MathMods :: Joint MSc

Semester1 Theory UAQ

Semester1 Theory UAQ

Theory  @  UAQ  30 ECTS credits

The first semester of the programme is focused on Mathematical Theory and is common to all students. It will be spent at UAQ - University of L'Aquila in Italy from September to March.

The University of L'Aquila has a longstanding tradition on the analysis of differential equations and dynamical systems with applications to engineering and social sciences. The goal of the first semester is to endow the student with advanced background in theoretical subjects such as Functional Analysis, Applied Partial Differential Equations, and Dynamical Systems. Functional Analysis prepares the student to work in "infinite dimensions", an essential feature to approach advanced methodologies in numerical analysis, approximation theory, optimization, stochastic analysis, in systematic and rigorous form. Applied Partial Differential Equations and Dynamical Systems are fundamental tools in mathematical modelling with applications to science and engineering (for example in fluid dynamics, life and social sciences, real world applications, and finance). These three units are typically touched only marginally in most of the applied sciences BSc curricula. One of the main goals of "MathMods" is to address them to students coming from Engineering and Physics BSc studies, thus reducing possible gaps with respect to BSc graduates in mathematics. The unit "Control System" plays the role of a selected "engineering-oriented" subject with a strong interface with dynamical systems, control theory, and basic harmonic analysis.

Below you can find information about the subjects for this semester.

  • Dynamical systems and bifurcation theory [6 credits]

    Dynamical systems and bifurcation theory

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

       

      The course is intended to introduce and develop an understanding of the concepts in nonlinear dynamical systems and bifurcation theory, and an ability to analyze nonlinear dynamic models of physical systems. The emphasis is to be on understanding the underlying basis of local bifurcation analysis techniques and their applications to structural and mechanical systems.

    • Topics

       

      Review of: first-order nonlinear ODE, first-order linear systems of autonomous ODE. Local theory for nonlinear dynamical systems: linearization, stable manifold theorem, stability and Liapunov functions, planar non-hyperbolic critical points, center manifold theory, normal form theory. Global theory for nonlinear systems: limit sets and attractors, limit cycles and separatrix cycles, Poincaré map. Hamiltonian systems. Poincaré-Bendixson theory. Bifurcation theory for nonlinear systems: structural stability, bifurcation at non-hyperbolic equilibrium points, Hopf bifurcations, bifurcation at non hyperbolic periodic orbits. Applications.

    • Prerequisites

       

      Ordinary differential equations

    • Books

       

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


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  • Functional analysis in applied mathematics and engineering [8 credits]

    Functional analysis in applied mathematics and engineering

    • ECTS credits 8
    • Semester 1
    • University University of L'Aquila
    • Objectives

       

      The objective of this course is to provide the main concepts and fundamental methods of Functional Analysis to enable a student to treat various concrete problems arising in applied mathematics and engineering like PDEs, integral equations, best approximation problems, control theory, calculus of variations, numerical analysis.

    • Topics

       

      Metric & Normed spaces. Uniform convergence and spaces of continuous functions. Compactness in infinite dimension. Lebesgue integral, Lp-spaces. Linear Operators, cornerstone theorems of FA. Dual Spaces, weak convergence. Hilbert Spaces, Fourier series. Adjoint operators. Compact operators. Introduction to spectral theory. Introduction to operator semigroups. Applications to PDE, integral equations, calculus of variations, and numerical analysis.

    • Prerequisites

       

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

    • 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
      • Lecture notes provided by the teacher.
      • 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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  • Applied partial differential equations [6 credits]

    Applied partial differential equations

    • 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 equation, Laplace equation, wave equation).

    • Topics

       

      Semilinear first order PDE's. Method of characteristics. Partial differential equations of second order. Classification, canonical forms. Well posed problems, IBV problems. The heat equation. Derivation, maximum principle, fundamental solution. Laplace's and Poisson's equations. Maximum principle, fundamental solution and Green's functions. The wave equation. One dimensional equation, fundamental solution and D'Alembert formula. Fundamental solution in three dimensions and strong Huygens' principle, Kirchoff’s formula. Method of descent and solution formula in two dimensions.


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  • Control systems [6 credits]

    Control systems

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

       

      The objective of this course is to introduce the students to the main aspects of control theory and engineering. In particular, the course will provide the students with mathematical tools for modeling, stability analysis and control design of linear dynamical control systems, with applications related to control of quadcopters and telecommunication networks.

    • Topics

       

      Dynamical models of linear time-invariant systems: transfer function and state space representations. Performance metrics for feedback control systems: transient and steady state response. The stability of feedback control systems. The Routh-Hurwitz stability criterion. The frequency response. Disturbance rejection. Design of basic controllers (Proportional, PD, PI, PID). Root Locus sketch and stabilization of minimum and non-minimum phase systems. Digital control: A/D and D/A converters, sampled-data systems, deadbeat and flat control. Elements of realisation theory. Output-feedback stabilisation in state-space representation: controllability, observability and the separation principle. Integral control. Elements of Matlab & Simulink tools for control systems. Experimental projects: attitude control and indoor localization of quadcopters. Control architecture and design of the SCADA Building Management System test-bed of the University of L'Aquila


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  • Italian Language and Culture for Foreigners (level A1) [4 credits]

    Italian Language and Culture for Foreigners (level A1)

    • ECTS credits 4
    • 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, by Luciana Ziglio and Giovanna Rizzo, published by Alma Edizioni, 2014, ISBN: 978-8861823181


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Our partners' addresses

University of L'Aquila, Italy (UAQ)

Department of Information Engineering, Computer Science and Mathematics, via Vetoio (Coppito), 1 – 67100 L’Aquila (Italy)

Autonomous University of Barcelona, Catalonia - Spain (UAB)

Departament de Matemàtiques, Edifici Cc - Campus UAB 08193 Bellaterra – Catalonia

Hamburg University of Technology, Germany (TUHH)

Institute of Mathematics
Schwarzenberg-Campus 3, Building E-10
D-21073 Hamburg - Germany

University of Nice - Sophia Antipolis, France (UNS)

Laboratoire J.A.Dieudonné
Parc Valrose, France-06108 NICE Cedex 2

Vienna Univ. of Technology, Austria (TUW)

Technische Universität Wien
Institute of Analysis & Scientific Computing
Wiedner Hauptstr. 8, 1040 Vienna - Austria

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