# Year 2 in Brno - Mathematical Engineering

• ## Multi-valued logic applications (4 credits)

• ECTS credits 4
• Semester 1
• University Brno University of Technology
• Prerequisites

Mathematical logic, fuzzy set theory.

• Objectives

The aim of the course is to provide students with information about the use of Multi-valued logic in technical applications.

• Topics

1. Multi-valued logic, formulae.

2. T-norms, T-conorms, generalized implications.

3. Linguistic variables and linguistic models.

4. Knowledge bases of expert systems.

5-6. Semantic interpretations of knowledge bases

7. Inference techniques and its implementation

8. Redundance a contradictions in knowledge bases

9. LMPS system

10. Fuzzification and defuzzification problem

11. Technical applications of multi-valued logic and fuzzy sets theory

12. Expert systems

13. Overview of AI methods

• Books

Jackson P.: Introduction to Expert Systems, Addison-Wesley 1999

• More information

The course is intended especially for students of mathematical engineering. It includes the theory of multi-valued logic, theory of linguistic variable and linguistic models and theory of expert systems based on these topics. Particular technical applications of these mathematical teories are included as a practice.

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• ## Financial Mathematics (4 credits)

• ECTS credits 4
• Semester 1
• University Brno University of Technology
• Prerequisites

The knowledge of Calculus and Linear Algebra together with probabilistic and statistical methods (including time series) as well as optimisation techniques within the framework of SOP and SO2 courses is required.

• Objectives

The basic concepts and models of financial problems are accompanied by the theory and simple examples.

• Topics

1. Basic concepts, money, capital and securities.

2. Simple and compound interest rate, discounting.

3. Investments, cash flows and its measures, time value of money.

4. Assets and liabilities, insurance.

5. Bonds, options, futures, and forwards.

6. Exchange rates, inflation, indices.

7. Portfolio optimization - classical model.

8. Postoptimization, risk, funds.

9. Twostage models in finance.

10. Multistage models in finance.

11. Scenarios in financial mathematics.

12. Modelling principles, identification of dynamic data.

13. Discussion on advanced stochastic models.

• Books

1. Dupačová,J. et al.: Stochastic Models for Economics and Finance, Kluwer, 2003.

• More information

The course presents basic financial models. It focuses on main concepts and computational methods. Several lectures are especially developed to make students familiar with optimization models.

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• ## Fuzzy Sets and Applications (4 credits)

• ECTS credits 4
• Semester 1
• University Brno University of Technology
• Prerequisites

Fundamentals of the set theory and mathematical analysis.

• Objectives

The course objective is to make students acquainted with basic methods and applications of fuzzy sets theory, that allows to model vague quantity of numerical and linguistic character, and subsequently systems and processes, which cannot be described with classical mathematical models. A part of the course is the work with fuzzy toolbox of software Matlab and shareware products.

• Topics

1. Fuzzy sets (motivation, basic notions, properties).

2. Operations with fuzzy sets (properties).

3. Operations with fuzzy sets (alfa cuts).

4. Triangular norms and co-norms, complements (properties).

5. Extension principle (Cartesian product, extension mapping).

6. Fuzzy numbers (definition, extension operations, interval arithmetic).

7. Fuzzy relations (basic notions, kinds).

8. Fuzzy functions (basic orders, fuzzy parameter, derivation, integral).

9. Linguistic variable (model, fuzzification, defuzzification).

10. Fuzzy logic (multiple value logic, extension).

11. Approximate reasoning and decision-making (fuzzy environment, fuzzy control).

12. Fuzzy probability (basic notions, properties).

13. Fuzzy models design for applications.

• Books

Klir, G. J. - Yuan, B.: Fuzzy Sets and Fuzzy Logic - Theory and Applications. New Jersey: Prentice Hall, 1995.

Zimmermann, H. J.: Fuzzy Sets Theory and Its Applications. Boston: Kluwer-Nijhoff Publishing, 1998.

• More information

The course is concerned with the fundamentals of the fuzzy sets theory: operations with fuzzy sets, extension principle, fuzzy numbers, fuzzy relations and graphs, fuzzy functions, linguistics variable, fuzzy logic, approximate reasoning and decision making, fuzzy control, fuzzy probability. It also deals with the applicability of those methods for modelling of vague technical variables and processes, and work with special software of this area.

