric geometry. Riesz theorem and adjoint operator.  Outline (exercises):  1. Gauss method of determination of inverse matrix. 2. Different methods of determinant calculation. 3. Evaluation of eigenvalues and eigenvectors, diagonalization. 4. Canonical transformation of a quadratic form, determination of character of the form and signature. 5. Examples of scalar products, GramSchmidt orthogonalization, orthonormal basis. 6. Metric geometry  calculation of distance and angles.
7. Riesz theorem and adjoint operator. Characterization of normal operators and their spectrum.  Goals:  Knowledge: Mastering of the concepts of theory of linear operators and matrices, especially in spaces equiped with a scalar product, and applications of linear algebra in metric geometry.
Skills:
Ability to use these findings in further studies not only in mathematical disciplines, but also in physics, economics etc.  Requirements:  Having passed the subject LAP.  Key words:  Determinant, eigenvalues and eigenvectors, diagonalization, quadratic and hermitian form, diagonalization, inverse operator, normal, hermitian and unitary operator.  References  Key references:
[1] Linear Algebra with Applications, PrenticeHall, Inc., Englewood Cliffs, New Jersey,1980
[2] C. W. Curtis : Linear Algebra, An Introductory Approach, SpringerVerlag, New York, Berlin, Heidelberg, Tokyo 1984.
Recommended references:
[3] P. Lancaster : Theory of Matrices, Academic Press, New York, London, 1969.


Linear Algebra B2  01LAB2 
Ambrož 
  
1+2 z,zk 
 
4 
Course:  Linear Algebra B2  01LAB2  Ing. Ambrož Petr Ph.D.    1+2 Z,ZK    4  Abstract:  The subject summarizes the most important notions and theorems related to the matrix theory, to the study of vector spaces with a scalar product and to the linear geometry.  Outline:  Matrices and systems of linear algebraic equations  determinants  scalar product and orthogonality  eigenvalues and eigenvectors of matrices  linear geometry in Euclidean space  Outline (exercises):  1. Solving systems of linear algebraic equations
2. Calculation of inverse matrices using the Gauss elimination
3. Permutations and determinants
4. Searching for orthogonal and orthonormal bases, application of the GramSchmidt orthogonalization method, calculation of orthogonal projections of vectors
4. Computation of eigenvalues and eigenvectors of matrices
5. Distinct descriptions of linear manifolds and convex sets, computation of intersections of linear manifolds
 Goals:  Knowledge:
Basic notions from the matrix theory, notions related to the scalar product and the linear geometry from the theoretical point of view.
Abilities:
Application of the knowledge in practical problems.  Requirements:  01LALA or 01LALB  Key words:  Matrices, systems of linear algebraic equations, determinants, scalar product and orthogonality, eigenvalues and eigenvectors of matrices, linear geometry in the Euclidean space  References  Key references:
[1] H. G. Campbell, Linear Algebra with Applications, PrenticeHall, Inc., Englewood Cliffs, New Jersey, 2nd edition, 1980
[2] C.W.Curtis, Linear Algebra, An Introductory Approach, SpringerVerlag, New York, Berlin, Heidelberg, Tokyo, 1974, 4th edition, 1984
Recommended references:
[3] P. Lancaster, Theory of Matrices, Academic Press, New York, London, 1969 

Linear Algebra 1  01LAL 
Dvořáková 
3+2 z 
  
2 
 
Course:  Linear Algebra 1  01LAL  doc. Ing. Dvořáková Lubomíra Ph.D.          Abstract:   Outline:   Outline (exercises):   Goals:   Requirements:   Key words:   References  

Linear Algebra A 1, Examination  01LALA 
Dvořáková 
 zk 
  
5 
 
Course:  Linear Algebra A 1, Examination  01LALA  Ing. Ambrož Petr Ph.D. / doc. Ing. Dvořáková Lubomíra Ph.D.          Abstract:   Outline:   Outline (exercises):   Goals:   Requirements:   Key words:   References  

Linear Algebra B 1, Examination  01LALB 
Dvořáková 
 zk 
  
3 
 
Course:  Linear Algebra B 1, Examination  01LALB  Ing. Ambrož Petr Ph.D. / doc. Ing. Dvořáková Lubomíra Ph.D.          Abstract:   Outline:   Outline (exercises):   Goals:   Requirements:   Key words:   References  

Linear Algebra with Applications  01LAWA 
Novotná 
  
2+0 zk 
 
2 
Course:  Linear Algebra with Applications  01LAWA  doc. Novotná Jarmila    2+0 ZK    2  Abstract:  The course deals with basic domains of linear algebra and their applications in economy and other disciplines. The language of instruction is English.  Outline:  Language of mathematics. Proofs. Systems of linear equations (methods of solving, applications). Matrices (matrix operations, matrix algebra, rank, introduction to linear transformations). Vectors (vectors in geometry, algebra of vectors, length and angle, lines and planes), vector spaces (vector spaces and subspaces, linear dependence and independence, basis, dimension).  Outline (exercises):   Goals:  Knowledge: basic knowledge of linear algebra and its applications. The course is run in English, students? functional bilingualism is developed. Skills: Application of basic knowledge from linear algebra. Communication about a nonlanguage subject in English.  Requirements:  English: competence A2 (Common European Framework of Reference for Languages)  Key words:  System of linear equations, matrix, vector space, applications, CLIL  References  Key references:
[1] W.K. Nicholson: Linear Algebra with Applications. Boston: PWS Publishing Company, 1993.
[2] D. Poole: Linear Algebra. A Modern Introduction. Brooks/Cole, Thomson Learning 2003.
Recommended references:
[3] P.M. Cohn: Classic Algebra. Wiley, 1991.


Linear Programming  01LIP 
Burdík 
2+2 z,zk 
  
4 
 
Course:  Linear Programming  01LIP  prof. RNDr. Burdík Čestmír DrSc.    2+1 Z,ZK    3  Abstract:  We study special problems about constrained extremum problems for multivariable functions (the function is linear and the constraint equations are given by linear equations and linear inequalities).  Outline:  Forms of the LP problem, duality. Linear equations and inequalities, convex polytope, basic feasible solution, complementary slackness. Algorithms: simplex, dual simplex, primal  dual, revised. Decomposition principle, transportation problem. Discrete LP (algorithm Gomory's). Application of LP in the theory of games  matrix games. Polynomial  time algorithms for LP (Khachian, Karmarkar).  Outline (exercises):  1. Optimality test of feasible solution, identify the extreme point of the set of admissible solutions, build a dual problem. 2. Simplex method, dual simplex method, primary  dual simplex method. 3. Gomory algorithm for integer programming. 4. Applications in game theory.  Goals:  Knowledge: The mathematical basis for systems of linear equations and inequalities. Skills: To be able to use memorized algorithms to solve specific problems of practice.  Requirements:  Basic course of Calculus and Linear Algebra (in the extent of the courses 01MA1, 01MAA24, 01LA1, 01LAA2 held at the FNSPE CTU in Prague).
 Key words:  Admissible and optimal solution, alkaline solution, the extreme point, the simplex method, weak complementarity, integer programming.  References  Key references:[1] George B. Dantzig and Mukund N. Thapa. 1997. Linear programming 1:Introduction. SpringerVerlag. [2] T.C.HU, 1970, Integer Programming and Network Flows, AddisonWeslley Publishing, Menlo Park
Recommended references: [3] George B. Dantzig and Mukund N. Thapa. 2003. Linear Programming 2: Theory and Extensions. SpringerVerlag, [4] Kattta G. Murty, Linear Programming, Wiley, 1983 

Logic in Informatics  01LOI 
Noguera 
  
2+0 zk 
 
2 
Course:  Logic in Informatics  01LOI  Mgr. Noguera Carles Ph.D.          Abstract:  Matematical logic, finite model theory, constraint satisfaction problems, modal logics, dynamic logics, intuitionism.  Outline:  The course will be composed by blocks taught by different lecturers (possibly partly in English) who are active researchers in their respective fields:
1.Classical propositional and predicate logic as a modelling tool. Finite model theory. Constraint satisfaction problems.
2.Modal logics and their applications in computer science.
3.Dynamic logics and formal verificiation of programs.
4.Intuitionism and constructivism.
 Outline (exercises):   Goals:  Acquired knowledge:
Basic notions and results of classical and nonclassical logics and their role in computer science.
Acquired skills:
Ability to apply results of mathematical logic to computer science.
 Requirements:  Previous basic knowledge of mathematical logic is recommended for the course.  Key words:   References  Compulsory literature:
[1] Mordechai BenAri. Mathematical Logic for Computer Science. Springer, 2012.
Optional literature:
[1] Johan van Benthem, Patrick Blackburn (eds.). Handbook of Modal Logic. Elsevier, 2006.
[2] David Harel, Dexter Kozen, Jerzy Tiuryn. Dynamic logic. MIT Press, 2000.
[3] Leonid Libkin. Elements of Finite Model Theory. Springer, 2004.


Logic for Mathematicians  01LOM 
Cintula 
  
2+0 zk 
 
2 
Course:  Logic for Mathematicians  01LOM  doc. Ing. Cintula Petr Ph.D.          Abstract:   Outline:   Outline (exercises):   Goals:   Requirements:   Key words:   References  

Calculus A2  01MAA2 
Pelantová 
  
4+4 z,zk 
 
10 
Course:  Calculus A2  01MAA2  prof. Ing. Pelantová Edita CSc.    4+4 Z,ZK    10  Abstract:  The subject is devoted mainly to the integral calculus of the real functions with one real variable and to the theory of the number series and the power series.  Outline:  Continuation of differential calculus: Taylor´s Polynomials, Taylor´s formula; Theory of integrals: primitives, definite integral (Riemann definition), techniques of integration and application of integrals; Infinite series: criteria of convergence, operations on series, absolute and conditional convergence, real and complex power series, the CauchyHadamard theorem, expansion of function into power series, summation of infinite series.  Outline (exercises):  Content of excercises consists in solving problems with emphasis on using theoretical results. The domains of problems: evaluation fo limits by the l´Hospital rule, uniform continuity, approximation of function by the Taylor polynomial, technics for determination of primitive functions, evaluation of volumes and areas, expansion of function into a power series  Goals:  Acquired knowledge: a rigorous construction of integral, to focus on properties of power series.
Acquired skills: application of the theoretical results in geometry, discrete mathematics and in physics.  Requirements:  Succesfull completion of the course Mathematical analysis I, i.e. familiarity with differential calculus.  Key words:  The Taylor polynomial, primitive function, the Riemann integral, series, convergence, power series  References  Obligatory:
[1] E. Pelantová: Matematická analýza II, skriptum ČVUT, 2007
[2] E.Pelantová, J.Vondráčková: Cvičení z matematické analýzy  Integrální počet a řady, skriptum ČVUT 2006
Optional:
[3] I. Černý, M. Rokyta: Differential and Integral Calculus of One Real Variable, Karolinum, Praha 1998
[4] I.Černý, Úvod do inteligentního kalkulu I, Academia 2005


Calculus A3  01MAA34 
Vrána 
4+4 z,zk 
4+4 z,zk 
10 
10 
Course:  Calculus A3  01MAA3  Ing. Fučík Radek Ph.D. / Ing. Vrána Leopold  4+4 Z,ZK    10    Abstract:  Function sequences and series, foundation of topology, and differential calculus of several variables.  Outline:  Function sequences and series: Pointwise and uniform convergence, interchange rules for limits, derivatives and integrals. Fourier's series, expansion of a function into trigonometrical series, tests for pointwise and uniform convergence of trigonometrical series, completeness of trigonometric system. Topology of normed linear space, compact, connected and complete sets, fixpoint theorem. Differential calculus of several variables: directional derivatives, partial and total derivatives, meanvalue theorems, extremum, manifolds, constrained extrema.  Outline (exercises):  Uniform convergence. Interchange rules. Expansion of a function into trigonometrical series.
Directional derivative. Total derivative. Local extrema.
 Goals:  To acquaint the students with the properties of function sequences and series, expansion of a function into trigonometrical series, with an introduction to the topology, and with foundation of differential calculus of several variables.  Requirements:  Basic Course of Calculus and Linear Algebra (in the extent of the courses 01MA1, 01MAA2, 01LA1, 01LAA2 held at the FNSP CTU in Prague).
 Key words:  Function sequences and series, Fourier's series, topological and metric space, compactness, connectness, completeness, total derivative, local extrema.
 References  Key reference: W.H.Fleming,Functions of Several Variables, AddisonWesley, Reading, MA, 1966.
Recommended references: Mariano Giaquinta, Giuseppe Modica, Mathematical Analysis  An Introduction to Functions of Several Variables, Birkhäuser, Boston, 2009

Course:  Calculus A4  01MAA4  Ing. Vrána Leopold    4+4 Z,ZK    10  Abstract:  Integration of functions of several variables, measure theory, foundation of differential and integral calculus on manifolds and complex analysis.
 Outline:  Lebesgue integral: Daniel?s construct, interchange rules, measurable sets and measurable functions. Fubini's theorem, theorem on changing variables. Parametrical integrals: Interchange theorems, Gamma and Beta functions. Differential forms: conservative, exact and closed form and their relations, potential. Line and surface integral: Green's, Gauss' and Stokes' theorem. Complex analysis: analytic functions, Cauchy's theorem, Taylor's expansion, Laurent's expansion, singularities, residue theorem.  Outline (exercises):  Smooth manifolds. Constrained extrems. Differential forms. Lebesgue integral in several variables. Use of Fubini's theorem and theorem on changing variables. Use of Gamma and Beta functions for computation of integrals. Computation of integrals
 Goals:  To acquaint the students with foundations of Lebesgue integration and with foundations of complex analysis and its use in applications.  Requirements:  Basic Course of Calculus and Linear Algebra (in the extent of the courses 01MA1, 01MAA23, 01LA1, 01LAA2 held at the FNSP CTU in Prague).  Key words:  Lebesgue integral, measurable functions and sets, Gamma and Beta functions, line and surface integral, divergence theorem, Cauchy's theorem, residue theorem.  References  Key reference: W.H.Fleming,Functions of Several Variables, AddisonWesley, Reading, MA, 1966.
Recommended references: Mariano Giaquinta, Giuseppe Modica, Mathematical Analysis  An Introduction to Functions of Several Variables, Birkhäuser, Boston, 2009


Calculus B2  01MAB2 
Pošta 
  
2+4 z,zk 
 
7 
Course:  Calculus B2  01MAB2  doc. Ing. Pošta Severin Ph.D.    2+4 Z,ZK    7  Abstract:  Basic calculus (real analysis, indefinite and definite integrals and series).
 Outline:  1. Antiderivative  basic properties, integration by parts, by substitution, antiderivative of rational and other elementary functions.
2. Newton and Riemann integrals, their relation, convergence of integral.
3. Some applications of integral  area of plane regions, length of a curve, volume and surface areas.
4. Infinite series  sum, basic properties, convergence of series with nonnegative terms, with arbitrary terms.
 Outline (exercises):  1. Antiderivatives. Integration by parts, by substitution.
2. Calculus of Riemann integrals.
3. Applications of integrals.
4. Infinite series and their convergence.  Goals:  The goal of this course is to manage basic techniques of computing indefinite and definite integrals and examining convergence of sequences.  Requirements:  Calculus 1 (01MANA or 01MANB).
 Key words:  integral calculus, real function, real variable, analysis, limit, antiderivative, Riemann integral, infinite series  References  Recommended references:
[1] T. Apostol: Mathematical Analysis, Addison Wesley, 1974.
[2] W. Rudin: Principles of Mathematical Analysis. McGrawHill, Mexico, 1980.


Calculus B3  01MAB34 
Krbálek 
2+4 z,zk 
2+4 z,zk 
7 
7 
Course:  Calculus B3  01MAB3  doc. Mgr. Krbálek Milan Ph.D.  2+4 Z,ZK    7    Abstract:  The course is devoted to functional sequences and series, theory of ordinary differential equations, theory of quadratic forms and surfaces, and general theory of metric spaces, normed and prehilbert?s spaces.  Outline:  1. Functional sequences and series  convergence range, criteria of uniform convergence, continuity, limit, differentiation and integration of functional series, power series, Series Expansion, Taylor?s theorem. 2. Ordinary differential equations  equations of first order (method of integration factor, equation of Bernoulli, separation of variables, homogeneous equation and exact equation) and equations of higher order (fundamental system, reduction of order, variation of parameters, equations with constant coefficients and special righthand side, Euler?s differential equation). 3. Quadratic forms and surfaces  regularity, types of definity, normal form, main and secondary signature, polar basis, classification of conic and quadric 4. Metric spaces  metric, norm, scalar product, neighborhood, interior and exterior points, boundary point, isolated and nonisolated point, boundary of set, completeness of space, Hilbert?s spaces.  Outline (exercises):  1. Functional sequences. 2. Functional series. 3. Power series 4. Solution of differential equations. 5. Quadratic forms. 6. Quadratic surfaces. 7. Metric spaces, normed and Hilbert?s spaces.  Goals:  Knowledge: Investigation of uniform convergence for functional sequences and series. Solution of differential equations. Classification of quadratic forms and surfaces. Classification of points of sets. Skills: Individual analysis of practical exercises.  Requirements:  Basic course of Calculus a Linear Algebra (in the extent of the courses 01MA1, 01MAB2, 01LA1, 01LAB2 held at the FNSPE CTU in Prague).  Key words:  Function sequences, function series, differential equations, quadratic forms, quadratics surfaces, metric spaces, norm spaces, preHilbert spaces  References  Key references:
[1] Robert A. Adams, Calculus: A complete course, 1999,
[2] Thomas Finney, Calculus and Analytic geometry, Addison Wesley, 1996
Recommended references:
[3] John Lane Bell: A Primer of Infinitesimal Analysis, Cambridge University Press, 1998
Media and tools: MATLAB 
Course:  Calculus B4  01MAB4  doc. Mgr. Krbálek Milan Ph.D.    2+4 Z,ZK    7  Abstract:  The course is devoted properties of functions of several variables, differential and integral calculus. Furthermore, the measure theory and theory of Lebesgue integral is studied.  Outline:  Differential calculus of functions of several variables  limit, continuity, partial derivative, directional partial derivative, total derivative and tangent plane, Taylor?s theorem, elementary terms of vector analysis, Jacobi matrix, implicit functions, regular mappings, change of variables, noncartesian coordinates, local and global extremes. Integral calculus of functions of several variables  Riemann?s construction of integral, Fubiny theorem, substitution of variables. Curve and surface integral  curve and curve integral of first and second kind, surface and surface integral of first and second kind, Green and Gauss and Stokes theorems. Fundamentals of measure theory  set domain, algebra, domain generated by the semidomain, sigmaalgebra, sets H_r, K_r and S_r, Jordan measure, Lebesgue measure. Abstract Lebesgue integral  measurable function, measurable space, fundamental system of functions, definition of integral, Levi and Lebesgue theorems, integral with parameter, Lebesgue integral and his connection to Riemann and Newton integral, theorem on substitution, Fubiny theorem for Lebesgue integral.  Outline (exercises):  1. Function of several variables (properties). 2. Function of several variables (differential calculus). 3. Function of several variables (integral calculus) 4. Curve and surface integral. 5. Measure Theory 6. Theory of Lebesgue integral.  Goals:  Knowledge: Investigation of properties for function of severable variables. Multidimensional integrations. Curve and surface integration. Theoretical aspects of measure theory and theory of Lebesgue integral. Skills: Individual analysis of practical exercises.  Requirements:  Basic course of Calculus a Linear Algebra (in the extent of the courses 01MA1, 01MAB2, 01MAB3, 01LA1, 01LAB2 held at the FNSPE CTU in Prague).  Key words:  Function of several variables, curve and surface integrals, measure theory, theory of Lebesgue integral  References  Key references:
[1] M. Giaquinta, G. Modica, Mathematical analysis  an introduction to functions of several variables, Birkhauser, Boston, 2009
Recommended references:
[2] S.L. Salas, E. Hille, G.J. Etger, Calculus (one and more variables), Wiley, 9th edition, 2002
Media and tools: MATLAB 

Mathematical Logic  01MAL 
Cintula 
2+1 z,zk 
  
4 
 
Course:  Mathematical Logic  01MAL  doc. Ing. Cintula Petr Ph.D.          Abstract:  Logic is in the same time an object studied by mathematics and the language used to formalize and study mathematics. The goal of the course is to introduce basic notion of results of classical mathematical logic.
1.Propositions, evaluation, tautologies, axioms, theorems, soundness, completeness, and decidability of Hilbert and Gentzen style propositional calculi.
2.Language of predicate calculus, terms, formulas, relational structures, satisfiability, truth, tautologies, axioms, theorems, soundness, model constructions.
3.Gödel completeness theorem, Skolem and Herbrand theorems.
4.The first and the second Gödel theorems on incompleteness of Peano arithmetics and undecidability of predicate calculus.
 Outline:  Logic is in the same time an object studied by mathematics and the language used to formalize and study mathematics. The goal of the course is to introduce basic notion of results of classical mathematical logic.
1.Propositions, evaluation, tautologies, axioms, theorems, soundness, completeness, and decidability of Hilbert and Gentzen style propositional calculi.
2.Language of predicate calculus, terms, formulas, relational structures, satisfiability, truth, tautologies, axioms, theorems, soundness, model constructions.
3.Gödel completeness theorem, Skolem and Herbrand theorems.
4.The first and the second Gödel theorems on incompleteness of Peano arithmetics and undecidability of predicate calculus.
 Outline (exercises):   Goals:  Knowledge:
Basic notions and results of classical propositional and predicate mathematical logic.
Skills:
Orientation in basics of mathematical logic and ability to use it in other disciplines.
 Requirements:   Key words:   References  Compulsory literature:
[1] V. Švejdar: Logika  neúplnost, složitost a nutnost. Academia, Praha 2002.
Optional literature:
[2] Nicholas J. J. Smith. Logic: The Laws of Truth. Princeton University Press, 2012.


Calculus 1  01MAN 
Pošta 
4+4 z 
  
4 
 
Course:  Calculus 1  01MAN  doc. Ing. Pošta Severin Ph.D.          Abstract:  Basic calculus (real analysis, functions of one real variable, differential calculus).
 Outline:  1. Basics of mathematical logic, equations and inequalities, goniometric functions, exponential and logarithmic functions, sums and products, induction.
2. Sets and mappings.
3. Real and complex sequence  limit, basic properties, limits of special sequences, number "e" and exponential function, some elementary functions.
4. Limit and continuity of functions of one real variable  basic properties.
5. Derivative of functions  basic properties.
6. Basic theorems of differential calculus. 7. Constructing graphs of functions.
 Outline (exercises):  1. Basic properties of functions and mappings.
2. Supremum, Infimum.
3. Limits of sequences.
4. Acculumation points.
5. Limits of real functions.
6. Continuity.
7. Derivative, graphs of real functions.
 Goals:  The goal of this course is to manage basic techniques of computing limits of sequences, limits of real functions of one real variable and of differential calculus.  Requirements:  No prerequisities.  Key words:  differential calculus, real function, real variable, continuity, limit, derivative  References  Recommended references:
[1] Apostol: Mathematical Analysis, Addison Wesley, 1974.
[2] W. Rudin: Principles of Mathematical Analysis. McGrawHill, Mexico, 1980.


Calculus A 1, Examination  01MANA 
Pošta 
 zk 
  
6 
 
Course:  Calculus A 1, Examination  01MANA  doc. Ing. Pošta Severin Ph.D.          Abstract:  Examination of knowledge about stuff lectured in the 01MAN course.  Outline:  Examination of knowledge about stuff lectured in the 01MAN course.  Outline (exercises):    Goals:  Verification of knowledge about stuff lectured in the 01MAN course.  Requirements:  No prerequisities.  Key words:  The keywords are given under the 01MAN course.  References  The source materials are given under the 01MAN course. 

