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Already enrolled at one of our partner universities?
As a local student, you can get access to the study paths below (specifically designed for international students). But you can also take advantage of other study possibilities. Check the >> local student section for more info.
Join the InterMaths
Joint MSc programme and gain a
double degree
in Applied and Interdisciplinary Mathematics
from two of our partner universities.
The InterMaths MSc programme is structured in 2 academic years (4 semesters) starting every September, studied in 2 different countries. Possible study paths are explained below.
The InterMaths programme assures the smooth credit transfer between partner universities and the awarding of a double MSc degree after earning 120 ECTS credits from a given list of courses ALL taught in English. Please read more in the programme details section about scholarships and fees.
View a chart showing the different grading scales in use at each partner institution and the corresponding grades (A, B, C...) and GPA.
Course Catalogue
The InterMaths structure for the next academic year includes spending Year 1 in L'Aquila (Italy) and Year 2 in one of the other partner universities. Note that the study paths for Year 2 in L'Aquila listed below are only available to local students who spend their Year 1 in their university of origin. It is indeed compulsory to spend each academic year in a different institution in order to gain a double MSc degree.
Browse the tabs below to learn more about our study pathways and their course units. Note that if you are a student of any of the partner universities, you should contact your local coordinator, as there may be also other alternative pathways available.
Year1 L'Aquila
Year 1 in L'Aquila  Scientific Computing
 1 Year
 Scientific Computing Pathway
 University of L'Aquila Place
 66 (min.*) ECTS Credits
 Read here Qualification
List of course units
* Students are required to earn 66 ECTS credits, at least, during their first year by successfully attending the following compulsory course units (Semester 1 and 2 amounting to 48 ECTS credits) and picking other 18 ECTS credits (minimum) from the elective ones listed below.
Semester 1

Applied partial differential equations (6 credits)
 ECTS credits 6
 Semester 1
 University University of L'Aquila
 Lecturer 1 Corrado Lattanzio

Objectives
Students will know basic of properties (existence, uniqueness, etc.) and techniques (characteristics, separation of variables, Fourier methods, Green's functions, similarity solutions, etc.) to solve basic PDEs (conservation laws, heat, Laplace, wave equations).

Topics
Integral curves and surfaces of vector fields. First order partial differential equations. Linear and quasi linear partial differential equations (PDEs) of first order. Method of characteristics. The initial value problem: existence and uniqueness. Development of shocks.
The CauchyKovalevsky theorem. Linear partial differential operators and their characteristic curves and surfaces. Methods for finding characteristic curves and surfaces. The initial value problem for linear first order equations in two independent variables. Holmgren's uniqueness theorem. Canonical form of first order equations. Classification and canonical forms of second order equations in two independent variables. Second order equations in two or more independent variables. The principle of superposition.
The divergence theorem and the Green's identities. Equations of Mathematical Physics.LAPLACE'S EQUATION AND HARMONIC FUNCTIONS Elementary harmonic functions. Separation of variables. Inversion with respect to circles and spheres. Boundary value problems associated with Laplace's equation. Representation theorem. Mean value property. Maximum principle. Harnack’s inequality and Liouville’s theorem. Wellposedness of the Dirichlet problem. Solution of the Dirichlet problem for the unit disc. Fourier series and Poisson's integral. Analytic functions of a complex variable and Laplace's equation in two dimensions. The Neumann problem.
GREEN'S FUNCTIONS. Solution to the Dirichlet problem for a ball in three dimensions. Further properties of harmonic functions. The Dirichlet problem in unbounded domains. Method of electrostatic images.
THE WAVE EQUATION. Cauchy problem. Energy method and uniqueness. Domain of dependence and range of influence. Conservation of energy. Onedimensional wave equation. D’Alembert formula. Characteristic parallelogram. Non homogeneous equation and Duhamel’s method. Multidimensional wave equation. Well posed problems. Fundamental solution (n=3) and strong Huygens’ principle. Kirchhoff formula. Method of descent. Poisson?s formula (n=2). Wave propagation in regions with boundaries. Uniqueness of solution of the initialboundary value problem. Separation of variables. Reflection of waves.
THE HEAT EQUATION. Heat conduction in a finite rod. Maximum principle and applications. Solution of the initialboundary value problem for the one dimensional heat equation. Method of separation of variables. The initial value problem for the one dimensional heat equation. Fundamental solution. Non homogeneous case and Duhamel’s method. Heat conduction in more than one space dimension.

Books
E. C. Zachmanoglou and Dale W. Thoe, lntroduction to Partial Differential Equations with Applications. Dover Publications, Inc.. 1986. ISBN 0486652513
L.C. Evans, Partial Differential Equations. American Mathematical Society. 2010. Second edition, ISBN13: 9780821849743
S. Salsa, Partial Differential Equations in Actions: from Modelling to Theory. SpringerVerlag Italia. 2008. ISBN 9788847007512
W. A. Strauss, Partial Differential Equations, Student Solutions Manual: An Introduction. John Wiley & Sons, LTD. 2008. Second edition, ISBN13: 9780470260715
W. A. Strauss, Partial Differential Equations: an introduction. John Wiley & Sons, LTD. 2007. Second edition, ISBN13 9780470054567
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Control Systems (6 credits)
 ECTS credits 6
 Semester 1
 University University of L'Aquila
 Lecturer 1 Alessandro D'Innocenzo

Objectives
The course provides the basic methodologies for modeling, analysis and controller design for continuoustime linear timeinvariant systems.

Topics
Frequency domain models of Linear Systems: Laplace Transform, Transfer Function, Block diagrams.
Time domain models of Linear Systems:State space representation. BIBO stability.
Control specifications for transient and steadystate responses. Polynomial and sinusoidal disturbances rejection.
The RouthHurwitz Criterion. PID controllers.
Analysis and controller design using the root locus.
Analysis and controller design using the eigenvalues assignment: controllability, observability, the separation principle.
Reference inputs in state space representations.
Controller design using MATLAB.
Advanced topics in control theory.

Books
R.C. Dorf, R.H. Bishop, Modern Control Systems. Prentice Hall. 2008. Eleventh Edition
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Dynamical Systems and Bifurcation Theory (6 credits)
 ECTS credits 6
 Semester 1
 University University of L'Aquila
 Lecturer 1 Bruno Rubino

Prerequisites
Ordinary Differential Equations

Topics
Linear systems of differential equations: uncoupled linear systems, diagonalization, exponentials of operators, the fundamental theorem for linear systems, planar linear systems, complex eigenvalues, multiple eigenvalues, stability theory, nonhomogeneous linear systems.
Local theory of nonlinear systems: initial value problem, hyperbolic equilibrium point, Stable Manifold Theorem. HartmanGrobman Theorem. Stability and Liapunov functions. Saddles, nodes, foci and centers. Nonhyperbolic critical points. Center manifold theory.
Global theory of nonlinear systems: limit set, attractor, limit cycle, Poincaré map, stable manifold theorem for periodic orbits, PoincaréBendixson theory. Mathematical background: Fundaments of perturbation analysis. The Multiple Scale Method. Basic concepts of bifurcation analysis: Bifurcation points, Linear codimension of a bifurcation, Imperfections, Fundamental path, Center Manifold Theory.
Basic mechanisms of multiple bifurcations: divergence, Hopf, nonresonant or resonant doubleHopf, DivergenceHopf, Doublezero bifurcation.

Books
Lawrence Perko, Differential equations and dynamical systems, SpringerVerlag, 2001
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Functional Analysis in Applied Mathematics and Engineering (9 credits)
 ECTS credits 9
 Semester 1
 University University of L'Aquila
 Lecturer 1 Marco Di Francesco

Prerequisites
Linear Algebra. Complex numbers. Differential and integral calculus of functions of real variables.

Topics
Basic functional analysis: normed and Banach spaces, Hilbert spaces, Lebesgue integral, linear operators, weak topologies, distribution theory, Sobolev spaces, fixed point theorems, calculus in Banach spaces, spectral theory.
Applications: ordinary differential equations, boundary value problems for partial differential equations, linear system theory, optimization theory.

Books
Ruth F. Curtain, A.J. Pritchard, Functional Analysis in Modern Applied Mathematics, Academic Press, 1977
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Italian Language and Culture for foreigners (level A1) (3 credits)
 ECTS credits 3
 Semester 1
 University University of L'Aquila
 Lecturer 1 Cinzia Di Martino
 Lecturer 2 Danilo Larivera
 Lecturer 3 Doriana Scarsella
 Objectives Students will reach a basic level of both written and spoken Italian (A1 level according to CEFR), and will acquire a smattering of Italian culture.
 Topics Greetings and introductions. Expressing likes and dislikes. Talking about daily activities. Understanding and using everyday expressions as well as basic phrases related to daily needs (buying something, asking for directions, ordering a meal). Interacting in a very simple way about known topics (family, nationality, home, studies). Italian gestures. Italian geography. Introduction to the most important Italian cities. Italian food.
 Books Nuovo Espresso 1, Alma Edizioni, ISBN: 9788861823181
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Semester 2

Big data models and algorithms (3 credits)
 ECTS credits 3
 Semester 2
 University University of L'Aquila
 Lecturer 1 TBD
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Data analytics and Data mining (6 credits)
 ECTS credits 6
 Semester 2
 University University of L'Aquila
 Lecturer 1 Fabrizio Rossi
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Parallel computing (6 credits)
 ECTS credits 6
 Semester 2
 University University of L'Aquila
 Lecturer 1 https://www.intermaths.eu/my/userprofile/

Topics
Linux/Unix OS and tools;
Basic Fortran (or C);
HPC architecture;
System Scheduler;
Message Passing Interface;
OpenMP;
GPU computing;
Applications: linear algebra, PDEs, ODEs.
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Italian Language and Culture for foreigners (level A2) (3 credits)
 ECTS credits 3
 Semester 2
 University University of L'Aquila
 Lecturer 1 Cinzia Di Martino
 Objectives Students will reach an elementary level of both written and spoken Italian (A2 level according to CEFR).
 Books Italian Espresso 1, Alma Edizioni, ISBN: 9788889237212. Civiltàpuntoit, di Marco Mezzadri e Linuccio Pederzani, Guerra Edizioni, ISBN: 9788855700160.
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Electives

Mechanics of Solids and Materials (6 credits)
 ECTS credits 6
 Semester 2
 University University of L'Aquila
 Lecturer 1 Amabile Tatone

Prerequisites
Some knowledge of linear algebra and basic notions in elementary mechanics of a pointwise body could be helpful.

