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The aim is to acquaint students with the basic problems of numerical mathematics. Thematic areas are:
• Systems of linear equations. Direct and basic iterative methods.
• Solving nonlinear equations and their systems
• Eigenvalue problem
• Approximation of functions
• Numerical quadrature
• Numerical methods of solving ordinary differential equations with initial and boundary conditions.

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The subject follows the Applied Mathematics and Numerical Methods I, the aim is to master methods of solving partial differential equations. Both elliptical and parabolic tasks will be solved. Less attention will be paid to hyperbolic problems. Problems of effective preconditioning of emerging systems of linear systems will also be addressed.

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Random sample. Idea of statistical inference. Random variables and their distribution. Normal distribution. Central limit theorem. Multiple distribution. Independence. Correlation. Theory of estimation. – point and interval estimate. Hypotheses testing. Test statistic and statistical decision. P-value. Simple linear regression – parameters estimation, hypotheses testing, prediction intervals, regression diagnostic. Simulation independent realizations of random variables.

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Multivariate normal distribution. Principal component analysis. Linear regression. Nonlinear regression. Bayes theorem. Bayesian parameters estimates. Bayesian inference in linear model. Time series and their frequency domain description. Kalman-Bucy filtr.
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Noncontact optical and electronic methods for measurement of macrotopography and microtopography of surfaces. Optical methods of deformation and displacement measurement. Noncontact vibration measurements and measurements of velocity and flow using optolelectronic techniques.

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Atomic structure of matter. Application of quantum mechanics at microscopic level. Interactions between particles.
Chemical bonds. Phases and aggregate states of matter. Phase equilibia and phase transitions (melting, solidification, evaporation). Phase diagrams.
Physical and chemical properties of solids and fluids (ideal and real gases/solutions, viscosity).
Physics and chemistry of surfaces. Adsorption, adhesion, wettability of surfaces (contact angles). Determination of surface tension and surface energy.
Hydrophobicity, hydrophilicity. Balance equations and fundamentals of phenomenological description of mass/energy transport. Diffusion, heat transport. Basic hydrodynamics.

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Atomic structure. Wave function and its interpretation. Schrodinger equation.
Chemical bonds (ionic, covalent, metallic, Van der Waals).
Aggregate states of matter (plasma, gas, liquid, solid state).
Structures of solids (crystalline, amorphous). Basic crystallography (symmetry, crystal lattice, reciprocal lattice, Miller indices).
Experimental determination of crystal structures (Bragg condition, diffractions-X-ray, neutron scattering, electron diffraction). Lattice defects (point, dislocations).
Types of materials (metals, ceramics, glass, polymers, composites, concrete) and their properties (mechanical, thermal, optical, electrical).

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Principles of quantum mechanics. Wave-particle duality, interpretation of wave function. Schrodinger equation.
Quantum structure of atoms. Excited atomic states. Induced and spontaneous electron transitions. Transition probabilities. Spectral lines.
X-rays, structure and composition of matter.
Physical principles of laser (generation of population inversion, types of lasers-semiconductor, liquid, gaseous). Applications in material science.
Basic principles of spectroscopic techniques (spectrometers, Raman spectroscopy) and samples preparation.
Physical principles of microscopy (optical, scanning microscopy, AFM).
Surface and interfacial forces (fluids, solids). Experimental determination of wetting angles and surface energies/tensions on atomic smooth-rough surfaces.

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Types of polymers (natural, artificial). Structure of polymers (amorphous, crystalline, fibres, elastomers). Input materials for polymers preparation.
Thermodynamical and kinetic aspects of polymerization. Chemical bonds in polymeric chains.
Physical and chemical properties of polymers (mechanical, thermal).
Electrospinning principle and NANOSPIDER equipment. Nanofibers vs. makroscopic matters-differential properties.
Modifications of polymer nanofibres (via plasmatic technologies,heterogeneous nucleation, bakteriocidity). Properties of polymer-based nanofibres thin films (hydrophobicity).
Application of polymer-based nanofibres in modern civil engineering, protection of cultural heritage and in environment (microfiltration, hydrophobicity,, bacteriocidity).
The visits of specialized labs (NANOSPIDER, Institute of Physics) are also expected.