Calculus B 1, Examination  01MANB 
Pošta 
 zk 
  
4 
 
Course:  Calculus B 1, Examination  01MANB  doc. Ing. Pošta Severin Ph.D.          Abstract:  Examination of knowledge about stuff lectured in the 01MAN course.  Outline:  Examination of knowledge about stuff lectured in the 01MAN course.  Outline (exercises):    Goals:  The goal of the course is to verify the knowledge about stuff lectured in the 01MAN course.  Requirements:  No prerequisities.  Key words:  The keywords are given under the 01MAN course.  References  The source materials are given under the 01MAN course. 

Markov processes  01MAPR 
Vybíral 
  
2+2 z,zk 
 
4 
Course:  Markov processes  01MAPR  doc. RNDr. Vybíral Jan Ph.D.          Abstract:   Outline:   Outline (exercises):   Goals:   Requirements:   Key words:   References  HÄGGSTRÖM, Olle, 2002. Finite Markov Chains and Algorithmic Applications. Cambridge Uni. Press.
PRÁŠKOVÁ, Zuzana; LACHOUT, Petr, 2012. Základy náhodných procesů I. 2. vyd. Matfyzpress.
RESNICK, Sydney I., 2005. Adventures in Stochastic Processes. 4. vyd. Birkhauser. 

Mathematical Statistics  01MAS 
Kůs 
  
2+0 zk 
 
3 
Course:  Mathematical Statistics  01MAS  Ing. Kůs Václav Ph.D.          Abstract:  The subject is devoted to usage of statistical methods studied in the course of Mathematical statistics. We deal with Fisher information matrix of statistical models, finding unbiased estimators with minimal variance, parameter estimation by method of moments and method of maximum likelihood, derivation of critical regions for hypothesis testing using the NeymanPearson lemma and likelihood ratio, confidence intervals and nonparametric density estimation.  Outline:  1. Unbiased minimum variance estimates, Fisher information matrix, RaoCramér inequality, Bhattacharrya inequality.
2. Moment estimators, Maximum likelihood principle, consistency, asymptotic normality and efficiency of MLE.
3. Testing of simple and composite hypotheses. The NeymanPearson lemma.
4. Uniformly most powerful tests. Randomized testing, generalized NeymanPearson lemma.
5. The likelihood ratio test, ttest, Ftest.
6. Nonparametric models, empirical distribution and density function, their properties.
7. Histogram and kernel density estimates (adaptive), properties.
8. Pearson goodness of fit test, KolmogorovSmirnov test.
9. Confidence sets and intervals, pivotal quantities, acceptance regions, Pratt theorem.  Outline (exercises):   Goals:  Knowledge:
In frame of the course, to provide students with the knowledge necessary for the following future subjects using stochastic models. To give a deeper insight into the field in the area of point statistical parameter estimation and testing statistical hypothesis in parametric and nonparametric probabilistic models.
Skills:
Basic statistical models and testing hypotheses processing. Orientation in majority of standard notions of the statistics and capabilities of practical applications in actual stochastic computations. Statistical data processing in statistical parametric and nonparametric model estimation and testing.  Requirements:  01MIP nebo 01PRST  Key words:  Unbiased estimators, information matrix, moment estimators, maximum likelihood principle, efficiency, statistical hypothesis, simple and composite hypotheses, most powerful tests, likelihood ratio test, nonparametric models, empirical distribution, histogram, kernel estimate, goodness of fitness, confidence sets and intervals.  References  Key references:
[1] Shao J., Mathematical Statistics, Springer, 1999.
Recommended references:
[2] Schervish M.J., Theory of Statistics, Springer, 1995.
[3] Lehmann E.L., Point Estimation, Wiley, N.Y., 1984.
[4] Lehmann E.L., Testing Statistical Hypotheses, Springer, N.Y., 1986.


Mathematical Statistics  Seminar  01MASC 
Hobza 
  
0+2 z 
 
2 
Course:  Mathematical Statistics  Seminar  01MASC  doc. Ing. Hobza Tomáš Ph.D.          Abstract:  The subject is devoted to practical use of statistical methods studied in the course Mathematical Statistics 01MAS. The tutorial deals with calculation of Fisher information matrix of statistical models, finding unbiased estimators with minimal variance, parameter estimation by method of moments and method of maximum likelihood, derivation of critical regions for hypothesis testing using the NeymanPearson lemma and likelihood ratio, calculation of confidence intervals and nonparametric density estimation.  Outline:  1. Transformations of random variables
2. Applications of the Law of large numbers nad the Central limit theorem
3. Calculation of Fisher information matrix of statistical models
4. Unbiased estimators with minimal variance
5. Parameter estimation by method of moments and method of maximum likelihood
6. Hypothesis testing using the NeymanPearson lemma
7. Likelihood ratio tests
8. Confidence intervals
9. Nonparametric density estimation  Outline (exercises):   Goals:  Acquired knowledge: Basic statistical methods for estimation of parameters of statistical models and testing statistical hypothesis about parameters of models.
Acquired skills: Application of statistical models and corresponding methods to practical problems and real data analysis.
 Requirements:  Basic course of Calculus and Probability (in the extent of the courses 01MAB3, 01MAB4 and 01MIP held at the FNSPE CTU in Prague).  Key words:  Unbiased estimators, information matrix, moment estimators, maximum likelihood estimators, hypothesis testing, likelihood ratio test, confidence intervals, nonparametric density estimates.  References  Compulsory literature:
[1] Shao J., Mathematical Statistics, Springer, 1999.
Optional literature:
[2] Lehmann E.L., Point Estimation, Wiley, N.Y., 1984.
[3] Lehmann E.L., Testing Statistical Hypotheses, Springer, N.Y., 1986.


Mathematics 1  01MAT12 
Fučík 
6 z 
6 z 
4 
4 
Course:  Mathematics 1  01MAT1  Ing. Fučík Radek Ph.D.  6 Z    4    Abstract:  The course is devoted to the study of the basics of calculus of one variable. It includes an introduction to differential and integral calculus, with particular emphasis on applications in practical problems.  Outline:  1. Functions and their properties.
2. Limits of functions.
3. Continuity.
4. The derivative, tangent to a curve, some differentiation formulas, derivatives of higher order.
5. Rolle's theorem, the mean value theorem (Lagrange). Extreme values, asymptotes, concavity and point of inflections, curve sketching.
6. The definite integral. The antiderivate function, indefinite integral, substitution, integration by parts. Newton's theorem, the area calculation. Primitive functions to trigonometric functions, mean integral.
7. The transcendental functions: logarithm function, e number, exponential function, hyperbolic functions.
8. Applications of the definite integral: the length of a curve, the volume and the area of a revolved curve.
 Outline (exercises):  1. Functions and their properties: domain of definition, range, inverse, absolute value, inequalities, quadratic inequalities, graphs, composition of functions, polynomials, division of polynomials.
2. Limits of functions: the limits of basic functions, the limits of trigonometric functions.
3. Continuity: The investigation of continuity of functions from the definition, identification of types of discontinuities.
4. Derivatives: derivative computation by definition, rules for derivatives of basic functions, tangents, higher order derivatives.
5. Rolle's theorem, the mean value theorem (Lagrange). Extreme values, asymptotes, concavity and point of inflections, curve sketching.
6. Integral calculus: the antiderivate functions, the method of substitution, the method of integration by parts, advanced techniques of integration of trigonometric functions, definite integrals, Newton's formula.
7. Transcendental functions: logarithm definition, characteristics, exponential, hyperbolic and trigonometric functions and their derivatives.
8. Applications of the definite integral: area under the graph of the function, length of a graph, volume and surface the area of a revolved curve.
 Goals:  Knowledge:
Elementary notions of mathematical analysis of the differential and integral calculus of functions of one real variable.
Abilities:
Understanding the basics of mathematical logic and mathematical analysis.  Requirements:   Key words:  Differential calculus, integral calculus, functions of one real variable, limits, extremes of functions.  References  Key references:
[1] Calculus, One Variable, S.L.Salas, Einar Hille, John Wiley and Sons, New York, Chichester, Brisbane, Toronto, Singapore, 1990 (6th edition), ISBN 0471517496
[2] Larson, Ron, and Bruce H. Edwards. Calculus of a single variable: Early transcendental functions. Cengage Learning, 2014.
[3] Pelantová, Edita, Vondráčková, Jana: Cvičení z matematické analýzy, ČVUT, Praha 2015
[4] Stewart, James. Single variable calculus: Early transcendentals. Nelson Education, 2015.

Course:  Mathematics 2  01MAT2  Ing. Fučík Radek Ph.D.    6 Z    4  Abstract:  The course, which is the continuation of Mathematics 1, is devoted to the integration techniques, improper Riemann integral, introduction to parametric curves (especially in polar coordinates), the basics of sequences and infinite series, and finally to the Taylor and power series and their applications.  Outline:  1. Integration techniques.
2. The improper integral and the convergence criteria.
3. Conic sections: ellipse, hyperbole, parable.
4. Polar coordinates.
5. Parametric curves: length of a curve, tangent to a curve, surfaces, volumes and surfaces of revolution.
6. Sequences: limits of sequences, important limits, the convergence criteria.
7. Series: the convergence criteria, absolute and nonabsolute convergence, alternating series.
8. Power series. Differentiation and integration of power series.
9. Taylor polynomial and Taylor series.  Outline (exercises):  1. Advanced integration techniques: integrals of rational functions, partial fractions, integration of trigonometric functions.
2. Improper Riemann integral: calculating improper integrals, convergence criteria.
3. Conic sections: circle, ellipse, hyperbole, parable, conic sections identification, description of conics through the distance between points and between a point and a line.
4. Polar coordinates: the transformation of points and equations between the cartesian and polar coordinates.
5. Parametric curves: length of a curve, tangent to the curve, surfaces, volumes and surfaces of revolution.
6. Properties of sets: finding suprema and infima of sets.
7. Sequences: limits of sequences, important limits, convergence criteria.
8. Infinite series: convergence criteria, absolute and relative convergence, alternating series.
9. Power series: convergence criteria, differentiation and integration of power series, sum of infinite series.
10. Taylor polynomials and Taylor series: the expansion of important functions in power series.
 Goals:  Knowledge:
Advanced integration techniques, improper Riemann integral, numerical sequences, and infinite power series.
Abilities:
Understanding the basics of mathematical logic and mathematical analysis. Taylor series expansion.  Requirements:  Mathematics 1.  Key words:  Differential calculus, integral calculus, functions of one variable, numerical sequences, infinite series, power series, Taylor series.  References  Key references:
[1] Calculus, One Variable, S.L.Salas, Einar Hille, John Wiley and Sons, New York, Chichester, Brisbane, Toronto, Singapore, 1990 (6th edition), ISBN 0471517496
[2] Larson, Ron, and Bruce H. Edwards. Calculus of a single variable: Early transcendental functions. Cengage Learning, 2014.
[3] Pelantová, Edita, Vondráčková, Jana: Cvičení z matematické analýzy, ČVUT, Praha 2015
[4] Stewart, James. Single variable calculus: Early transcendentals. Nelson Education, 2015.


Mathematics 3  01MAT34 
Dvořáková, Krejčiřík, Tušek 
2+2 z,zk 
2+2 z,zk 
4 
4 
Course:  Mathematics 3  01MAT3  doc. Mgr. Krejčiřík David DSc.  2+2 Z,ZK    4    Abstract:  The subject summarises the most important notions and theorems related to the study of finitedimensional vector spaces.  Outline:  1. Vector spaces;
2. Linear span and independence;
3. Basis and dimension;
4. Linear transformations;
5. Operator equations;
6. Scalar product and orthogonality;
7. Linear functionals and adjoint;
8. Matrices;
9. Determinants;
10. Spectrum;
11. Matrix exponential;
12. Quadratic forms.  Outline (exercises):  0. Complex numbers;
1. Examples of vector spaces and subspaces;
2. Linear dependence of vectors  problem with parametres.
3. Selection of basis vectors from a set of generators, completing a basis;
4. Injectivity and kernel of a linear mapping;
5. Examples of scalar products and orthogonalization process;
6. Examples of linear functionals and construction of adjoint mappings;
7. Operations with matrices and construction of the matrix of a linear mapping;
8. Working with determinants, computation of the inverse matrix;
9. Eigenvalues and eigenfunctions of matrices;
10. Construction of matrix exponential;
11. Properties of quadratic forms.  Goals:  Knowledge: Learning basic concepts of linear algebra necessary for a proper understanding of related subjects, such as analysis of functions of several variables, numerical mathematics, and so on. Skills: Applications of theoretical concepts and theorems in continuing subjects.  Requirements:  Basic high school mathematics.  Key words:  Vector space, subspace, linear dependence, basis, dimension, linear transformations, matrices, trace, determinant, orthogonality, spectrum, eigenvalues, eigenvectors, quadratic form, matrix exponential.  References  Key references:
[1] S. Axler: Linear algebra done right, Springer, New York 2014
Recommended references:
[2] J. Kopáček, Matematika pro fyziky II, UK, Praha, 1989.
[3] Lecture notes on the hompeage of the lecturer.

Course:  Mathematics 4  01MAT4  Ing. Tušek Matěj Ph.D.    2+2 Z,ZK    4  Abstract:  Linear and nonlinear differential equations of the first order. Linear differential equations of higher order with constant coefficients. Multivariable calculus and its applications.  Outline:  1. Linear differential equations of the first order 2. Nonlinear differential equation of the first order 3. Exact and homogeneous equations. 4. Linear differential equations of higher order 5. Linear differential equation with constant coefficients 6. Quadratic forms 7. Limit and continuity of multivariable functions 8. Multivariable calculus 9. Total differential 10. Implicit function 11. Change of variables 12. Extreme values of multivariable functions 13. Multidimensional Riemann integral 14. Fubini theorem and substitution theorem.
 Outline (exercises):  1. Linear differential equations of the first order 2. Nonlinear differential equation of the first order 3. Linear differential equations of higher order 4. Linear differential equation with constant coefficients 5. Limit and continuity of multivariable functions 6. Implicit function 7. Extreme values of multivariable functions 8. Multidimensional Riemann integral 9. Fubini theorem and substitution theorem.
 Goals:  Knowledge: To learn how to solve some elementary classes of differential equations, especially LDE. To become familiar with multivariable calculus.
Abilities: To apply the knowledge above to particular problems in engineering.  Requirements:  Basis course in single variable calculus and linear algebra (in the extent of the courses at FNSPE, CTU in Prague: 01MAT1, 01MAT2, 01MAT3).  Key words:  Differential equations, multivariable calculus.  References  key references:
[1] J. Marsden, A. Weinstein: Calculus III, Springer, 1985.
recommneded references:
[2] W. Rudin: Principles of Mathematical Analysis, McGrawHill, 1976.
[3] J. Stewart: Multivariable Calculus, 8th Edition, Brooks Cole, 2015.


Mathematics, Examination 1  01MATZ12 
Fučík 
 zk 
 zk 
2 
2 
Course:  Mathematics, Examination 1  01MATZ1  Ing. Fučík Radek Ph.D.   ZK    2    Abstract:   Outline:   Outline (exercises):   Goals:   Requirements:   Key words:   References  
Course:  Mathematics, Examination 2  01MATZ2  Ing. Fučík Radek Ph.D. / Ing. Tušek Matěj Ph.D.     ZK    2  Abstract:   Outline:   Outline (exercises):   Goals:   Requirements:   Key words:   References  

Mathematical Methods in Biology and Medicine  01MBI 
Klika 
2+1 kz 
  
3 
 
Course:  Mathematical Methods in Biology and Medicine  01MBI  doc. Ing. Klika Václav Ph.D.  2+1 KZ    3    Abstract:  Spatially independent models; enzyme kinetics; excitable system; reactiondiffusion equations; travelling waves; pattern formation; conditions for Turing instability, the effect of domain size; the concept of stability in PDEs, spectrum of a linear operator, semigroups  Outline:  (ODEs)
1. Spatially independent models: single and multispecies interacting models including their analysis (discrete and continuous)
jednodruhové a vícedruhové interagující modely včetně jejich analýzy (diskrétní i spojité)
2. Enzyme kinetics (law of mass action) and nonequilibrium thermodynamics
3. Excitable systems  a model for nerve pulses (FitzhughNagumo); theory of bifurcations and dynamical systems
(PDEs)
4. The influence of space (reactiondiffusion equations)
5. Diffusion equation  derivation, solution, possible modification, penetration depth, longrange diffusion
6. Travelling waves
7. Pattern formation  diffusiondriven instability (Turing instability), the effect of domain size
8. Concept of stability in evolution equations in form of partial differential equations, connection to spectrum and brief touching upon theory of semigroups  Outline (exercises):  Outline of excercises follows outline of the course. For analysis of models and eventual plotting of results and solutions, symbolic mathematical programs will be used (as Mathematica, Maple).  Goals:  Knowledge:
To gain deeper insight into acquired knowledge and concepts from the whole study by their usage in constructing and analysis of models in biology.
Skills:
deeper insight into acquired knowledge and terms from study; formulation and analysis of models  Requirements:  Course of Calculus, Linear Algebra, The equations of mathematical physics. Further, functional analysis is recommended. (In the extent of the courses 01MA1, 01MAA24, 01LA1, 01LAA2, 01MMF or 01RMF, 01FA held at the FNSPE CTU in Prague).  Key words:  mathematical biology; discrete, continuous and spatial models; reactiondiffusion models; Turing instability  References  Key references:
[1] L. EdelsteinKeshet  Mathematical Models in Biology, SIAM, 2005
[2] F. Maršík  Biotermodynamika, Academia, 1998
[3] G. de Vries, T. Hillen, M. Lewis, J. Muller, B. Schonfisch  A Course in Mathematical Biology, SIAM, 2006
[4] J D Murray  Mathematical Biology: I. An Introduction, Springer, 2002
[5] J D Murray  Mathematical Biology II: Spatial Models and Biomedical Applications, Springer, 2014
[6] J Crank  The mathematics of diffusion. Oxford university press, 1979.
Recommended references:
[1] J. Keener, J. Sneyd  Mathematical Physiology, I: Cellular Physiology, Springer, 2009
[2] W. Rudin  Analyza v komplexním a reálném oboru, Academia, Praha 2003


Mathematics for Particle Systems  01MCS 
Krbálek 
2+1 kz 
  
3 
 
Course:  Mathematics for Particle Systems  01MCS  doc. Mgr. Krbálek Milan Ph.D.          Abstract:  Keywords:
Asymptotic Expansions, Balanced Distributions, Dyson gases, Particle Chain, Statistical Rigidity, Nonlinear PDE
 Outline:  1. Special Functions
2. Asymptotic Methods
3. Class of Balanced Distributions
4. Dyson Gases
5. Poissonian and SemiPoissonian Systems
6. Particle Chains and Associated Statistical Properties
7. Theory of Statistical Rigidity
8. Nonlinear PDE
9. Integral Equations with Hermitian Kernel  Outline (exercises):   Goals:  Acquired knowledge:
Students learn to predict some advanced statistical properties of particle chains with specific type of mutual interactions.
Acquired skills:
Derivation of asymptotic properties, Derivation of stochastic properties of particle chains.  Requirements:   Key words:   References  Compulsory literature:
[1]M.L. Mehta, Random Matrices (Third edition), New York: Academic, 2004
[2]E.T. Copson. Asymptotic Expansions. Cambridge University Press, Cambridge, England, 1965.
[3]V.S. Vladimirov, Equation of mathematical physics, Marcel Dekker INC, New York 1971
Optional literature:
[4]M. Krbálek, Theoretical predictions for vehicular headways and their clusters, J. Phys. A: Math. Theor. 46 (2013), 4451011
[5] M. Krbálek, Equilibrium distributions in a thermodynamical traffic gas, J. Phys. A: Math. Theor. 40 (2007), 58135821


Modelling of Extreme Events  01MEX 
Krbálek, Kůs 
  
2+0 zk 
 
2 
Course:  Modelling of Extreme Events  01MEX  Ing. Kůs Václav Ph.D.    2+0 ZK    2  Abstract:  The course is devoted to extremal events models, it means thus events which occur with very low probability, but with significant influence on behaviour of described model. We deal with fluctuation of random sums and fluctuation of maxima, further distributions for modeling extremal events and various models will be introduced and applied. Theoretical results will be applied on real data.  Outline:  1. Motivation example from agregated net traffic, solvation (machine learning), onoff approximation.
2. Distribution free inequalities (Cantelli, Chernoff, Hoeffding,...).
3. Nonparametric density estimators and their tails (adaptive kernel estimator, data transformed based methods), semiparametric estimation (Barron).
4. Distribution for modelling extremal values, heavy tailed probability distribution, generalized Pareto, loggama, lognormal, heavytailed Weibull, generalized Gumbel distribution of extremal values  parameter estimates of these distributions and their asymptotic properties.
5. PP and QQ plots for filtration of true distribution of extremal values, ME  mean excess function, its empirical estimator and usage.
6. Return period of the (insuarance) event, ordered statistics, Gumbel method.
7. Fluctuation of random sums, stable and alphastable distributions, spectral representation of stable distribution.
8..Fluctuation of random maxima: Gumbel, Fréchet and Weibull distribution as the limit distributions of maximal value of iid variables, large deviations, FisherTippett law.
9. Maximum domain attraction (MDA)  range of stable weak convergence of maxima Mn, application on distribution and expected value of POT models.
10. Models with subexponential distribution for heavy tailed distributions, class of functions R_alpha, with regularly varying of the order alpha in infinity, Karamat theorem.
11. Application on flood data, assurance (cumulative number of insured accidents), many examples.  Outline (exercises):   Goals:  Knowledge:
Mathematical models for the events which occur with very low probability, but with significant influence on behaviour of described model  different distributions for modeling extremal events, GEV, GPD, properties of mentioned models, fluctuation of random sums, maximum domain of attraction, POT methods.
Skills:
Application of given methods and models on real data with the aim to predict.
 Requirements:  01MIP nebo 01PRST. 01MAS.  Key words:  Agregated net traffic estimation, tail estimation, distributionfree inequalities, nonparametric and semiparametric estimators, fluctuation of random sums, fluctuation of random maxima, alphastabile distributions, maximum domailn attraction, GEV, Gumbel distribution, Weibull distribution, Generalized Pareto Distribution (GPD).  References  Key references:
[1] P. Embrechts, C. Klüppelberg, T. Mikosch, Modelling Extremal Events, New York Springer, 1997.
Recommended references:
[2] S. Coles, An Introduction to Statistical Modeling of Extreme Values SpringerVerlag London, 2001.
[3] P. Embrechts, H. Schmidli, Modelling of extremal events in insurance and finance, New York, Springer, 1994. 

Measure and Probability  01MIP 
Kůs 
4+2 z,zk 
  
6 
 
Course:  Measure and Probability  01MIP  doc. Ing. Hobza Tomáš Ph.D. / Ing. Kůs Václav Ph.D.          Abstract:  The subject is devoted to the introduction to Theory of probability on measuretheoretic level for discrete models, continuous distributions and general distributions of random variables. We deal with the examples of distributions including multidimensional Gaussian distribution and their properties. Further the (non)+integral characteristics of random variables (E, Var,...), convergence modes (Lp, P, a.s., D) and variants of limit theorems are derived (LLN, CLT).  Outline:  1. Axioms of probability space, sigmafields, probability measure.
2. Dependent and independent events. Borel sets, measurable functions, random variables and probability distributions.
3. RadonNikodym theorem. Discrete and absolutely continuous distributions, examples.
4. Product measure, integral w.r.t. probability measure.
5. Expectation of random variables, moments and central moments.
6. Lp space, Schwarz inequality, Chebyshev inequality, covariance.
7. Characteristic function and its properties, applications.
8. Almost sure convergence, in Lp, convergence in probability.
9. Law of large numbers (Chebyshev, Kolmogorov,...).
10. Weak convergence, its properties, Lévy theorem, Slutsky lemma.
11. Central limit theorems (CLT), LindebergFeller fundamental CLT, Lindeberg condition, BerryEsseen theorem.
12. The multivariate normal distribution with its properties.
13. Cochran's theorem and the independence of the sample mean and sample variance, populations, natural extensions in sample space, the existence of independently distributed sequences.  Outline (exercises):  Tasks solvation in the areas:
1. Axioms of probability space
2. Dependent and independent events.
3. Particular discrete distributions, examples (Binomial, Poisson, Pascal, Geometric, Hypergeometric, Multinomial distribution).
4. Particular absolute continuous distributions, examples (Uniform, Gamma, Beta, Normal, Exponential,...).
5. Distributions based on transformations (Student, Chisquared, FisherSnedecer) and quantiles.
6. Computations of characteristic functions, expectations and moments of particular distributions.
7. Covariance and Correlation of selected random variables.
8. Law of large numbers and Central limit theorems  asymptotics and usefulness.
9. Two dimensional normal distribution.  Goals:  Knowledge:
In frame of the basic course in Probability on measuretheoretic level, to provide students with the knowledge necessary for the following future subjects using probability and stochastic models. To give a deeper insight into the field.
Skills:
Orientation in majority of standard notions of the probability theory and capabilities of practical applications in actual probabilistic computation.
 Requirements:  01MAA34 or 01MAAB34.  Key words:  Measure, probability, events, random variables, distributions, expectation, characteristic function, convergence, limit theorems, multivariate Gaussian distribution.  References  Key references:
[1] Rényi A., Foundations of probability, HoldenDay Inc., San Francisco, 1970.
[2] Taylor J.C., An Introduction to Measure and Probability, Springer, 1997.
Recommended references:
[3] Jacod J., Protter P., Probability Essentials, Springer, 2000.
[4] Schervish M.J., Theory of Statistics, Springer, 1995. 

