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Already enrolled at one of our partner universities?

If you are 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.

magnifier11Course 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.

Year 1 L'Aquila

Interdisciplinary Mathematics

Year 1 in L'Aquila - Interdisciplinary Mathematics

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List of course units

Semester 1

  • Applied partial differential equations (6 credits)

    • ECTS credits 6
    • Semester 1
    • University University of L'Aquila
    • 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 Cauchy-Kovalevsky 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. Well-posedness 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. One-dimensional wave equation. D’Alembert formula. Characteristic parallelogram. Non homogeneous equation and Duhamel’s method. Multi-dimensional 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 initial-boundary value problem. Separation of variables. Reflection of waves.

      THE HEAT EQUATION. Heat conduction in a finite rod. Maximum principle and applications. Solution of the initial-boundary 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 0-486-65251-3

      L.C. Evans, Partial Differential Equations. American Mathematical Society. 2010. Second edition, ISBN-13: 978-0821849743

      S. Salsa, Partial Differential Equations in Actions: from Modelling to Theory. Springer-Verlag Italia. 2008. ISBN 978-88-470-0751-2

      W. A. Strauss, Partial Differential Equations, Student Solutions Manual: An Introduction. John Wiley & Sons, LTD. 2008. Second edition, ISBN-13: 978-0470260715

      W. A. Strauss, Partial Differential Equations: an introduction. John Wiley & Sons, LTD. 2007. Second edition, ISBN-13 978-0470-05456-7


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  • Dynamical Systems and Bifurcation Theory (6 credits)

    • ECTS credits 6
    • Semester 1
    • University University of L'Aquila
    • 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. Hartman-Grobman 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, Poincaré-Bendixson 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 double-Hopf, Divergence-Hopf, Double-zero bifurcation.

    • Books

      Lawrence Perko, Differential equations and dynamical systems, Springer-Verlag, 2001


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  • Functional analysis (6 credits)

    • ECTS credits 6
    • Semester 1
    • University University of L'Aquila
    • Objectives

       

      Learn the fundamental structures of Functional Analysis.

      Get familiar with the main examples of functional spaces, in particular with the theory of Hilbert spaces and Lebesgue spaces.

      Get familiar with the basic notions of operator theory. Be able to frame a functional equation in an abstract functional setting.

    • Topics

      Lebesgue Measure and Integration.

      L^p Spaces.

      Basic of Topological Vector Spaces, Normed and Banach Spaces, Linear Operators and linear functionals.

      Hilbert Spaces.

      Weak topology, Weak * topology, weak compactness.

      Applications of Baire Category in Functional Analysis: Uniform Boundedness, Open Mapping, Closed Graph, Inverse Mapping.

      Banach and Hilbert adjointness, self-adjointness.

      Compact Operators.

      Riesz Fredholm spectral theory.

    • Books

      Terence Tao, An introduction to measure theory.. American Mathematical Society, Providence, RI, ISBN: 978-0-8218-6919-2 . 2011.

      Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations.. Universitext. Springer, New York,. 2011. xiv+599 pp. ISBN: 978-0-387-70913-0

      Alberto Bressan, Lecture notes on functional analysis. With applications to linear partial different. Graduate Studies in Mathematics, 143. American Mathematical Society, Providence, RI,. 2013. xii+250 pp. ISBN: 978-0-8218-8771-4.

      Michael Reed, Barry Simon, Methods of modern mathematical physics. I. Functional analysis. Second edition. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York,. 1980. xv+400 pp. ISBN: 0-12-585050-6.

      Stein, Elias M.; Shakarchi, Rami , Real analysis. Measure theory, integration, and Hilbert spaces. Princeton Lectures in Analysis, III. Princeton University Press, Princeton, NJ,. 2005. xx+402 pp. ISBN: 0-691-11386-6


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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
    • Topics

      Linux/Unix OS and tools;

      Basic Fortran (or C);

      HPC architecture and libraries;

      Application (ex ODEs, PDEs, elastodynamics).


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  • Italian Language and Culture for foreigners (level A1) (3 credits)

    • ECTS credits 3
    • Semester 1
    • University University of L'Aquila
    • 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

  • Combinatorics and cryptography (6 credits)

    • ECTS credits 6
    • Semester 2
    • University University of L'Aquila
    • 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, Diffie-Helman 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 Reed-Solomon 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


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  • Foundations of advanced geometry (6 credits)

    • ECTS credits 6
    • Semester 2
    • University University of L'Aquila
    • 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, n-dimensional 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 Borsuk-Ulam theorem, the Seifert-Van 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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  • Complex Analysis (6 credits)

    • ECTS credits 6
    • Semester 2
    • University University of L'Aquila
    • 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
    • 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, Ornstein-Uhlenbeck 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. Springer-Verlag.