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Fundamentals of geometric, physical and quantum optics. Relativistic optics. Fundamentals of physical electronics. Lasers, laser beams and thein applications. Modern part of optics and thein applications in science and engineering (adaptive optics, gradient index optics, nonlinear optics, acoustooptics, electrooptics, etc.). Sources and detectors of optical radiation. Physical principles of modern optical elements and instruments with applications in science and engineering.

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Geometric and diffraction theory of optical imaging. Fundamentals of radiometry, photometry and colorimetry. Transfer properties of optical systems. Deconvolution techniques in spatial and spectral domain. Digital methods of image processing.

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Maxwell's equations. Constitutive relations. Boundary conditions. Linear and nonlinear electromagnetic media. Electromagnetic waves. Polarization, interference and diffraction of electromagnetic waves. Radiation and detection of electromagnetic waves.

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Basic terminology, definitions, principles and postulates of equilibrium thermodynamics.
Thermodynamical system, phase, aggregate state of matter. State equations.
Gibbs model of phase interface. Thermodynamical equilibrium conditions.
Ehrenfest classification of phase changes. 1st order phase transitions (Clausius-Clapeyron equation, nucleation). Condensation, solidification, melting, sublimation.
Surfaces. Surface energy and surface tension. Young-Laplace equation. Experimental determination of surface tension/energy.
Fundamentals of small systems thermodynamics.Porous systems.
Introduction to linear nonequilibrium thermodynamics.Generalized forces and fluxes.Balance equations for mass, impulse and energy.

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Structure of matter. Modeling of processes at various temporal/spatial levels of description. Fundamentals of probability theory (distribution functions, discrete/continuous variables, Stirling approximation). Introduction to statistical physics. Fluctuations. Boltzmann distribution (microstates, physical interpretation). Statistical ensembles (microcanonical, canonical, grandcanonical). Translational, rotational and vibrational partition functions. Elements of statistical thermodynamics. Determination of macroscopic characteristics of fluids and solid states (energy, heat capacity, potentials). Kinetic theory of gases (mean free path, pressure, effusivity).

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Transport of mass and energy.
Particles diffusion in fluids and solid states. Statistical and phenomenological description. Fick law, diffusion equation, analytical solutions. Diffusion in small systems.
Heat transfer. Fourier law, heat conduction equation, analytical solutions. Heat conduction in small systems.
Modern theory of phase transitions. Homogeneous and heterogeneous nucleation. Nucleation rate. Nucleation of water molecules in atmosphere-condensation. Formation of solid clusters in metastable liquids. Modeling of a very first stage of hydratation processes.

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The course Applied Chemistry brings information in a branch of classical and modern Chemistry. The goal of this course is to improve chemical knowledge of postgraduate students and show them the possibilities of chemical approach to solve their projects. The course comprises some thematic branches, namely chemical analysis, separatory, optical and electrical methods. In the branch of chemical analysis the classical and modern approach will be compared, it means qualitative and quantitative analysis. The electrical methods include conductometry, TDR technique and high temperature measurements. The principle of the separatory methods will be illustrated due to liquid chromatography. The optical methods will be presented by optical microscopy, ED XRDF and IR spectrometry. Finally, the possibilities of particle size and distribution determination will be solved, using sewing method and laser analysis.

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The course Applied Chemistry – practical laboratory lessons follows the theoretical classes of Applied chemistry course. According to the themes the practical laboratory measurement will be performed. Students will be familiarized with devices operation, possibilities of outputs and useful applications. In the branch of chemical analysis the classical and modern approach will be compared. The electrical methods include high temperature dilatometry and conductometry. The separatory method will be presented using liquid chromatography. ED XRDF and IR spectrometry will represent the optical methods. Finally, the particle size measurement using laser analyser will be realized.

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Sound propagation, basic acoustic quantities, frequency-dependent characteristics.
Spatial acoustics (sound level measurements, reverberation time, …)
Description of instruments used for acoustic measurements - generators, sensors, amplifiers, analyzers and recording devices.
Experimental determination of acoustic properties of building materials - attenuation of sound propagation through building materials, description of measuring apparatus.
Non-destructive measurements of materials properties by means of acoustic and ultrasonic methods (measurements of elastic modulus, velocity of waves propagation, attenuation of waves).
Ultrasonic defectoscopy (identification of cracks and cavities in materials, determination of inhomogeneities, ...)