Management, Communication a Innovation  01MKI 
Rubeš 
0+1 z 
  
1 
 
Course:  Management, Communication a Innovation  01MKI  Rubeš Přemysl          Abstract:  Keywords
Career, Work, Emplyoment, Resume, CV, Cover Latter, Business, Management, Sales, Communication, Motivation, Happinness, Team management, Startups, Technology, Global companies, Presentation, Effectivity, Productivity.
 Outline:  1.Motivation, fulfillment, happiness, success, career progress
2.Selfdiscipline, effective work habits, timemanagement, productivity
3.Resume writing, cover later, selfpresentation, interviews, application process
4.Expectation management, delegation, performance management
5.Effective communication, presentation skills, sales, networking, building relationships
6.Market, global companies, technological trends, business analysis and modelling
7.Mathematics in practices, use in the modern tech trends
8.Entrepreneurship, building and managing business, financing, investors
9.Decision theory, Behavioral Economics, Neurosciences
 Outline (exercises):   Goals:  Acquired knowledge:
Scientific findings on motivation, happiness, selfdiscipline, habits, human brain, decision making, learning, communication, management and leadership. Overview of global market, possibilities for future career, use of knowledge acquired on FJFI in the real world, modern technological trends and their connection to mathematics. Basic concepts of team and project management, and running business.
Acquired skills:
Make informed decision about future careers choices based on motivation, skills and knowing all the possibilities offered by current market. Clear communication, selfpresentation, resume, cover letters, business meetings, interviews. Apply basic timemanagement, selfdiscipline tools, effective work habits.
 Requirements:   Key words:   References  Compulsory literature:
[1] Andy Grove: High Output Management, Knopf Doubleday Publishing Group, 1995
[2] Carol Dweck: Mindset, Random House Publishing, 2006
Optional literature:
[3] Daniel Kahneman: Thinking Fast and Slow, 2011
[4] Dan Pink: Drive, Riverhead books, 2011
[5] Angela Duckworth: Grit, Scribner, 2016
[6] Stephen Covey: 7 Habits of Highly Effective People, Simon&Schuster, 2013


Method of Finite Volumes  01MKO 
Beneš 
1+1 kz 
  
2 
 
Course:  Method of Finite Volumes  01MKO  prof. Dr. Ing. Beneš Michal  1+1 KZ    2    Abstract:  The subject is devoted to the numerical solutions of linear partial differential equations of first and second order using the finite difference and the finite volume methods. The lecture discusses the basic properties of numerical methods for solving elliptic, parabolic and hyperbolic equations, the modified equation and the numerical viscosity.
 Outline:  Finite difference method (FDM) for linear conservation law equation (explicit, implicit, upwind). Spectral criterion, CFL condition, stability of numerical schemes. Finite volume method (FVM) for nonlinear conservation law equation (LaxWendroff, LaxFriedrichs, RungeKutta, predictorcorrector, MacCormack). FVM for multidimensional conservation law equations (extension of the given numerical schemes to FVM  triangles, quadrilaterals). Compositional schemes, FVM for NavierStokes equations for compressible and incompressible fluids (artificial viscosity method). Discussion and presentation of problems solved by students in research projects.  Outline (exercises):   Goals:  Finite difference and finite volume methods and their application to elliptic, parabolic and hyperbolic equations.
Skills:
Application of FVM to solve the NavierStokes equations.  Requirements:  Basic course of Calculus, Linear Algebra and Ordinary Differential Equations (in the extent of the courses 01MA1, 01MAA24, 01LA1, 01LAA2, 01NM held at the FNSPE CTU in Prague).
 Key words:  Finite difference method, Finite volume method, Elliptic PDE, Hyperbolic PDE, Parabolic PDE, NavierStokes equations  References  Key references:
[1] Blazek, J.: Computational Fluid Dynamics: Principles and Applications, 2005, Elsevier.
[2] Ferziger, J. H., Peric M.: Computational methods for fluid dynamics, 1996, Springer.
[3] J.W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Springer Science & Business Media, 2013
Recommended references:
[4] M. Feistauer: Mathematical Method in Fluid Dynamics, Longman, 1993
[5] Chung, T.J.: Computational Fluid Dynamics, 2002, Cambridge University Press 

Finite Element Method  01MKP 
Beneš 
  
2 zk 
 
3 
Course:  Finite Element Method  01MKP  prof. Dr. Ing. Beneš Michal    2 ZK    3  Abstract:  The course is devoted to the mathematical theory of the finite element method numerically solving boundaryvalue and initialboundaryvalue problems for partial differential equations. Mathematical properties of the method are explained. The approximation error estimates are derived.  Outline:  1. Weak solution of boundaryvalue problem for an elliptic partial differential equation.
2. Galerkin method
3. Basics and features of the FEM
4. Definition and common types of finite elements.
5. Averaged Taylor polynomial
6. Local and global interpolant
7. BrambleHilbert lemma
8. Global interpolation error
9. Mathematical features of the FEM and details of use
10. Examples of software packages based on FEM
 Outline (exercises):  Exercise is merged with the lecture and contains examples of problem formulation, examples on function bases, examples related to the interpolation theory and examples of software packages based on FEM, in particular.
 Goals:  Knowledge:
Weak formulation of boundaryvalue and initialboundaryvalue problems for partial differential equations, Galerkin method, basics of FEM, error estimates, applications.
Skills:
Formulation of given problem into the form convenient for FEM, method implementation, application, explanation of results and error assessment.
 Requirements:  Basic course of Calculus, Linear Algebra and Numerical Mathematics, variational methods (in the extent of the courses 01MA1, 01MAA24, 01LA1, 01LAA, NM, or 01MA1, 01MAB24, 01LA1, 01LAB2, NMET, VAME held at the FNSPE CTU in Prague).  Key words:  Boundaryvalue problems for partial differential equations, finiteelement method, Galerkin method, BrambleHilbert lemma, interpolation error.  References  Key references:
[1] S. C. Brenner a L. Ridgway Scott, The mathematical theory of finite element methods, New York, Springer 1994
[2] P.G. Ciarlet, The finite element method for elliptic problems, Amsterdam, NorthHolland, 1978
[3] V. Thomée, The Galerkin finite element methods for parabolic problems, LNM 1054, Berlin, Springer, 1984
[4] S. A. Ragab, H. E. Fayed, Introduction to Finite Element Analysis for Engineers, CRC Press, Taylor Francis, 2017
Recommended references:
[5] P. Grisvard, Elliptic problems in nonsmooth domains, Boston, Pitman, 1985
[6] K. Rektorys, Variational methods in engineering and mathematical physics, Praha, Academia 1999 (translated to English)
Media and tools:
Computer training room with OS Windows/Linux and software package FEM 

Mathematical Models of Traffic Systems  01MMDS 
Krbálek 
  
2+2 z,zk 
 
4 
Course:  Mathematical Models of Traffic Systems  01MMDS  doc. Mgr. Krbálek Milan Ph.D.          Abstract:  Basic macroscopic quantities and relationships among them. Fundamental relation of vehicular modelling. Microscopic description of traffic and discussion of statistical nature of microquantities. Headwaydistributions and relationships among them. Special functions for theory of vehicular microstructure. Theorem on saddle point approximation. Discussion on empirical knowledge on macroscopic and microscopic phenomena of vehicular traffic. Methods for evaluations of traffic data. Classification of traffic models. LighthillWhitham model and associated solutions. ColeHopf transformation. Cauchy problem and its solution in generalized functions. Burgers PDE. Cellular traffic models: NaSchmodel, FukuiIschibaschi model and models with exclusion rules. Theoretical solution of TASEP model. CFmodels. Formulation of interaction dynamics in CFmodels. Numerical representation of models. Thermodynamic traffic models. Interaction potentials. Analytical solutions for basic variants of models. Derivation of clearance distribution. Class of balanced distributions and its properties. Criteria for acceptance of traffic headwaydistributions. Statistical rigidity a NVstatistics. Rigidity of Poissonian processes. Cluster function. Derivation of general formula for statistical rigidity. Analysis of statistical rigidity for traffic models.
 Outline:   Outline (exercises):  1. Macro and microquantities in empirical traffic data.
2. Traffic macromodels based on description of microstructure.
3. Properties of balanced distributions.
4. Model TASEP and associated statistical description based on Matrix Product Ansatz.
5. Steadystate solution of thermodynamical traffic model.
6. Headway distribution and its properties.
7. Balancing particle system and its description.
8. Statistical rigidity.  Goals:  Knowledge: Theoretical formulation of traffic models, their analytical solutions and statistical prediction of their microstructure.
Skills: Statistical analysis of vehicular data or data from numerical realizations of traffic models.  Requirements:   Key words:   References  [1] D. Helbing, Traffic and related selfdriven manyparticle systems, Rev. Mod. Phys. 73 (2001), 1067
[2] Li, L., Chen, X.M., Vehicle headway modeling and its inferences in macroscopic/microscopic traffic flow theory: A survey,Transportation Research Part C 76 (2017) 170
[3] D. Chowdhury, L. Santen, and A. Schadschneider, Physics Reports 329 (2000), 199
[4] N. Rajewski, L. Santen, A. Schadschneider, M. Schreckenberg: The asymmetric exclusion process: comparison of update procedures, Journal of statistical physics 92 (1998), 151
[5] M. Krbálek, Equilibrium distributions in a thermodynamical traffic gas, J. Phys. A: Math. Theor. 40 (2007), 5813


Mathematical Methods in Fluid Dynamics 1  01MMDT1 
Neustupa 
  
2+0 z 
 
2 
Course:  Mathematical Methods in Fluid Dynamics 1  01MMDT1  prof.Ing. Fořt Jaroslav CSc.  2+0 Z    2    Abstract:  The contents of the course is the introduction to mathematical methods in fluid dynamice. Concretely: mathematical modelling of fundamentals physical laws by means of partial differential equations, formulation of associated boundary or initialboundary value problems for various type sof fluids as well as various type sof flows, properties and some speciál solutions of these problems.  Outline:  1. Kinematice of fluids  the rate of deformation tensor, Reynolds? transport formula, compressible or incompressible flow, respectively fluid. 2. Volume and surface forces in the fluid, stress tensor. 3. Stokesian fluid and its special cases: ideal and Newtonian fluid. 4. Basic conservation laws (of mass, momentum, energy) and their mathematical modeling (equation of continuity, Euler and NavierStokes equations, equation of energy). 5. Second law of thermodynamics and ClausiusDuhem inequality. 6. Examples of simple solutions of the NavierStokes equations. 7. Laws of similarity. 8. Turbulent flows. 9. Boundary layer. 10. Basic qualitative properties of the NavierStokes equations  strong and weak solutions, questions of existence and uniqueness in steady and nonsteady case.  Outline (exercises):   Goals:  To learn basic principles of mathematical modelling in fluid dynamics, to learn and understand mathematical models of various type sof flows (compressible or incompressible, viscous or nonviscous, laminar or turbulent, etc.), to learn about basic methods and results in the field of qualitative properties of the NavierStokes equations.  Requirements:  Basic courses of calculus and differential equations (in the extent of the courses 01DIFR, 01MA1, 01MAA24, 01RMF held at the FNPE
CTU in Prague).
 Key words:  rate of deformation tensor, stress tensor, Stokesian fluid, ideal fluid, Newtonian fluid, equation of kontinuity, Rulet equations, NavierStokes equations, turbulent flows, boundary layer.  References  Key references:
[1] J.Neustupa: Lecture notes on mathematical fluid mechanics.
Recommended references:
[2] G.K.Batchelor: An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge 1967.
[3] G.Gallavotti: Foundations of Fluid Mechanics, Springer 2002.
[4] W.M.Lai, D.Rubin and E.Krempl: Introduction to Continuum Mechanics. Pergamon Press, Oxford 1978.
[5] L.D.Landau and E.M.Lifschitz: Fluid Mechanics. Pergamon Press, Oxford 1959.
[6] Y.Nakayama and R.F.Boucher: Introduction fo Fluid Mechanics. Elsevier 2000.
[7] W.Noll: The Foundations of Classical Mechanics in the Light of Recent Advances in Continuum Mechanics, The Axiomatic Method. North Holland, Amstedram 1959.
[8] J. Serrin: Mathematical Principles of Classical Fluid Mechanics. In Handbuch der Physik VIII/1, ed.~C.~Truesdell and S.~Flugge, Springer, Berlin 1959.
[9] R.Temam and A.Miranville: Mathematical Modelling in Continuum Mechanics. Cambridge University Press, Cambridge 2001.
[10] G.Truesdell and K.R.Rajagopal: An Introduction to the Mechanics of Fluids. Birkhauser 2000.


Mathematical Modelling of Nonlinear Systems  01MMNS 
Beneš 
2 zk 
  
3 
 
Course:  Mathematical Modelling of Nonlinear Systems  01MMNS  prof. Dr. Ing. Beneš Michal  2 ZK    3    Abstract:  The course consists of basic terms and results of the theory of finite and infinitedimensional dynamical systems generated by evolutionary differential equations, and description of bifurcations and chaos. Second part is devoted to the explanation of basic results of the fractal geometry dealing with attractors of such dynamical systems.  Outline:  I.Introductory comments
II.Dynamical systems and chaos
1.Basic definitions and statements
2.Finitedimensional dynamical systems and geometric theory of ordinary differential equations
3.Infinitedimensional dynamical systems and geometric theory of ordinary differential equations
4.Bifurcations and chaos; tools of the analysis
III.Mathematical foundations of fractal geometry
1.Examples; relation to the dynamicalsystems theory
2.Topological dimension
3.General measure theory
4.Hausdorff dimension
5.Attempts to define a geometrically complex set
6.Iterative function systems
IV.Conclusion  Application in mathematical modelling
 Outline (exercises):  Exercise makes part of the contents and is devoted to solution of particular examples from geometric theory of differential equations, linearization and Lyapunovfunction method, bifurcation analysis and fractal sets.  Goals:  Knowledge:
Deterministic dynamical systems, chaotic state description, geometric theory of ordinary and partial differential equations, theoretical fundaments of fractal geometry.
Skills:
Application of linearization method and Lyapunovfunction method in fixedpoint stability analysis, bifurcation analysis, stability of periodic trajectory, charakteristics of fractal sets and their dimension.
 Requirements:  Basic course of Calculus, Linear Algebra and Ordinary Diferential Equations, Functional Analysis, Variational Methods (in the extent of the courses 01MA1, 01MAA24, 01LA1, 01LAA2, DIFR, or 01MA1, 01MAB24, 01LA1, 01LAB2, FA1, VAME held at the FNSPE CTU in Prague).  Key words:  Evolutionary differential equations, dynamical systems, attractors, bifurcations and chaos, topological and Hausdorff dimension, iterative function systems.  References  Key references:
[1] F.Verhulst, Nonlinear Differential Equations and Dynamical Systems, SpringerVerlag, Berlin 1990
[2] M.Holodniok, A.Klíč, M.Kubíček, M.Marek, Methods of analysis of nonlinear dynamical models, Academia, Praha 1986
[3] G.Edgar, Measure, Topology and Fractal Geometry, Springer Verlag, Berlin 1989
[4] K. Falconer, Fractal Geometry  Mathematical Foundations and Applications, J. Wiley and Sons, Chichester, 2014
Recommended references:
[5] D.Henry, Geometric Theory of Semilinear Parabolic Equations, Springer Verlag, Berlin 1981
[6] R.Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer Verlag, Berlin 1988
[7] G.C. Layek, An Introduction to Dynamical Systems and Chaos, Springer Verlag, Berlin 2015
Media and tools:
Course web page with selected motivation exaamples. 

Mathematical Models of Groundwater Flow  01MMPV 
Mikyška 
  
2+0 kz 
 
2 
Course:  Mathematical Models of Groundwater Flow  01MMPV  doc. Ing. Mikyška Jiří Ph.D.    2+0 KZ    2  Abstract:  The course provides an overview of computational methods for selected groundwater flow problems. The first part of the course is devoted to mathematical formulations of these problems. The second part is aimed at selected numerical methods, emphasizing implementation issues related to these methods.
 Outline:  1. Basic terminology and quantities, Darcy's law and its extensions.
2. Derivation of basic equations, classical formulation of the fluid flow problem in saturated zone.
3. Brief introduction to the theory of Sobolev spaces.
4. Weak formulation of the secondorder elliptic boundary value problems.
5. Existence and uniqueness of the weak solution.
6. Finite Element Method (FEM) for steadystate flow equation in saturated domain.
7. Implementation problems related to FEM. Assembling of the equations, treatment of the boundary conditions.
8. Formulation of a nonstationary problem and its numerical solution by means of method of lines.
9. Discussion of alternative methods of time discretization, several special techniques.
10. Finite Volume Method (FVM) on dual mesh for parabolic equations.
11. Comparison of FEM with FVM, relation between these two methods.
12. Computer demonstrations of several simulation tools.  Outline (exercises):   Goals:  Darcy's law, balance equations, formulaton of the flow problem in the saturated zone, finite element method for an elliptic boundary value problem, extension of the method to the initialboundary value problem for a parabolic equation, assembling of the finite element systems, treatment of boundary conditions, mass lumping.
Skills: correct formulation of boundary value problems for elliptic partial differential equations, application of the finite element method including computer implementation of the method.  Requirements:  Basic course of Calculus, Linear Algebra and Ordinary Differential Equations (in the extent of the courses 01MA1, 01MAA24, 01LA1, 01LAA2, 01NM held at the FNSPE CTU in Prague).  Key words:  Darcy law, flow in the saturated zone, weak solution, Sobolev spaces, finite element method, partial differential equation of elliptic and parabolic type, finite volume method.  References  Key references:
[1] J. Bear, A. Verruijt: Modelling Groundwater Flow and Polution, D. Reidel Publishing Company, Dordrecht, Holland, 1990.
Recommended references:
[2] P.S. Huyakorn, G. F. Pinder, Computational Methods in Subsurface Flow, Academic Press, 1983
Media and tools:
A computer with OS Linux, C language compiler and UG library. 

Methods for Sparse Matrices  01MRM 
Mikyška 
2+0 zk 
  
2 
 
Course:  Methods for Sparse Matrices  01MRM  doc. Ing. Mikyška Jiří Ph.D.    2+0 ZK    2  Abstract:  The course is aimed at utilization of sparse matrices in direct methods for solution of large systems of linear algebraic equations. The course will cover the decomposition theory for symmetric and positive definite matrices. Theoretic results will be further applied for solution of more general systems. Main features of the methods and common implementation issues will be covered.  Outline:  1. Sparse matrices and their representation in a computer.
2. Algorithm of the Choleski decomposition for a symmetric and positive definite matrix.
3. Description of structure of sparse matrices and creation of fillin during the Choleski decomposition.
4. Influence of the matrix ordering, algorithms RCM, minimum degree, nested dissection, frontal method.
5. More general systems.
6. Iterative methods and preconditioning, analysis of the stationary methods, regular decompositions.
7. Examples of simple preconditioning, preconditioning of the conjugate gradient method.
8. Incomplete LU decomposition (ILU), color ordering.
9. Multigrid methods  analysis of the Richards iteration on a model example.
10. Multigrid methods  nested iterations, methods on 2 meshes, Vcycle, Wcycle, Full Multigrid method.
11. Demonstration of selected methods on computers.  Outline (exercises):   Goals:  Methods of representation of the sparse matrices in a computer, creation of fillin in the Choleski decomposition of a symmetric positive definite matrix, elimination trees, effect of matrix ordering, more general systems, iterative methods and preconditioning, stationary iterative methods, incomplete LU decomposition, introduction to the multigrid methods.
Skills: Application of the above mentioned methods to solve systems of linear algebraic equations originating from the discretization of elliptic and parabolic problems by the finite difference or finite element methods.  Requirements:  Basic course of Calculus, Linear Algebra, Numerical Methods and Numerical Linear Algebra (in the extent of the courses 01MA1, 01MAA24, 01LA1, 01LAA2, 01NM, 01PNLA held at the FNSPE CTU in Prague).  Key words:  Sparse matrices, Choleski decomposition, fillin, matrix ordering, iterative methods, preconditioning, incomplete LU decomposition, multigrid methods.  References  Key references:
[1] Y. Saad: Iterative Methods for Sparse Linear Systems, Second Edition, SIAM, 2003.
Recommended references:
[2] A. George, J. W. Liu: Computer Solution of Large Sparse Positive Definite Systems, PrenticeHall, Englewood Cliffs, NJ, 1981.
[3] A. Greenbaum: Iterative Methods for Solving Linear Systems, Society for Industrial and Applied Mathematics, Philadelphia 1997
[4] W. L. Briggs, Van E. Henson, S. F. McCormick, A Multigrid Tutorial, Second Editon, SIAM, 2000.
Media and tools:
A computer with OS Linux and software Octave. 

Theory of Random Processes  01NAH 
Vybíral 
3+0 zk 
  
3 
 
Course:  Theory of Random Processes  01NAH  doc. RNDr. Vybíral Jan Ph.D.  3+0 ZK    3    Abstract:  The course is devoted in part to the basic notions of the general theory of random processes and partially to the theory of stationary processes and sequences both weakly and strongly stationary ones.  Outline:  Notion of a random peocess, Kolmogorovˇs theorem, properties of trajectories, elements of stochastic analysis, random derivative and random integral, Wiener process, Karhunen´s theorem and spectral resolution of a random process, weak stationarity, spectral density and a linear process, ergodic theorem for weakly stationary processes, question of prediction for weakly stationary processes andsequences, strong stationarity, ergodic theorems for strongly stationary processes.  Outline (exercises):   Goals:  Basic elements of rheora of random processes, notion of a random integral, theory of weakly stationary and strongly stationary processes. Skills: Mainly application of theory of weakly stationary processes in ingineering practice.  Requirements:  Basic course of Calculus and Linear Algebra, Course of Probability Theory and Mathematical Statistics  Key words:  random process and random sequence, staochastic analzsis and random integral, spectral resolution, weak and strong stationarity, prediction, ergodic theorems.  References  Key references:
M.B. Priestley: Spectral Analysis and Time Series, Academic Press, 1981
Recommended references:
P.Z.Peebles: Probabilty, Random Variables and Random Signal Principles, McGrawHill , 2001


Nonlinear Programming  01NELI 
Fučík 
3+0 zk 
  
4 
 
Course:  Nonlinear Programming  01NELI  prof. RNDr. Burdík Čestmír DrSc.  3+0 ZK    4    Abstract:  Nonlinear optimization problems find their application in may areas of applied mathematics. The lecture covers the basics of mathematical programming theory with emphasis on convex optimization and basic methods for unconstrained and constrained optimization. The lecture is supplemented by illustrative examples.  Outline:  1. Mathematical programming: introduction, overview of basic optimization problems, linear and nonlinear programming, weak and strong Lagrange duality,
2. Summary of the required mathematical apparatus: pseudoinverse matrix, least squares method, conjugate gradient method
3. Convex sets and functions, basic properties and examples, operations preserving convexity
4. Unconstrained optimization problems
5. Constrained optimization tasks
6. Algorithms unconstrained optimization problems
7. Algorithms constrained optimization tasks: overview of basic methods, penalty methods, inner point methods, logarithmic barrier function
 Outline (exercises):   Goals:  Knowledge:
Mathematical basis of nonlinear optimization.
Abilities:
Use of nonlinear optimization algorithms in practice.  Requirements:  Basic course of Calculus and Linear Algebra.
 Key words:  Nonlinear optimization, convex sets, convex functions, Lagrange duality, KarushKuhnTuckerovy conditions, unconstrained optimization, optimization with constraints.  References  Key references:
[1] Bertsekas, Dimitri P., and Athena Scientific. Convex optimization algorithms. Belmont: Athena Scientific, 2015.
[2] Nesterov, Yurii. Lectures on convex optimization. Vol. 137. Springer, 2018.
[3] Jeter, Melvyn. Mathematical programming: an introduction to optimization. Routledge, 2018.
Recommended references:
[3] Stephen Boyd and Lieven Vandenberghe, Convex optimization, Cambridge University Press 2004
[4] Li, Li. Selected Applications of Convex Optimization. Vol. 103. Springer, 2015. 