Objectives
To get familiar with kinematics of continuum, a suitable notion of force distribution, a general method delivering balance equations in continuum mechanics, the formal way of describing material properties and energy balance mainly for solid matter.

Topics
Placements and motions. Rigid and affine motions. Deformation gradient, stretch and rotation. Stretching and spin. Test velocity fields and force distributions. Working and stress. Working balance principle. Balance equations. Frame indifference principle. Affine bodies. Cauchy continuum. Cauchy stress and PiolaKirchhoff stress.
Material response. Material objectivity. Symmetry group and isotropy. Elastic and hyperelastic materials. Strain energy function. Constraints and reactive stress. Incompressibility. MooneyRivlin and neoHookean materials. Dissipative stress and dissipation principle. Fluids and solids. A general scheme for describing growth and relaxation via KronerLee decomposition. Remodeling forces and stress. Eshelby tensor. Viscoelasticity.
Numerical simulations with Comsol Multiphysics.

Books
C. Truesdell, A First Course in Rational Continuum Mechanics. Academic Press. 1977.
M. Gurtin, An Introduction to Continuum Mechanics. Academic Press. 1981.
P. Chadwick, Continuum Mechanics: Concise Theory and Problems. Dover Books on Physics. 1976.
 Link https://www.disim.univaq.it/didattica/content
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Combinatorics and cryptography (6 credits)
 ECTS credits 6
 Semester 2
 University University of L'Aquila
 Lecturer 1 Norberto Gavioli

Objectives
The student will be requested to have a good preparation on the presented topics, and to be able to implement some of the algorithms in a programming language

Topics
Abstract: Basic cryptograpy and coding theory will be developed. Some protocols and algorithms will be discussed focusing an security and data integrity.
Programme: Elementary arithmetics: Integers, divisibility, prime numbers, Euclidean division and g.c.d., Congruence classes, Chinese remainder theorem, cyclic and abelian groups, Lagrange theorem, Euler theorem, the structure of invertible classes mod p^n, Fields with p elements, polynomials, Euclidean division and g.c.d., Congruence classes of polynomials, Finite fields, primitive elements and polynomials, Legendre/Jacoby symbols and quadratic reciprocity. Cryptography: Classical cryptosystems: Shift cyphers, Vigenère Chipher, Substitution Chiper, One time pads, LFSR Data Encryption Standard: Simplified DES and differential cryptanalysis, Attacks, password encryption RSA: the algorithm, Attacks, Primality testing, the public key concept. Discrete logarithms: Bit commitment, DiffieHelman Key exchange, ELGAMAL Hash function: SHA, birthday attacks Digital signatures: RSA signatures, Hashing and signing, DSA Error correcting codes: Binary block codes, distance and correction of errors, classical bounds, linear codes, cyclic codes, Hamming codes, BCH and ReedSolomon codes.

Books
[1] Wade Trappe, Lawrence C. Washington, Introduction to cryptography: with coding theory 2nd ed.. Pearson Prentice Hall. 2006.
[2] https://www.disim.univaq.it/didattica/content.php?corso=424&pid=88&did=0&lid=en
 Link https://www.disim.univaq.it/didattica/content
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Complex Analysis (6 credits)
 ECTS credits 6
 Semester 2
 University University of L'Aquila
 Lecturer 1 Margherita Nolasco

Prerequisites
Knowledge of all topics treated the Mathematical Analysis courses in the first and second year: real function of real variables, limits, differentiation, integration; sequences and series of funcions; ordinary differential equations.

Objectives
Knowledge of basic topics of complex analysis: elementary functions of complex variable, differentiation, integration and main theorems on analytic functions.
Ability to use such knowledge in solving problems and exercises

Topics
Complex numbers. Sequences. Elementary functions of complex numbers. Limits, continuity. Differentiation. Analytic functions. Armonic functions.
Contour integrals. Cauchy's Theorem. Cauchy's integral formula. Maximum modulus theorem. Liuville's theorem.
Series representation of analytic functions. Taylor's theorem. Laurent's series and classification of singularities.
Calculus of residues. The residue theorem. Application in evaluation of integrals. Rouche's theorem.
Conformal mappings. Main theorems. Fractional linear transformations.
Fourier transform for L^1 functions. Applications. Fourier transform for L^2 functions. Plancherel theorem.
Laplace transform and applications.

Books
J.E. Marsden, M.J. Hoffman, Basic complex analysis. Freeman New York.
W. Rudin, Real and complex analysis. Mc Graw Hill.
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Stochastic processes (6 credits)
 ECTS credits 6
 Semester 2
 University University of L'Aquila
 Lecturer 1 Ida Germana Minelli

Objectives
The course aims to give an introduction to the theory of stochastic processes with special emphasis on applications and examples. On successful completion of this module the students should become familiar with some of the most known classes of stochastic processes (such as martingales, markov processes, diffusion processes) and to acquire both the mathematical tools and intuition for being able to describe systems with randomness evolving in time in terms of a probability model and to analyze it charcterizing some of its properties.

Topics
Stochastic Processes: definition, finite dimensional distributions, stationarity, sample path spaces, construction, examples.
Filtrations, stopping times, conditional expectation.
Markov processes: definition, main properties and examples. Birth and death processes.
Poisson process with applications on queueing models.
Martingales: definition, main properties and examples. Branching processes.
Brownian motion: definition, construction and main properties.
Brownian Bridge, Geometric Brownian Motion, OrnsteinUhlenbeck process.
Ito integral and stochastic differential equations. Applications and examples.

Books
P. Billingsley, Probability and measure. John Wiley and Sons.
G. Grimmett, D. Stirzaker, Probability and random Processes. Oxford University Press.
B. Oksendal, Stochastic Differential Equations. SpringerVerlag.
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Network optimization (6 credits)
 ECTS credits 6
 Semester 2
 University University of L'Aquila
 Lecturer 1 Fabrizio Rossi

Objectives
Ability to recognize and model network optimization problems as Integer Linear Programming problems. Knowledge of fundamental algorithmic techniques for solving large scale Integer Linear Programming problems. Knowledge of commercial and open source Integer Linear Programming solvers.

Topics
1. Formulations of Integer and Binary Programs: The Assignment Problem; The Stable Set Problem; Set Covering, Packing and Partitioning; Minimum Spanning Tree; Traveling Salesperson Problem (TSP); Formulations of logical conditions.
2. Mixed Integer Formulations: Modeling Fixed Costs; Uncapacitated Facility Location; Uncapacitated Lot Sizing; Discrete Alternatives; Disjunctive Formulations.
3. Optimality, Relaxation and Bounds. Geometry of R^n: Linear and affine spaces; Polyhedra: dimension, representations, valid inequalities, faces, vertices and facets; Alternative (extended) formulations; Good and Ideal formulations.
4. LP based branchandbound algorithm: Preprocessing, Branching strategies, Node and variable selection strategies, Primal heuristics.
5. Cutting Planes algorithms. Valid inequalities. Automatic Reformulation: Gomory's Fractional Cutting Plane Algorithm. Strong valid inequalities: Cover inequalities, lifted cover inequalities; Clique inequalities; Subtour inequalities. Branchandcut algorithm.
6. Software tools for Mixed Integer Programming.
7. Lagrangian Duality: Lagrangian relaxation; Lagrangian heuristics.
8. Network Problems: formulations and algorithms. Constrained Spanning Tree Problems; Constrained Shortest Path Problem; Multicommodity Flows; Symmetric and Asymmetric Traveling Salesman Problem; Vehicle Routing Problem Steiner Tree Problem; Network Design. Local Search Tabu search and Simulated Annealing MIP based heuristics.
9. Heuristics for network problems: local search, tabu search, simulated annealing, MIP based heuristics.

Books
L.A. Wolsey, Integer Programming. Wiley. 1998.
 Link http://www.di.univaq.it/rossi/networkdesign.html
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Numerical methods for linear algebra and optimisation (6 credits)
 ECTS credits 6
 Semester 2
 University University of L'Aquila
 Lecturer 1 TBD
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Year2 L'Aquila
Interdisciplinary Mathematics
Year 2 in L'Aquila  Interdisciplinary Mathematics
 2 Year
 Interdisciplinary Mathematics Pathway
 University of L'Aquila Place
 60* ECTS Credits
 Read here Qualification
 Not available if you spent your Year 1 in L'Aquila Note
List of course units
*Students are required to earn 60 ECTS credits, at least, during their second year by successfully attending the following compulsory course units (Semester 1 and 2 amounting to 48 ECTS credits) and picking other 12 ECTS credits (minimum) from the elective ones listed below.
Semester 1

Advanced Analysis 1 (6 credits)
 ECTS credits 6
 Semester 1
 University University of L'Aquila
 Lecturer 1 Corrado Lattanzio

Prerequisites
Basic notions of functional analysis, functions of complex values, standard properties of classical solutions of semilinear first order equations, heat equation, wave equation, Laplace and Poisson's equations.

Objectives
Knowledge of mathematical methods that are widely used by researchers in the area of Applied Mathematics, as Sobolev Spaces, distributions. Application of this knowledge to a variety of topics, including the basic equations of mathematical physics and some current research topics about linear and nonlinear partial differential equations.

