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
    • Topics

       

      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

    • 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

       

      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
    • 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, Laplace, wave equations). 

    • Topics

       

      Linear first order PDE's. Method of characteristics. The Burgers' equation. Shocks and rarefaction waves. Riemann problem for scalar conservation laws. Partial differential equations of second order. Well posed problems, IBV problems. The heat equation. Derivation, maximum principle, fundamental solution. The Fourier transform method. 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 dimension and strong Huygens' principle, Kirchoff's formula.


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  • Control systems [6 credits]

    Control systems

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

       

      Introduction to linear systems. Structural properties: controllability and observability. Introduction to Lyapunov stability theory. Introduction to control systems. The Laplace transform. The Nyquist criterium. Frequency based methods. The steady state and the transient responses. Basic control actions. The locus root method.


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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

University of Hamburg, Germany (UHH)

Department of Mathematics
Bundesstr. 55 20146 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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