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Influence of the environment on the structure and properties of materials, their aging and degradation. Chemical deterioration of materials. Concrete carbonation, corrosion of metals. Degradation of natural materials and polymers. Protection of materials against environmental impact.
Lectures:
1. Influence of CO2 on building materials, concrete carbonation
2. Degradation and rehabilitation of concrete
3. Electrochemistry
4. Corrosion of metal materials
5. Biodegradation, wood rehabilitation
6. Aging and degradation of polymers

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The aim of the subject is to provide to students detailed knowledge in the field of current trends in materials used in
construction industry and also in materials applied historically in older and culture heritage valuable buildings. The scope of the subject comprises description of building materials and interpretation of their properties and performance in relation to their structure and composition. Within the frame of Materials Engineering course, the students will summarize their knowledge in materials behaviour and dependence of their mechanical-physical parameters on exterior effects and climate conditions changes. The students will also gain knowledge and skills in the field of materials research and actual and latest trends in materials basis for building industry.

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Thermal conductivity of gases. Dynamical theory of crystal lattice. Heat capacity of materials. Conduction and radiation heat transfer in materials. Heat transfer equation. Thermal field. Measurement methods of thermal diffusivity, thermal conductivity, and heat capacity of solids, fluids, and gases. Impulse measurement methods. Temperature sensors. Linear and volumetric thermal expansion of solids, fluids, and gases. Thermal expansion coefficient of isotropic and anisotropic materials.

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Solution of steady-state and transient processes using finite element method
Computer implementation of finite element method
Programming of finite element method problems in C language
Methods of solving nonlinear problems
Convergence of finite element method, error estimate
Solution of problems involving phase change and chemical reactions
Computational modeling of one-dimensional problems
Computational modeling of multi-dimensional problems
Computational modeling of multi-dimensional problems using parallel solvers

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1. Kinematics of continuum thermomechanics
2. Forces, work, and power in continuum thermomechanics
3. Global balance laws of continuum thermomechanics
4. Local balance laws of mass and momentum
5. Local balance laws of kinetic, potential, and mechanical energy
6. Thermodynamic postulates and thermodynamic laws
7. Thermodynamic potentials
8. Continuum without irreversible processes, model of thermoelastic continuum
9. Local balance law of internal energy
10. Local balance law of total energy
11. Local balance law of entropy
12. Unified form of balance laws in thermomechanics
13. Fundamentals of the theory of mixtures, balance law of mass of a mixture component

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1. Description of porous medium
2. Water vapor transport in porous medium
3. Knudsen diffusion and surface diffusion in porous medium
4. Liquid water transport in porous medium
5. Phase changes of water in porous medium
6. Convective models of moisture transport
7. Diffusion models of moisture transport
8. Construction of constitutive equations using methods of irreversible thermodynamics
9. Thermodynamic model of coupled heat and moisture transport
10. Diffusion models of coupled heat and moisture transport
11. Convective models of coupled heat and moisture transport
12. Coupled heat, moisture, and chemical compounds transport
13. Effect of electric field on heat and moisture transport

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The students will be introduced to the method of digital image correlation (DIC) and its use in experimental mechanics. Hardware requirements will be discussed along with the introduction of essential algorithms and post-processing of results. The students will be actively engaged in experimental measurements and processing of results. Those interested in programming will be involved in development of open-source DIC codes. Besides DIC, the students will be introduced to numerical modeling in order to comprehend the meaning of the experimentally obtained data and become able to analyze them critically. The introduction of high-speed cameras and their use in experimental mechanics is also within the scope of the subject.

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The course is intended for students who did not have the opportunity to study basic goals, tasks and elementary means of an experimental analysis during the course of the bachelor’s and master’s degree study. Within the course, students will familiarize with basic procedures and principles of the experimental analysis of building and civil engineering structures. The interpretation of the problems will include the overview of testing methods used to determine basic material properties, the description of experiments focused on observation of climate loads, the examples of verification and identification of theoretical models based on experimental results, the experiments realized on physical models for estimation of wind effects in wind tunnels and for investigation of earthquake effect on shake tables, the long term monitoring of building and civil engineering structures. The interpretation will further include the principles of preparation, realization and evaluation of static load tests realized on structural elements or whole structures, the basic methods used for an analysis of measured data obtained during dynamic tests, the principles of preparation, realization and evaluation of dynamic tests including an experimental modal analysis and a dynamic load test, the principles of experiments focused on evaluation and assessment of vibration effects on building structures from the view of the load capacity limit state and on users of building structures from the view of the serviceability limit state, the demonstration of several practical tasks.