Topics
Distributions. Locally integrable functions. The space of test function D(U). Distributions. Distributions associated to Locally integrable functions. Singular distributions. Examples. Operations on distributions: sum, products times functions, change of variables, restrictions, tensor product. Differentiation and his properties; comparison with classical derivatives. Differentiation of jump functions. Partition of unity. Support of a distribution; compactly supported distributions.
Convolution. Convolution in Lp spaces. Regularity of the convolution. Regularizing sequences and smoothing by means of convolutions. Convolution between distributions and regularization of distributions. Denseness of D(U) in D'(U).
Sobolev spaces. Definition of weak derivatives and his motivation. Sobolev spaces Wk,p(U) and their properties. Interior and global approximation by smooth functions. Extensions. Traces. Embeddings theorems: GagliardoNirenbergSobolev inequality and Embedding theorem for p < n. Embedding theorem for p = n. Hölder spaces. Morrey inequality. Embedding theorem for p > n. Sobolev inequalities in the general case. Compact embeddings: RellichKondrachov theorem, Poincaré inequalities. Characterization of the dual space H1.
Second order parabolic equations. Definition of parabolici operator. Weak solutions for linear parabolici equations. existence of weak solutions: Galerkin approximation, construction of approximating solutions, energy estimates, existence and uniqueness of solutions. Existence of solutions of viscous scalar conservation laws.
First order nonlinear hyperbolic equations. Scalar conservation laws: derivation, examples. Weak solutions, RankineHugoniot conditions, entropy conditions. L1 stability, uniqueness and comparison for weak entropy solutions. Convergence of the vanishing viscosity and existence of the weak, entropy solution. Riemann problem. Definition of hyperbolic system. Quasilinear hyperbolic systems, symmetric and symmetrizable systems. Existence of solutions: approximations, a priori estimate, local existence of classical solutions.

Books
V.S. Vladimirov, Equations of Mathematical Physics. Marcel Dekker, Inc..
C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics. Springer.
L.C. Evans, Partial Differential Equations. AMS.
M.E. Taylor, Partial Differential Equations, Nonlinear equations. Springer.
H. Brezis, Sobolev Spaces and Partial Differential Equations. Springer.
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Mathematical fluid dynamics (6 credits)
 ECTS credits 6
 Semester 1
 University University of L'Aquila
 Lecturer 1 Donatella Donatelli

Prerequisites
Basic notions of functional analysis, functions of complex values, standard properties of the heat equation, wave equation, Laplace and Poisson's equations.

Objectives
This course is designed to give an overview of fluid dynamics from a mathematical viewpoint and to introduce students to the mathematical modeling of fluid dynamic type. At the end of the course students will be able to perform a qualitative and quantitative analysis of solutions for particular fluid dynamics problems and to use concepts and mathematical techniques learned from this course for analysis of other partial differential equations.

Topics
Derivation of the governing equations: Euler and NavierStokes.
Eulerian and Lagrangian description of fluid motion; examples of fluid flows.
Vorticity equation in 2D and 3D.
Dimensional analysis: Reynolds number, Mach Number, Frohde number.
From compressible to incompressible models.
Fluid dynamic modeling in various fields: biofluids, atmosphere and ocean, astrophysics.
Existence of solutions for viscid and inviscid fluids.
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High performance computing laboratory and applications to differential equations (6 credits)
 ECTS credits 6
 Semester 1
 University University of L'Aquila
 Lecturer 1 Nicola Guglielmi

Topics
Linux/Unix OS and tools;
Basic Fortran (or C);
HPC architecture and libraries;
Application (ex ODEs, PDEs, elastodynamics).
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Semester 2

Advanced Analysis 2 (6 credits)
 ECTS credits 6
 Semester 2
 University University of L'Aquila
 Lecturer 1 Margherita Nolasco

Prerequisites
A good knowledge of the basic arguments of a course of Functional Analysis, in particular, a good knowledge of the theory of Lebesgue's integral and the L^p spaces.
The first module of the course, in particular a good knowledge of the theory of distributions and Sobolev spaces.

Objectives
Aim of the course is the knowledge of advanced techniques of mathematical analysis and in particular the basic techniques of the modern theory of the partial differential equations.

Topics
Abstract Measure theory.
AC and BV functions.
Fourier transforms.
Second order elliptic equations.
Variational methods.
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Master's thesis (UAQ) (24 credits)
 ECTS credits 24
 Semester 2
 University University of L'Aquila

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 Students work on their Master's Thesis over the 4th semester following their agreement with their thesis advisor.
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Electives

Combinatorics and cryptography (6 credits)
 ECTS credits 6
 Semester 2
 University University of L'Aquila
 Lecturer 1 Norberto Gavioli

Objectives
The student will be requested to have a good preparation on the presented topics, and to be able to implement some of the algorithms in a programming language

Topics
Abstract: Basic cryptograpy and coding theory will be developed. Some protocols and algorithms will be discussed focusing an security and data integrity.
Programme: Elementary arithmetics: Integers, divisibility, prime numbers, Euclidean division and g.c.d., Congruence classes, Chinese remainder theorem, cyclic and abelian groups, Lagrange theorem, Euler theorem, the structure of invertible classes mod p^n, Fields with p elements, polynomials, Euclidean division and g.c.d., Congruence classes of polynomials, Finite fields, primitive elements and polynomials, Legendre/Jacoby symbols and quadratic reciprocity. Cryptography: Classical cryptosystems: Shift cyphers, Vigenère Chipher, Substitution Chiper, One time pads, LFSR Data Encryption Standard: Simplified DES and differential cryptanalysis, Attacks, password encryption RSA: the algorithm, Attacks, Primality testing, the public key concept. Discrete logarithms: Bit commitment, DiffieHelman Key exchange, ELGAMAL Hash function: SHA, birthday attacks Digital signatures: RSA signatures, Hashing and signing, DSA Error correcting codes: Binary block codes, distance and correction of errors, classical bounds, linear codes, cyclic codes, Hamming codes, BCH and ReedSolomon codes.

Books
[1] Wade Trappe, Lawrence C. Washington, Introduction to cryptography: with coding theory 2nd ed.. Pearson Prentice Hall. 2006.
[2] https://www.disim.univaq.it/didattica/content.php?corso=424&pid=88&did=0&lid=en
 Link https://www.disim.univaq.it/didattica/content
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Foundations of advanced geometry (6 credits)
 ECTS credits 6
 Semester 2
 University University of L'Aquila
 Lecturer 1 Anna Tozzi

Objectives
The goal of this course is to provide the motivations, definitions and techniques for the translation of topological problems into algebraic ones, hopefully easier to deal with. On successful completion of this module, the student should understand the fundamental concepts of algebraic geometry and should be aware of potential applications of algebraic topological invariants in other fields as theoretical physics , including the computational fluid mechanics and electrodynamics.

Topics
General topology: topological spaces and continuous maps, induced, quotient and product topology, metric spaces, Hausdorff spaces, compact spaces, connected spaces, paths and path connected spaces
Manifolds and surfaces: the pancake problems, ndimensional manifolds, surfaces and classification of surfaces.
Homotopy: Retracts and contractible spaces, paths and multiplication, the fundamental group, the fundamental group of the circle.
Covering spaces: the fundamental group of a covering space, the fundamental group of a orbit space, lifting theory and existence theorems, the BorsukUlam theorem, the SeifertVan Kampen theorem, the fundamental group of a surface.
Introduction to singular homology: standard and simplicial simplexes.

Books
Czes Kosniowski, A first course in algebraic topology. Cambridge University Press. 1980.
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Mathematical models for collective behaviour (6 credits)
 ECTS credits 6
 Semester 1
 University University of L'Aquila
 Lecturer 1 Debora Amadori

Objectives
Aim of the course is to present some mathematical models currently used in the analysis of collective phenomena, such as vehicular and pedestrian traffic, and flocking phenomena. Emphasis will be given to the mathematical treatment of specific problems coming from real world applications.

Topics
Macroscopic traffic models. LWR model, its derivation. Fundamental diagrams. The Riemann problem, examples. Second order models for traffic flow: PayneWhitham model, description, drawbacks; AwRascle model, shocks description, domains of invariance, instabilities near vacuum.
Theory: systems of conservation laws, strict hyperbolicity, RankineHugoniot conditions; Lax admissibility condition. The Riemann problem for systems: the linear case; GNL and LD fields; rarefactions and contact discontinuities. BV functions, examples and properties. A compactness theorem.
Wave front tracking algorithm: approximate rarefactions, possible types of interactions. Bounds on number of waves and on total variation. Compactness of approximate solutions. The initialboundary value problem on the half line: boundary Riemann problem, interactions with the boundary, control of the total variation by means of a Lyapunovtype functional. The Toll gate problem.
Networks, basic definitions, conservation of the flux. Examples. Distributions along the roads, maximization of the flux. Riemann problem on a junction composed by 2 incoming roads and 2 outgoing roads. The case of 2 incoming roads and 1 outgoing road: the "right of way" rule. Junction between one incoming and one outgoing road, different fluxes.
Pedestrian flow: normal and panic situation. Macroscopic models for evacuation, conservation of "mass", eikonal equation. The Hughes model for pedestrian flow. The eikonal equation: non uniqueness, viscosity solutions, relation with vanishing viscosity approximation. The Hughes model in one space dimension. Curve of turning points, RankineHugoniot conditions. The case of constant initial density and of symmetric initial data; conservation of the left and right mass; an example with mass exchange across the turning point. Macroscopic models for pedestrian flow that include: knowledge of a preferred path, discomfort from walking along walls, tendency of avoiding high densities of pedestrian in a neighborhood (nonlocal term of convolution type), angle of vision, obstacle in the domain. Linearized stability around a smooth solution.
Introduction to the theory of flocking. Examples: Krause model for opinion dynamics, CuckerSmale model, model for attractionrepulsion phenomena. The CuckerSmale flocking model: basic properties, estimates on the kinetic energy. A "flocking theorem": proof by bootstrapping method (Ha and Tadmor). Some drawbacks of the model. Introduction to the kinetic limit for flocking: the Nparticle distribution function, Liouville equation, marginal distribution, continuity equation. The formal meanfield limit: a Vlasovtype equation.

Books
M.D. Rosini, Macroscopic models for vehicular flows and crowd dynamics: theory and applications. Springer. 2013. http://link.springer.com/book/10.1007/9783319001555/page/1
M. Garavello, B. Piccoli, Traffic flow on networks. Conservation laws models. AIMS Series on Applied Mathematics. 2006. http://www.aimsciences.org/books/am/AMVol1.html
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Biomathematics (6 credits)
 ECTS credits 6
 Semester 1
 University University of L'Aquila
 Lecturer 1 Marco Di Francesco
 Lecturer 2 Cristina Pignotti

Prerequisites
Basic calculus and analysis (differential and integral calculus with functions of many variables).
Ordinary differential equations.
Basics in finite dimensional dynamical systems.
Elementary methods for the solution of linear partial differential equations (separation of the variables).