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  • Numerical methods for differential equations (6 credits)

    • ECTS credits 6
    • Semester 2
    • University University of L'Aquila
    • Objectives

      Goals of the course:
      Give the mathematical instruments to handle with optimization problems and differential
      equations. The course consists of 6 credits and lasts 60 hours.

      Expected learning results:
      Being able to solve numerically, both for the theoretical aspects and for the implementation
      issues, general problems arising in differential modeling.

    • Topics

      Numerical methods for the Cauchy problem. Ons step methods. Stability theory.
      Stiff problems and differential-algebraic problems. Numerical methods for boundary value problems.
      Numerical methods for elliptic and parabolic PDEs.

    • Books

      E. Hairer, S.P. Norsett and G. Wanner, Solving ordinary differential equations. I.
      Nonstiff problems. Second edition. Springer Verlag.
      E. Hairer and G. Wanner, Solving ordinary differential equations. I.
      Stiff and differential-algebtraic problems. Second edition. Springer Verlag.
      P. Henrici, Discrete variable methods in ordinary differential equations. Ed. John Wiley.
      J.D. Lambert, Computational methods in ordinary differential equations. Ed. John Wiley.


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  • Data analytics and Data driven decision (6 credits)

    • ECTS credits 6
    • Semester 2
    • University University of L'Aquila

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  • Italian Language and Culture for foreigners (level A2) (3 credits)

    • ECTS credits 3
    • Semester 2
    • University University of L'Aquila
    • 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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Scientific Computing

Year 1 in L'Aquila - Scientific Computing

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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
    • 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 Cauchy-Kovalevsky 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. Well-posedness 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. One-dimensional wave equation. D’Alembert formula. Characteristic parallelogram. Non homogeneous equation and Duhamel’s method. Multi-dimensional 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 initial-boundary value problem. Separation of variables. Reflection of waves.

      THE HEAT EQUATION. Heat conduction in a finite rod. Maximum principle and applications. Solution of the initial-boundary 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 0-486-65251-3

      L.C. Evans, Partial Differential Equations. American Mathematical Society. 2010. Second edition, ISBN-13: 978-0821849743

      S. Salsa, Partial Differential Equations in Actions: from Modelling to Theory. Springer-Verlag Italia. 2008. ISBN 978-88-470-0751-2

      W. A. Strauss, Partial Differential Equations, Student Solutions Manual: An Introduction. John Wiley & Sons, LTD. 2008. Second edition, ISBN-13: 978-0470260715

      W. A. Strauss, Partial Differential Equations: an introduction. John Wiley & Sons, LTD. 2007. Second edition, ISBN-13 978-0470-05456-7


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  • Control Systems (6 credits)

    • ECTS credits 6
    • Semester 1
    • University University of L'Aquila
    • Objectives

      The course provides the basic methodologies for modeling, analysis and controller design for continuous-time linear time-invariant 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 steady-state responses. Polynomial and sinusoidal disturbances rejection.

      The Routh-Hurwitz 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
    • 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. Hartman-Grobman 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, Poincaré-Bendixson 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 double-Hopf, Divergence-Hopf, Double-zero bifurcation.

    • Books

      Lawrence Perko, Differential equations and dynamical systems, Springer-Verlag, 2001


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  • Functional Analysis in Applied Mathematics and Engineering (9 credits)

    • ECTS credits 9
    • Semester 1
    • University University of L'Aquila
    • 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
    • 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


Electives

  • Mechanics of Solids and Materials (6 credits)

    • ECTS credits 6
    • Semester 2
    • University University of L'Aquila
    • 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 Piola-Kirchhoff stress.

      Material response. Material objectivity. Symmetry group and isotropy. Elastic and hyperelastic materials. Strain energy function. Constraints and reactive stress. Incompressibility. Mooney-Rivlin and neo-Hookean materials. Dissipative stress and dissipation principle. Fluids and solids. A general scheme for describing growth and relaxation via Kroner-Lee 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.