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The major goal of the course is to expand knowledge about experimental analysis of building and civil engineering structures obtained during master’s or doctoral degree study. Within the course, students will familiarize with the basic design of the static and dynamic experiments applied on building and civil engineering structures, relative sensors, absolute sensors, strain gauges, principles of strain measurement by means of strain gauges, basics of estimating measurement uncertainty, experiments realized on physical models, basics of the similarity theory, model laws, experimental methods for axial tensile force determination in rods, cables and stays, static and dynamic load tests and long term monitoring realized on building and civil engineering structures illustrated on practical examples (real reasons for realization, arrangement of experiments, ways of processing data, basic conclusions), the demonstration of practical tasks.

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Fundamentals of transmission a reflexion optical microscopy. Polarization of light and its application in the phase study of the materials. The sample preparation for microscopical research.
Fundamentals of scannig electron microscopy and microanalysis. Electron sources and eletron interaction with matter, detection of secondary signals and interpretation of secondary emissions. Scannig (SEM) a transmission electron microscopy (TEM), elementary microanalysis (EDS/WDS) a electron diffraction (BESD-O.I.M.). The outline of the most applications SEM, ESEM, EDS, WDS, O.I.M). Implementation of SEM and EDS in material research. The sample preparation.
X-ray phase diffraction and structural analysis. The fundamentals of XRD analysis and its application in the structural and phase exploration of construction materials. Phase identiffication, preffered orientational textural arangement and XRD textural analysis of stress and deformation. The sample preparation.

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The course will cover analytical methods for multiscale modeling of heterogenous materials, with emphasis on:
1. Introduction, overview of governing equations of elasticity, tensor notation, and averaging
2. Minimum energy principles, material symmetries
3. Elementary theory of overall moduli, concentration factors, Voigt-Reuss bounds
4. Exact solution for two-phase composites, idea of improved bounds
5. Eshelby problem
6. Approximate evaluation of overall moduli: dilute approximation, self-consistent method, Mori-Tanaka method
7. Improved bounds on overall moduli: Hashin-Shtrikman bounds
8. Thermo-elasticity
9. Extension to stationary transport processes

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The course will cover numerical methods for multiscale modeling of heterogenous materials, with emphasis on:
1. Overview of the finite element method for elasticity and heat conduction
2. Introduction to the method of asymptotic expansion for heat conduction and elasticity
3. First-order computational homogenization for elasticity
4. First-order computational homogenization for heat conduction and thermo-elasticity
5. Homogenization nonlinear problems -- application to non-linear conduction and elasticity
6. Two-scale simulations -- basic principles and implementation strategy, applications
Reduced-order models, combining computational homogenization with micromechanics

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The course is devoted to the measurements and modeling of basic laboratory tests using the finite element method. Attention is concentrated on the description of nonlinear response of soil with the help of traditional material models. Knowledge gained from the modeling of simple laboratory tests will be exploited in the analysis of selected geotechnical structures. All numerical simulations will be performed employing the GEO5 FEM software package.
Topics covered in individual lectures:
1. week: Material behavior at a material point, stress-strain relationship, modulus of elasticity, Poisson number, bulk modulus, oedometric modulus, invariants of stress and strain tensors, plastic strain.
2. week: Introduction to theory of plasticity, yield surface, stress return mapping, Mohr-Coulomb model.
3. week: Laboratory – running oedometric test.
4. week: General stiffness method, introduction to FEM – application to beams.
5. week: Laboratory – running simple shear test.
6. week: Selected plasticity models - Drucker-Prager model, Cam-clay model.
7. week: Finite elements – three-noded triangle, linear FEM models.
8. week: Solution of nonlinear problems in FEM, Newton-Raphson method.
9. week: Formulation of numerical model of oedometric and triaxial laboratory test.
10. week: Models of simple geotechnical structures (excavation of construction ditch, sheeting and retaining walls, slope stability analysis).
11. week: Laboratory – completing all measurements, removing samples from laboratory devices, evaluating collected data.
12. week: Calibrating material models based on the measured data and data available in literature.
13. week: Course evaluation