Objectives
1) To learn the basics in the mathematical modelling of population dynamics.
2) To provide a mathematical description of ODE models in population dynamics and the intepretation of the qualitative behaviour of the solutions.
3) To get the basic notions in mathematical models in epidemiology and reaction kinetics.
4) To learn the mathematical modelling of population models in heterogeneous environment, described by partial differential equations.
5) To deal with advanced models in biology such as chemotaxis models and structured dynamics equations.
6) To get a sound background in reaction diffusion phenomena, Turing instability, and pattern formation.

Topics
Continuous Population Models for Single Species. Continuous Growth Models. Delay models. Linear Analysis of Delay Population Models: Periodic Solutions.
Continuous models for Interacting Populations. PredatorPrey Models: LotkaVolterra Systems. Realistic Predator–Prey Models. Competition Models: Principle of Competitive Exclusion. Mutualism or Symbiosis.
Reaction kinetics. Enzyme Kinetics: Basic Enzyme Reaction. Transient Time Estimates and Nondimensionalisation. MichaelisMenten QuasiSteady State Analysis.
Dynamics of Infectious Diseases: Epidemic Models and AIDS. Simple Epidemic Models (SIR, SI) and Practical Applications. Modelling Venereal Diseases. AIDS: Modelling the Transmission Dynamics of the Human Immunodeficiency Virus (HIV).
Timespace dependent models: PDEs in biology. Diffusion equations. Diffusion and Random walk. The gaussian distribution. Smoothing and decay properties of the diffusion operator. Nonlinear diffusion.
Reaction–diffusion models for one single species. Diffusive Malthus equation and critical patch size. Travelling waves. Fisher–Kolmogoroff equation.
Reaction–diffusion systems. Multi species waves in PredatorPrey Systems. Turing instability and spatial patterns.
Chemotaxis modelling. Diffusion vs. Chemotaxis: stability vs. instability. Diffusion vs. Chemotaxis: stability and blow–up. Chemotaxis with nonlinear diffusion. Models with maximal density.
Nonlocal interaction models in biology. Mathematical models of swarms. Approximation with interacting particle systems. Asymptotic behaviour.
Structured population dynamics. An example in ecology: competition for resources. Continuous traits. Evolutionary stable strategy in a continuous model.

Books
James D. Murray, Mathematical Biology I: an introduction. Springer.
James D. Murray, Mathematical Biology II: Spatial models and biomedical applications . Springer.
Benoit Perthame, Transport equations in biology. Birkaeuser.
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Stochastic processes (6 credits)
 ECTS credits 6
 Semester 2
 University University of L'Aquila
 Lecturer 1 Ida Germana Minelli

Objectives
The course aims to give an introduction to the theory of stochastic processes with special emphasis on applications and examples. On successful completion of this module the students should become familiar with some of the most known classes of stochastic processes (such as martingales, markov processes, diffusion processes) and to acquire both the mathematical tools and intuition for being able to describe systems with randomness evolving in time in terms of a probability model and to analyze it charcterizing some of its properties.

Topics
Stochastic Processes: definition, finite dimensional distributions, stationarity, sample path spaces, construction, examples.
Filtrations, stopping times, conditional expectation.
Markov processes: definition, main properties and examples. Birth and death processes.
Poisson process with applications on queueing models.
Martingales: definition, main properties and examples. Branching processes.
Brownian motion: definition, construction and main properties.
Brownian Bridge, Geometric Brownian Motion, OrnsteinUhlenbeck process.
Ito integral and stochastic differential equations. Applications and examples.

Books
P. Billingsley, Probability and measure. John Wiley and Sons.
G. Grimmett, D. Stirzaker, Probability and random Processes. Oxford University Press.
B. Oksendal, Stochastic Differential Equations. SpringerVerlag.
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Kinetic and hydrodynamic models (6 credits)
 ECTS credits 6
 Semester 1
 University University of L'Aquila
 Lecturer 1 https://www.intermaths.eu/my/userprofile/

Prerequisites
Mathematical Analysis, Fourier transform.

Objectives
This course provides an introduction to the classical kinetic theory of gases and the principles of kinetic modeling.
A special focus is given to the derivation of hydrodynamic equations from kinetic models by means of nonperturbative techniques and to the analysis of numerical schemes for the simulation of fluid flows.
On successful completion of this module the student has the knowledge on the basic principles and the simulation strategies of kinetic models.

Topics
Boltzmann equation and the principles of kinetic description.
Kinetic models: BGK,Maxwell molecules, Vlasov equation and FokkerPlanck equation.
The closure problem and methods of reduced description: ChapmanEnskog expansion, Grad's Moment method.
Nonperturbative techniques in kinetic theory: the method of the slow invariant manifold.
Overview on Lattice Boltzmann models.
Monte Carlo simulations of lattice gas models.
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Time series and prediction (6 credits)
 ECTS credits 6
 Semester 1
 University University of L'Aquila
 Lecturer 1 Umberto Triacca

Objectives
The course is an introduction to Time Series Analysis and Forecasting. The level is the firstyear graduate in Mathematics with a prerequisite knowledge of basic inferential statistical methods.
The aim of the course is to present important concepts of time series analysis (stationarity of stochastic processes, ARIMA models, forecasting etc.). At the end of the course, the student should be able to select an appropriate ARIMA model for a given time series.

Topics
Stochastic processes (some basic concepts)
Stationary stochastic processes
Autocovariance and autocorrelation functions
Ergodicity of a stationary stochastic process
Estimation of moment functions of a stationary process
ARIMA models
Estimatiom of ARIMA models
Building ARIMA models
Forecasting from ARIMA models

Books
[1]Time Series Analysis Univariate and Multivariate Methods, 2nd Edition, W. W. Wei, 2006, Addison Wesley.
[2] Time Series Analysis, J. Hamilton, 1994, Princeton University Press.
[3] Time Series Analysis and Its Applications with R Examples, Shumway, R. and Stoffer, D., 2006, Springer.
[4]Introduction to Time Series and Forecasting. Second Edition, P. Brockwell and R. Davis, 2002, Springer.
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Mathematical economics and finance (6 credits)
 ECTS credits 6
 Semester 1
 University University of L'Aquila
 Lecturer 1 Massimiliano Giuli

Prerequisites
I assume familiarity with vector and topological spaces, and with the standard model of the real numbers. I assume that you know the basic facts about metric spaces, normed and seminormerd spaces, Banach and Hilbert spaces.

Objectives
On successful completion of this course, the student should:
 Know the fundamental fixed point theorems for setvalued maps and the basic existence results for equilibrium problems and variational inequalities.
 Explain some interconnections among these various results.
 Apply this analysis to game and economic theory

Topics
Sperner’s lemma
The KnasterKuratowskiMazurkiewicz lemma
Brouwer's fixed point theorem
Variational inequalities and equilibrium problems
Generalized monotonicity and convexity
BrézisNirenbergStampacchia theorem and Fan's minimax principle
Continuity of correspondences
Browder, Kakutani and FanGlicksberg fixed point theorems
GaleNikaidoDebreu theorem
Nash equilibrium of games and abstract economies
Walrasian equilibrium of an economy
An application to traffic network
 Link https://www.disim.univaq.it/didattica/content
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Scientific Computing
Year 2 in L'Aquila  Scientific Computing
 2 Year
 Scientific Computing Pathway
 University of L'Aquila Place
 60 ECTS Credits
 Read here Qualification
 Not available if you spent your Year 1 in L'Aquila Note
List of course units
Semester 1

Advanced Analysis 1 (6 credits)
 ECTS credits 6
 Semester 1
 University University of L'Aquila
 Lecturer 1 Corrado Lattanzio

Prerequisites
Basic notions of functional analysis, functions of complex values, standard properties of classical solutions of semilinear first order equations, heat equation, wave equation, Laplace and Poisson's equations.

Objectives
Knowledge of mathematical methods that are widely used by researchers in the area of Applied Mathematics, as Sobolev Spaces, distributions. Application of this knowledge to a variety of topics, including the basic equations of mathematical physics and some current research topics about linear and nonlinear partial differential equations.

Topics
Distributions. Locally integrable functions. The space of test function D(U). Distributions. Distributions associated to Locally integrable functions. Singular distributions. Examples. Operations on distributions: sum, products times functions, change of variables, restrictions, tensor product. Differentiation and his properties; comparison with classical derivatives. Differentiation of jump functions. Partition of unity. Support of a distribution; compactly supported distributions.
Convolution. Convolution in Lp spaces. Regularity of the convolution. Regularizing sequences and smoothing by means of convolutions. Convolution between distributions and regularization of distributions. Denseness of D(U) in D'(U).
Sobolev spaces. Definition of weak derivatives and his motivation. Sobolev spaces Wk,p(U) and their properties. Interior and global approximation by smooth functions. Extensions. Traces. Embeddings theorems: GagliardoNirenbergSobolev inequality and Embedding theorem for p < n. Embedding theorem for p = n. Hölder spaces. Morrey inequality. Embedding theorem for p > n. Sobolev inequalities in the general case. Compact embeddings: RellichKondrachov theorem, Poincaré inequalities. Characterization of the dual space H1.
Second order parabolic equations. Definition of parabolici operator. Weak solutions for linear parabolici equations. existence of weak solutions: Galerkin approximation, construction of approximating solutions, energy estimates, existence and uniqueness of solutions. Existence of solutions of viscous scalar conservation laws.
First order nonlinear hyperbolic equations. Scalar conservation laws: derivation, examples. Weak solutions, RankineHugoniot conditions, entropy conditions. L1 stability, uniqueness and comparison for weak entropy solutions. Convergence of the vanishing viscosity and existence of the weak, entropy solution. Riemann problem. Definition of hyperbolic system. Quasilinear hyperbolic systems, symmetric and symmetrizable systems. Existence of solutions: approximations, a priori estimate, local existence of classical solutions.

Books
V.S. Vladimirov, Equations of Mathematical Physics. Marcel Dekker, Inc..
C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics. Springer.
L.C. Evans, Partial Differential Equations. AMS.
M.E. Taylor, Partial Differential Equations, Nonlinear equations. Springer.
H. Brezis, Sobolev Spaces and Partial Differential Equations. Springer.
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Mathematical fluid dynamics (6 credits)
 ECTS credits 6
 Semester 1
 University University of L'Aquila
 Lecturer 1 Donatella Donatelli

Prerequisites
Basic notions of functional analysis, functions of complex values, standard properties of the heat equation, wave equation, Laplace and Poisson's equations.