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• ## Mathematical Methods in Fluid Dynamics (4 credits)

• ECTS credits 4
• Semester 1
• University Brno University of Technology
• Prerequisites

Evolution partial differential equations, functional analysis, numerical methods for partial differential equations.

• Objectives

The course is intended as an introduction to the computational fluid dynamics. Considerable emphasis will be placed on the inviscid compressible flow: namely, the derivation of Euler equations, properties of hyperbolic systems and an introduction of several methods based on the finite volumes. Methods for computations of viscous flows will be also studied, namely the pressure-correction method and the spectral element method. Students ought to realize that only the knowledge of substantial physical and mathematical aspects of particular types of flows enables them to choose an effective numerical method and an appropriate software product. The development of individual semester assignement constitutes an important experience enabling to verify how the subject matter was managed.

• Topics

1. Material derivative, transport theorem, mass, momentum and energy conservation laws.

2. Constitutive relations, thermodynamic state equations, Navier-Stokes and Euler equations, initial and boundary conditions.

3. Traffic flow equation, acoustic equations, shallow water equations.

4. Hyperbolic system, classical and week solution, discontinuities.

5. The Riemann problem in linear and nonlinear case, wave types.

6. Finite volume method in one and two dimensions, numerical flux.

7. Local error, stability, convergence.

8. The Godunov's method, flux vector splitting methods: the Vijayasundaram, the Steger-Warming, the Van Leer.

9. Viscous incompressible flow: finite volume method for orthogonal staggered grids, pressure correction method SIMPLE.

10. Pressure correction method for colocated variable arrangements, non-orthogonal and unstructured meshes.

11. Stokes problem, spectral element method.

12. Steady Navier-Stokes problem, spectral element method.

13. Unsteady Navier-Stokes problem.

• Books

R.J. LeVeque: Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002.

E.F. Toro: Riemann Solvers and Numerical Methods for Fluid Dynamics, A Practical Introduction, Springer, Berlin, 1999.

S.V. Patankar: Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York, 1980.

J.H. Ferziger, M. Peric: Computational Methods for Fluid Dynamics, Springer-Verlag, New York, 2002.

M.O. Deville, P.F. Fischer, E.H. Mund: High-Order Methods for Incompressible Fluid Flow. Cambridge University Press, Cambdrige, 2002.

A. Quarteroni, A. Valli: Numerical Approximatipon of Partial Differential Equations. Springer-Verlag, Berlin, 1994.

• More information

Basic physical laws of continuum mechanics: laws of conservation of mass, momentum and energy. Theoretical study of hyperbolic conservation laws, particularly of Euler equations that describe the motion of inviscid compressible fluids. Numerical modelling based on the finite volume method. Numerical modelling of incompressible flows: Navier-Stokes equations, pressure-correction method, spectral element method.

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• ## Fundamentals of Optimal Control Theory (4 credits)

• ECTS credits 4
• Semester 1
• University Brno University of Technology
• Prerequisites

Linear algebra, differential and integral calculus, ordinary differential equations, mathematical programming, calculus of variations.

• Objectives

The aim of the course is to explain basic ideas and results of the optimal control theory, demonstrate the utilized techniques and apply these results to solving practical variational problems.

• Topics

1. The scheme of variational problems and basic task of optimal control theory.

2. Maximum principle.

3. Time-optimal control of an uniform motion.

4. Time-optimal control of a simple harmonic motion.

5. Basic results on optimal controls.

6. Variational problems with moving boundaries.

7. Optimal control of systems with a variable mass.

8. Optimal control of systems with a variable mass (continuation).

9. Singular control.

10. Energy-optimal control problems.

11. Variational problems with state constraints.

12. Variational problems with state constraints (continuation).

13. Solving of given problems.

• Books

[1] Pontrjagin, L. S. - Boltjanskij, V. G. - Gamkrelidze, R. V. - Miščenko, E. F.: Matematičeskaja teorija optimalnych procesov, Moskva, 1961.

[2] Lee, E. B. - Markus L.: Foundations of optimal control theory, New York, 1967.