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  • Combinatorics and cryptography (6 credits)

    • ECTS credits 6
    • Semester 2
    • University University of L'Aquila
    • 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, Diffie-Helman 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 Reed-Solomon 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


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  • Complex Analysis (6 credits)

    • ECTS credits 6
    • Semester 2
    • University University of L'Aquila
    • 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
    • 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, Ornstein-Uhlenbeck 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. Springer-Verlag.


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  • Network optimization (6 credits)

    • ECTS credits 6
    • Semester 2
    • University University of L'Aquila
    • 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 branch-and-bound 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. Branch-and-cut 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.


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  • Numerical methods for linear algebra and optimisation (6 credits)

    • ECTS credits 6
    • Semester 2
    • University University of L'Aquila

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Year 2 L'Aquila

Interdisciplinary Mathematics

Year 2 in L'Aquila - Interdisciplinary Mathematics

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  • 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
    • 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: Gagliardo-Nirenberg-Sobolev 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: Rellich-Kondrachov theorem, Poincaré inequalities. Characterization of the dual space H-1.

      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, Rankine-Hugoniot 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
    • 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 Navier-Stokes.

      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
    • 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
    • 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
    • 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, Diffie-Helman 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 Reed-Solomon 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


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  • Foundations of advanced geometry (6 credits)

    • ECTS credits 6
    • Semester 2
    • University University of L'Aquila
    • 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, n-dimensional 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 Borsuk-Ulam theorem, the Seifert-Van 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
    • 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: Payne-Whitham model, description, drawbacks; Aw-Rascle model, shocks description, domains of invariance, instabilities near vacuum.

      Theory: systems of conservation laws, strict hyperbolicity, Rankine-Hugoniot 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 initial-boundary value problem on the half line: boundary Riemann problem, interactions with the boundary, control of the total variation by means of a Lyapunov-type 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, Rankine-Hugoniot 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, Cucker-Smale model, model for attraction-repulsion phenomena. The Cucker-Smale 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 N-particle distribution function, Liouville equation, marginal distribution, continuity equation. The formal mean-field limit: a Vlasov-type equation.

    • Books

      M.D. Rosini, Macroscopic models for vehicular flows and crowd dynamics: theory and applications. Springer. 2013. http://link.springer.com/book/10.1007/978-3-319-00155-5/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
    • 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: Lotka-Volterra 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. Michaelis-Menten Quasi-Steady 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).

      Time-space 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 Predator-Prey 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
    • 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, Ornstein-Uhlenbeck 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. Springer-Verlag.


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  • Kinetic and hydrodynamic models (6 credits)

    • ECTS credits 6
    • Semester 1
    • University University of L'Aquila
    • 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 non-perturbative 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 Fokker-Planck equation.

      The closure problem and methods of reduced description: Chapman-Enskog expansion, Grad's Moment method.

      Non-perturbative 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
    • Objectives

      The course is an introduction to Time Series Analysis and Forecasting. The level is the first-year 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
    • 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 set-valued 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 Knaster-Kuratowski-Mazurkiewicz lemma

      Brouwer's fixed point theorem

      Variational inequalities and equilibrium problems

      Generalized monotonicity and convexity

      Brézis-Nirenberg-Stampacchia theorem and Fan's minimax principle

      Continuity of correspondences

      Browder, Kakutani and Fan-Glicksberg fixed point theorems

      Gale-Nikaido-Debreu theorem

      Nash equilibrium of games and abstract economies

      Walrasian equilibrium of an economy

      An application to traffic network


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Scientific Computing

Year 2 in L'Aquila - Scientific Computing

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  • 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
    • 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: Gagliardo-Nirenberg-Sobolev 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: Rellich-Kondrachov theorem, Poincaré inequalities. Characterization of the dual space H-1.

      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, Rankine-Hugoniot 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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  • Kinetic and hydrodynamic models (6 credits)

    • ECTS credits 6
    • Semester 1
    • University University of L'Aquila
    • 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 non-perturbative 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 Fokker-Planck equation.

      The closure problem and methods of reduced description: Chapman-Enskog expansion, Grad's Moment method.

      Non-perturbative 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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  • Mathematical fluid dynamics (6 credits)

    • ECTS credits 6
    • Semester 1
    • University University of L'Aquila
    • 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 Navier-Stokes.

      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
    • 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)


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Semester 2

  • 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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Year 2 Brno

Year 2 in Brno - Mathematical Engineering

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List of course units

Semester 1

  • Multi-valued logic applications (4 credits)

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

      Mathematical logic, fuzzy set theory.