Neural Networks and their Applications 1  01NEUR1 
Hakl, Holeňa 
  
2+0 zk 
 
2 
Course:  Neural Networks and their Applications 1  01NEUR1           Abstract:  Keywords:
Neural networks, data separation, functional approximation, supervised learning
 Outline:  1.Basic concepts of artificial neural networks.
2.Most common kinds artificial neural networks.
3.Basic numerical methods for neural networks learning.
4.Network design and architecture optimization techniques.
5.Overview of basic types of problems solved by neural networks.
6.Working with artificial neural networks in the Matlab and ROOT.  Outline (exercises):   Goals:  Acquired knowledge:
Basic concepts, features and models of neural networks.
Acquired skills:
Orientation in the art, the ability to use models of artificial neural networks for solving practical problems in the field of approximation of functions, separation of sets and time series prediction.
 Requirements:   Key words:   References  Compulsory literature:
[1] R. Rojas. Neural Networks ? A Systematic Introduction. Springer. 1991
Optional literature:
[2] B.D. Ripley. Pattern Recognition and Neural Networks. Cambridge University Press. 1996


Theoretical Fundamentals of Neural Networks  01NEUR2 
Hakl, Holeňa 
2+0 zk 
  
3 
 
Course:  Theoretical Fundamentals of Neural Networks  01NEUR2           Abstract:  Keywords:
Functional approximation, supervised learning, VapnikChervonenkisdimension
 Outline:  1.Approach to artificial neural networks from the theory of function approximation.
2.Approach to artificial neural networks from the probability theory.
3.Analysis of the solvability of selected tasks neural network models.
4.Qualitative measure of neural networks (VCdimension, pseudodimension, sensitivity dimension).
5.Theoretical background of neural networks learning.
6.Selected advanced classification applications of artificial neural networks.
 Outline (exercises):   Goals:  Acquired knowledge:
The theoretical foundation for the study of the properties and potential of artificial neural networks models.
Acquired skills:
Advanced ability to analyze the appropriateness and effectiveness of artificial neural networks models for practical applications. The fundamental basis for the expansion of theoretical knowledge enabling greater understanding and development of artificial intelligence.
 Requirements:  Some selected topics in this lecture are closely related to the content of the lecture "Probabilistic learning models" that presents these selected topics in a much broader and deeper form.  Key words:   References  Compulsory literature:
[1] J. Šíma, R. Neruda. Teoretické otázky neuronových sítí. Matfyzpress. 1996
Optional literature:
[2] M. Anthony, P. L. Bartlett. Neural Network Learning: Theoretical foundations. Cambridge university Press, 2009.
[3] M. Vidyasagar. A theory of Learning and Generalization. Springer 1997.
[4] V. Roychowdhury, KY. Siu, A. Orlitsky. Theoretical advances in neural computation and learning. Kluwer Academic Publishers. 1994.
[5] H. White. Artificial Neural Networks: Approximation and Learning Theory. Blackwell Publishers. Cambridge. 1992.


Design of Experiments  01NEX 
Franc, Hobza 
2+1 kz 
  
4 
 
Course:  Design of Experiments  01NEX  Ing. Franc Jiří Ph.D. / doc. Ing. Hobza Tomáš Ph.D.  2+1 KZ    4    Abstract:  For processes of any kind that have measurable inputs and outputs, Design of Experiments (DOE) methods help us in the optimum selection of inputs for experiments, and in the analysis of results. The course consists of selected methods of DOE such as: completely randomized design, randomized block design, Latin squares design and two level factorial experiments.  Outline:  1. Introduction to design of experiments and analysis of their results
2. Completely randomized singlefactor design: introduction of model with fixed effects, tests of equality of means, choice of the sample size, check of suitability of model, test of equality of variances, transformation to obtain homoscedasticity, model with random effects, estimates of model parameters, and confidence intervals
3. Methods of multiple comparisons: LSD method, Bonferroni method, Scheffé method, Tukey method
4. Randomized block design: definition of model, test of equality of effects, power of test, choice of the sample size, estimate of the lost values
5. Latin and GraecoLatin squares designs: test of equality of effects, verification of suitability of model, residua, multiple comparisons
6. Two level factorial experiments: statistical models and their properties for 2^2, 2^3 a 2^k designs  Outline (exercises):  1. Statistical hypothesis testing
2. Comparison of several treatment meana  analysis of variance
3. Randomized block design
4. Latin and GraecoLatin square design
5. Factorial experiments  Goals:  Knowledge:
Basic notions and principles of design and analysis of experiments.
Skills:
Application to solution of practical problems, i.e. ability to design an experiment for a concrete problem and to do its statistical evaluation.  Requirements:  Basic course of Calculus and Probability (in the extent of the courses 01MAB3, 01MAB4 and 01PRST held at the FNSPE CTU in Prague).  Key words:  Design of experiments, completely randomized experiment, randomized block experiment, multiple comparison, Latin squares, GraecoLatin squares, two level factorial experiment.  References  Key references:
[1] D. C. Montgomery: Design and analysis of experiments, Wiley 2008
Recommended references:
[2] J. Antony: Design of Experiments for Engineers and Scientists, ButterworthHeinemann, 2003


Numerical Mathematics 1  01NMA1 
Oberhuber 
4+0 zk 
  
4 
 
Course:  Numerical Mathematics 1  01NMA1  Ing. Oberhuber Tomáš Ph.D.          Abstract:  The course introduces to numerical methods for solving the basic problems arising from technical and research problems. The accent is put on a good understanding of the root of theoretical methods.  Outline:  1. Recapitulation of necessary concepts from linear algebra and functional analysis.
2. Direct and iterative methods for solving linear algebraic equations. Matrix inversion.
3. Solving the partial eigenvalue problem.
4. Solution of the full eigenvalue problem.
5. Solving the equation f (x) = 0
6. Systems of nonlinear algebraic and transcendental equations.
7. Interpolation functions by polynomials.
8. Numerical calculation of derivatives.
9. Numerical calculation of integral
 Outline (exercises):   Goals:  Knowledge: Correct understanding of the theoretical basis for numerical algorithms is accented. Skills: Applications of numerical methods for solution of basic mathematical tasks originated from technical or scientific problems.  Requirements:   Key words:  Direct methods, iterative methods, eigenvalue problem, systems of equations, interpolation, numerical calculation of integrals  References  Key references:
[1] A. Quarteroni, R. Sacco, F. Saleri: Numerical Mathematics. SpringerVerlag 2000
[2] J.W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Springer Science & Business Media, 2013
Recommended references:
[3] A. S. Householder: The Theory of Matrices in Numerical Analysis. Blaisdell Publishing Company 1965 

Numerical Methods 2  01NME2 
Beneš 
  
2+0 kz 
 
2 
Course:  Numerical Methods 2  01NME2  prof. Dr. Ing. Beneš Michal    2+0 KZ    2  Abstract:  The course is devoted to numerical solution of boundaryvalue problems and intialboundaryvalue problems for ordinary and partial differential equations. It explains methods converting boundaryvalue problems to initialvalue problems and finitedifference methods for elliptic, parabolic and firstorder hyperbolic partial differential equations.  Outline:  I.Numerical solution of ordinary differential equations  boundaryvalue problems
1.Shooting method
2Method of transformation of a boundaryvalue problem
3.Method of finite differences
4.Solution of nonlinear equations
II.Numerical solution of partial differential equations of the elliptic type
1.Finitedifference method for linear secondorder equations
2.Convergence and the error estimate
3.Method of lines
III.Numerical solution of partial differential equations of the parabolic type
1.Method of finite differences for onedimensional problems
2.Method of finite differences for higherdimensional problems
3.Method of lines
IV.Numerical solution of hyperbolic conservation laws
1.Formulation and properties of hyperbolic conservation laws
2.Simplest finitedifference methods
 Outline (exercises):   Goals:  Knowledge:
Numerical methods based on transformation of a boundaryvalue problem to an initialvalue problem, finitedifference method for ODE's and PDE's.
Skills:
Application of given methods in particular examples in physics and engineering including computer implementation and error assessment.  Requirements:  Basic course of Calculus, Linear Algebra and Ordinary Differential Equations (in the extent of the courses 01MA1, 01MAB24, 01LA1, 01LAB2, 12NMET held at the FNSPE CTU in Prague).
 Key words:  Boundaryvalue problems and initialboundaryvalue problems for differential equations, shooting methods, finitedifference methods, energy methods giving properties of numerical schemes, explicit and implicit methods, conservation laws.  References  Key references:
[1] A.A. Samarskij, Theory of Difference Schemes, CRC Press, New York, 2001
[2] J.W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Springer Science & Business Media, 2013
[3] R.J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady State and Time Dependent Problems, SIAM, 2007
[4] R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002
Recommended references:
[5] E. Godlewski a P.A. Raviart, Numerical approximation of hyperbolic systems of conversation laws, New York, Springer 1996
Media and tools:
Computer training room with Windows/Linux and programming languages C, Pascal, Fortran. 

Numerical Mathematics 2  01NUM2 
Beneš 
  
2+1 z,zk 
 
3 
Course:  Numerical Mathematics 2  01NUM2  prof. Dr. Ing. Beneš Michal    2+1 Z,ZK    3  Abstract:  The course is devoted to numerical solution of boundaryvalue problems and intialboundaryvalue problems for ordinary and partial differential equations. It explains methods converting boundaryvalue problems to initialvalue problems and finitedifference methods for elliptic, parabolic and firstorder hyperbolic partial differential equations.  Outline:  I.Numerical solution of ordinary differential equations  boundaryvalue problems
1.Shooting method
2Method of transformation of a boundaryvalue problem
3.Method of finite differences
4.Solution of nonlinear equations
II.Numerical solution of partial differential equations of the elliptic type
1.Finitedifference method for linear secondorder equations
2.Convergence and the error estimate
3.Method of lines
III.Numerical solution of partial differential equations of the parabolic type
1.Method of finite differences for onedimensional problems
2.Method of finite differences for higherdimensional problems
3.Method of lines
IV.Numerical solution of hyperbolic conservation laws
1.Formulation and properties of hyperbolic conservation laws
2.Simplest finitedifference methods
 Outline (exercises):  1.Taylor expansion in the context of difference formulas with particular properties
2.Normalized conversion method
3.Nonlinear difference schemes.
4.Definition of the weak solution of an elliptic boundaryvalue problem.
5.Relation of difference approximations and of the finitevolume method
 Goals:  Knowledge:
Numerical methods based on transformation of a boundaryvalue problem to an initialvalue problem, finitedifference method for ODE's and PDE's.
Skills:
Application of given methods in particular examples in physics and engineering including computer implementation and error assessment.  Requirements:  Basic course of Calculus, Linear Algebra and Ordinary Differential Equations (in the extent of the courses 01MA1, 01MAA24, 01LA1, 01LAA2, 01NM held at the FNSPE CTU in Prague).  Key words:  Boundaryvalue problems and initialboundaryvalue problems for differential equations, shooting methods, finitedifference methods, energy methods giving properties of numerical schemes, explicit and implicit methods, conservation laws.  References  Key references:
[1] A.A. Samarskij, Theory of Difference Schemes, CRC Press, New York, 2001
[2] I. Babuška, M. Práger, E. Vitásek, Numerical Processes in Differential Equations, Wiley, London 1966
[3] R.J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady State and Time Dependent Problems, SIAM, 2007
[4] R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002
[5] J.W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Springer Science & Business Media, 2013
Recommended references:
[6] R.J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady State and Time Dependent Problems, SIAM, 2007
[7] E. Godlewski a P.A. Raviart, Numerical approximation of hyperbolic systems of conversation laws, New York, Springer 1996
Media and tools:
Computer training room with Windows/Linux and programming languages C, Pascal, Fortran. 

Parallel Algorithms and Architectures  01PAA 
Oberhuber 
  
3 kz 
 
4 
Course:  Parallel Algorithms and Architectures  01PAA  Ing. Oberhuber Tomáš Ph.D.    3 KZ    4  Abstract:  This course deals with the parallel data processing. It is important in situations when one processing unit (CPU) is not powerful enough to finish given task in reasonable time. When designing parallel algorithms, good knowledge of the parallel architectures is important. Therefore these architectures are studied as a part of this course too.  Outline:  1. Introduction
2. Sequential architectures
3. Parallel architectures
4. Communication networks
5. Communication operations
6. Introduction to CUDA
7. OpenMP / MPI
8. Analysis of parallel algorithms
9. Sorting algorithms
10. Matrix algorithms
11. Graph algorithms, MonteCarlo methods 12. Combinatorial search
 Outline (exercises):  1. Programming in CUDA
2. OpenMP / MPI
3. Analysis of parallel algorithms
4. Sorting algorithms
5. Matrix algorithms
6. Graph algorithms, MonteCarlo methods
7. Combinatorial search
 Goals:  Knowledge:
Parallel architectures, basic types of parallel architectures, communication in parallel architectures, programming standards OpenMP, MPI, CUDA/OpenCL, sorting algorithms, matrix algorithms, graph algorithms, MonteCarlo methods, analysis of the parallel algorithms.
Skills:
The students will learn how to choose appropriate parallel architecture for given problem, how to design proper parallel algorithm, analyze it and derive its efficiency and how to implement the parallel algorithm.  Requirements:  Basic algorithms, programing in C/C++.  Key words:  Parallel architectures, parallel algorithms, arhitectures with shared memory, architectures with distributed memory, interconnection networks, basic communication operations, OpenMP, MPI, GPGPU, sorting, matrices, graphs, numerical computations, graph algorithms, MonteCarlo methods, combinatorial search.
 References  Key references:
Grama A., Karypis G., An Introduction to Parallel Computing: Design and Analysis of Algorithms
Recommended references:
CUDA Programming guide
Media and tools:
Computer lab 

Advanced Algorithmization  01PALG 
Oberhuber 
2 kz 
  
2 
 
Course:  Advanced Algorithmization  01PALG  Ing. Oberhuber Tomáš Ph.D.          Abstract:  Keywords:
String algorithms, graph algorithms, dynamic programming, suffix tress, graph cuts, numerical methods for solution of partial differential equations.
 Outline:  1. Pancake sort
2. Turnpike problem
3. String algorithms  motives finding, strings alignment, suffix trees
4. Graph algorithms in image processing  graph cuts
5. Graph algorithms for solution of PDEs  methods for solution of the HamiltonJacobi equation
 Outline (exercises):   Goals:  Knowledge:
Pancake sort, Turnpike problem, motives finding, dynamic programming for string comparison, suffix trees for searching of substrings, graph algorithms in image processing, graph algorithms and numerical methods for solution of the HamiltonJacobi equation.
Skills:
Students will learn new techniques for design of algorithms and how to apply knowledge from the graph theory and numerical mathematics for a construction of particular algorithms.
 Requirements:   Key words:   References  Compulsory literature:
[1] R. Sedgewick, Algorithms in C++: Graph algorithms, 2002, AddisonWesley.
[2] N. C. Jones, P. A. Pevzner, An introduction to bioinformatics, MIT Press, 2004.
Optional literature:
[3] W.K. Sung, Algorithms in bioinformatics  a practical introduction, CRC Press, 2010.


Modern Theory of Partial Differential Equations  01PDR 
Tušek 
  
2+0 zk 
 
2 
Course:  Modern Theory of Partial Differential Equations  01PDR  Ing. Tušek Matěj Ph.D.          Abstract:  Sobolev spaces, continuous and compact embedding theorems, trace theorem.
Elliptic PDE of Second Order, LaxMilgram theorem, regularity, maximum principle, harmonic functions.  Outline:  1. Sobolev spaces
1.1 Definition, completeness, examples
1.2 Continuous and compact embedding theorems
1.3 Trace theorem
2. Weak solution (importance, derivation of the weak formulation)
3. Elliptic PDE of Second Order
3.1 Existence and uniqueness of weak solutions (LaxMilgram theorem)
3.2 Regularity of weak solutions
3.3 Relation to the calculus of variations, Poincaré inequality
3.4 Maximum principle for classical and weak solutions  Outline (exercises):   Goals:  Acquired knowledge: fundamental facts about Sobolev spaces; weak solution and its importance; theorems on existence, uniqueness, and regularity of weak solutions of partial differential equations (PDE) of the second order; maximum principle
Acquired skills: derivation of the weak formulation, understanding the relation to the classical theory, to get ready for selfstudy of other important cases (such as evolution equations)  Requirements:  Basic knowledge of theory of distributions and functional analysis.  Key words:  partial differential equations, Sobolev spaces, elliptic regularity, maximum principle  References  Compulsory literature:
[1] Evans L.C.: Partial Differential Equations, 2nd ed., American Mathematical Society, 2010.
Optional literature:
[2] Protter M.H., Weinberger H.F.: Maximum Principles in Differential Equations, Springer, 1984.
[3] Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order, Springer, 2001 (reprint).
[4] Adams R.A.: Sobolev Spaces, Academic Press, 1975. 

Programming of Peripherals Devices  01PERI 
Čulík 
2+0 z 
  
2 
 
Course:  Programming of Peripherals Devices  01PERI  Ing. Čulík Zdeněk  2+0 Z    2    Abstract:  Memory organization, input and output ports, computer bus. Software libraries for computer peripherals, 3D graphic libraries.
Principles of peripherals device drivers.  Outline:  1. Memory and I/O addressing
2. Interrupt requests and interrupt controllers
3. Keyboard (BIOS, I/O ports, principles of simple keyboard driver), serial communication, video adapters.
4. Examples of OpenGL graphical programs. Introduction to Open Inventor library
5. Disk devices (IDE and SCSI interfaces)
6. Overview of device drivers for Windows and Linux operating systems
7. Real time operating systems.  Outline (exercises):   Goals:  Knowledge:
Overview of methods used for peripheral programming. Introduction to software libraries for concrete peripherals devices.
Skills:
To develop software application which optimally uses hardware resources.  Requirements:   Key words:  Peripherals devices, device drivers.  References  [1] A. Rubini, J. Corbet: Linux Device Drivers, O Reilly, 2001
[2] D. Shreiner, T. Davis, M, Woo, J. Neider: OpenGL Programming Guide: The Official Guide to Learning OpenGL, Pearson Education, 2003
[3] T. Shanley, D. Anderson: PCI System Architecture, AddisonWesley, 1999
[4] Friedheim Schmidt: The SCSI Bus and IDE Interface: Protocols, Applications and Programming, AddisonWesley, 1997
[5] http://oss.sgi.com/projects/inventor/


Mainframe Programming  01PMF 
Oberhuber 
  
2 z 
 
2 
Course:  Mainframe Programming  01PMF  Ing. Oberhuber Tomáš Ph.D.    2 Z    2  Abstract:  In this course the basics of programming in z/OS are explained namely the programming in assembler. Basic instructions, macros, I/O operations, DLL library loading and some other topics are discussed.  Outline:  1. Introduction to assembleru
2. Structure of instructions
3. Data types
4. Inputs and outputs
5. Data conversions
6. Tables and loops
7. Logical operations
8. Subroutines
9.10. Macros
11.12. Dynamic modules  Outline (exercises):  2. Structure of instructions
3. Data types
4. Inputs and outputs
5. Data conversions
6. Tables and loops
7. Logical operations
8. Subroutines
9.10. Macros
11.12. Dynamic modules  Goals:  Knowledge:
Structure of the assembler instructions, data types, inputs and outputs, data conversions, tables and loops, logical operations, subroutines, macros, dynamic modules.
Skills:
Student will learn how to write simple programs in assembleru for the system z/OS. The student will be able to easier understand specialize training courses of software developing companies.  Requirements:  Basic operations with the mainframe on the level of the course Introduction to the mainframe.  Key words:  Mainframe, z/OS, assembler, HLASM, macros.  References  Key references:
K. McQuillen, A. Prince, MVS Assembler Language, 1987, Mike Murach.
Recommended references:
IBM, IBM System/370, Principles of Operation, IBM, 1975.
Media and tools:
Computer lab, mainframe account. 

Probabilistic Learning Models  01PMU 
Hakl 
2+0 zk 
  
2 
 
Course:  Probabilistic Learning Models  01PMU  Ing. Hakl František CSc.  2+0 ZK    2    Abstract:  Introduction into the theory PAC learning model, VCdimension of finite sets, Sauer, Cover and Radon's lemma,
VCdimension of composed mappings, application of VCdimension for lower bound of necessary patterns, analysis of
properties of delta rule based learning processes, PAC learning model extensions and PAO learning, Fourier
coefficients search for Boolean functions.  Outline:  1. PAC learning introduction
2. Concepts and concept classes
3. PAC learning in finite case
4. VapnikChervonenkis dimension (Sauer's lemma, Cover's lemma, Radon's lemma)
5. VCdimension of finite set systems
6. VCdimension of union and intersection
7. VCdimension of linear concepts
8. Application of Cover?s lemma
9. VapnikChervonenkis dimension of composed mapping
10. Pattern complexity and VCdimension
11. Minimal number of patterns and PAC learning
12. Delta rule learning algorithm
13. Lover bound for maximal steps of delta rule algorithm
14. Polynomial learning and pattern dimension
15. Almost optimal solution of the cover set problem
16. Polynomial learning and conceptual complexity
17. Probabilistic learning algorithms
18. Probabilistic approximation of Fourier expansion
19. Probabilistic search of Fourier coefficients of Boolean functions
20. Probably approximately optimal learning  Outline (exercises):   Goals:  Make the students acquainted with theoretical and mathematical fundamentals of PAClike learning algorithms.  Requirements:   Key words:  Probably approximately correct learning, VapnikCervonenkis dimension, pattern complexity, delta rule algorithm,
cover set problems, probabilistic Fourier coefficient search  References  Key references:
F. Hakl, M. Holeňa. Úvod do teorie neuronových sítí. Ediční středisko ČVUT, Praha, 1997.
Recommended references:
Vwani Roychowdhury, KaiYeung Siu, Alon Orlitsky. Theoretical Advances in Neural Computation and Learning. Kluwer
Academic Publishers, 1994.
Martin Anthony and Norman Biggs. Computational Learning Theory. Press
Syndicate of the University of Cambridge, 1992.
A. Blumer, A. Ehrenfeucht, D. Haussler, and M. K. Warmuth. Learnabil
ity and the VapnikChervonenkis Dimension. Journal of the Association for
Computing Machinery, 36:929965, oct 1989. 