Objectives
This course is designed to give an overview of fluid dynamics from a mathematical viewpoint and to introduce students to the mathematical modeling of fluid dynamic type. At the end of the course students will be able to perform a qualitative and quantitative analysis of solutions for particular fluid dynamics problems and to use concepts and mathematical techniques learned from this course for analysis of other partial differential equations.

Topics
Derivation of the governing equations: Euler and NavierStokes.
Eulerian and Lagrangian description of fluid motion; examples of fluid flows.
Vorticity equation in 2D and 3D.
Dimensional analysis: Reynolds number, Mach Number, Frohde number.
From compressible to incompressible models.
Fluid dynamic modeling in various fields: biofluids, atmosphere and ocean, astrophysics.
Existence of solutions for viscid and inviscid fluids.
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High performance computing laboratory and applications to differential equations (6 credits)
 ECTS credits 6
 Semester 1
 University University of L'Aquila
 Lecturer 1 Nicola Guglielmi

Topics
Linux/Unix OS and tools;
Basic Fortran (or C);
HPC architecture and libraries;
Application (ex ODEs, PDEs, elastodynamics).
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Machine learning (6 credits)
 ECTS credits 6
 Semester 1
 University University of L'Aquila
 Lecturer 1 Pasquale Caianiello
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Semester 2

Kinetic and hydrodynamic models (6 credits)
 ECTS credits 6
 Semester 1
 University University of L'Aquila
 Lecturer 1 https://www.intermaths.eu/my/userprofile/

Prerequisites
Mathematical Analysis, Fourier transform.

Objectives
This course provides an introduction to the classical kinetic theory of gases and the principles of kinetic modeling.
A special focus is given to the derivation of hydrodynamic equations from kinetic models by means of nonperturbative techniques and to the analysis of numerical schemes for the simulation of fluid flows.
On successful completion of this module the student has the knowledge on the basic principles and the simulation strategies of kinetic models.

Topics
Boltzmann equation and the principles of kinetic description.
Kinetic models: BGK,Maxwell molecules, Vlasov equation and FokkerPlanck equation.
The closure problem and methods of reduced description: ChapmanEnskog expansion, Grad's Moment method.
Nonperturbative techniques in kinetic theory: the method of the slow invariant manifold.
Overview on Lattice Boltzmann models.
Monte Carlo simulations of lattice gas models.
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Master's thesis (UAQ) (30 credits)
 ECTS credits 30
 Semester 2
 University University of L'Aquila

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 Students work on their Master's Thesis over the 4th semester following their agreement with their thesis advisor.
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Year2 Brno
Year 2 in Brno  Mathematical Engineering
 2 Year
 Mathematical Engineering Pathway
 Brno University of Technology Place
 60 ECTS Credits
 Read here Qualification
List of course units
Semester 1

Multivalued logic applications (4 credits)
 ECTS credits 4
 Semester 1
 University Brno University of Technology
 Lecturer 1 Miloslav Druckmüller

Prerequisites
Mathematical logic, fuzzy set theory.

Objectives
The aim of the course is to provide students with information about the use of Multivalued logic in technical applications.

Topics
1. Multivalued logic, formulae.
2. Tnorms, Tconorms, generalized implications.
3. Linguistic variables and linguistic models.
4. Knowledge bases of expert systems.
56. 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 multivalued logic and fuzzy sets theory
12. Expert systems
13. Overview of AI methods

Books
Jackson P.: Introduction to Expert Systems, AddisonWesley 1999

More information
The course is intended especially for students of mathematical engineering. It includes the theory of multivalued 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.
 Link http://www.fme.vutbr.cz/studium/predmety/predmet.html?pid=158628
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Financial Mathematics (4 credits)
 ECTS credits 4
 Semester 1
 University Brno University of Technology
 Lecturer 1 Pavel Popela

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.
 Link http://www.fme.vutbr.cz/studium/predmety/predmet.html?pid=158642
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Fuzzy Sets and Applications (4 credits)
 ECTS credits 4
 Semester 1
 University Brno University of Technology
 Lecturer 1 Zdeněk Karpíšek

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 conorms, 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 decisionmaking (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: KluwerNijhoff 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.
 Link http://www.fme.vutbr.cz/studium/predmety/predmet.html?pid=158643
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Mathematical Methods in Fluid Dynamics (4 credits)
 ECTS credits 4
 Semester 1
 University Brno University of Technology
 Lecturer 1 Libor Čermák

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 pressurecorrection 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, NavierStokes 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 StegerWarming, 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, nonorthogonal and unstructured meshes.
11. Stokes problem, spectral element method.
12. Steady NavierStokes problem, spectral element method.
13. Unsteady NavierStokes 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, McGrawHill, New York, 1980.
J.H. Ferziger, M. Peric: Computational Methods for Fluid Dynamics, SpringerVerlag, New York, 2002.
M.O. Deville, P.F. Fischer, E.H. Mund: HighOrder Methods for Incompressible Fluid Flow. Cambridge University Press, Cambdrige, 2002.
A. Quarteroni, A. Valli: Numerical Approximatipon of Partial Differential Equations. SpringerVerlag, 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: NavierStokes equations, pressurecorrection method, spectral element method.
 Link http://www.fme.vutbr.cz/studium/predmety/predmet.html?pid=158649
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Fundamentals of Optimal Control Theory (4 credits)
 ECTS credits 4
 Semester 1
 University Brno University of Technology
 Lecturer 1 Jan Cermak

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. Timeoptimal control of an uniform motion.
4. Timeoptimal 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. Energyoptimal 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. Timeoptimal control of linear problems. Problems with state constraints. Singular control. Applications.
 Link https://www.fme.vutbr.cz/studium
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Reliability and Quality (4 credits)
 ECTS credits 4
 Semester 1
 University Brno University of Technology
 Lecturer 1 Josef Bednář

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 noncensored 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 0471316482
Ireson, Grant W. Handbook of Reliability Engineering and Management.Hong Kong :McGrawHill,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.
 Link http://www.fme.vutbr.cz/studium/predmety/predmet.html?pid=158662
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Semester 2

Analysis of Engineering Experiment (4 credits)
 ECTS credits 4
 Semester 2
 University Brno University of Technology
 Lecturer 1 Zdeněk Karpíšek

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.Oneway analysis of variance.
2.Twoway 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.
 Link http://www.fme.vutbr.cz/studium/predmety/predmet.html?pid=158675
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Modern methods of solving differential equations (5 credits)
 ECTS credits 5
 Semester 2
 University Brno University of Technology
 Lecturer 1 Jan Franců

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. LaxMildgam 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.
 Link http://www.fme.vutbr.cz/studium/predmety/predmet.html?pid=158636
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Data Visualisation (4 credits)
 ECTS credits 4
 Semester 2
 University Brno University of Technology
 Lecturer 1 Dalibor Martišek

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 2D and 3D 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 isosurfaces. Algorithms, which construct surfaces – marching cubes and volume algorithms  ray casting, ray tracing.
 Link http://www.fme.vutbr.cz/studium/predmety/predmet.html?pid=158668
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Geometric Algorithms and Cryptography (4 credits)
 ECTS credits 4
 Semester 2
 University Brno University of Technology
 Lecturer 1 https://www.intermaths.eu/my/userprofile/
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Mathematical Structures (4 credits)
 ECTS credits 4
 Semester 2
 University Brno University of Technology
 Lecturer 1 Josef Šlapal

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.
 Link https://www.fme.vutbr.cz/studium
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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 thirtyminute 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 thirtyminute 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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Year2 Gdansk
Year 2 in Gdansk  Mathematics for new materials design
 2 Year
 Mathematics for new materials design Pathway
 Gdansk University of Technology Place
 60 ECTS Credits
 Read here Qualification
List of course units
Semester 1

Quantum simulations with particles (6 credits)
 ECTS credits 6
 Semester 1
 University Gdansk University of Technology
 Lecturer 1 Maciej Bobrowski

Prerequisites
Base knowledge on physics, mathematics and chemistry.

Objectives
Purposes:
 Pass the knowledge on application of quantum methods for issues of change of electronic structure present in molecules and crystals.
 Teaching axioms of quantum mechanics and their applications.
 Teaching of commonly utilized quantum methods based on wave functions and electron densities: HF, CI, MCSCF, CC, MPn, DFT.
 Teaching of utilization of commonly applied basis sets in quantum calculations

Topics
Application of quantum methods in cases of solving of electronicstructure change for systems of molecules and crystals, axioms of quantum mechanics and their applications, commonly utilized quantum methods based on wave functions and electron densities: HF, CI, MCSCF, MPn, CC, DFT, basis sets.