• More information

The course familiarises students with basic methods used in the modern control theory. This theory is presented as a remarkable example of the interaction between practical needs and mathematical theories. Also dealt with are the following topics: Optimal control. Pontryagin's maximum principle. Time-optimal control of linear problems. Problems with state constraints. Singular control. Applications.

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• ## Reliability and Quality (4 credits)

• ECTS credits 4
• Semester 1
• University Brno University of Technology
• Prerequisites

Mastering basic and advanced methods of probability theory and mathematical statistics is assumed.

• Objectives

The course objective is to make students majoring in Mathematical Engineering acquainted with methods of the reliability theory for modelling and assessing technical systems reliability, with methods of mathematical statistics used for quality control of processing, and with a personal project solution using statistical software.

• Topics

Basic notions of objects reliability. Functional characteristics of reliability. Numerical characteristics of reliability. Probability distributions of time to failure. Truncated probability distributions of time to failure, mixtures of distributions. Calculating methods for system reliability. Introduce to renewal theory, availability. Estimation for censored and non-censored samples. Stability and capability of process. Process control by variables and attributes (characteristics, charts). Statistical acceptance inspections by variables and attributes (inspection kinds). Special statistical methods (Pareto analysis, tolerance limits). Fuzzy reliability.

• Books

Montgomery, Douglas C.:Introduction to Statistical Quality Control /New York :John Wiley & Sons,2001. 4 ed. 796 s. ISBN 0-471-31648-2

Ireson, Grant W. Handbook of Reliability Engineering and Management.Hong Kong :McGraw-Hill,1996. 1st Ed. nestr. ISBN 0070127506

• More information

The course is concerned with the reliability theory and quality control methods: functional and numerical characteristics of lifetime, selected probability distributions, calculation of system reliability, statistical methods for measure lifetime date, process capability analysis, control charts, principles of statistical acceptance procedure. Elaboration of project of reliability and quality control out using the software Statistica and Minitab.

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• ## Analysis of Engineering Experiment (4 credits)

• ECTS credits 4
• Semester 2
• University Brno University of Technology
• Prerequisites

Descriptive statistics, probability, random variable, random vector, random sample, parameters estimation, hypotheses testing, and regression analysis.

• Objectives

The course objective is to make students majoring in Mathematical Engineering and Physical Engineering acquainted with important selected methods of mathematical statistics used for a technical problems solution.

• Topics

1.One-way analysis of variance.

2.Two-way analysis of variance.

3.Regression model identification.

4.Nonlinear regression analysis.

5.Regression diagnostic.

6.Nonparametric methods.

7.Correlation analysis.

8.Principle components.

9.Factor analysis.

10.Cluster analysis.

11.Continuous probability distributions estimation.

12.Discrete probability distributions estimation.

13.Stochastic modeling of the engineering problems.

• Books

Ryan, T. P.: Modern Regression Methods. New York : John Wiley, 2004.

Montgomery, D. C. - Renger, G.: Applied Statistics and Probability for Engineers. New York: John Wiley & Sons, 2003.

Hahn, G. J. - Shapiro, S. S.: Statistical Models in Engineering. New York: John Wiley & Sons, 1994.

• More information

The course is concerned with the selected parts of mathematical statistics for stochastic modeling of the engineering experiments: analysis of variance (ANOVA), regression models, nonparametric methods, multivariate methods, and probability distributions estimation. Computations are carried out using the software as follows: Statistica, Minitab, and QCExpert.

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• ## Modern methods of solving differential equations (5 credits)

• ECTS credits 5
• Semester 2
• University Brno University of Technology
• Prerequisites

Differential and integral calculus of one and more real variables, ordinary and partial differential equations, functional analysis, function spaces, probability theory.

• Objectives

The aim of the course is to provide students an overview of modern methods applied for solving boundary value problems for differential equations based on function spaces and functional analysis including construction of the approximate solutions.