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The course is devoted to the modeling of time-dependent problems in geomechanics. Advanced laboratory measurements in gallery Josef will be accompanied by numerical modeling using the finite element method. The course covers 4 different topics: (i) Heat transport, (ii) Steady state ground water flow, (iii) Transient ground water flow, (iv) Coupled mechanical and ground water flow in fully saturated deformable soil body – consolidation. Numerical modeling will be performed employing the GEO5 FEM and SIFEL software packages.
Topics covered in individual lectures:
1-4 weeks: One-day course on measurements of transport parameters – gallery Josef. Laboratory measurements in permeameter, measurement of the coefficient of thermal conductivity and thermal capacity on a rock sample. Setting up in situ experiment and measurements kick off. The measured data will be collected and gradually evaluated throughout the whole semester.
5-7 weeks: Modeling of heat transport – theoretical formulation of stationary and nonstationary heat transport, boundary conditions, FEM formulation, methods to solve nonstationary transport problem (time integration), modeling a selected task using FEM (program SIFEL).
8-10 weeks: Modeling of ground water flow – theoretical formulation of stationary and nonstationary ground water flow, Darcy law, continuity equation, boundary conditions, transition zone, formulation of finite elements, Modeling a selected task using FEM (program GEO 5).
11-13: Modeling consolidation – theoretical formulation of fully coupled transport of ground water in a deformable soil body assuming fully saturated soils, formulation of finite elements, Modeling a selected task using FEM (program GEO 5).

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1. Microstructure of heterogeneous materials and their description
2. Image and microstructure analysis
3. SEM scanning electron microscopy methods and analytical techniques
4. Practical demonstration of SEM and measurement (lab.)
5. Nanoindentation and small volume properties
6. Evaluation of elastic and viscoelastic parameters
7. Practical demonstration of nanoindentor and measurement (lab.)
8. Spherical indentation, plastic material parameters
9. Principles of nanomechanical analysis of heterogeneous materials
10. Deconvolution and homogenization on heterogeneous systems
11. AFM microscopy for 3D surface mapping
12. Practical demonstration of AFM and measurement (lab.)
13. Material scales links, multi-scale modeling

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The objective of the course is to deliver an introduction to numerical methods for solving partial differential equations, with particular focus on finite element method. It is suitable for students without previous knowledge in the field. It consists of the two main parts:
- overview and derivation of fundamental equations for theory of elasticity and heat transfer, introduction to method of weighted residuals, strong and weak solution, choice of approximation and weight functions.
- application of finite element and finite difference method to solution of selected problems from engineering practice (1D elasticity, beams, grids on elastic foundation, plates on elastic foundation, 1D and 2D stationary and transient heat transfer).
The students will not only understand theoretical aspects of the methods, but will use and further develop prototype implementations in Matlab to understand the algorithmic aspects of the methods. During the seminars, the students will individually or in a small teams solve selected problems, interpret and discuss results.

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The objective of the course is to extend basic knowledge of numerical methods for solving PDEs and particularly finite element method towards their advanced applications in engineering. The course will focus on problems of geometrically and materially nonlinear static (theoretical framework, linearization, algorithmic aspects, solution methods – direct and indirect control, plasticity and damage based models). Introduction to Isogeometric analysis, eXtended finite element method, mesh generation and efficient methods for solution sparse linear systems.
The students will not only understand theoretical aspects of the methods, but will use and further develop prototype implementations in Matlab to understand the algorithmic aspects of the methods. During the seminars, the students will individually or in a small teams solve selected problems, interpret and discuss results.

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The course focuses at systematic description of nonlinear mechanical behavior of homogeneous and heterogeneous materials: Formulation of constitutive equations of fundamental material models (elastoplastic, viscoelastic, progressive damage). Mathematical models of heterogeneous materials (fundamentals of mesomechanics). Fundamentals of linear fracture mechanics (stress intensity factor, energetic criterion of local crack stability, other criteria). Fundamentals of nonlinear fracture mechanics (crack with localized plastic zone, cohesive crack model, size effect). Fundamentals of the theory of fatigue processes.

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This course covers the fundamentals of tensor algebra and calculus and demonstrates the power of tensor notation applied to formulation and solution of engineering problems. Selected examples cover solid and fluid mechanics, as well as heat and mass transport problems. The first part of the course is devoted to the definition of tensors, understood as linear mappings, to algebraic operations with tensors, to tensor fields and their differentiation, and to transformations between volume and surface integrals based on the Green and Gauss theorems. In the second part, it is shown how these mathematical tools enable an elegant description and analysis of various physical problems, with focus on applications in civil and structural engineering.
The classes combine lectures and seminars, with emphasis on problems assigned as homework, which form the basis of presentations and discussions in class. The objective is not only to transfer specific knowledge, but also to develop the students‘ aptitude for independent thinking and critical analysis. At the same time, mastering of tensorial notation by the students will greatly facilitate their future reading of modern scientific literature in many fields of research.