Books
1. Frank Jensen, Introduction to Computational Chemistry, Wydawnictwo Wiley, 2007,
2. YungKuo Lim, Problems and Solutions on Quantum Mechanics, Wydawnictwo World Scientific, 2005,
3. C. J. Ballhausen, H. B. Gray, Molecular Orbital Theory, Wydawnictwo W. A. Benjamin Inc. 1964,
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Computer modelling and design of materials (5 credits)
 ECTS credits 5
 Semester 1
 University Gdansk University of Technology
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Mechanics of composites and metamaterials (6 credits)
 ECTS credits 6
 Semester 1
 University Gdansk University of Technology
 Lecturer 1 https://www.intermaths.eu/my/userprofile/
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Introduction to low dimensional systems and nanotechnology (6 credits)
 ECTS credits 6
 Semester 1
 University Gdansk University of Technology

Objectives
Gaining knowledge on fundamentals of nanotechnology and lowdimensional systems and experimental methods

Topics
1. Fundamentals of Nanotechnology (M. Gazda)
a. Synthesis of nanomaterials: examples of processes topdown and bottomup;
b. Imaging nanomaterials: e.g. scanning probe microscopy, electron microscopy;
c. Examples of nanomaterials: e.g. graphene, Ag and Au nanoparticles, quantum dots;
d. Surface and its importance
e. Structure, structural phase transition and morphology of nanomaterials;
f. Selected mechanical and thermal properties of nanomaterials and nanostructured materials;
g. Selected electronic, optical and magnetic properties of nanomaterials;
h. Examples of applications of nanomaterials . 
Books
Nanoscopic Materials
Sizedependent PhenomenaEmil Roduner
Institute of Physical Chemistry, University of Stuttgart,
Stuttgart, Germany The Royal Society of Chemistry 2006
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Classical simulations with particles (6 credits)
 ECTS credits 6
 Semester 1
 University Gdansk University of Technology
 Lecturer 1 Jacek Dziedzic

Prerequisites
Basic concepts of classical physics  force, acceleration, potential. Basic knowledge of calculus (Riemann's
integral, minimisation of a function, partial and total derivatives). 
Objectives
We introduce the basics of physics of materials, with particular attention to the relationships between atomic
structure and macroscopic physical properties. Classical computational particle methods are covered, mainly
the molecular dynamics (MD) approach  its basic theory (integration of e.o.m.) and practicalities (potentials,
boundary conditions, initialisation, neighbourhood, cutoff radius) followed by a brief tour of more advanced
concepts of MD (rigid molecules, shell model, constrained dynamics, thermostats, barostats, Ewald method). 
Topics
Section devoted to physics of materials:
Crystalline and glassy materials (shortrange, mediumrange and longrange order, radial and angular
distribution functions); thermodynamics of phase transitions; glass transition; gels (classification and
applications); quasicrystals; liquid crystals; auxetics.
Basic concepts of crystallography (Bravais lattice, primitive and elementary cell, simple and complex
lattice, Miller indices, etc.); symmetry operations; crystallographic point groups and space groups; models of
amorphous systems (CRN, RCP, randomcoil); reciprocal lattice and its properties; conditions
for Bragg’s diffraction and Laue diffraction.
Bonding in crystals (ionic, covalent, metallic, molecular and hydrogen); binding energies (lattice sums,
Madelung energy, the Evjen method and Ewald method); fluctuationdissipation effects.
Structural defects: point defects (Schottky, Frenkel, substitutions, vacancies, intercalations); line defects
(screw and edge dislocations, Frank network, mechanisms of dislocation generation, relationship with the
strength of materials), planar defects (lowangle boundaries, stacking faults, twinning).
Defects in the electronic structure (plasmons, excitons, polarons, magnons, Fcenters).
Lattice vibrations (mono and diatomic chain, optical and acoustic branches, dispersion relations); normal
vibrations; models of lattice heat capacity (classical, Einstein, Debye); the most significant
anharmonic effects.
Principles of the Drude model, electrical conductivity of metals, magnetoresistive effect and the Hall
effect.
The Fermi gas of free electrons, the FermiDirac distribution, Fermi level and chemical potential, degenerate
and nondegenerate gas, density of states, WiedemannFranz law.
Thermoemission and cold emission from metal to vacuum; contact voltage.
The model principles of the band theory; Bloch’s theorem; classification of solids on the basis of the band
theory; effective mass and quasimomentum.
Dependence of electrical conductivity on temperature in semiconductors and metals (due to changes
in the carrier densities and in the relaxation time). Deviations from Ohm’s law (collisional ionisation, Zener
effect, PooleFrenkel effect, field dependence of relaxation time).
Section devoted to the molecular dynamics method:
Motivation behind computational approaches to nanotechnology, continuum and particle methods, classical
and quantumbased methods, scaling of computational effort.
The molecular dynamics method, its advantages and limitations. Conservation of energy in Newtonian
mechanics. Phase space and trajectories.
Periodic, open and mixed boundary conditions, minimum image convention, quasiinifinity, limitations of
PBCs. Cutoff radius and its consequences. Hockney’s linked cells and Verlet neighbour list.
Initializing an MD simulation (positions, velocities), skew start, equilibration.
Integration of the equations of motion. Verlet, leapfrog and predictorcorrector methods. Sources of error in
integrating the equations of motion.
Visualization in MD, calculating macroscopic quantities (energy, temperature, virial, pressure, specific heat,
RDF, ADF, S(k), MSD, D(T)).
Potential and its relationship with force. General and particular forms of potentials. Selected potentials: LJ,
BornMayer, harmonic, Morse, StillingerWeber, SuttonChen, GAFF, AMOEBA).Polarizability and shell models (Cochran, Fincham).
Constrained dynamics, formal approach, SHAKE, RATTLE, QSHAKE.
(Optionally): Rigid molecules in MD simulations, Euler angles, rotation matrix, vector transformations,
quaternions.
Coulombic interactions in MD, Ewald method.
NVT and NpT ensembles, primitive thermostats, ESM and CSM thermo and barostats: Andersen,
Berendsen, HooverEvans, NoseHoover, NoseAndersen, ParrinelloRahman.
(Optionally): Hybrid (QM/MM) methods.
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Ethics in nanotechnology (1 credits)
 ECTS credits 1
 Semester 1
 University Gdansk University of Technology
 Lecturer 1 Marek Chmielewski

Prerequisites
Not required

Objectives
The aim of the course is the answer on the question of ethics influence on the accuracy of the science
investigation procedure and presentation in the public results of the research and measurement results. 
Topics
The content of the course is the analysis and verification of existing codes of the ethics in the subjects of the
research and development in science. Understanding and analyzing the ethic code in the field of
nanotechnology. The analysis is also the history and evolution of content included within the applicable
codex. In addition, the lecture will be analyzed as current controversial statements and publications in the
field of science and especially nanotechnology. 
Books
The Ethics of Nanotechnology, Andrew Chen
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Semester 2

Humanities and social science course 1 (2 credits)
 ECTS credits 2
 Semester 2
 University Gdansk University of Technology
 Lecturer 1 https://www.intermaths.eu/my/userprofile/
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Humanities and social science course 2 (1 credits)
 ECTS credits 1
 Semester 2
 University Gdansk University of Technology
 Lecturer 1 TBD
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Methodology of scientific research (1 credits)
 ECTS credits 1
 Semester 2
 University Gdansk University of Technology
 Lecturer 1 Jarosław Rybicki

Objectives
The course aims at presentation of modern methods of thinking applied in science and technology

Topics
1. INTRODUCTION. Ontological, psychological, semiotic, theorycognitive terminology. Formal logic.
Philosophy of logic. Methodology vs. science. Science vs. logic.
2. PHENOMENOLOGICAL METHOD. Objectivity of phenomenologists. Return to "issue in itself", intuitive
cognition.
3. SEMIOTIC METHODS. Sign and its three dimensions. Formalism. Essence of formalism  calculation.
Application of calculation to nonmathematical subjects. Validation of formalism. Eidetic and operational
sense. Models. Artificial language. Syntactic rules of sense. Construction of language. Atomic and molecular
expressions. Notion of syntactic category. Functors and arguments. Examples of syntactic nonsense.
Semantic functions and levels Two semantic functions of sign. Designation and significance. Semantic
levels. Language and metalanguage. Semantic meaning and verifiability. Rule of verifiability. Verification
levels: technical possibility, physical possibility, logical possibility, transempirical possibility. Principle of
intersubjectivity. Verifiability of general clauses.
4. AXIOMATIC METHOD. Structure of indirect cognition. Law and rule. Two basic forms of inference:
deduction and reduction. Reliable and unreliable rules of inference. Concept of axiomatic system. Structure
of axiomatic clause system. Requirements for axiomatic system. Constitutional system. Progressive and
regressive deduction. Mathematical logic. Methodological significance. Implication and derivability. Definition
and creation of concepts. Basic types of definition. Real and nominal definitions. Syntactic and semantic
definitions. Analytical and synthetic definitions. Types of syntactic definitions: clear definitions, contextual
definitions, recursive definitions, definitions by axiomatic system. Semantic deictic definitions. Real
definitions. Application of axiomatic method. Axiomatization of logic of HilbertAckermann clauses.
5. REDUCTION METHODS. Historical introductory remarks. Concept and division of reduction. Concept of
verification and explanation. Regressive reduction. Reduction sciences. Structure of natural sciences.
Observation clauses. Progress in natural sciences. Verification of hypotheses. Experience and thinking.
Types of explanatory sentences. Causal explanation and teleological explanation. Cooccurrence laws and
functional laws. Deterministic laws and statistical laws. Authentic and nonauthentic induction. Division of
induction. Primary and secondary induction. Qualitative and quantitative induction. Deterministic and
statistical induction. Enumerative and eliminatory induction. Postulates of determinism, closed system,
relationship between laws, simplicity. 
Books
J. M. Bocheński, Współczesne metody myślenia, wydawnictwo "Wdrodze", Poznań (1992)
Supplementary Literature
K. Popper, Logika odkrycia naukowego, PWN (1983)
M. Grzegorczyk, Logika matematyczna, PWN (1979)
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Master's thesis (GUT) (26 credits)
 ECTS credits 26
 Semester 2
 University Gdansk 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.
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Year2 Katowice
Mathematical modeling
Year 2 in Katowice  Mathematical modeling
 2 Year
 Mathematical modelling Pathway
 University of Silesia in Katowice Place
 63 ECTS Credits
 Read here Qualification
List of course units
Semester 1

Computational mathematics (3 credits)
 ECTS credits 3
 Semester 1
 University University of Silesia in Katowice
 Lecturer 1 Przemysław Koprowski

Topics
The aim of Computational mathematics course is to teach students how to use computational (both numerical and symbolic) methods in applications coming from various branches of mathematics.
The course covers the following subjects:
1. Polynomial algorithms: squarefree factorization, polynomial factorization over finite fields, factorization of rational polynomials, monomial orders and Groebner bases;
2. Elimination theory: elimination with Groebner bases, classical elimination with resultants;
3. Inifinite summation and Gosper's algorithm;
4. Numerical integration: MonteCarlo algorithm.
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Applied Graph Theory (6 credits)
 ECTS credits 6
 Semester 1
 University University of Silesia in Katowice
 Lecturer 1 Ekaterina Shulman

Topics
The course establishes the fundamental concepts of the graph theory and shows several applications in various topics. In particular, the famous problems of the graph theory will be discussed: Minimum Connector Problem, Hall's Marriage Theorem, the Assignment Problem, the Network Flow Problem, the Committee Scheduling Problem, the Four Color Problem, the Traveling Salesman Problem.