• Topics

1. Motivation. Overview of selected means of functional analysis.

2. Lebesgue spaces, generalized functions, description of the boundary.

3. Sobolev spaces, different approaches, properties. Imbedding and trace theorems, dual spaces.

4. Weak formulation of the linear elliptic equations.

5. Lax-Mildgam lemma, existence and uniqueness of the solutions.

6. Variational formulation, construction of approximate solutions.

7. Linear and nonlinear problems, various nonlinearities. Nemytskiy operators.

8. Weak and variational formulations of the nonlinear equations.

9. Monotonne operator theory and its applications.

10. Application of the methods to the selected equations of mathematical physics.

11. Introduction to Stochastic Differential Equations. Brown motion.

12. Ito integral and Ito formula. Solution of the Stochastic differential equations.

13. Reserve.

• Books

S. Fučík, A. Kufner: Nonlinear Differential Equations, Nort Holland, 1980.

K. Rektorys: Variational Methods in Mathematics, Science and Engineering, Dordrecht, D. Reidel Publ. Comp., 1980.

J. Nečas: Direct Methods in the Theory of Elliptic Equations, Springer, Heidelberg 2012.

B. Oksendal: Stochastic Differential Equations, Springer, Berlin 2000.

• More information

The course yields overview of modern methods for solving differential equations based on functional analysis. It deals with the following topics: Survey of spaces of functions with integrable derivatives. Linear elliptic equations: the weak and variational formulation of boundary value problems, existence and uniqueness of the solution, approximate solutions and their convergence. Characteristics of the nonlinear problems. Weak and variational formulation of the nonlinear coercive problems, existence of the solution. Application to the selected nonlinear equations of mathematical physics. Introduction to stochastic differential equations.

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• ## Data Visualisation (4 credits)

• ECTS credits 4
• Semester 2
• University Brno University of Technology
• Prerequisites

Students are expected to be familiar with basic programming techniques and their implementation in Borland Delphi, and with basic 2D and 3D graphic algorithms (colour systems, projection, curves and surfaces construction)

• Objectives

Students will be made familiar with basic methods of 3D data reconstruction and conditions for their use.

• Topics

1) Curves defined by equation f(x,y)=0, surfaces defined by equation f(x,y,z)=0 – pixel algorithm.

2) Curves defined by equation f(x,y)=0 – grid algorithm.

3) Surfaces defined by equation f(x,y,z)=0 – marching cubes algorithm.

4) Contour lines of surface.

5) Surface visualisation using the palette.

6) 2D visualisation of 3D data grid.

7) 3D visualisation of 3D data grid using marching cubes algorithm.

8) 3D filters.

9) 3D visualisation using volume methods – ray casting.

10) 2D reconstruction of confocal microscope outputs.

11) 3D reconstruction of confocal microscope outputs.

12) 2D reconstruction of Visible Human Project data.

13) 3D reconstruction of Visible Human Project data.

• Books

Martišek, K.: Adaptive filters for 2-D and 3-D Digital Images Processing, FME BUT Brno, 2012

• More information

The course is lectured in winter semester in the fourth year of mathematical engineering study. It familiarises students with basic principles of basic algorithm of computer modelling of 2D and 3D data, namely of scalar fields. Lecture summary: Construction of implicit curves and surfaces, contour lines and iso-surfaces. Algorithms, which construct surfaces – marching cubes and volume algorithms - ray casting, ray tracing.

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• ## Geometric Algorithms and Cryptography (4 credits)

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• ## Mathematical Structures (4 credits)

• ECTS credits 4
• Semester 2
• University Brno University of Technology
• Prerequisites

Students are expected to know the mathematics taught within the bachelor's study programme and the graph theory taught in the master's study programme.

• Objectives

The aim of the course is to show the students possibility of a unified perspective on seemingly different mathematical subjects.

• Topics

1. Sets and classes

2. Mathematical structures

3. Isomorphisms

4. Fibres

5. Subobjects

6. Quotient objects

7. Free objects

8. Initial structures

9. Final structures

10. Cartesian product

11. Cartesian completeness

12. Functors

13. Reflection and coreflection

• Books

[1] Jiří Adámek, Theory of Mathematical Structures, D. Reidel Publ. Company, Dordrecht, 1983.

[2] A.Adámek, H.Herrlich. G.E.Strecker: Abstract and Concrete Categories, John Willey & Sons, New York, 1990

• More information

The course will familiarise students with basic concepts and results of the theory of mathematical structures. A number of examples of concrete structures will be used to demonstrate the exposition.