Advanced Methods of Numerical Linear Algebra  01PNLA 
Mikyška 
2+0 zk 
  
3 
 
Course:  Advanced Methods of Numerical Linear Algebra  01PNLA  doc. Ing. Mikyška Jiří Ph.D.  2+0 ZK    3    Abstract:  Representation of real numbers in computers, behaviour of rounding errors during numerical computations, sensitivity of a problem, numerical stability of an algorithm. We will analyse sensitivity of the eigenvalues of a given matrix and sensitivity of roots of systems of linear algebraic equations. Then, the backward analysis of these problems will be performed. The second part of the course is devoted to the methods of QRdecomposition, least squares problem, and to several modern Krylov subspace methods for the solution of systems of linear algebraic equations and the Lanczos method for approximation of the eigenvalues of a symmetric square matrix.  Outline:  1. Introduction, basic terminology, representation of numbers in computers
2. Standard arithmetics IEEE, behaviour of rounding errors in computations in finite precision arithmetics, forward and backward analysis
3. Similarity transforms, Schur's theorem, measurement of the distances between spectra of two matrices
4. Theorem on sensitivity of the spectra of general matrices
5. Sensitivity of eigenvalues of diagonalizable and normal matrices, backward analysis of the eigenvalue problem
6. Sensitivity of roots of systems of linear algebraic equations, backward analysis of the solutions to the systems of algebraic equations
7. QRdecompositions and orthogonal transformations
8. Householder transform
9. GrammSchmidt orthogonalization process
10. Krylov space methods  introduction, Arnoldi's algorithm, method of generalized minimal residual (GMRES) for solution of systems of linear algebraic equations
11. Lanczos algorithm, approximation of eigenvalues of a symmetric matrix
12. Overview of the Krylov space methods for solution of systems of linear algebraic equations
13. Preconditioning of the iterative methods, examples of simple preconditioners
 Outline (exercises):   Goals:  Floating point arithmetics, rounding errors in the finite precision arithmetics, backward analysis and its application to estimation of the approximation error, sensitivity and backward analysis of matrix spectra and solution of systems of the linear algebraic equations, methods for QR decomposition, Arnoldi algorithm, basic Krylov subspace methods for solution of systems of linear algebraic equations (GMRES, CG, MinRes, BiCG, QMR), and the Lanczos method for approximation of eigenvalues of a symmetric matrix.
Skills: To choose a suitable method for solution of a system of linear algebraic equations or evaluation of a spectrum of a given matrix and to estimate error of the obtained approximation.  Requirements:  Basic course of Calculus, Linear Algebra and Ordinary Differential Equations (in the extent of the courses 01MA1, 01MAA24, 01LA1, 01LAA2, 01NM held at the FNSPE CTU in Prague).  Key words:  Floating point arithmetics, rounding errors, sensitivity, numerical stability, backward analysis, QR decomposition and orthogonal transformations, least squares problem, Krylov subspace methods.  References  Key references:
[1] D. S. Watkins: Fundamentals of Matrix Computations, J. Willey, New York, 1991
Recommended references:
[2] B. N. Parlett: Symmetric Eigenvalue Problem, Prentice Hall, Engl. Cliffs, 1988
[3] G. H. Golub, C. F. van Loan: Matrix Computations, John Hopkins, 1997.


Advanced Numerical Methods  01PNM 
Beneš 
  
2+0 kz 
 
2 
Course:  Advanced Numerical Methods  01PNM  prof. Dr. Ing. Beneš Michal    2+0 KZ    2  Abstract:  The course is devoted to advanced numerical solution of boundaryvalue problems and intialboundaryvalue problems for ordinary and partial differential equations. It explains the shooting method, advanced finitedifference methods and finitevolume method for nonlinear elliptic, parabolic and firstorder hyperbolic partial differential equations.  Outline:  I.Numerical solution of ordinary differential equations  boundaryvalue problems
1.Shooting method
2Method of finite differences for nonlinear equations
II.Numerical solution of partial differential equations of the elliptic type
1.Finitedifference method for nonlinear secondorder equations
2.Convergence and the error estimate
3.Finite volume method
III.Numerical solution of partial differential equations of the parabolic type
1.Method of finite differences for nonlinear evolution problems
2.Method of lines
3. Finite volume method
IV.Numerical solution of hyperbolic conservation laws
1.Formulation and properties of hyperbolic conservation laws
2.Simplest finitedifference methods
3. Finite volume method
 Outline (exercises):   Goals:  Knowledge:
Numerical methods for nonlinear boundaryvalue problems, finitedifference method for ODE's and PDE's, finitevolume method.
Skills:
Application of given methods in particular examples in physics and engineering including computer implementation and error assessment.  Requirements:  Basic course of Calculus, Linear Algebra and Ordinary Differential Equations (in the extent of the courses 01MA1, 01MAB24, 01LA1, 01LAB2, 12NME1 held at the FNSPE CTU in Prague).
 Key words:  Boundaryvalue problems and initialboundaryvalue problems for differential equations, shooting methods, finitedifference methods, energy methods giving properties of numerical schemes, explicit and implicit methods, conservation laws, finite volume method.  References  Key references:
[1] A.A. Samarskij, Theory of Difference Schemes, CRC Press, New York, 2001
[2] J.W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Springer Science & Business Media, 2013
[3] R.J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady State and Time Dependent Problems, SIAM, 2007
[4] R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002
Recommended references:
[5] E. Godlewski a P.A. Raviart, Numerical approximation of hyperbolic systems of conversation laws, New York, Springer 1996
Media and tools:
Computer training room with Windows/Linux and programming languages C, Pascal, Fortran. 

Computer Graphics 1  01POGR12 
Strachota 
2 z 
2 z 
2 
2 
Course:  Computer Graphics 1  01POGR1  Ing. Strachota Pavel Ph.D.  2 Z    2    Abstract:  The first part of the twosemester "Computer Graphics" course is devoted to the specifics of digital display devices spanning from history up to the state of the art technologies. Further, a survey of fundamental problems in 2D computer graphics is given together with their solutions. Focus is put on mathematical description of problems and explanation of the corresponding algorithms using knowledge previously obtained in a variety of subjects available at FNSPE. The final part of the course covers the applications of computer graphics approaches in the process of authoring scientific documents and presentations.  Outline:  1. Computer graphics hardware
2. Human vision, color perception and representation
3. Raster graphics algorithms
4. Computational geometry
5. Image transforms (interpolation, warping, morphing)
6. Formats and algorithms for image data compression and storage
7. Graphical user interfaces
8. WWW and multimedia technologies
9. Computer graphics in scientific document authoring
10. Technology of digital photography  Outline (exercises):  The exercises are integrated in the lectures and are devoted to solving the simpler of the particular problems in 2D computer graphics, e.g. digital dithering algorithms, boundary fill, convex hull determination, LZW compression, etc.  Goals:  Knowledge:
Proper grasp of the fundamental problems of 2D graphics as well as a notion of cutting edge contemporary technologies. Solid theoretical and practical foundations for further development of computer graphics methods and their customization to particular needs.
Skills:
Immediate ability to apply the approaches of computer graphics in multimedia presentations, scientific visualization and data processing. Complex design and implementation of the corresponding software instruments. Capability to produce highquality outputs of scientific research (articles, slides, posters) by means of professional typesetting technologies.  Requirements:    Key words:  Display devices, GPU, color spaces, raster graphics algorithms, computational geometry, warping, morphing, graphical image formats, data compression, GUI, multimedia, data visualization, digital photography.  References  Key references:
[1] J. F. Hughes, A. van Dam, M. McGuire, D. F. Sklar, J. D. Foley, S. K. Feiner, K. Akeley: Computer Graphics: Principles and Practice (3rd ed.), Addison Wesley, 2014.
Recommended references:
[2] Žára, Beneš, Sochor, Felkel: Moderní počítačová grafika. Computer Press, Praha, 2005.
[3] J. Vince: Mathematics for Computer Graphics. Springer Verlag, London, 2006.
[4] E. Pazera: Focus on SDL. Premier Press, Cincinnati, 2003.
Media and tools:
Computer lab with Windows/Linux OS and C, C++, Java, C# programming languages, MS Visual Studio, Qt development framework. SDL library. 
Course:  Computer Graphics 2  01POGR2  Ing. Oberhuber Tomáš Ph.D. / Ing. Strachota Pavel Ph.D.    2 Z    2  Abstract:  The second part of the twosemester "Computer Graphics" course begins with a brief introduction to signal theory in the context of aliasing  a phenomenon ubiquitous in computer graphics. Further, a well structured survey of fundamental problems in 3D computer graphics is given together with their solutions, from the description of a 3D scene to its realistic rendering. Focus is put on mathematical description of problems and explanation of the corresponding algorithms using knowledge previously obtained in a variety of subjects available at FNSPE. The algorithm implementation aspect such as data structures design etc. is also a matter of concern. In the last lecture, a number of theoretical concepts are demonstrated using Blender, an opensource 3D modeling and rendering software instrument.  Outline:  1. Introduction to signal theory
2. The objectives of 3D computer graphics
3. Representing curves and surfaces
4. Solid modeling
5. Procedurebased modeling techniques
6. Matrix transforms
7. Projections
8. Visible surface determination
9. Illumination and shading
10. Texturing
11. Ray tracing and photorealistic rendering methods
12. Modeling and rendering 3D scenes in Blender  Outline (exercises):  The exercises are integrated in the lectures and are devoted to solving the simpler of the particular problems in 3D computer graphics, e.g. cubic spline rasterization, algorithms for regularized Boolean operations on octrees, fractal terrain modeling by means of the Terragen software tool, geometrical transforms in homogeneous coordinates, silhouette algorithm in visible surface determination, the elementary version of the ray tracing method, etc.  Goals:  Knowledge:
Proper grasp of the fundamental problems of 3D graphics as well as a notion of cutting edge contemporary technologies. Solid theoretical and practical foundations for further development of computer graphics methods and their customization to particular needs.
Skills:
Immediate ability to apply the approaches of computer graphics in multimedia presentations, scientific visualization and data processing. Complex design and implementation of the corresponding software instruments.  Requirements:  Completing the "Computer graphics 1 (01POGR1)" course is strongly recommended though not strictly obligatory.  Key words:  Signal theory, aliasing, curves and surfaces, solid modeling, procedural and fractal modeling, projections, visible surface determination, illumination and shading, ray tracing, radiosity, photon maps.  References  Key references:
[1] J. F. Hughes, A. van Dam, M. McGuire, D. F. Sklar, J. D. Foley, S. K. Feiner, K. Akeley: Computer Graphics: Principles and Practice (3rd ed.), Addison Wesley, 2014.
Recommended references:
[2] Žára, Beneš, Sochor, Felkel: Moderní počítačová grafika. Computer Press, Praha, 2005.
[3] A. S. Glassner: An Introduction to Ray Tracing. Morgan Kaufmann Publishers, San Francisco, 2002.
[4] M. F. Cohen, J. R. Wallace: Radiosity and Realistic Image Synthesis. Morgan Kaufmann Publishers, San Francisco, 1993.
[5] P. Prusinkiewicz, A. Lindenmayer: The Algorithmic Beauty of Plants. Springer Verlag, 1990.
Media and tools:
Computer lab with Windows/Linux OS and C, C++, Java, C# programming languages, MS Visual Studio, Qt development framework. SDL library, OpenGL and DirectX APIs, Blender, 3dsMax.


Programmer's Practicum  01PROP 
Oberhuber 
0+2 z 
  
2 
 
Course:  Programmer's Practicum  01PROP  Ing. Klement Vladimír Ph.D.  0+2 Z    2    Abstract:  The purpose of this course is to acquire good programming habits which will help in writing of clean code, i.e. such that is easy to comprehend by others and suitable for adding new functionality. Using specific examples, the students get familiar with naming conventions, and continue through writing project documentation, principles of defensive programming, debugging, up to creating objectoriented design, design patterns and refactoring.  Outline:  I. The basics of writing clean code
1. Formatting
2. Data structures
3. Naming of variables
4. Rules for writing functions
5. Handling of errors, exceptions
6. Comments
II. Objectoriented design
1. Namespaces
2. Class organization
3. Inheritance and abstraction
4. Special types of classes
III. Code development
1. Coding conventions
2. Specification and design
3. Testing of code
4. Refactoring
5. Documentation  Outline (exercises):  The excercise is an integral part of the course, its contents is given by the subject's sylabus.  Goals:  Knowledge:
How to write clean code, coding conventions. Principles of defensive programming, code management and guidelines for refactoring. How to write documentation. Code structure and functionality, creating of consistent blocks and their testing. Objectoriented design, open for changes. Development of clean and comprehensible code.
Skills:
The student learns to write transparent code, which is easy to understand by other developers, flexible in terms of adding new functionality and easy for debugging.  Requirements:  C/C++ programming, objectoriented programming  Key words:  Clean code, coding conventions, defensive programming, test driven development, objectoriented design, refactoring, documentation.  References  Key references:
[1] R.C. Martin, Clean Code: A Handbook of Agile Software Craftmanship, Prentice Hall 2009
[2] S. McConnell, Code Complete, Second Edition, Microsoft Press, 2004
Recommended references:
M. Fowler, Refactoring: Improving the Design of Existing Code, AddisonWesley, 2002


Probability and Statistics  01PRST 
Hobza 
3+1 z,zk 
  
4 
 
Course:  Probability and Statistics  01PRST  doc. 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, goodnessoffit 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, goodnessoffit tests.  References  Key references:
[1] H. G. Tucker: An introduction to probability and mathematical statistics. Academic Press, 1963
[2] H. PishroNik: Introduction to Probability, Statistics, and Random Processes, Kappa Research, LLC, 2014
Recommended references:
[3] J. Shao: Mathematical statistics, Springer, 2003


Probability and Statistics B  01PRSTB 
Hobza 
3+1 kz 
  
4 
 
Course:  Probability and Statistics B  01PRSTB  doc. Ing. Hobza Tomáš Ph.D.  3+1 KZ    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, goodnessoffit 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, goodnessoffit tests  References  Key references:
[1] H. G. Tucker: An introduction to probability and mathematical statistics. Academic Press, 1963
Recommended references:
[2] J. Shao: Mathematical statistics, Springer, 1999


LaTeX  Publication Instrument  01PSL 
Ambrož 
  
0+2 z 
 
2 
Course:  LaTeX  Publication Instrument  01PSL  Ing. Ambrož Petr Ph.D.    0+2 Z    2  Abstract:  The course is devoted to the basics and facilities of computer typography, particularly to the system LaTeX  Outline:  1) LaTeX  philosophy of the programme, typesetting of the text, paragraphs
2) Typesetting and structuring of the documents, tables in LaTeX
3) Mathematics in LaTeX
4) Advanced mathematical constructions
5) Graphics, figures, bibliographic data and databases
6) Beamer  package for typesetting transparencies
 Outline (exercises):  1) Installation of the LaTeX system
2) Structuring on the document, text and paragraphs
3) Itemization environments, tables
4) Mathematics in LaTeX
5) AMSLaTeX package
6) References
7) Including the graphics files  Goals:  Knowledge:
Computer typography, facilities of the LaTeX system.
Skills:
Use of the system LaTeX for typesetting of a (typographically fair) document.  Requirements:   Key words:  Typography, LaTeX.  References  References:
[1] J. Rybička, LaTeX pro začátečníky, Konvoj, 1999.
[2] T. Oetiker et al.,
The Not So Short Introduction to LaTeX2e,
www.ctan.org/texarchive/info/lshort/english/lshort.pdf
Recommended references:
[3] H. Kopka, P.W. Daly, Guide to LaTeX, AddisonWesley Professional, (2003)
Teching tools:
Computer lab (Windows/Unix) with LaTeX system. 

Windows Programming  01PW 
Čulík 
2+0 z 
  
2 
 
Course:  Windows Programming  01PW  Ing. Čulík Zdeněk  2+0 Z    2    Abstract:  Simple graphical programs for MS Windows. Basic editing controls. File input and output. User defined components, dynamic type identification and reflection.  Outline:  1. Graphical user interface development in C# language
2. Programming of basic controls
3. Functions and classes for image processing
4. Storing information in XML format
5. Access to relational databases
6. Programing simple components for Visual Studio development environment
7. Principles of dynamic data type identification and using of reflection in graphical development environments
 Outline (exercises):   Goals:  Knowledge:
Programming language C#, platform .NET, graphical user applications for MS Windows.
Skills:
Design and develop application using C# programming language.  Requirements:   Key words:  Win32, .Net, C#, Visual Studio.  References  [1] C. Petzold, Windows Programming, Microsoft Press, 1996
[2] C. Petzold, Programming Microsoft Windows Forms, Microsoft Press, 2005
[3] C. Petzold, .NET Book Zero, http://www.charlespetzold.com/dotnet/
[4] http://msdn.microsoft.com/ 

Regression Data Analysis  01REAN 
Franc, Víšek 
2+2 z,zk 
  
4 
 
Course:  Regression Data Analysis  01REAN  Ing. Franc Jiří Ph.D. / prof.RNDr. Víšek Jan Ámos CSc.          Abstract:  Key words:
Regression model, crosssectinal and panel data, classical and robust estimators.  Outline:  Linear model, the least squares, estimator minimizing sum of absolute values of residuals. The best linear unbiased estimator of regression coefficients  orthogonality condition and sphericality (homoscedasticity), consistency. Asymptotic normality of the estimator of regression coefficients. The best unbiased estimator of regression coefficients. Coefficient of determination, role of intercept, significance of explonatory variables. Confidence intervals testing submodels, Chow test. Statistical packages, possibilities, inputs and outputs, reliability, interpretation of results, White test of heteroscedasticity, index plot. Normality test, Theil residuals, tests of good fit, KS test, normal plot. Colinearity, condition number, FarrarGlauber test, redundancy, ridge regression, estimator with linear restrictions. AR, MA, AR(I)MA, invertibilty and stationarity conditions. Smoothing the linear envelope of trends, moving averages. Seasonal and cyclic components, randomness test. Efficient estimate of AR(1), MA(1), AR(2), MA(2), PraisWinston, ochraneOrcutt. Robust regression, Mestimators, qualitative and quantitative robustness, influence function, influential points (outliers, leverage points). The least median of squares, the trimmed least squares and the least trimmed squares, the weighted least squares and the least weighted squares, algorithms, aplications. Philosophical ideas of mathematical modelling.  Outline (exercises):  Exercises in regression will be held in agreement with the lecture and its aim is the application of the regression methods in R.
Introduction to R, linear model, Least Squares estimation, residuals, submodel, ANOVA, tests of model assumptions, normality, independence, QQ plots, multicollinearity, logistic regression, nonlinear regression, transformations, robust methods in regression
 Goals:  Acquired knowledge:
To continue in statistical lectures and to offer one of the most powerful tool for data modelling. To make students familiar with theoretical and practical aspects of topic and open them the point of view of statistician and econometrician, classical and robust approach.
Acquired skills:
Independent application of regression methods to epmirical data.
 Requirements:   Key words:   References  Key references:
[1] Statistical data analysis, in Czech. Publishing house of the Czech Technical University in Prague, 1997. (187 pages, ISBN 8001017354)
Recommended references:
[2] Atkinson, A.C. (1985): Plots, Transformations and Regression: An Introduction to Graphical Methods of Diagnostic Regression Analysis. Oxford: Claredon Press.
[3] Baltagi, B. H. (2001): A Companion to Theoretical Econometrics,Massachusetts, Oxford: Blackwell.
[4] Drapper, N. R., Smith, H (1998): Applied Regression Analysis, New York: J.Wiley.
[5] Judge, G. G., W. E. Griffiths, R. C. Hill, H. Lütkepohl, T. C. Lee (1982): Introduction to the Theory and Practice of Econometrics. New York: J.Wiley.
[6] Wooldridge,J.M. (2001): Econometric Analysis of Cross section and Panel Data. The MIT Press, Cambridge, Massachusetts, U.S.A. and London, England.


The Equations of Mathematical Physics  01RMF 
Klika 
4+2 z,zk 
  
6 
 
Course:  The Equations of Mathematical Physics  01RMF  doc. Ing. Klika Václav Ph.D.  4+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, 01MAA24, 01LA1, 01LAA2, 01VYMA held at the FNSPE
CTU in Prague).  Key words:  Mathematical methods in physics, distributions, integral transfomations, partial differential equations  References  Key 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 

Image Processing and Pattern Recognition 1  01ROZ1 
Flusser, Zitová 
  
2+2 zk 
 
4 
Course:  Image Processing and Pattern Recognition 1  01ROZ1  doc. RNDr. Zitová Barbara Ph.D.    2+2 ZK    4  Abstract:  An introductory course on image processing and pattern recognition. Major attention is paid to image sampling and quantization, image preprocessing (noise removal, contrast stretching, sharpening, and deblurring, Wiener filtering, blind deconvolution), edge detection, morphology and geometric transformations and warping. Numerous applications and experimental results are presented in addition to the theory.  Outline:  image sampling and quantization, Shannon theorem, aliasing
basic image operations, histogram, contrast stretching, noise removal, image sharpening
linear filtering in the spatial and frequency domains, convolution, Fourier transform
edge detection, corner detection
image degradations and their modelling, inverse and Wiener filtering, restoration of motionblurred and outoffocus blurred images
image segmentation
mathematical morphology
image registration and matching
 Outline (exercises):  Image visualisation and Matlab basics
Fourier transform
Noise and denoising methods
Edge detectors and histogram equalisation
Image registration
Morphology
 Goals:  to teach students introduction to image processing  Requirements:  Passing linear algebra and calculus  Key words:  image analysis, edge detection, denoising methods, image preprocessing and registration, morphology  References  Key references:
Gonzales R. C., Woods R. E., Digital Image Processing (3rd ed.), AddisonWesley, 2008
ecommended references:
Pratt W. K.: Digital Image Processing (3rd ed.), John Wiley, New York, 2001
Media and tools: a complete set of lecture slides and excersises are available at http://zoi.utia.cas.cz/ROZ1


Image Processing and Pattern Recognition 2  01ROZP2 
Flusser 
2+1 zk 
  
4 
 
Course:  Image Processing and Pattern Recognition 2  01ROZP2  Ing. Flusser Jan DrSc.          Abstract:  The course is a continuation of ROZ1. Major attention is paid to features for shape description and recognition, and to general pattern recognition techniques. Numerous applications and experimental results are presented in addition to the theory.  Outline:  features for description and recognition of 2D shapes
invariant features, Fourier descriptors, moment invariants, differential invariants
statistical pattern recognition, supervised and nonsupervised classification, NN classifier, linear classifier, Bayessian classifier
clustering in a feature space, iterative and hierarchical methods
dimensionality reduction of a feature space
 Outline (exercises):  simple 2D visual features
Fourier descriptors
basic classification algorithms
cluster analysis  simple examples
 Goals:  to teach students the basics of object recognition  Requirements:  passing of ROZ1  Key words:   References  Duda R.O. et al., Pattern Classification, (2nd ed.), John Wiley, New York, 2001
Media and tools: a complete set of lecture slides and excersises are available at http://zoi.utia.cas.cz/ROZ2 

Special Functions and Transformations in Image Analysis  01SFTO 
Flusser 
  
2+0 zk 
 
2 
Course:  Special Functions and Transformations in Image Analysis  01SFTO  Ing. Flusser Jan DrSc.    2+0 ZK    2  Abstract:  The course broadens topics of the courses ROZ1 and ROZ2. Main attention will be paid to several special functions and transformations (especially moment functions and wavelet transform) and their use in selected tasks of image processing  edge detection, noise removal, recognition of deformed objects, image registration, image compression, etc. Both the theory and practical applications will be discussed.  Outline:  geometric moments, definitions and basic properties
complex moments
moment invariants to rotation and scaling
moment invariants to affine transform
moment invariants to convolution/blurring and combined invariants
orthogonal polynomials and orthogonal moments (Legendre moments,
FourierMellin moments, Zernike moments)
discrete moments and their effective calculation
introduction to wavelet transform (WT)
edge and corner detection by means of the WT
image denoising by means of the WT
image registration by means of the WT
waveletbased image compression (block quantizing)
other applications of the WT in image processing
 Outline (exercises):   Goals:  theory of moments and its application in digital image processing. An introduction to vavelet theory and an application of wavelets in digital image processing. An practical application of introduced algorithms (edge detection, denoising, image registration, object recognition, compression).  Requirements:  passing of ROZ1 and ROZ2  Key words:  moment theory, wavelets, object recognition, denoising, image compression, image registration  References  Key references:
Jan Flusser, Tomáš Suk and Barbara Zitová, Moments and Moment Invariants in Pattern Recognition, Wiley & Sons Ltd., 2009 (317 pp., ISBN 9780470699874)
Recommended references:
S. Mallat: A Wavelet Tour of Signal Processing, Academic Press, 2008
Media and tools: a complete set of lecture slides and excersises are available at http://zoi.utia.cas.cz/ PGR013 

Computer Networks 1  01SITE12 
Minárik 
1+1 z 
1+1 z 
2 
2 
Course:  Computer Networks 1  01SITE1  Ing. Minárik Miroslav  1+1 Z    2    Abstract:  Understanding the history and present network (LAN, WAN, use the principles and technologies). Architecture of reference model ISO/OSI. Network protocols, practical exercises with TCP/IP communications. Internet services  mail, remote access, www. Secure communication, tunneling.
Directory services, certificates, certification authorities, public key infrastructure (PKI). Use in practice. Network security  firewalls (packet filters, proxies, gateways, NAT, DMZ), practical exercises.
(According to the interest  the serial control lines, modems)
 Outline:  1st Past and present of computer networks. Topology, used principles and technologies.
2nd The reference model ISO/OSI
3rd Network protocols, TCP/IP communication
4th Internet services. Remote access, electronic mail (formats, transmission, access to the mailbox)
5th Security of services, tunneling
 Outline (exercises):  1st Access to email, formatting and transmission
2nd Secure encrypted communication channel, tunneling
3rd TCP / IP communication (option C, C + +, Java, etc.)
4th Remote access (telnet, ssh, XWindows, Remote Desktop, VNC)
 Goals:  Knowledge: use of secure channels, principles of electronic mail, directory services and their use, public key infrastructure, firewall principles.
Skills: Build a secure transmission channel, work with certificates, basic routing and firewall settings.
 Requirements:  Programming basics and Algorithmization (in the extent of the courses ZPRO, ZALG held at the FNSPE
CTU in Prague).
 Key words:  Formatting and transmission of electronic mail (MIME, SMTP, IMAP, POP).
Secure communications (encryption, ssh, ssl, stunnel).
TCP / IP communications.
Directory services (LDAP, LDIF).
Public Key Infrastructure, an electronic signature.
Firewall.
 References  Key references:
[1] Scott Oaks, Java security, O'Reilly, 2001
Recommended references:
[3] William R. Cheswick, Steven M. Bellovin, Aviel D. Rubin, "Firewalls and Internet security: repelling the wily hacker?, ADDISONWESLEY, 2003
[4] Gert De Laet, Gert Schauwers, "Network security fundamentals?, Cisco Press, 2004
[5] William Stallings, "Cryptography and Network Security: Principles and Practice?, Prentice Hall, 2006
Interneto:
[6] http://www.protocols.com/
[7] standardy "RequestForComments? (http://www.ietf.org/)
Media and tools:
Computer training room with Windows/Linux and programming languages Java, C, C++, Pascal.