Books
1. Bollobas B., Modern Graph Theory, SpringerVerlag, 2001.
2. Diestel G. T., Graph Theory, SpringerVerlag, 1997, 2000.
3. Foulds L. R., Graph Theory Applications, SpringerVerlag, 1992
4. Hartland G., Zhang P., A First Course in Graph Theory (Dover Books on Mathematics), 2012.
5. Matousek J., Nesetril J., An invitation to discrete mathematics, Oxford, 2008.
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Mathematical methods in physics (6 credits)
 ECTS credits 6
 Semester 1
 University University of Silesia in Katowice
 Lecturer 1 Jerzy Dajka

Prerequisites
Basic linear algebra is enough. A bit of number theory can be useful but not necessary.

Objectives
My aim is to present mathematical methods for quantum information processing. As in most applications it is enough to work with qubits and systems of qubits, mathematical methods originate from linear algebra, which is usually one of first curses taught. It makes quantum information accessible for very 'fresh' students. I would like to convince students that quantum information processing is useful, interesting, counterintuitive, sometimes seemingly as mysterious as the Schroedinger cat.

Topics
Mathematical formalism of quantum mechanics.
Postulates of quantum mechanics.
Quantum information: quantum gates, nogo theorems, measurement.
Quantum entanglement: mathematical basis.
Selected applications: teleportation, dense coding.
Quantum cloning and applications.
Basic protocols for quantum cryptography: BB84, B92.
Quantum nonlocality: Bell and LeggettGarg inequalities, contextuality.
Dynamics of quantum systems, open quantum systems.
Quantum error correction.

Books
Quantum Computation and Quantum Information by Michael A. Nielsen & Isaac L. Chuang
Lecture notes by John Preskill http://www.theory.caltech.edu/people/preskill/ph229/
 Link http://zft.us.edu.pl/dajka
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Statistics (3 credits)
 ECTS credits 3
 Semester 1
 University University of Silesia in Katowice
 Lecturer 1 Agnieszka Kulawik

Topics
The aim of the Statistics unit is to get a deep knowledge on constructing statistical models and making statistical analysis, and to improve the skills of using statistical computer packages.
The contents of this unit are the following:
1. Organising statistical analysis: collecting and data, their analysis and graphical description.
2. Linear and nonlinear statistical models – estimation theory and statistical hypotheses testing.
3. Applications of linear and nonlinear statistical models in econometrics and financial mathematics.
4. Parametric tests of significance involving two or more samples.
5. Conformity tests.
6. Nonparametric tests of significance involving two or more samples.
7. Applications of statistical computer software to estimation and statistical testing
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Wavelet transforms (6 credits)
 ECTS credits 6
 Semester 1
 University University of Silesia in Katowice
 Lecturer 1 Janusz Morawiec

Topics
The main goal of the lecture is to present basic properties of wavelet transforms and some methods of construction of wavelet bases. We will pay special attention to these wavelet transforms which have used to the analysis and the synthesis of sound signals. We also will pay special attention to structures of bases with special properties which have used to the data compression in digital transmissions.

Books
[1] C.K. Chui, An Introduction to Wavelets, Academic Press, Boston, 1992.
[2] I. Daubechies, The wavelet transform, Timefrequency localization and signal analysis, IEEE Trans. Inform. Theory 36 (1990), 9611005.
[3] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philidelphia, 1992.
[4] C. Heil, D. Walnut, Continuous and discrete wavelet transforms, SIAM Review 31 (1989), 628666.
[5] G. Kaiser, A Friendly Guide to Wavelets, Birkhauser, Boston, 1994.
[6] D. Kozlow, Wavelets. A tutorial and a bibliography, Rendiconti dell’Instituto di Matematica dell’Universita di Trieste, 26, supplemento (1994).
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Workshop on Problem solving (2 credits)
 ECTS credits 2
 Semester 1
 University University of Silesia in Katowice
 Lecturer 1 Radosław Czaja
 Lecturer 2 Anna Brzeska
 Topics The main aim of the module Problem Workshops is to acquaint students with chosen branches of mathematics with applications to knowledge domains such as: economics, biology, physics, chemistry, and computer science. Additional aims are: training analytical skills (for example, constructing mathematical models of chosen problems from applied sciences), training methodological skills (for example, use of available technology to prepare a project or analysis), training cognitive skills (for example, an analysis of data or source content given in a form of articles or manuals, also in a foreign language) and training skills of teamwork (for example, work in small groups during and outside the workshop).
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Polish language and culture for foreigners (level A1) (3 credits)
 ECTS credits 3
 Semester 1
 University University of Silesia in Katowice
 Lecturer 1 Aleksandra Achtelik
 Lecturer 2 Małgorzata Nieużyła
 Topics The aim of the module is to develope all language skills (listening, reading, speaking and writing) and to prepare students for quite easy communication in Polish, necessary while studying in Poland. Students acquire not only linguistic and communicative competence, but also sociocultural: they get to know selected aspects of Polish culture, basic habits and holidays celebrated in Poland, taking into account the pragmatic and sociolinguistic efficiency. Programme includes basic communication situations: greetings and farewells, shopping, ordering food, traveling, etc.
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Semester 2

Applications of the theory of functional equations (6 credits)
 ECTS credits 6
 Semester 2
 University University of Silesia in Katowice
 Lecturer 1 Roman Ger

Topics
Applications in Geometry:
1. Joint characterization of Euclidean, hyperbolic and elliptic geometries.
2. Characterizations of the cross ratio.
3. A description of certain subsemigroups of some Lie groups.
Applications in Functional Analysis:
1. Analytic form of linearmultiplicative functionals in the Banach algebra of integrable functions on the real line.
2. A characterization of strictly convex spaces.
3. Some new characterizations of inner product spaces.
4. BirkhoffJames orthogonality.
5. Addition theorems in Banach algebras; operator semigroups.

Books
1. J. Aczel & J. Dhombres, Functional equations in several variables, Cambridge University Press, Cambridge, 1989. 2. J. Aczel & S. Gołąb, Funktionalgleichungen der Theorie der Geometrischen Objekte, PWN, Warszawa, 1960. 3. J. Dhombres, Some aspects of functional equations, Chulalongkorn Univ., Bangkok, 1979. 4. D. Ilse, I. Lehman and W. Schulz, Gruppoide und Funktionalgleichungen, VEB Deutscher Verlag der Wissenschaften, Berlin, 1984. 5. M. Kuczma, An introduction to the theory of functional equations and inequalities, Polish Scientific Publishers & Silesian University, WarszawaKrakówKatowice, 1985.
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Dynamical systems on measures  physical and biological models (6 credits)
 ECTS credits 6
 Semester 2
 University University of Silesia in Katowice
 Lecturer 1 Henryk Gacki
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Collective project (4 credits)
 ECTS credits 4
 Semester 2
 University University of Silesia in Katowice
 Lecturer 1 Radosław Wieczorek

Objectives
In this module the students, divided into teams consisting of several people, implement projects associated with the given problem.

Topics
The project consists of several phases:
1. Planning for the project. The allocation of roles and responsibilities in the team.
2. Review of available literature on the given matter.
3. Analysis of the problem, seeking methods of its solution.
4. Implementation of the solution. This phase, depending on the project, should include elements such as the analysis of empirical data, calibration, simulation and testing of the solution.
5. Preparation of the final report and presentation of results. Both the final effect and the individual phases of the project are assessed. Laboratory classes serve to current reporting and discussing work progress, and give the opportunity of obtaining assistance in the project implementation.
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Master's Thesis at US

Master’s Thesis (US) (18 credits)
 ECTS credits 18
 Semester 2
 University University of Silesia in Katowice

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 University of Silesia in Katowice also following two local courses: Seminar 1 (1st semester, 4 credits) and Seminar 2 (2nd semester, 14 credits).
Seminar 1 (1st semester, 4 credits): The module is aimed for skills, both spoken and written, precise mathematical language, to formulate and justify mathematical content of the topic related to the Master’s theses. Due to the nature of the module is expected that the curriculum will be closely related to the topics of the Master’s theses.
Seminar 2 (2nd semester, 14 credits): The module is aimed for skills, both spoken and written, precise mathematical language, including understanding the role of proof in mathematics. Due to the nature of the module is expected that the curriculum will be closely related to the module content Seminar 1.
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Mathematics in finance and economics
Year 2 in Katowice  Mathematics in finance and economics
 2 Year
 Mathematics in finance and economics Pathway
 University of Silesia in Katowice Place
 63 ECTS Credits
 Read here Qualification
List of course units
Semester 1

Computational mathematics (3 credits)
 ECTS credits 3
 Semester 1
 University University of Silesia in Katowice
 Lecturer 1 Przemysław Koprowski

Topics
The aim of Computational mathematics course is to teach students how to use computational (both numerical and symbolic) methods in applications coming from various branches of mathematics.
The course covers the following subjects:
1. Polynomial algorithms: squarefree factorization, polynomial factorization over finite fields, factorization of rational polynomials, monomial orders and Groebner bases;
2. Elimination theory: elimination with Groebner bases, classical elimination with resultants;
3. Inifinite summation and Gosper's algorithm;
4. Numerical integration: MonteCarlo algorithm.
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Mathematical methods in physics (6 credits)
 ECTS credits 6
 Semester 1
 University University of Silesia in Katowice
 Lecturer 1 Jerzy Dajka

Prerequisites
Basic linear algebra is enough. A bit of number theory can be useful but not necessary.

Objectives
My aim is to present mathematical methods for quantum information processing. As in most applications it is enough to work with qubits and systems of qubits, mathematical methods originate from linear algebra, which is usually one of first curses taught. It makes quantum information accessible for very 'fresh' students. I would like to convince students that quantum information processing is useful, interesting, counterintuitive, sometimes seemingly as mysterious as the Schroedinger cat.

Topics
Mathematical formalism of quantum mechanics.
Postulates of quantum mechanics.
Quantum information: quantum gates, nogo theorems, measurement.
Quantum entanglement: mathematical basis.
Selected applications: teleportation, dense coding.
Quantum cloning and applications.
Basic protocols for quantum cryptography: BB84, B92.
Quantum nonlocality: Bell and LeggettGarg inequalities, contextuality.
Dynamics of quantum systems, open quantum systems.
Quantum error correction.