    • Objectives

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

    • Topics

      1. Multi-valued logic, formulae.

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

      3. Linguistic variables and linguistic models.

      4. Knowledge bases of expert systems.

      5-6. Semantic interpretations of knowledge bases

      7. Inference techniques and its implementation

      8. Redundance a contradictions in knowledge bases

      9. LMPS system

      10. Fuzzification and defuzzification problem

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

      12. Expert systems

      13. Overview of AI methods

    • Books

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

    • More information

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


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  • Financial Mathematics (4 credits)

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

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

    • Objectives

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

    • Topics

      1. Basic concepts, money, capital and securities.

      2. Simple and compound interest rate, discounting.

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

      4. Assets and liabilities, insurance.

      5. Bonds, options, futures, and forwards.

      6. Exchange rates, inflation, indices.

      7. Portfolio optimization - classical model.

      8. Postoptimization, risk, funds.

      9. Twostage models in finance.

      10. Multistage models in finance.

      11. Scenarios in financial mathematics.

      12. Modelling principles, identification of dynamic data.

      13. Discussion on advanced stochastic models.

    • Books

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

    • More information

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


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  • Fuzzy Sets and Applications (4 credits)

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

      Fundamentals of the set theory and mathematical analysis.

    • Objectives

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

    • Topics

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

      2. Operations with fuzzy sets (properties).

      3. Operations with fuzzy sets (alfa cuts).

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

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

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

      7. Fuzzy relations (basic notions, kinds).

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

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

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

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

      12. Fuzzy probability (basic notions, properties).

      13. Fuzzy models design for applications.

    • Books

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

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

    • More information

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


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  • Mathematical Methods in Fluid Dynamics (4 credits)

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

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

    • Objectives

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

    • Topics

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

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

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

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

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

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

      7. Local error, stability, convergence.

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

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

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

      11. Stokes problem, spectral element method.

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

      13. Unsteady Navier-Stokes problem.

    • Books

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

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

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

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

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

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

    • More information

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


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  • Fundamentals of Optimal Control Theory (4 credits)

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

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

    • Objectives

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

    • Topics

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

      2. Maximum principle.

      3. Time-optimal control of an uniform motion.

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

      5. Basic results on optimal controls.

      6. Variational problems with moving boundaries.

      7. Optimal control of systems with a variable mass.

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

      9. Singular control.

      10. Energy-optimal control problems.

      11. Variational problems with state constraints.

      12. Variational problems with state constraints (continuation).

      13. Solving of given problems.

    • Books

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

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

    • More information

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


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  • Reliability and Quality (4 credits)

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

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

    • Objectives

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

    • Topics

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

    • Books

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

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

    • More information

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


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Semester 2

  • Analysis of Engineering Experiment (4 credits)

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

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

    • Objectives

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

    • Topics

      1.One-way analysis of variance.

      2.Two-way analysis of variance.

      3.Regression model identification.

      4.Nonlinear regression analysis.

      5.Regression diagnostic.

      6.Nonparametric methods.

      7.Correlation analysis.

      8.Principle components.

      9.Factor analysis.

      10.Cluster analysis.

      11.Continuous probability distributions estimation.

      12.Discrete probability distributions estimation.

      13.Stochastic modeling of the engineering problems.

    • Books

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

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

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

    • More information

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


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  • Modern methods of solving differential equations (5 credits)

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

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

    • Objectives

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

    • Topics

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

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

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

      4. Weak formulation of the linear elliptic equations.

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

      6. Variational formulation, construction of approximate solutions.

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

      8. Weak and variational formulations of the nonlinear equations.

      9. Monotonne operator theory and its applications.

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

      11. Introduction to Stochastic Differential Equations. Brown motion.

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

      13. Reserve.

    • Books

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

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

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

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

    • More information

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


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  • Data Visualisation (4 credits)

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

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

    • Objectives

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

    • Topics

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

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

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

      4) Contour lines of surface.

      5) Surface visualisation using the palette.

      6) 2D visualisation of 3D data grid.

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

      8) 3D filters.

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

      10) 2D reconstruction of confocal microscope outputs.

      11) 3D reconstruction of confocal microscope outputs.

      12) 2D reconstruction of Visible Human Project data.

      13) 3D reconstruction of Visible Human Project data.