Course:  Computer Networks 2  01SITE2  Ing. Minárik Miroslav    1+1 Z    2  Abstract:  Understanding the history and present network (LAN, WAN, use the principles and technologies). Architecture of reference model ISO/OSI. Network protocols, practical exercises with TCP/IP communications. Internet services  mail, remote access, www. Secure communication, tunneling.
Directory services, certificates, certification authorities, public key infrastructure (PKI). Use in practice. Network security  firewalls (packet filters, proxies, gateways, NAT, DMZ), practical exercises.
(According to the interest  the serial control lines, modems)
 Outline:  6th Network and computer security (firewalls: packet filter, proxies, gateways, NAT), virtual private network
7th Directory services, identify real world entities, ASN1, LDAP, LDIF
8th Certificates, CA, public key infrastructure
9th Electronic Signature
 Outline (exercises):  5th Access to the directory service, LDAP LDIF
6th Simple CA based on OpenSSL
7th Encryption, digital signature (the Java JCE)
8th Interconnection networks, routing, firewall (filtering, NAT)
 Goals:  Knowledge: use of secure channels, principles of electronic mail, directory services and their use, public key infrastructure, firewall principles.
Skills: Build a secure transmission channel, work with certificates, basic routing and firewall settings.
 Requirements:  Programming basics and Algorithmization (in the extent of the courses ZPRO, ZALG held at the FNSPE
CTU in Prague).  Key words:  Formatting and transmission of electronic mail (MIME, SMTP, IMAP, POP).
Secure communications (encryption, ssh, ssl, stunnel).
TCP / IP communications.
Directory services (LDAP, LDIF).
Public Key Infrastructure, an electronic signature.
Firewall.
 References  Key references:
[1] Scott Oaks, Java security, O'Reilly, 2001
Recommended references:
[3] William R. Cheswick, Steven M. Bellovin, Aviel D. Rubin, "Firewalls and Internet security: repelling the wily hacker?, ADDISONWESLEY, 2003
[4] Gert De Laet, Gert Schauwers, "Network security fundamentals?, Cisco Press, 2004
[5] William Stallings, "Cryptography and Network Security: Principles and Practice?, Prentice Hall, 2006
Internet:
[6] http://www.protocols.com/
[7] standardy "RequestForComments? (http://www.ietf.org/)
Media and tools:
Computer training room with Windows/Linux and programming languages Java, C, C++, Pascal.


System Reliability and Clinical Experiments  01SKE 
Kůs 
  
2+0 kz 
 
3 
Course:  System Reliability and Clinical Experiments  01SKE  Ing. Kůs Václav Ph.D.    2+0 KZ    3  Abstract:  The main goal of the subject is to provide the mathematical principles of reliability theory and techniques of survival data analysis, reliability of component systems, asymptotic methods for reliability, concept of experiments under censoring and their processing in clinical trials (lifetime models). The techniques are illustrated and tested within practical examples originating from lifetime material experiments and clinical trials.  Outline:  1. Reliability function, mean time before failure, hazard rate, conditional reliability, mean rezidual life.
2. Systems with monotone hazard rate and their characteristics, TTT transformation and its usage.
3. Binomial, exponential distribution, Poisson process, Weibull disttribution and its flexibility, practical examples.
4. Generalized Gamma and Erlang distribution, Rayleigh distribution, Inverted Gaussian, BirnbaumSaundersův model.
5. Component systems reliability analysis, serial, parallel, koon systems, bridge systems, pivotal decomposition.
6. Repairable and renewal systems, perfect and imperfect switching.
7. Asymptotics for minimum time before failure, serialparallel systems, Gumbel distribution.
8. Lifetime data  censoring (type I, type II, random, mixed), maximum likelihood and Bayesian estimates of the systems under censoring.
9. Nonparametric approach, KaplanMeier estimate of reliability, Nelson estimate of cumulative hazard rate.
10. Cox proporcional hazard model, its properties, PH assumption testing, usage, examples.
11. Applications to the data from clinical research, case studies in biometry, particular data processing.  Outline (exercises):   Goals:  Knowledge:
Extension of the statistical procedures for objets reliability analysis with random effects and their applications in stochastic survival tasks.
Skills:
Orientation in various stochastic reliability multicomponent systems and their properties.  Requirements:  01MAS or 01PRST  Key words:  Reliability function, hazard rate, Weibull distribution, component systems, asymptotic methods, censoring, applications, clinical trials.  References  Key references:
[1] Rausand M., Hoyland A., System Reliability Theory: Models, Statistical Methods, and Applications, Second Ed., Willey, 2004.
Recommended references:
[2] Kleinbaum D.G., Survival Analysis, Springer, 1996.
[3] Lange N, et al., Case studies in Biometry, Wiley, 1994.
[4] Kovalenko I.N., Kuznetsov N.Y., Pegg P.A., Mathematical theory of reliability of time dependent systems with practical applications, Wiley, 1997. 

Seminar on Calculus B1  01SMB12 
Krbálek 
0+2 z 
0+2 z 
2 
2 
Course:  Seminar on Calculus B1  01SMB1  doc. Mgr. Krbálek Milan Ph.D.  0+2 Z    2    Abstract:  The course is devoted to support the lectures of Calculus B3.  Outline:  Physical applications of theory of differential equation, general properties of metric, norm and prehilbert spaces, Hilbert spaces of functions.  Outline (exercises):  Physical applications of theory of differential equation, general properties of metric, norm and prehilbert spaces, Hilbert spaces of functions.  Goals:  Knowledge: Application of mathematical theory to the practical tasks. Skills: Individual analysis of practical exercises.  Requirements:  Basic course of Calculus a Linear Algebra (in the extent of the courses 01MA1, 01MAB2, 01LA1, 01LAB2 held at the FNSPE CTU in Prague).  Key words:  Solution of differential equations, metric spaces, normed and Hilbert?s spaces.  References  Key references:
[1] Robert A. Adams, Calculus: A complete course, 1999,
[2] Thomas Finney, Calculus and Analytic geometry, Addison Wesley, 1996
Recommended references:
[3] John Lane Bell: A Primer of Infinitesimal Analysis, Cambridge University Press, 1998
Media and tools: MATLAB 
Course:  Seminar on Calculus B2  01SMB2  doc. Mgr. Krbálek Milan Ph.D.    0+2 Z    2  Abstract:  The course is devoted to support the lectures of Calculus B4.  Outline:  Regular mappings in two or threedimensional space, analytical forms of tangent hyperplanes to quadrics and pseudoquadrics, volumes of chosen bodies, derivative of integral with parameter, application of measure theory and theory of Lebesgue integral.  Outline (exercises):  Regular mappings in two or threedimensional space, analytical forms of tangent hyperplanes to quadrics and pseudoquadrics, volumes of chosen bodies, derivative of integral with parameter, application of measure theory and theory of Lebesgue integral.  Goals:  Knowledge: Application of mathematical theory to the practical tasks. Skills: Individual analysis of practical exercises.  Requirements:  Basic course of Calculus a Linear Algebra (in the extent of the courses 01MA1, 01MAB2, 01MAB3, 01LA1, 01LAB2 held at the FNSPE CTU in Prague).  Key words:  Function of several variables, measure theory, theory of Lebesgue integral  References  Key references:
[1] M. Giaquinta, G. Modica, Mathematical analysis  an introduction to functions of several variables, Birkhauser, Boston, 2009
Recommended references:
[2] S.L. Salas, E. Hille, G.J. Etger, Calculus (one and more variables), Wiley, 9th edition, 2002
Media and tools: MATLAB 

Statistical methods with applications  01SME 
Hobza 
  
2+0 kz 
 
2 
Course:  Statistical methods with applications  01SME  doc. Ing. Hobza Tomáš Ph.D.          Abstract:  The course consists of selected methods of statistical data analysis such as: linear regression and correlation, analysis of variance, nonparametric methods, contingency tables, simulation of random variables and their application. The aim is to illustrate the use of statistical procedures on examples. Solutions of concrete examples by use of statistical software are also included.  Outline:  1. Hypothesis testing and godnesoffit tests
2. Linear regression and correlation
3. Oneway and twoway analysis of variance
4. Nonparametric tests  sign and rank tests, Wilcoxon test, KruskalWallis test
5. Contingency tables  tests of independence and homogeneity
6. Simulation of random variables  Outline (exercises):   Goals:  Knowledge:
Basic statistical procedures for data analysis, nonparametric methods.
Skills:
Application of theoretically studied statistical procedures to practical problems of data analysis including use of these methods on computer in the MATLAB environment.
 Requirements:  Basic course of Calculus and Probability (in the extent of the courses 01MAB3, 01MAB4 and 01MIP held at the FNSPE CTU in Prague).  Key words:  Hypothesis testing, goodnessoffit tests, linear regression, ANOVA, nonparametric tests, contingency tables.  References  Key references:
[1] Shao, Jun: Mathematical Statistics, Springer, New York 1999
Recommended references:
[2] J.P. Marques de Sá: Applied statistics using SPSS, STATISTICA, MATLAB and R, Springer, 2007


Modern Trends in Corporate Information Technologies  01SMF 
Oberhuber 
  
2 z 
 
2 
Course:  Modern Trends in Corporate Information Technologies  01SMF  Ing. Oberhuber Tomáš Ph.D.    2 Z    2  Abstract:  The course is devoted to mainframe administration basics. After introduction to mainframe hardware the following lectures covers security, transaction systems, virtualization and nonrelational databases
in the mainframe environment.  Outline:  1. Mainframe hardware.
2. Security in the mainframe environment (SAF, RACF).
3. Transaction systems (CICS).
4. Virtualization (History, Basic terms, Virtualization concept, Mainframe hardware
virtualization).
5. Nonrelational databases.
 Outline (exercises):  1. Transaction systems
2. Nonrelational databases  Goals:  Knowledge: Basic overview of the mainframe
administration technics and technology.
Skills: Orientation in the area
of mainframe system administration.  Requirements:  Basics of operating systems, mainframe and databases.  Key words:  Mainframe, system administration, system security, transaction systems, virtualization, nonrelational databases.  References  Key references:
IBM, Introduction to the New Mainframe: z/OS Basics, IBM, 2005.
Recommended references:
IBM, Introduction to the New Mainframe: Security, IBM, 2006.
IBM, Introduction to the New Mainframe: z/VM Basics, IBM, 2003.
Media and tools:
Computer training room with Windows/Linux. 

Software Seminar 1  01SOS12 
Čulík 
0+2 z 
0+2 z 
2 
2 
Course:  Software Seminar 1  01SOS1  Ing. Čulík Zdeněk  0+2 Z    2    Abstract:  Java, Java Beans,
Assembly language programming for microprocessors Intel 80x86  Outline:  1. Introduction to Java programming language
2. Java Beans components
3. Assembly language programming for microprocessors Intel 80x86
4. Registers, memory addressing
5. Instruction set, instruction codes
6. Procedure call, numeric coprocessor, MMX instructions
7. Virtual memory (80386)
8. CISC and RISC processor architectures, 64bit microprocessors
 Outline (exercises):  1. Simple application written in Java programming language
2. Java data types, comparison with other programming languages
3, Introduction to graphical user interface design using Swing library
4. Classes and methods
5. Arrays in Java, differences between implementations of arrays in Java, C and Pascal
5. Interfaces, data model for JList component
7. Trees and JTree graphical component
8. Dynamic type identification  reflection and introspection
9. File input and output
10. Registers and simple Intel 80x86 instructions
11. Debugging on machine instruction level
12. Subroutines and parameter passing conventions
13. Translation of some specific high level programming language construction to machine code  Goals:  Knowledge:
Introduction to Java programming language.
Differences between Java and C++. Overview of Intel 80x86 microprocessor architecture.
Skills:
Development of simple Java Application.  Requirements:   Key words:  Java, assembly language.  References  [1] B.Eckel, Thinking in Java (4th Edition), Prentice Hall, 2006
[2] http://mindview.net/Books
[3] http://developer.intel.com 
Course:  Software Seminar 2  01SOS2  Ing. Čulík Zdeněk    0+2 Z    2  Abstract:  Graphical libraries GTK+ and Qt. Development of graphical user interface using C and C++ programming languages. Portable applications for Unix like operating systems, especially for Linux systems. Portability to Microsoft Windows.  Outline:  1. Introduction to graphical user interface programming in Linux (GTK+ and Qt library)
2. Development of simple application for GTK library. Object oriented framework Qt
3. Basic user interface controls
4. Response to user events
5. Compilation of applications under Linus operating system.  Outline (exercises):  1. Source code for simple GTK application
2. Compilation and linking
3. Programming callback routines as a response on user events
4. Designing user interface using Glage
5. Minimal application for Qt graphics library
6. Qt signals and slots
7. Using Qt Designer and Creator
8. Widgets for lists, tables and trees
9. KDE desktop environment and KDevelop application
 Goals:  Knowledge:
Structure of GTK and Qt graphical user interface libraries used in Unix based operating systems.
Skills:
Write a C or C++ application with graphical user interface for Linux operating system.  Requirements:   Key words:  Qt, GTK, Linux.  References  [1] J. Blanchette, M. Summerfield, C++ GUI Programming with Qt 4, 2nd Edition, Prentice Hall, 2008
[2] H. Pennington, GTK+ /Gnome Application Development, Sams, 1999
[3] M. Summerfield, Rapid GUI Programming with Python and Qt, Prentice Hall, 2007
[4] http://qt.nokia.com
[5] http://library.gnome.org/devel
[6] http://www.gtk.org 

Geometrical aspects of spectral theory  01SPEC 
Krejčiřík 
  
2+0 zk 
 
2 
Course:  Geometrical aspects of spectral theory  01SPEC           Abstract:  1. Motivations. The crisis of classical physics and the rise of quantum mechanics. Mathematical formulation of quantum theory. Spectral problems in classical physics.
2. Elements of functional analysis. The discrete and essential spectra. Sobolev spaces. Quadratic forms. Schrödinger operators.
3. Stability of the essential spectrum. Weyl's theorem. Bound states. Variational and perturbation methods.
4. The role of the dimension of the Euclidean space. Criticality versus subcriticality. The Hardy inequality. Stability of matter.
5. Geometrical aspects. Glazman's classification of Euclidean domains and their basic spectral properties.
6. Vibrational systems. The symmetric rearrangement and the FaberKrahn inequality for the principal frequency.
7. Quantum waveguides. Elements of differential geometry: curves, surfaces, manifolds. Effective dynamics.
8. Geometrically induced bound states and Hardytype inequalities in tubes.
 Outline:   Outline (exercises):   Goals:  Acquired knowledge:
The goal of the lecture is to acquaint the students with spectral methods in the theory of linear differential operators coming both from modern as well as classical physics, with a special emphasis put on geometrically induced spectral properties.
Acquired skills:
Mastering of advanced methods of spectral theory of selfadjoint operators; variational tools, partial differential equations, geometric analysis, Sobolev spaces.
 Requirements:   Key words:  Schrödinger operators; Hardy inequality; Effective dynamics  References  Key references
[1] B. Davies, Spectral theory and differential operators, Cambridge University Press, 1995.
[2] A. Henrot, Extremum problems for eigenvalues of elliptic operators, Frontiers in Mathematics, Birkhäuser, Basel, 2006.
[3] M. Reed and B. Simon, Methods of modern mathematical physics, IIV, Academic Press, New York, 19721978.
Recommended references:
[1] W. O. Amrein, A. Boutet de Monvel and V. Georgescu, C0 groups, commutator methods and spectral theory of Nbody Hamiltonians, Progress in Math. Ser., vol. 135, Birkhäuser, 1996.
[2] D. E. Edmunds and W. D. Evans, Spectral theory and differential operators, Oxford University Press, 1987.
[3] L. C. Evans, Partial Differential Equations, Amer. Math. Soc., 2010.


Social Systems and Their Simulations  01SSI 
Hrabák, Krbálek 
2+1 kz 
  
4 
 
Course:  Social Systems and Their Simulations  01SSI  Ing. Hrabák Pavel Ph.D.          Abstract:  The course is devoted to the issue of social systems modeling. That includes stochastic methods and methods of statistical physics for description and analytical solution of social interaction systems, implementation of particular models and comparison of the computer simulations results with the empirical data.  Outline:  1. Interdisciplinary aspects of quantitative sociodynamics, basic terminology,
2. Model classification, basic tools for simulation,
3. Cellular automata and interacting particle systems,
4. TASEP, NagelSchreckenberg model, Floorfield model,
5. Multilane komunications and cellular traffic models,
6. ODE based models,
7. Carfollowing models,
8. Socialforce model of room evacuation,
9. Parametre calibration and validation,
10. Fundamental diagrams
11. Experimental studies,
12. Stationary state characteristics of models.  Outline (exercises):  1. Computer simulation of particular models of social system,
2. The steadystate solution of chosen models,
3. Gaining and processing empirical data.  Goals:  Znalosti:
Matematický popis systému se sociální interakcí,
Přehled modelů užívaných pro simulaci sociálních systémů,
Použití stochastických metod a metod statistické fyziky pro jejich popis.
Schopnosti:
Implementace modelů na výpočetní technice,
Zpracování a porovnání výsledků simulací s empirickými daty.  Requirements:  Basic course in Probability and mathematical statistics, statiscal physics, analysis of chaotic systems and programing in MATLAB (in the extent of the courses 01PRST, 01SM, 02TSFA, 01CHAOS, 18MTL held at the FNSPE CTU in Prague).  Key words:   References  Key references:
[1] D. Helbing, Quantitative Sociodynamics: Stochastic Methods and Models of Social Interaction Processes, Kluwer Academic, Dordrecht, 1995.
[2] A. Schadschneider, D. Chowdhury, K. Nishinari: Stochastic transport in conplex systems, Elsevier BV., Oxford, 2011.
[3] W. Weidlich, Sociodynamics  a systematic approach to mathematical modelling in the social sciences, CRC Press, 2000.. 