Books
Quantum Computation and Quantum Information by Michael A. Nielsen & Isaac L. Chuang
Lecture notes by John Preskill http://www.theory.caltech.edu/people/preskill/ph229/
 Link http://zft.us.edu.pl/dajka
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Decision Making Techniques and Tools (6 credits)
 ECTS credits 6
 Semester 1
 University University of Silesia in Katowice
 Lecturer 1 Paweł Błaszczyk
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Statistics (3 credits)
 ECTS credits 3
 Semester 1
 University University of Silesia in Katowice
 Lecturer 1 Agnieszka Kulawik

Topics
The aim of the Statistics unit is to get a deep knowledge on constructing statistical models and making statistical analysis, and to improve the skills of using statistical computer packages.
The contents of this unit are the following:
1. Organising statistical analysis: collecting and data, their analysis and graphical description.
2. Linear and nonlinear statistical models – estimation theory and statistical hypotheses testing.
3. Applications of linear and nonlinear statistical models in econometrics and financial mathematics.
4. Parametric tests of significance involving two or more samples.
5. Conformity tests.
6. Nonparametric tests of significance involving two or more samples.
7. Applications of statistical computer software to estimation and statistical testing
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Wavelet transforms (6 credits)
 ECTS credits 6
 Semester 1
 University University of Silesia in Katowice
 Lecturer 1 Janusz Morawiec

Topics
The main goal of the lecture is to present basic properties of wavelet transforms and some methods of construction of wavelet bases. We will pay special attention to these wavelet transforms which have used to the analysis and the synthesis of sound signals. We also will pay special attention to structures of bases with special properties which have used to the data compression in digital transmissions.

Books
[1] C.K. Chui, An Introduction to Wavelets, Academic Press, Boston, 1992.
[2] I. Daubechies, The wavelet transform, Timefrequency localization and signal analysis, IEEE Trans. Inform. Theory 36 (1990), 9611005.
[3] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philidelphia, 1992.
[4] C. Heil, D. Walnut, Continuous and discrete wavelet transforms, SIAM Review 31 (1989), 628666.
[5] G. Kaiser, A Friendly Guide to Wavelets, Birkhauser, Boston, 1994.
[6] D. Kozlow, Wavelets. A tutorial and a bibliography, Rendiconti dell’Instituto di Matematica dell’Universita di Trieste, 26, supplemento (1994).
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Workshop on Problem solving (2 credits)
 ECTS credits 2
 Semester 1
 University University of Silesia in Katowice
 Lecturer 1 Radosław Czaja
 Lecturer 2 Anna Brzeska
 Topics The main aim of the module Problem Workshops is to acquaint students with chosen branches of mathematics with applications to knowledge domains such as: economics, biology, physics, chemistry, and computer science. Additional aims are: training analytical skills (for example, constructing mathematical models of chosen problems from applied sciences), training methodological skills (for example, use of available technology to prepare a project or analysis), training cognitive skills (for example, an analysis of data or source content given in a form of articles or manuals, also in a foreign language) and training skills of teamwork (for example, work in small groups during and outside the workshop).
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Polish language and culture for foreigners (level A1) (3 credits)
 ECTS credits 3
 Semester 1
 University University of Silesia in Katowice
 Lecturer 1 Aleksandra Achtelik
 Lecturer 2 Małgorzata Nieużyła
 Topics The aim of the module is to develope all language skills (listening, reading, speaking and writing) and to prepare students for quite easy communication in Polish, necessary while studying in Poland. Students acquire not only linguistic and communicative competence, but also sociocultural: they get to know selected aspects of Polish culture, basic habits and holidays celebrated in Poland, taking into account the pragmatic and sociolinguistic efficiency. Programme includes basic communication situations: greetings and farewells, shopping, ordering food, traveling, etc.
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Semester 2

Applications of the theory of functional equations (6 credits)
 ECTS credits 6
 Semester 2
 University University of Silesia in Katowice
 Lecturer 1 Roman Ger

Topics
Applications in Geometry:
1. Joint characterization of Euclidean, hyperbolic and elliptic geometries.
2. Characterizations of the cross ratio.
3. A description of certain subsemigroups of some Lie groups.
Applications in Functional Analysis:
1. Analytic form of linearmultiplicative functionals in the Banach algebra of integrable functions on the real line.
2. A characterization of strictly convex spaces.
3. Some new characterizations of inner product spaces.
4. BirkhoffJames orthogonality.
5. Addition theorems in Banach algebras; operator semigroups.

Books
1. J. Aczel & J. Dhombres, Functional equations in several variables, Cambridge University Press, Cambridge, 1989. 2. J. Aczel & S. Gołąb, Funktionalgleichungen der Theorie der Geometrischen Objekte, PWN, Warszawa, 1960. 3. J. Dhombres, Some aspects of functional equations, Chulalongkorn Univ., Bangkok, 1979. 4. D. Ilse, I. Lehman and W. Schulz, Gruppoide und Funktionalgleichungen, VEB Deutscher Verlag der Wissenschaften, Berlin, 1984. 5. M. Kuczma, An introduction to the theory of functional equations and inequalities, Polish Scientific Publishers & Silesian University, WarszawaKrakówKatowice, 1985.
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Mathematics of finance, discrete models (6 credits)
 ECTS credits 6
 Semester 2
 University University of Silesia in Katowice
 Lecturer 1 Maciej Sablik
 Topics In our lecture we present an introduction to the mathematics of finance, and in particular the models with discrete time. We are going to discuss, among others, the following questions: mathematical finance in one period, the fundamental theorem of asset pricing, the multiperiod market model, arbitrage opportunities and martingale measures, binomial trees and the CRR model, introduction to optimal stopping and American options, risk measures, indifference valuation and optimal derivative design, optimal risk transfer in principal agent games, bonds and contracts for bonds, contracts swap and swaptions, contracts cap and floor, models with infinite set of simple events.
 Books The lecture will be based on a book by Stanley R. Pliska Introduction to Mathematical Finance: Disrete Time Models Blackwell Publishing Ltd, Oxford 2004.
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Collective project (4 credits)
 ECTS credits 4
 Semester 2
 University University of Silesia in Katowice
 Lecturer 1 Radosław Wieczorek

Objectives
In this module the students, divided into teams consisting of several people, implement projects associated with the given problem.

Topics
The project consists of several phases:
1. Planning for the project. The allocation of roles and responsibilities in the team.
2. Review of available literature on the given matter.
3. Analysis of the problem, seeking methods of its solution.
4. Implementation of the solution. This phase, depending on the project, should include elements such as the analysis of empirical data, calibration, simulation and testing of the solution.
5. Preparation of the final report and presentation of results. Both the final effect and the individual phases of the project are assessed. Laboratory classes serve to current reporting and discussing work progress, and give the opportunity of obtaining assistance in the project implementation.
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Master's Thesis at US

Master’s Thesis (US) (18 credits)
 ECTS credits 18
 Semester 2
 University University of Silesia in Katowice

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 University of Silesia in Katowice also following two local courses: Seminar 1 (1st semester, 4 credits) and Seminar 2 (2nd semester, 14 credits).
Seminar 1 (1st semester, 4 credits): The module is aimed for skills, both spoken and written, precise mathematical language, to formulate and justify mathematical content of the topic related to the Master’s theses. Due to the nature of the module is expected that the curriculum will be closely related to the topics of the Master’s theses.
Seminar 2 (2nd semester, 14 credits): The module is aimed for skills, both spoken and written, precise mathematical language, including understanding the role of proof in mathematics. Due to the nature of the module is expected that the curriculum will be closely related to the module content Seminar 1.
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Year2 Lviv
Year 2 in Lviv  Applied Mathematics
 2 Year
 Applied Mathematics Pathway
 Ivan Franko National University of Lviv Place
 60 ECTS Credits
 Read here Qualification
List of course units
Semester 1

Optimization of complex systems (6 credits)
 ECTS credits 6
 Semester 1
 University Ivan Franko National University of Lviv
 Lecturer 1 https://www.intermaths.eu/my/userprofile/
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Modern programming technologies (4.5 credits)
 ECTS credits 4.5
 Semester 1
 University Ivan Franko National University of Lviv
 Lecturer 1 https://www.intermaths.eu/my/userprofile/
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Algorithms and data structures (6 credits)
 ECTS credits 6
 Semester 1
 University Ivan Franko National University of Lviv
 Lecturer 1 https://www.intermaths.eu/my/userprofile/
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Pattern recognition (4.5 credits)
 ECTS credits 4.5
 Semester 1
 University Ivan Franko National University of Lviv
 Lecturer 1 https://www.intermaths.eu/my/userprofile/
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Foreign language for scientific publications (3 credits)
 ECTS credits 3
 Semester 1
 University Ivan Franko National University of Lviv
 Lecturer 1 https://www.intermaths.eu/my/userprofile/
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Scientific seminar (3 credits)
 ECTS credits 3
 Semester 1
 University Ivan Franko National University of Lviv
 Lecturer 1 https://www.intermaths.eu/my/userprofile/
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Course project (3 credits)
 ECTS credits 3
 Semester 1
 University Ivan Franko National University of Lviv
 Lecturer 1 https://www.intermaths.eu/my/userprofile/
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Semester 2

Research internship (12 credits)
 ECTS credits 12
 Semester 2
 University Ivan Franko National University of Lviv
 Lecturer 1 https://www.intermaths.eu/my/userprofile/

Objectives
The aim of Industrial Internship is to engage the student in commertial projects, usually connected with mathematical modelling or software development.

More information
Depending on student's interests he/she can be temporarily enrolled at IT company, scientific institute, university or other organization which deals with mathematical, computer modelling, simulation or similar problems. Lviv has a wide range of possibilities, hosting over 200 IT companies with nearly 15000 of employees, a dozen of universities and over 30 scientific institutes.
http://ami.lnu.edu.ua/en/students/career
http://www.nas.gov.ua/EN/Structure/Pages/geoPosition.aspx
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Master's thesis (IFNUL) (18 credits)
 ECTS credits 18
 Semester 2
 University Ivan Franko National University of Lviv
 Lecturer 1 https://www.intermaths.eu/my/userprofile/

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.
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