    • Books

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

    • More information

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


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  • Geometric Algorithms and Cryptography (4 credits)


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  • Mathematical Structures (4 credits)

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

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

    • Objectives

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

    • Topics

      1. Sets and classes

      2. Mathematical structures

      3. Isomorphisms

      4. Fibres

      5. Subobjects

      6. Quotient objects

      7. Free objects

      8. Initial structures

      9. Final structures

      10. Cartesian product

      11. Cartesian completeness

      12. Functors

      13. Reflection and coreflection

    • Books

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

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

    • More information

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


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

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

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

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

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

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

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

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

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

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

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Year 2 Katowice

Mathematical modeling

Year 2 in Katowice - Mathematical modeling

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List of course units

Semester 1

  • Computational mathematics (3 credits)

    • ECTS credits 3
    • Semester 1
    • University University of Silesia in Katowice
    • 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: square-free 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: Monte-Carlo algorithm.


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  • Applied Graph Theory (6 credits)

    • ECTS credits 6
    • Semester 1
    • University University of Silesia in Katowice
    • 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, Springer-Verlag, 2001.
      2. Diestel G. T., Graph Theory, Springer-Verlag, 1997, 2000.
      3. Foulds L. R., Graph Theory Applications, Springer-Verlag, 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
    • 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, counter-intuitive, sometimes seemingly as mysterious as the Schroedinger cat.

    • Topics

      Mathematical formalism of quantum mechanics.

      Postulates of quantum mechanics.

      Quantum information: quantum gates, no-go 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 Leggett-Garg 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/


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  • Statistics (3 credits)

    • ECTS credits 3
    • Semester 1
    • University University of Silesia in Katowice
    • 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 non-linear statistical models – estimation theory and statistical hypotheses testing.

      3. Applications of linear and non-linear statistical models in econometrics and financial mathematics.

      4. Parametric tests of significance involving two or more samples.

      5. Conformity tests.

      6. Non-parametric 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
    • 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, Time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 36 (1990), 961-1005.

      [3] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philidelphia, 1992.

      [4] C. Heil, D. Walnut, Continuous and discrete wavelet transforms, SIAM Review 31 (1989), 628-666.

      [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
    • 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 team-work (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
    • 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
    • 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 linear-multiplicative 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. Birkhoff-James 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, Warszawa-Kraków-Katowice, 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

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  • Collective project (4 credits)

    • ECTS credits 4
    • Semester 2
    • University University of Silesia in Katowice
    • 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

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List of course units

Semester 1

  • Computational mathematics (3 credits)

    • ECTS credits 3
    • Semester 1
    • University University of Silesia in Katowice
    • 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: square-free 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: Monte-Carlo algorithm.


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  • Mathematical methods in physics (6 credits)

    • ECTS credits 6
    • Semester 1
    • University University of Silesia in Katowice
    • 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, counter-intuitive, sometimes seemingly as mysterious as the Schroedinger cat.

    • Topics

      Mathematical formalism of quantum mechanics.

      Postulates of quantum mechanics.

      Quantum information: quantum gates, no-go 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 Leggett-Garg 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/


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  • Decision Making Techniques and Tools (6 credits)

    • ECTS credits 6
    • Semester 1
    • University University of Silesia in Katowice

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  • Statistics (3 credits)

    • ECTS credits 3
    • Semester 1
    • University University of Silesia in Katowice
    • 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 non-linear statistical models – estimation theory and statistical hypotheses testing.

      3. Applications of linear and non-linear statistical models in econometrics and financial mathematics.

      4. Parametric tests of significance involving two or more samples.

      5. Conformity tests.

      6. Non-parametric 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
    • 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, Time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 36 (1990), 961-1005.

      [3] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philidelphia, 1992.

      [4] C. Heil, D. Walnut, Continuous and discrete wavelet transforms, SIAM Review 31 (1989), 628-666.

      [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
    • 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 team-work (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
    • 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
    • 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 linear-multiplicative 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. Birkhoff-James 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, Warszawa-Kraków-Katowice, 1985.


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  • Mathematics of finance, discrete models (6 credits)

    • ECTS credits 6
    • Semester 2
    • University University of Silesia in Katowice
    • 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 multi-period 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
    • 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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Year 2 Lviv

Year 2 in Lviv - Applied Mathematics

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List of course units

 

Semester 1

Semester 2

  • Research internship (12 credits)

    • ECTS credits 12
    • Semester 2
    • University Ivan Franko National University of Lviv
    • 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://itcluster.lviv.ua/en

      http://www.nas.gov.ua/EN/Structure/Pages/geoPosition.aspx

       


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  • Master's thesis (LNU) (18 credits)

    • ECTS credits 18
    • Semester 2
    • University Ivan Franko National University of Lviv
    • 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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