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• ## Master's thesis (BUT) (15 credits)

• ECTS credits 15
• Semester 2
• University Brno University of Technology
• Objectives
The topic of the thesis can be proposed to the student by the local InterMaths coordinator or by the student him/herself. In any case, the InterMaths executive committee is the responsible to approve the thesis project before its formal start. The taste and expectations of the students are respected whenever possible. The local InterMaths coordinator in the hosting institution is the responsible to provide an academic advisor to the student, although proposals from the students will always be heard in this respect.

In some cases, after the agreement with the local InterMaths coordinator, the thesis topic can be related to a problem proposed by a private company. In this case, a tutor will be designated by the company as responsible person of the work of the student, especially if he/she is eventually working in the facilities of the company; however, the academic advisor is, in any case, the responsible to ensure the progress, adequacy and scientific quality of the thesis. The necessary agreements between the university and the company will be signed in due time, according to the local rules, in order that academic credits could be legally obtained during an internship, and the students be covered by the insurance against accidents outside the university.

NOTE: Although the thesis is scheduled for the 4th semester, some preliminary work may be anticipated due to the local rules - such as preliminary local courses in the 3rd semester, ensuring that the student can follow the main courses of the 3rd semester without problems. In this point, the personalised attention to the students has to be intensified, and decisions taken case by case.

• More information
In addition to previously mentioned, inludes the Master's Thesis at Brno University of Technology also following 4 local courses: Diploma Project 1 (1st semester, 4 credits), Diploma Project 1 (1st semester, 4 credits) Diploma Project 2 (2nd semester, 6 credits), Diploma Seminar 2 (2nd semester, 3 credits).

Diploma Project 1 (1st semester, 4 credits): Students will proceed in preparing their Master's Thesis so that they could be finished in the next semester. Leadership of Master's Thesis - It is given individually by the supervisor of the Master Thesis. The work on the Master Thesis will be checked by supervisors. If the supervisor is not satisfied with a student's result, the student will be assigned extra work to intensify the effort. Specific literature related to the Master's Thesis topic recommended by a supervisor. In the course, students are instructed by their supervisors how to use scientific literature, how to solve problems connected with their Master's Thesis and how to create a software on PC for preparing their Master's Thesis.

Diploma Seminar 1 (1st semester, 2 credits): The goal of the seminar is to teach students about how to present mathematical results to a broader (mathematical) audience. This will prepare them for their performance during the defence of the Master's Thesis. Acquaint students with formal and contentual aspect of professional reports. Exploitation and quotation of literature. Form of report: presentations, reports. In the course of the seminars, students report (in a form of a thirty-minute lecture) on their results obtained in working out the Master's Thesis.

Diploma Project 2 (2nd semester, 6 credits): Students will work out the project of their Master's Thesis so that they could be finished before the end of the semester. Supervised student's work on Master's Thesis. The work on the diploma theses will be checked by supervisors. If the supervisor is not satisfied with a student's results, the student will be assigned extra work to intensify the effort. In the course students are instructed by their supervisors how to use scientific literature, how to solve problems connected with their diploma theses and how to create a software on PC for preparing their diploma theses. Project specifications from industrial companies are appreciated.

Diploma Seminar 2 (2nd semester, 3 credits): The goal of the seminar is to teach students about how to present mathematical results to a broader (mathematical) audience. This will prepare them for their performance during the defence of the Master's Thesis. Topics Seminars 1.-13.: In every week, one seminar will be organized at which individual students will refer their diploma theses in such a way that all students will be given one chance during the semester. The theses will be discussed by the audience immediately after they are referred. In the course of the seminars, students report (in a form of a thirty-minute lecture) on their results obtained in working out the Master's Thesis.

http://www.fme.vutbr.cz/studium/predmety/predmet.html?pid=158638
http://www.fme.vutbr.cz/studium/predmety/predmet.html?pid=158664
http://www.fme.vutbr.cz/studium/predmety/predmet.html?pid=158639
http://www.fme.vutbr.cz/studium/predmety/predmet.html?pid=158637

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#### Programme Structure

»Year1 L'Aquila: IntMat | SciCom
»Year2 L'Aquila: IntMat | SciCom
»Year2 BrnoMath. Engineering
»Year2 Katowice: MatMod | MatFin
»Year2 LvivApplied Maths

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