Seminar of Contemporary Mathematics 1  01SSM12 
Pelantová, Tušek 
0+2 z 
0+2 z 
2 
2 
Course:  Seminar of Contemporary Mathematics 1  01SSM1  prof. Ing. Pelantová Edita CSc. / Ing. Tušek Matěj Ph.D.  0+2 Z    2    Abstract:  This seminar provides a different approach to those fields of mathematics that are included in curriculum but also to those that are not part of basic courses of mathematics.  Outline:  1. Definition of Eudox' real numbers, 2. Kurzweil integral, 3. Nonstandard analysis, 4. Probabilistic methods in combinatorics, 5. Distributional properties of sequences, 6. Groebner basis, 7. Solving differential equations using symmetry methods, 8. On simplicial partitions. This seminar is partially covered by hosting collaborators of Dept of Mathematics.  Outline (exercises):  The subject is a seminar.
This seminar provides a different approach to those fields of mathematics that are included in curriculum but also to those that are not part of basic courses of mathematics.  Goals:  Knowledge:
Overview in modern trends in mathematics.
Skills: During working on simple problems and recherche on selected topic, students get acquianted with basis of scientific work.  Requirements:  Knowledge of mathematical analysis, linear and general algebra in the scope of bachelor's degree programme Mathematical Modelling at FNSPE, CTU in Prague.  Key words:  Modern trends in mathematics.  References  Presentations of lecturers which are published on web pages of the subject and other study materials are recommended on individual basis according to the selected to pick for home work.
Obligatory:
[1] P.J. Davis, R. Hersh, The mathematical experience, Birkhauser Boston, 1981
Recommended:
[2] M. Aigner, G. M. Ziegler, Proofs from THE BOOK, SpringerVerlag, 2004 
Course:  Seminar of Contemporary Mathematics 2  01SSM2  doc. Ing. Klika Václav Ph.D. / prof. Ing. Pelantová Edita CSc.    0+2 Z    2  Abstract:  This seminar provides a different approach to those fields of mathematics that are included in curriculum but also to those that are not part of basic courses of mathematics.  Outline:  1. Symbolic dynamics.
2. Nonstandard numeration systems.
3. Paralel algorithms.
4. Symetries of differential equations and applications
5. Integration factors, first integrals of differential equations.
This seminar is partially covered by hosting collaborators of Dept of Mathematics.  Outline (exercises):  The subject is a seminar.
This seminar provides a different approach to those fields of mathematics that are included in curriculum but also to those that are not part of basic courses of mathematics.  Goals:  Knowledge:
Overview in modern trends in mathematics.
Skills:
During working on simple problems and recherche on selected topic, students get acquianted with basis of scientific work.  Requirements:  Knowledge of mathematical analysis and linear algebra in the scope of bachelor's degree programme Mathematical Modelling at FNSPE, CTU in Prague.  Key words:  Modern trends in mathematics.  References  Presentations of lecturers which are published on web pages of the subject and other study materials are recommended on individual basis according to the selected topic for home work.
Obligatory:
[1] P.J. Davis, R. Hersh, The mathematical experience, Birkhauser Boston, 1981
Recommended:
[2] M. Aigner, G. M. Ziegler, Proofs from THE BOOK, SpringerVerlag, 2004 

Stochastic Methods  01STOM 
Franc 
2+0 kz 
  
2 
 
Course:  Stochastic Methods  01STOM  Ing. Franc Jiří Ph.D.          Abstract:  Keywords:
Markov processes, transition probabilities, stationary distribution, hitting probabilities, transition rates, Poisson process, queuing theory.  Outline:  1.Introduction to stochastic systems, homogeneity, stationarity, simulation of Bernoulli process and simple random walk.
2.Analysis of random walk and simulation of gambler?s ruin problem.
3. Discretetime Markov chains I, transition probability, ChapmanKolmogorov theorem, classification of states, recurrent and transient states.
4.Discretetime Markov chains II, Ergodic theorem, stationary distribution.
5.Discretetime Markov chains III, hitting probabilities, Reversibility, Branching processes, Simulation of Ehrenfest and Bernoulli processes of diffusion.
6.Continuoustime Markov Chain I, transition functions
7.Continuoustime Markov Chain II, Kolmogorov backward and forward equations, Limiting probabilities and stationary distribution, Balance equations.
8.Birth and Death process:
9.Poisson process.
10.Renewal process.
11.Queueing theory
12.Markov Chain Monte Carlo.
 Outline (exercises):   Goals:  Acquired knowůledge:
Understanding the limit behavior of stochastic systems in the connection with the state classification and will be able to construct the transition probabilities matrix (transition rates) based on given information.
Acquired skills:
Application of given methods in particular examples in physics and engineering.  Requirements:   Key words:   References  Key references:
[1] Grimmett, G., Stirzaker, D.: Probability and Random Processes, Oxford Uni. press, 2001.
[2] Lefebvre, M.: Applied Stochastic Processes, Springer, 2000.
Recommended references:
[1] Prášková, Z., Lachout, P.: Základy náhodných procesů, Karolinum 1998.
[2] Norris, J. R.: Markov Chains, Cambridge Uviversity Press 1997.
[3] Häggström, O.: Finite Markov chains and algorithmic applications, Cambridge Uviversity Press 2002.
[4] Ching, WaiKi: Markov chains: models, algorithms and applications, Springer 2006.
Working environment:
R, Matlab


Statistical Decision Theory  01STR 
Kůs 
  
2+0 zk 
 
2 
Course:  Statistical Decision Theory  01STR  Ing. Kůs Václav Ph.D.    2+0 ZK    2  Abstract:  The subject is devoted to the statistical techniques for general decision procedures based on optimization of suitable stochastic criterion, their mutual comparisons with respect to their properties and applicability.  Outline:  1. General principles of classical statistics.
2. Loss and risk functions, decision functions, optimal strategies.
3. Bayes and minimax solutions, admissibility principle and its consequences within classical statistics.
4. Convex loss functions, properties of Bayes estimates.
5. Unbiasedness, sufficiency, RaoBlackwell theorem and its applications, UMVUE estimators.
6. Minimum distance estimates.
7. Computational aspects for Bayesian methods, numerical procedures, approximative calculations.
8. Examples from the survival data analysis under random censoring experimental scheme.  Outline (exercises):   Goals:  Knowledge:
Extension of the decision makinng principles with random effects and their application in optimization tasks.
Skills:
Orientation in various stochastical approaches and their properties. Practical task solvations within risk models and numerical treatment.
 Requirements:  01MAS or 01PRST, recommended 01MIP.  Key words:  Loss functions, optimal strategies, Bayesian risk, minimax solution, admissibility, aproximative calculation.  References  Key references:
[1] Berger J.O., Statistical Decision Theory and Bayesian Analysis, Springer, N.Y., 1985.
Recommended refernces:
[2] Fishman G.S., Monte Carlo, Springer, 1996. 

Student's Scientific Conference  01SVK 
Mikyška 
  
5 dní z 
 
1 
Course:  Student's Scientific Conference  01SVK  doc. Ing. Mikyška Jiří Ph.D.          Abstract:   Outline:   Outline (exercises):   Goals:   Requirements:   Key words:   References  

Number Theory  01TC 
Masáková, Pelantová 
  
4+0 zk 
 
4 
Course:  Number Theory  01TC  prof. Ing. Masáková Zuzana Ph.D.    2+0 ZK    4  Abstract:  The subject is devoted to elementary number theory and to fundamentals of transcendental and algebraic theory.
 Outline:  1. Algebraic number fields, field isomorphisms
2. Diophantic equations, Pell's equation
3. Rational approximations, continued fractions
4. Algebraic and transcendental numbers
5. Rings of integers in algebraic number fields and divisibility
6. Applications for diophantic equations and in geometry
7. Number representation in noninteger bases, finite and periodic expansions  Outline (exercises):   Goals:  Knowledge: overview of fundamental tools of elementary and algebraic numer theory
Skills: application of methods for solution of concrete problems
 Requirements:  Knowledge of analysis and algebra (both linear and abstract) in the extent of bachelor's curriculum of mathematical modeling at FNSPE.  Key words:  Algebraic number, number field, transcendent number, continued fraction, diphantine equation, betaexpansions  References  Obligatory:
[1] Z. Masáková, E. Pelantová, Teorie čísel, Skriptum ČVUT 2010.
Optional:
[2] E. B. Burger, R. Tubbs, Making transcendence transparent, SpringerVerlag 2004.
[3] M. Křížek, F. Luca, L. Somer, 17 Lectures on Fermat Numbers, SpringerVerlag 2001. 

Game Theory  01TEH 
Kroupa 
  
2+0 zk 
 
2 
Course:  Game Theory  01TEH  doc. Ing. Kroupa Tomáš Ph.D.          Abstract:  1. Representation of games: extensive, strategic and coalitional.
2. Pure and mixed strategies. Nash equilibrium.
3. Information model of games. Correlated equilibrium.
4. Algorithms for the equilibrium computation.
5. Behavioral strategies. Kuhn's theorem.
6. Subgame perfect equilibrium. Backward induction.
7. Introduction to evolutionary games.
8. Coalitional games and their solutions.
9. Core.
10. Shapley value.
11. Banzhaf power index and ShapleyShubik power index.
 Outline:  1. Representation of games: extensive, strategic and coalitional.
2. Pure and mixed strategies. Nash equilibrium.
3. Information model of games. Correlated equilibrium.
4. Algorithms for the equilibrium computation.
5. Behavioral strategies. Kuhn's theorem.
6. Subgame perfect equilibrium. Backward induction.
7. Introduction to evolutionary games.
8. Coalitional games and their solutions.
9. Core.
10. Shapley value.
11. Banzhaf power index and ShapleyShubik power index.  Outline (exercises):   Goals:  Acquired knowledge: basic mathematical forms of games, their solutions and computational algorithms
Acquired skills: familiarity with gametheoretic models and their application in economics and computer science
 Requirements:   Key words:   References  Compulsory literature:
Maschler M., Solan E., Zamir S.: Game theory, Cambridge University Press, 2013
Optional literature:
von Neumann J., Morgenstern O.: Theory of Games and Economic Behavior, Princeton University Press, Princeton, New Jersey, 1944


Matrix Theory  01TEMA 
Pelantová 
  
2+0 z 
 
3 
Course:  Matrix Theory  01TEMA  prof. Ing. Pelantová Edita CSc.    2+0 Z    3  Abstract:  The subject deals mainly with:
1) similarity of matrices and canonical forms of matrices
2) PerronFrobenius theory and its applications
3) tensor product
4) Hermitian and positive semidefinite matrices
 Outline:  1. The Jordan Theorem and transformation of matrix into its canonical form, invariant subspaces.
2. Canonical forms of matrices with real resp. rational entries.
3. Relation between matrices and graphs
4 Nonnegative matrices and the PerronFrobenius theorem, stochastic matrices.
5. The tensor product of matrices and its properties.
6. Hermitian matrices, the interlacing theorem
7. Positive semidefinite matrices, the Hadamard inequality
 Outline (exercises):   Goals:  Acquired knowledge: fundamental results in the theory of canonical forms of matrices, in the PerronFrobenius theory for nonnegative matrices, the spectral theory for the hermitian matrices and the tensor product of matrices.
Acquired skills: applications of these results in the graph theory, for group representations, in the algebraic number theory, in numerical analysis.
 Requirements:  Successful completion of courses Linear algbera and General algebra.  Key words:  Canonical forms of matrix, similarity of matrices, Perron Frobenius theorem, positive semidefinite matrices, stochastic matrices, matrix norms, tensor product  References  Obligatory:
[1] Fuzhen Zhang: Matric Theory, Springer 2011
[2] M. Fiedler, Special Matrices and Their Applications in Numerical Mathematics. Second Edition. Dover Publications, Inc., Mineola, U.S.A., 2008.
Optional:
[3] Shmuel Friedland, Matrices  algebra, analysis and applications, World Scientific 2016. 

Information Theory  01TIN 
Hobza 
2+0 zk 
  
2 
 
Course:  Information Theory  01TIN  doc. Ing. Hobza Tomáš Ph.D.  2+0 ZK    2    Abstract:  Information theory explores the fundamental limits of the representation and transmission of information. We will focus on the definition and implications of (information) entropy, the source coding theorem, and the channel coding theorem. These concepts provide a vital background for researchers in the areas of data compression, signal processing, controls, and pattern recognition.  Outline:  1. Information source and entropy, joint and conditional entropy, information divergence, informations and their relation to entropy
2. Jensen inequality and the methods of convex analysis, sufficient statistics and data processing theorem
3. Fano inequality and CramérRao inequality, asymptotic equipartition property of memoryless sources
4. Entropy rates of information sources, stationary and Markov sources
5. Data compression, Kraft inequality for instantaneous and uniquely decodable codes, Huffman codes
6. Capacity of noisy channels, Shannon theorem about transmissibility of a source through a channel  Outline (exercises):   Goals:  Knowledge:
Basic notions and principles of information theory.
Skills:
Ability of application of acquired knowledge to solution of practical problems such as finding optimal Huffman codes, calculation of stacionary distribution of Markov chains, calculation of information channel capacity.  Requirements:  Basic course of Calculus and Probability (in the extent of the courses 01MAA3, 01MAA4 and 01MIP held at the FNSPE CTU in Prague).  Key words:  Entropy, information, information divergence, Fano?s inequality, Markov chains, entropy rate, data compression, Huffman code, instantaneous code, Kraft inequality, asymptotic equipartition property.  References  Key references:
[1] Cover, T. M., Thomas, J. A.: Elements of information theory. John Wiley & Sons, NewYork 2012.
Recommended references:
[2] Stone, J.V.: Information Theory  A Tutorial Introduction. Sebtel Press, Sheffield 2015.
[3] Csiszár, I., Körner, J.: Information theory  coding theorems for discrete memoryless systems. Cambridge University Press, Cambridge 2016.


Theory of Codes  01TKO 
Pelantová 
  
2 zk 
 
2 
Course:  Theory of Codes  01TKO  prof. Ing. Pelantová Edita CSc.    2 ZK    2  Abstract:  Algebraic methods used in error detecting and error correcting codes.  Outline:  Error detecting and error correcting codes, minimum distance of a code, the Hamming bound.
Codes with the best parameters, the Hadamard matrices, Levenshtein theorem.
Linear codes: generator and parity check matrices, standard decoding, Hamming codes, Golay code, cyclic codes, BCH codes, ReedMuller codes.  Outline (exercises):   Goals:  To acquaint students with using results of linear and general algebra for creating error detecting and error correcting codes and their decoding methods.  Requirements:  Basic results and techniques of the linear and general algebra, particularly, of finite fields.  Key words:  Code, linear codes, parameters of codes and their inequalities, cyclic codes, BCH codes, decoding algorithms.  References  Obligatory:
Blahut R.E.: Theory and Practice of Error Control Codes. AddisonWesley, Massachusetts, 1984.
Optional:
F.J. MacWilliams and N.J.A. Sloane, The Theory of ErrorCorrecting Codes, NorthHolland: New York, NY, 1978.


Theory of Codes  01TKOB 
Pelantová 
  
2+0 zk 
 
2 
Course:  Theory of Codes  01TKOB  prof. Ing. Pelantová Edita CSc.    2+0 ZK    2  Abstract:  Algebraic methods used in error detecting and error correcting codes.  Outline:  Error detecting and error correcting codes, minimum distance of a code, the Hamming bound.
Codes with the best parameters, the Hadamard matrices, Levenshtein theorem.
Linear codes: generator and parity check matrices, standard decoding, Hamming codes, Golay code, cyclic codes, BCH codes, ReedMuller codes.  Outline (exercises):   Goals:  To acquaint students with using results of linear and general algebra for creating error detecting and error correcting codes and their decoding methods.  Requirements:  Basic results and techniques of the linear and general algebra, particularly, of finite fields.  Key words:  Code, linear codes, parameters of codes and their inequalities, cyclic codes, BCH codes, decoding algorithms.  References  Obligatory:
Blahut R.E.: Theory and Practice of Error Control Codes. AddisonWesley, Massachusetts, 1984.
Optional:
F.J. MacWilliams and N.J.A. Sloane, The Theory of ErrorCorrecting Codes, NorthHolland: New York, NY, 1978.


Random Matrix Theory  01TNM 
Vybíral 
2+0 zk 
  
2 
 
Course:  Random Matrix Theory  01TNM  doc. Mgr. Krbálek Milan Ph.D. / doc. RNDr. Vybíral Jan Ph.D.          Abstract:  1. Wigner matrices and their level density. Semicircle Law.
2. Classes of random matrices.
3. Classes GOE(2) and GUE(2) and associated level spacing distribution.
4. Joint spectral denisty for classes GOE and GUE.
5. Poissonian matrices.
6. Unfolding and unfolding theorem. Procedure of unfolding and respective methodology. .
7. Dyson's gases. Metropolis algorithm for detection of steady states.
8. Statistical rigidity for basic classes of random matrices.
 Outline:  1. Wigner matrices and their level density. Semicircle Law.
2. Classes of random matrices.
3. Classes GOE(2) and GUE(2) and associated level spacing distribution.
4. Joint spectral denisty for classes GOE and GUE.
5. Poissonian matrices.
6. Unfolding and unfolding theorem. Procedure of unfolding and respective methodology. .
7. Dyson's gases. Metropolis algorithm for detection of steady states.
8. Statistical rigidity for basic classes of random matrices.
 Outline (exercises):   Goals:  We introduce basic classes of random matrices and their properties. We describe the unfolding procedure theoretically and show how shoud be applied in case of random matrix spectra. We introduce numerical approaches leading to the Level Spacing Distribution and Statistical Rigidity.  Requirements:   Key words:   References  M.L. Mehta: Random Matrices 3rd edition, Academic Press, New York (2004)
F. Haake: Quantum Signatures of Chaos, Springer Berlin (1992)
R. Scharf, F.M. Izrailev, Dyson?s Coulomb gas on circle and intermediate eigenvalue statistic, J. Phys A: Math. Gen. 23 (1990), 963
M. Krbálek and P. Šeba, Statistical properties of the city transport in Cuernavaca (Mexico) and random matrix ensembles, J. Phys. A: Math. Theor. 33 (2000), L229 

Topology  01TOP 
Burdík 
2+0 zk 
  
2 
 
Course:  Topology  01TOP  prof. 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, 01MAA24, 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.  References  Key references: [1] John L. Kelly, (Springer, 1975, 315 pp., ISBN10:0387901256), [2] Bourbaki; Elements of Mathematics: General Topology, AddisonWesley(1966).
Recommended references: [3] Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0486434796. [4] Basener, William (2006). Topology and Its Applications (1st ed.).Wiley. ISBN 0471687553. 

Introduction to semigroup theory  01TPG 
Klika 
2+0 z 
  
3 
 
Course:  Introduction to semigroup theory  01TPG           Abstract:  It is known that a system of linear ordinary differential equations can be solved by virtue of the matrix exponential. However, the extension to partial differential equations is not straightforward. For example in the case of heat equation the matrix is replaced by Laplace operator which is not bounded and the series for the exponential will not converge. Moreover, solutions of the heat equation exist in general only for positive times and hence the solution operator can be at best a semigroup. The aim of the course is to provide a mathematical foundation for these types of problems and extend the concept of stability from ordinary differential equations, which is again in relation to spectrum of a linear operator.  Outline:  1. Exponential of a matrix, bounded operator and possible extensions to unbounded operators.
2. Strongly continuous semigroups.
3. Uniformly continuous semigroups.
4. Analytic semigroups.
5. Semigroup generators.
6. HilleYoshida theorem.
7. LumerPhillips theorem.
8. Notions of stability.
9. Application to selected problems: relationship between spectrum and stability, exponential of unbounded operator.  Outline (exercises):   Goals:  Knowledge: Theory of semigroups and its application to studying stability of partial differential operators (incl. the relation to spectrum).
Skills: Construction and identification of an exponential of concrete bounded and unbounded operators.  Requirements:  Functional analysis (01FA1, 01FA2), equations of mathematical physics (01RMF), modern theory of partial differential equations (01PDR).  Key words:  exponential of operator, semigroup, generator of semigroup, HilleYoshida theorem, stability, spectrum of operator  References  Key references:
1. K J Engel, R Nagl, A Short: Course on Operator Semigroups, Springer, New York, 2006.
2. A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
Recommended references:
3. L C Evans: Partial Differential Equations, 2nd ed., Amer. Mat. Soc., Providence, 2010.
4. J A Goldstein: Semigroups of Linear Operators and Applications, Second Edition, Courier Dover Publications, 2017. 

Complexity Theory  01TSLO 
Majerech 
3+0 zk 
  
3 
 
Course:  Complexity Theory  01TSLO  Majerech Vladan Ph.D.  3+0 ZK    3    Abstract:  The course is devoted to incorporation of complexity questions during algorithm development, introduction to NP completeness and generally to complexity classes of deterministic or nondeterministic Turing machines bounded by time or space. Emphasis is placed on mutual relations among these classes. Aside from nondeterministic classes we examine probability classes. Class of interactive protocols is presented at the end of lecture course.  Outline:  1.Complexity dimensions  expected, randomized, amortized; basic data structures 2. Divide & conquer  recurrence, Strassen algorithm, sorting (+lower bound), median algorithm, prune and search 3. Fibonacci heaps, Dijkstra algorithm, minimum spanning tree  Fredman+Tarjan algorithm, Kruskal algorithm and DFU. 4. NPcompleteness and basic transforms (SAT, tiling, clique) 5. Further NP complete examples (Hamiltonian paths and cycles, knapsack) Fully polynomial approximation scheme for the knapsack problem 6. Turing machines, linear compression and speedup, tape reductions, universal machines 7. Constructability, inclusions among complexity classes, hierarchy theorems 8. Padding argument, Borodin?s theorem, Blum?s theorem 9. Generalised nondeterminism and probability classes 10. Polynomial hierarchy, complete problems. 11. Interactive protocols  Outline (exercises):   Goals:  Learn to incorporate complexity questions during algorithms development, learn to think about lower bounds of problem?s complexities. Learn basic relations among complexity classes.  Requirements:   Key words:  complexity, NP completeness, algorithm  References  Key references:
[1] J. L. Balcázar, J. Díaz, J Gabarró: "Structural Complexity I", Springer  Verlag Berlin Heidelberg New York London Paris Tokyo 1988
Recommended references:
[2] Hopcroft, Ullmann "Introduction to Automata Theory and Computing", ISBN 020102988X 

Introduction to Mainframe  01UMF 
Oberhuber 
2 z 
  
2 
 
Course:  Introduction to Mainframe  01UMF  Ing. Oberhuber Tomáš Ph.D.  2 Z    2    Abstract:  In this course we teach the mainframe architecture. We explain how to operate the system z/OS, how to start a job using the JCL and we explain some differences when programming in C/C++ for z/OS:  Outline:  1. Introduction to the mainframe
2. Memory management in z/OS
3. Data sets in z/OS
4. ISPF  user interface
5. JES  system for job processing
6.10. JCL  scripting language
11. Programming v C/C++
12. Rexx  Outline (exercises):  1. ISPF  user interface
2. JCL  scripting language
3. Programming in C/C++
4. Programming in REXX
 Goals:  Knowledge:
Understanding the differences between common servers and mainframes, hardware of zSeries, z/OS operating system, data sets, working with ISPF, writing of JCL scripts, programming in C/C++.
Skills:
Student is able to work with ISPF, he can cerate and manage data sets, write JCL scripts and compile and link programs written in C/C++. The student understands what requirements are posed on highly reliable systems.  Requirements:  Basics of operating systems, basic knowledge of Unix/Windows, programming in C/C++.  Key words:  Mainframe, z/OS, z/Serie, JCL, ISPF, C/C++. Rexx.  References  Key references:
IBM, Introduction to the new mainframe, IBM, 2005.
Recommended references:
IBM, ABCs of z/OS System Programming Volume 13, IBM, 2004.
Media and tools:
Computer lab, account on the mainframe system. 

Probabilistic Models of Artificial Intelligence  01UMIN 
Vejnarová 
2+0 kz 
  
2 
 
Course:  Probabilistic Models of Artificial Intelligence  01UMIN  Doc. RNDr. Vejnarová Jiřina CSc.  2+0 KZ    2    Abstract:  The course is devoted to the survey of methods used for uncertainty processing in the field of artificial inteligence. The main attention is paid to socalled graphical Markov models, particularly to Bayesian networks.  Outline:  1. Introduction to artificial intelligence: problem solving, state spaces, search for solution, algorithm A star, optimality of solution. 2. Uncertainty in artificial intelligence: uncertainty in expert systems, pseudobayesian uncertainty processing in Prospector. 3. Imprecise probabilities: capacities, lower an upper probabilities, coherence, belief functions, possibility measures, credal sets. 4. Conditional independence and its properties: factorization lemma, block independence lemma. 5.Graphical Markov properties: pairwise, local and global Markov properties. 6. Triangulated graphs: graph decomposition, maximum cardinality search, perfect ordering of nodes and cliques, triangularization of graphs, running intersection property, junction trees. 7. Bayesian networks: consistency of distribution represented by a Bayesian network, dependence structure. 8. Computation in Bayesain networks: Shachter algorithm, transformation of Bayesian networks to decomposable models, message passing in junction trees.  Outline (exercises):   Goals:  Uncertainty models in artificial inteligence and methods of its processing. Skills: Application of given methods to particular problems.  Requirements:  Basic course of probabilitz and mathematical statistics  Key words:  artificial intelligence, uncertainty, imprecise probabilities, conditional independence, graphical Markov properties, decomposable graphs, Bayesian networks.  References  key references:
[1] P. Hájek, T. Havránek, R. Jiroušek, Uncertain Information Processing in Expert Systems, CRC Press 1992.
key references:
[1] P. Hájek, T. Havránek, R. Jiroušek, Uncertain Information Processing in Expert Systems, CRC Press 1992. 