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The covered material splits into three blocks: (i) Important relations and theorems necessary in the area of the theory of reliability and mathematical statistics, (ii) Analytical and simulation methods to analyze reliability of structures, (iii) Advanced methods or reliability analysis exploiting the Bayesian inference in conjunction with MCMC simulation.
List of lectures:
1. Basic relations, definitions and notation, 2. Selected probability distributions and important inequalities, 3. Transformation of probability density function (one and more variables), 4. Reliability of simple structures, 5. Evolution of reliability in time, 6. Reliability and solution methods, 7. Renewable systems, 8. Reflection of the theory in EC standards, 9. Analytical methods to address reliability, 10. Simulation methods, 11. Monte Carlo type simulation, 12. MCMC sampling (Markov chain-Monte Carlo, Bayesian statistical method).

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In this course, which is taught exclusively in English, attention is paid to the structure of a scientific or technical paper, to grammatical and stylistic aspects and to the creative scientific writing process from manuscript preparation up to its publication (including the selection of an appropriate journal and the manuscript submission and review process). Other topics covered in the course include effective search for and processing of information sources in a network environment, exploitation of library, open-access and other resources and tools, citation rules and publication ethics. Students get acquainted with citation managers, manuals of style, typesetting rules and tools for the preparation of a technical manuscript in LaTeX. Basic information on bibliometric tools and evaluation of scientific output is also provided.

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Témata seminářů s tematicky zaměřenými přednáškami:
• Approaches and testing for evaluation of advanced aggregate characteristics (incl. fine particles and fillers)
• Principles of mix design and assessment of mixtures in hydraulic binders and cold recycled mixtures
• Bituminous binders and approaches in advanced testing (performance-based testing, functional approach for evaluation, rheological models)
• Utilization of rheological testing of bituminous binders by dynamic shear rheometer (DSR)
• Measurement of technical characteristic and properties by applying DSR in case of bituminous binders (stiffness, fatigue, creep) and MSCR test
• Design and composition of compacted asphalt mixtures and mastic asphalt including their material characteristics
• Functional testing of asphalt mixtures – principles, advantages and the fundamentals for composite characterization
• Pavement cement concrete (resistance to cycling effects of water and frost, thixotrophy etc.)
Laboratorní praktika:
• Fundamental tests for bituminous binders including dynamic viscosity and force duktility
• Testing on dynamic shear rheometer
• Testing on bending beam rheometer, including artificial laboratory ageing of bituminous binders
• Dynamic performance-based tests executed on universal testers (stiffness, creep, fatigue, dynamic modules, etc.)
• Practice of design and optimization (mix composition) for composites using bituminous or hydraulic binders foreseen for road structures
• Execution of defined laboratory tasks with focus on determination of advanced bitumen and asphalt mix characteristics or properties related to pavement mixtures bond by hydraulic binders

Knowledge of English and optional language is required for all programs.

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The course is aimed at making doctoral degree students familiar with the basic genres of the academic style and prepare them for real-life communication situations, such as e.g. the presentation of their own research and development achievements, writing grant applications, common correspondence, writing abstracts, etc. The course should also assist in the preparation for the examination in English, which is a compulsory part of doctoral degree study. The course is not compulsory, it is not completed by granting a credit or passing an examination.

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The course in Czech for foreigners is aimed at the university students’ needs for mastering written and spoken language with the basic inventory of linguistic structures needed for making oneself understood in common situations of everyday practical life; advanced students develop the ability of independent work with a simple technical text. The course is not compulsory, it is not completed by granting a credit or passing an examination.

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The doctoral degree student who wants to pass an examination in French can choose from the elective courses offered by the Department of Languages. The course is not compulsory.

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The doctoral degree student who wants to pass an examination in German can choose from the elective courses offered by the Department of Languages. The course is not compulsory.

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The doctoral degree student who wants to pass an examination in Russian can choose from the elective courses offered by the Department of Languages. The course is not compulsory.

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The doctoral degree student who wants to pass an examination in Spanish can choose from the elective courses offered by the Department of Languages. The course is not compulsory.