Introduction to Object Programming  01UOP 
Čulík 
0+2 zk 
  
2 
 
Course:  Introduction to Object Programming  01UOP  Ing. Čulík Zdeněk          Abstract:  Object oriented programming languages. Object oriented programming libraries for graphics, databases and distributed systems.  Outline:  1. Evolution of object oriented programming languages
2. Type inheritance, encapsulation, polymorphism
3. Interfaces in C++ and Java
4. Templates and generic constructions
5. Design patterns
6. Objects and graphical user interface
7. 3D graphics, Open Inventor
8. Distributed systems: CORBA, COM, DBus
9. Object oriented databases
10. History: Simula 67, Smalltalk, Ada
11. Programming language Python  Outline (exercises):   Goals:  Knowledge:
Evolution of object oriented programming languages. Objects and modern software technologies.
Skills:
Design object oriented application. Develop application which uses object oriented libraries.  Requirements:   Key words:  Programming languages, object oriented programming, C++, Java, Python  References  [1] B. Stroustrup: The C++ Programming Language, 3rd Edition, AddisonWesley, 1997
[2] B. Stroustrup: The Design and Evolution
[3] B.Eckel: Thinking in Java (4th Edition), Prentice Hall, 2006
[4] M.Lutz: Learning Python: Powerful ObjectOriented Programming, O'Reilly Media, 2009 

Introduction to Riemannian geometry  01URG 
Krejčiřík 
2+0 zk 
  
2 
 
Course:  Introduction to Riemannian geometry  01URG           Abstract:  This lecture is intended for an advanced undergraduate having already taken a basic course on topological and differential manifolds. In addition to understanding the geometric meaning of curvature and its intimate relationship to topology, the student will learn the basic apparatus of Riemannian geometry suitable for further study of modern parts of mathematics and mathematical physics. Possible extension of this lecture is the geometric analysis of partial differential equations on Riemannian manifolds.  Outline:  1. Motivation. The notion of curvature in classical theories of curves and surfaces.
2. Review of basic tools. Tensors, manifolds and vector bundles.
3. Riemannian metrics. The volume element and integration. Constantcurvature model spaces.
4. Connections. Covariant derivatives of tensor fields. Parallel transport along curves. Geodesics.
5. Riemannian geodesics. The LeviCivita connection. The exponential map. Normal coordinates. Geodesics of the model spaces.
6. Geodesics and distance. Geodesics as lengthminimising curves. First variation formula. The Gauss lemma. Completeness and the HopfRinow theorem.
7. Curvature. Local invariants of Riemannian metrics. The curvature tensor. Flat manifolds. The Ricci and scalar curvatures.
8. Riemannian submanifolds. The second fundamental form. Hypersurfaces in the Euclidean space, the Gauss curvature and Theorema Egregium. Sectional curvatures.
9. The GaussBonnet theorem. The Umlaufsatz and the GaussBonnet formula. The Euler characteristic of a topological manifold.
10. Jacobi fields. The Jacobi equation. Conjugate points. The second variation formula.
11. Curvature and topology. Comparison theorems. The CartanHadamard and Bonnet's theorems.  Outline (exercises):   Goals:  Knowledge:
Learning the basic apparatus of Riemannian geometry suitable for further study of modern parts of mathematics and mathematical physics. A particular goal of the lecture is understanding the geometric meaning of curvature and its intimate relationship to topology.
Skills:
Routine work with tensorial and variational calculus on manifolds, computation of the connection and curvature tensor from the metric, solving the differential equations for geodesics and Jacobi fields, integration on manifolds.  Requirements:  Basic courses on analysis on manifolds and topology (01DPV, 01TOP).  Key words:  Riemannian geometry, metric, connection, geodesic, curvature.  References  Key references:
1. J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, Springer, 1997.
Recommended references:
2. M. P. do Carmo, Riemannian geometry, Birkhauser 1992.
3. O. Kovalski, Úvod do Riemannovy geometrie, Univerzita Karlova, 1995.
4. M. Spivak, A Comprehensive Introduction to Differential Geometry, Volumes IV, Publish or Perish, 1999.
5. P. Petersen: Riemannian Geometry. Springer, 2016.


Introduction to Computer Science  01UTI 
Ambrož, Masáková 
  
2+0 kz 
 
2 
Course:  Introduction to Computer Science  01UTI  Ing. Ambrož Petr Ph.D.    2+0 KZ    2  Abstract:  Fundamental notions of computer science: algorithms, various types of automata, introduction to information theory and coding theory.  Outline:  Algorithms and algorithmically computable functions, algorithmically definable sets. Markov?s normal algorithms, Turing machines, pushdown storage automata, finite automata.
Sequential machines, analysis, synthesis and minimization. Introduction to information theory and coding theory.
 Outline (exercises):   Goals:  To acquaint the students with fundamentals parts of computer science.  Requirements:   Key words:  Algorithm, automaton, entropy, coding.  References  povinná
H.R. Lewis, C.H.Papadimitriou: Elements of the Theory of Computation. PrenticeHall, Englewood Cliffs, New Jersey, 1981.
doporučená
Z. Manna: Mathematical Theory of Computation. Mc.GrawHill,Inc. 1974.


Variational Methods  01VAM 
Beneš 
2 zk 
  
3 
 
Course:  Variational Methods  01VAM  prof. Dr. Ing. Beneš Michal  2 ZK    3    Abstract:  The course is devoted to the methods of classical variational calculus  functional extrema by Euler equations, second functional derivative, convexity or monotonicity. Further, it contains investigation of quadratic functional, generalized solution, Sobolev spaces and variational problem for elliptic PDE's.  Outline:  1. Functional extremum, Euler equations.
2. Conditions for functional extremum.
3. Theorem on the minimum of a quadratic functional.
4. Construction of minimizing sequences and their convergence.
5. Choice of basis.
6. Sobolev spaces.
7. Traces. Weak formulation of the boundary conditions.
8. Vellipticity. LaxMilgram theorem.
9. Weak solution of boundaryvalue problems.  Outline (exercises):  Exercise makes part of the contents and is devoted to solution of particular examples in variational calculus  shortest path, minimal surface area, bending rod, CahnHilliard phasetransition theory etc.  Goals:  Knowledge:
Classical variational calculus  conditions for existence of functional extrema, Euler equations, extremum of quadratic functional, generalized solution of operator equation, Sobolev spaces and weak solution of boundary value problems for elliptic PDE.
Skills:
Analysis of functional extrema, solution of common problems of variational calculus and determination of solution properties.  Requirements:  Basic course of Calculus, Linear Algebra and Numerical Mathematics, variational methods (in the extent of the courses 01MA1, 01MAA24, 01LA1, 01LAA, 01NM, 01FA12 held at the FNSPE CTU in Prague).
 Key words:  Variational calculus, Gâteaux derivative, Fréchet derivative, functional extrema, convexity, monotonicity, quadratic functional, Sobolev spaces, weak solution, LaxMilgram theorem.  References  Key references:
[1] S. V. Fomin, R. A. Silverman, Calculus of variations, Courier Dover Publications, Dover 2000
[2] K. Rektorys, Variational Methods In Mathematics, Science And Engineering, Springer, Berlin, 2001
Recommended references:
[3] B. Dacorogna, Introduction to the Calculus of Variations, Imperial College Press, London 2004
[4] B. Van Brunt, The calculus of variations, Birkhäuser, Basel 2004
[5] E. Giusti, Direct methods in the calculus of variations, World Scientific, Singapore 2003
[6] B. S. Mordukhovich, Variational Analysis and Applications, Springer International Publishing, 2018 

Variational Methods B  01VAMB 
Beneš 
2 kz 
  
2 
 
Course:  Variational Methods B  01VAMB  prof. Dr. Ing. Beneš Michal  2 KZ    2    Abstract:  The course is devoted to the methods of classical variational calculus  functional extrema by Euler equations, second functional derivative, convexity or monotonicity. Further, it contains investigation of quadratic functional, generalized solution, Sobolev spaces and variational problem for elliptic PDE's.
 Outline:  1. Functional extremum, Euler equations.
2. Conditions for functional extremum.
3. Theorem on the minimum of a quadratic functional.
4. Construction of minimizing sequences and their convergence.
5. Choice of basis.
6. Sobolev spaces.
7. Traces. Weak formulation of the boundary conditions.
8. Vellipticity. LaxMilgram theorem.
9. Weak solution of boundaryvalue problems.  Outline (exercises):  Exercise makes part of the contents and is devoted to solution of particular examples in variational calculus  shortest path, minimal surface area, bending rod, CahnHilliard phasetransition theory etc.  Goals:  Knowledge:
Classical variational calculus  conditions for existence of functional extrema, Euler equations, extremum of quadratic functional, generalized solution of operator equation, Sobolev spaces and weak solution of boundary value problems for elliptic PDE.
Skills:
Analysis of functional extrema, solution of common problems of variational calculus and determination of solution properties.  Requirements:  Basic course of Calculus, Linear Algebra and Numerical Mathematics, variational methods (in the extent of the courses 01MA1, 01MAB24, 01LA1, 01LAB2, 12NMET held at the FNSPE CTU in Prague).
 Key words:  Variational calculus, Gâteaux derivative, Fréchet derivative, functional extrema, convexity, monotonicity, quadratic functional, Sobolev spaces, weak solution, LaxMilgram theorem.  References  Key references:
[1] S. V. Fomin, R. A. Silverman, Calculus of variations, Courier Dover Publications, Dover 2000
[2] K. Rektorys, Variational Methods In Mathematics, Science And Engineering, Springer, Berlin, 2001
Recommended references:
[3] B. Dacorogna, Introduction to the Calculus of Variations, Imperial College Press, London 2004
[4] B. Van Brunt, The calculus of variations, Birkhäuser, Basel 2004
[5] E. Giusti, Direct methods in the calculus of variations, World Scientific, Singapore 2003
[6] B. S. Mordukhovich, Variational Analysis and Applications, Springer International Publishing, 2018 

Selected Topics in Functional Analysis  01VPF 
Šťovíček 
2+2 z,zk 
  
4 
 
Course:  Selected Topics in Functional Analysis  01VPF  prof. Ing. Šťovíček Pavel DrSc.          Abstract:  Keywords:
Banach spaces, Hilbert spaces, Linear operators, Fourier transform, semigroups of operators  Outline:  1. Basic notions in topology and measure theory
2. Basic inequalities, convex functions
3. Banach spaces, spaces of bounded linear operators
4. Hilbert spaces, projectors, RadonNikodym theorem
5. HahnBanach theorem
6. Weak topology and convergence
7. Fourier transform and applications
8. Semigroups of operators
9. Applications in stochastic processes  Outline (exercises):   Goals:  Ackquired knowledge::
Basic properties of linear operators in Banach and Hilbert spaces, meaning and use of Fourier transform.
Acquired skills:
Application of knowledge in particular examples.
 Requirements:  Basic course of Calculus and Linear Algebra (in the extent of the courses 01MA, 01MAA24, 01LAP, 01LAA2 held at the FNSPE CTU in Prague).  Key words:   References  Key references:
[1] Blank, Exner, Havlíček: Hilbert Space Operators in Quantum Physics, Springer, 2008.
Recommended references:
[2] M. Reed, B. Simon: Methods of Modern Mathematical Physics I.IV., Academic Press, N. Zealand, 19721979
[3] Bobrowski: Functional Analysis for Probability and Stochastic Processes, An Introduction, New York, 2005 

Research Project 1  01VUAM12 
Hobza 
0+6 z 
0+8 kz 
6 
8 
Course:  Research Project 1  01VUAM1  doc. Ing. Hobza Tomáš Ph.D.  0+6 Z    6    Abstract:  Research project on the selected topic under the supervision. Supervision and regular checking of the research project under preparation.  Outline:  Research yearlong project on the selected topic under the supervision.  Outline (exercises):   Goals:  Student participation in research.
Knowledge: particular research theme depending on a given project topic.
Abilities: working unaided on a given task, understanding the problem, producing the original specialist text.
 Requirements:  Bachelor thesis BP12, individual.
The ability of independent students research work and skills.
 Key words:  Research project, research, development, mathematical and computer science models, applications.  References  Individual  according to the given references. 
Course:  Research Project 2  01VUAM2  doc. Ing. Hobza Tomáš Ph.D.    0+8 KZ    8  Abstract:  Research project on the selected topic under the supervision. Supervision and regular checking of the research project under preparation.  Outline:  Research yearlong project on the selected topic under the supervision.  Outline (exercises):   Goals:  Student participation in research.
Knowledge: particular research theme depending on a given project topic.
Abilities: working unaided on a given task, understanding the problem, producing the original specialist text.
 Requirements:  Bachelor thesis BP12, individual.
The ability of independent students research work and skills.
 Key words:  Research project, research, development, mathematical and computer science models, applications.  References  Individual  according to the given references. 

Research Project 1  01VUMM12 
Hobza 
0+6 z 
0+8 kz 
6 
8 
Course:  Research Project 1  01VUMM1  doc. Ing. Hobza Tomáš Ph.D.  0+6 Z    6    Abstract:  Research project on the selected topic under the supervision. Supervision and regular checking of the research project under preparation.  Outline:  Research yearlong project on the selected topic under the supervision.  Outline (exercises):   Goals:  Student participation in research.
Knowledge: particular research theme depending on a given project topic.
Abilities: working unaided on a given task, understanding the problem, producing the original specialist text.
 Requirements:  Bachelor thesis BP12, individual.
The ability of independent students research work and skills.  Key words:  Research project, research, development, mathematical models, applications.  References  Individual  according to the given references. 
Course:  Research Project 2  01VUMM2  doc. Ing. Hobza Tomáš Ph.D.    0+8 KZ    8  Abstract:  Research project on the selected topic under the supervision. Supervision and regular checking of the research project under preparation.  Outline:  Research yearlong project on the selected topic under the supervision.  Outline (exercises):   Goals:  Student participation in research.
Knowledge: particular research theme depending on a given project topic.
Abilities: working unaided on a given task, understanding the problem, producing the original specialist text.
 Requirements:  Bachelor thesis BP12, individual.
The ability of independent students research work and skills.  Key words:  Research project, research, development, mathematical models, applications.  References  Individual  according to the given references. 

Research Project 1  01VUSI12 
Hobza 
0+6 z 
0+8 kz 
6 
8 
Course:  Research Project 1  01VUSI1  doc. Ing. Hobza Tomáš Ph.D.  0+6 Z    6    Abstract:  Research project on the selected topic under the supervision. Supervision and regular checking of the research project under preparation.  Outline:  Research yearlong project on the selected topic under the supervision.  Outline (exercises):   Goals:  Student participation in research.
Knowledge: particular research theme depending on a given project topic.
Abilities: working unaided on a given task, understanding the problem, producing the original specialist text.
 Requirements:  Bachelor thesis BP12, individual.
The ability of independent students research work and skills.
 Key words:  Research project, research, development, mathematical and computer science models, applications.  References  Individual  according to the given references. 
Course:  Research Project 2  01VUSI2  doc. Ing. Hobza Tomáš Ph.D.    0+6 KZ    8  Abstract:  Research project on the selected topic under the supervision. Supervision and regular checking of the research project under preparation.  Outline:  Research yearlong project on the selected topic under the supervision.  Outline (exercises):   Goals:  Student participation in research.
Knowledge: particular research theme depending on a given project topic.
Abilities: working unaided on a given task, understanding the problem, producing the original specialist text.
 Requirements:  Bachelor thesis BP12, individual.
The ability of independent students research work and skills.
 Key words:  Research project, research, development, mathematical and computer science models, applications.  References  Individual  according to the given references. 

Selected Topics in Mathematics  01VYMA 
Mikyška 
  
2+2 z,zk 
 
4 
Course:  Selected Topics in Mathematics  01VYMA  doc. Ing. Mikyška Jiří Ph.D.    2+2 Z,ZK    4  Abstract:  Fourier series: complete orthogonal systems, expansion of functions into Fourier series, trigonometric Fourier series and their convergence. Complex analysis: derivative of holomorphic functions, integral, Cauchy's theorem, Cauchy's integral formula, singularities, Laurent series, residue theorem.  Outline:  1. Theory of Fourier series in a general Hilbert space, complete orthogonal systems, Bessel inequality, Parseval equality.
2. Fourier series in L2, trigonometric system, Fourier coefficients, Bessel inequality, Parseval equality, expansion of a function into trigonometric series.
3. Criteria of convergence of Fourier series.
4. Analysis of complex functions: derivative, analytical functions, CauchyRiemann conditions.
5. Contour integral of complex functions of a complex variable, theorem of Cauchy, Cauchy's integral formula.
6. Expansion of an analytic function into a power series, isolated singularities, Laurent expansion, residue theorem.  Outline (exercises):  1. Summary of properties of function series, investigation of the uniform convergence of function series.
2. Fourier series in a general Hilbert space, GrammSchmidt ortogonalization, ortogonal polynomials.
3. Trigonometric system in L2. Expansions of trigonometric functions into trigonometric Fourier series, investigation of convergence of the trigonometric series. Summation of some series using the Fourier expansions.
4. Elementary functions of complex variables: polynomials, exponential function, goniometric functions, complex logarithm
5. Analysis in a complex domain: continuity, derivative, CauchyRiemann conditions.
6. Evaluation of contour integrals of complex functions of a complex variable, applications of the Cauchy theorem, Cauchy integral formula and residue theorem.  Goals:  Expansion of functions to the Fourier series and investigation of their convergence, application of theory of analytic functions for evaluation of curve integrals in complex plane and evaluation of some types of definite integral of real functions of a real variable.
Skills: to use expansions of functions into a Fourier series to evaluate sums of some series, evaluation of definite integrals using the theory of functions of complex variable.  Requirements:  Basic Calculus (in the extent of the courses 01MA1, 01MAA23, or 01MAB23 held at the FNSPE CTU in Prague).  Key words:  Sequences and series of functions,
Fourier series, complex analysis.  References  Key references:
[1] J. DunningDavies, Mathematical Methods for Mathematicians, Physical Scientists and Engineers, John Wiley and Sons Inc., 1982.
Recommended references:
[2] A. S. Cakmak, J. F. Botha, and W. G. Gray, Computational and Applied Mathematics for Engineering Analysis, SpringerVerlag Berlin, Heidelberg, 1987. 

Analysis and Processing of Diagnostic Signals  01ZASIG 
Převorovský 
  
3+0 zk 
 
3 
Course:  Analysis and Processing of Diagnostic Signals  01ZASIG  Ing. Převorovský Zdeněk CSc.          Abstract:  Digital signal processing, signal transformations and filtrations, spectral and timefrequency analysis  Outline:  1.Systems and signals continuous and discrete in time. Time and amplitude discretization. Sampling theorem.
2.Description and properties of systems; linearity, stability, causality, time invariance.
3.Programming in MATLAB with Signal and Wavelet Toolboxes.
4.Harmonic signals, delta distribution. Convolution and correlation.
5.Laplace and Fourier transform. Transfer function, impulse and system response.
6.Hilbert transform, analytical signals. Ztransform and digital filtering, FIR and IIR filters.
7.Signal envelope and parameterization. Timereversal techniques. Nonlinear methods.
8.Stochastic signal processing, statistical parameters, noise analysis.
9.Signal arrival detection, localization and recognition of sources.
10.Timefrequency analysis, Fourier window and wavelet transform.  Outline (exercises):   Goals:  Acquired knowledge:
systém analysis and modern techniques of analog and digital signal processing used in physics, measurement, information, and related fields. Description of systems and signals in diverse reprezentations based on integral transforms and their discrete equivalents. Frequency analysis, parameterization, filtration, and transfer of signals. Working with MATLAB Signal and Wavelet Toolbox.
Acquired skills:
Methods and algorithms of digital data processing and analysis.
 Requirements:   Key words:   References  Compulsory literature:
[1] Davídek V., Sovka P.: Číslicové zpracování signálů a implementace. (FEL ČVUT, Praha 1999),
[2] Smith S.W.: The Scientist and Engineer's Guide to Digital Signal Processing. (California Tech. Publishing,
San Diego 1999; www.DSPguide.com)
[3] McClellan J., Burrus C.S., Oppenheim A.V, et al: Computer based Excersises for Signal Processing using
MATLAB (PrenticeHall, MathWorks, 1998)
Optional literature:
[4] Porat B.: A Course in Digital Signal Processing. (MATLAB based books, J.Wiley& Sons, Inc., 1997;
www.mathworks.com/support/books/)
[5] Hrdina Z., Vejražka F.: Signály a soustavy. (FEL ČVUT, Praha 2001)
[6] Vích R., Smékal Z. : Číslicové filtry. (ACADEMIA, Praha, 2000)
[7] Oppenheim A.V., Schaffer R.W.: Discrete Time Signal Processing. (PrenticeHall, Englewood Cliffs, New
Jersey,1990)
[8] Mertins A.: Signal Analysis  Wavelets, Filter Banks, TimeFrequency Transforms. (John Wiley & Sons,
Chichester, N.Y.,1999)
[9] http://ocw.mit.edu/6003F11


Generalized Linear Models and Application  01ZLIM 
Hobza 
  
2+1 zk 
 
3 
Course:  Generalized Linear Models and Application  01ZLIM  doc. Ing. Hobza Tomáš Ph.D.  2+1 ZK    3    Abstract:  In this course we will consider a series of statistical models which generalize classical linear models with normally distributed objective variables. This course consists of the theory of generalized linear models (GLM), outline of the algorithms used for GLM estimation, and explanation how to determine which algorithm to use for a given data analysis.  Outline:  1. Generalized linear models: exponential family, regularity conditions, score function
2. Estimation of parameters: maximum likelihood estimates, numerical methods used for their calculation, NewtonRaphson, Fisherscoring
3. Testing of models: asymptotical distribution of the score function and the MLE estimates, models comparisons, residual analysis
4. Analysis of covariance (ANCOVA), elements of matrix algebra, general model of analysis of covariance, one factor ANCOVA
5. Models for binary data: uniform model, logistic model, normal model, Gumbel model
6. Poisson regression: Poisson distribution, univariate and multivariate Poisson regression, tests and residua, Poisson model for small area estimation
7. Multivariate logistic regression: multivariate logit model, tests about estimated parameters, residua, logit area model  Outline (exercises):  1. Estimation of parameters, maximum likelihood estimates, numerical methods used for their calculation, NewtonRaphson, Fisherscoring
2. Testing of models, models comparisons, residual analysis
3. Analysis of covariance (ANCOVA)
4. Logistic regression
5. Poisson regression
6. Multivariate logistic regression.  Goals:  Knowledge:
Generalized linear statistical models and methods for estimation of their parameters.
Skills:
Application of theoretically studied statistical procedures to practical problems of data analysis including demonstration of use of these methods on computer in the MATLAB or R environment.  Requirements:  Basic course of Calculus and Probability (in the extent of the courses 01MAB3, 01MAB4 and 01PRST held at the FNSPE CTU in Prague).  Key words:  Generalized linear model, score function, analysis of covariance, logistic regression, Poisson regression, residua.  References  Key references:
[1] A.J. Dobson: An Introduction to Generalized Linear Models. London: Chapman and Hall, 1990
Recommended references:
[2] J.K. Lindsey: Applying Generalized Linear Models. Springer Verlag, 1998


Introduction to Operating Systems  01ZOS 
Čulík 
  
2+0 z 
 
2 
Course:  Introduction to Operating Systems  01ZOS  Ing. Čulík Zdeněk    2+0 Z    2  Abstract:  Introduction to structure of operating systems. Processes, thread, memory management. Synchronization of multi=threaded applications. Memory mapped files.  Outline:  1. Introduction to operating systems (kernel structure, security)
2. Processes and threads (creation and termination of processes and threads, thread scheduling and priority).
3. Thread synchronization (critical sections, semaphores)
4. Memory management (virtual memory, memory mapped files)
5. Introduction to distributed systems (RPC  remote procedure call, CORBA and COM architecture)
6. TCP/IP network communication (packet routing, DNS service)
 Outline (exercises):   Goals:  Knowledge:
Structure of operating system, low level file handle manipulation, process and thread creation.
Skills:
Develop multithreaded application.  Requirements:   Key words:  processes, threads, memory managment  References  [1] A. S. Tanenbaum: Operating Systems: Design And Implementation, Prentice Hall, Englewood Cliffs, 1987
[2] W. Stallings, Operating Systems: Internals and Design Principles, Prentice Hall, 2005
[3] J. M. Richter: Advanced Windows, Microsoft Press, Redmond, 1997
[4] A. Rubini, J. Corbet: Linux Device Drivers, O'Reilly, 2001
[5] D. Bovet, M. Cesati, A. Oram: Understanding the Linux Kernel, O'Reilly, 2001


Introduction to Computer Security 1  01ZPB1 
Vokáč 
  
1+1 z 
 
2 
Course:  Introduction to Computer Security 1  01ZPB1  Ing. Vokáč Petr          Abstract:   Outline:   Outline (exercises):   Goals:   Requirements:   Key words:   References  

Introduction to Computer Security 2  01ZPB2 
Vokáč 
1+1 z 
  
2 
 
Course:  Introduction to Computer Security 2  01ZPB2  Ing. Vokáč Petr          Abstract:   Outline:   Outline (exercises):   Goals:   Requirements:   Key words:   References  

Introduction to Graph Theory  01ZTG 
Ambrož 
4+0 zk 
  
4 
 
Course:  Introduction to Graph Theory  01ZTG  Ing. Ambrož Petr Ph.D.          Abstract:   Outline:   Outline (exercises):   Goals:   Requirements:   Key words:   References  
