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.

  • 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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  • 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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  • Real and Functional Analysis [6 credits]

    Real and Functional Analysis

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

       

      Introducing basic tools of advanced real analysis such as metric spaces, Banach spaces, Hilbert spaces, bounded operators, weak convergences, compact operators, weak and strong compactness in metric spaces, spectral theory, in order to allow the student to formulate and solve linear ordinary differential equations partial differential equations, classical variational problems, and numerical approximation problems in an "abstract" form.

    • Topics

       

      • Metric spaces, normed linear spaces. Topology in metric spaces. Compactness.
      • Spaces of continuous functions.  Convergence of function sequences. Approximation by polynomials. Compactness in spaces of continuous functions. Arzelà's theorem. Contraction mapping theorem.
      • Crash course on Lebesgue meausre and integration. Limit exchange theorema. Lp spaces. Completeness of Lp spaces.
      • Introduction to the theory of linear bounded operators on Banach spaces. Bounded operators. Dual norm. Examples. Riesz' lemma. Norm convergence for bounded operators.
      • Hilbert spaces. Elementary properties. Orthogonality. Orthogonal projections. Bessel's inequality. Orthonormal bases. Examples.
      • Bounded operators on Hilbert spaces. Dual of a Hilbert space. Adjoin operator, self-adjoint operators, unitary operators. Applications. Weak convergence on Hilbert spaces. Banach-Alaoglu's theorem.
      • Introduction to spectral theory. Compact operators. Spectral theorem for self-adjoint compact operators on Hilbert spaces. Hilbert-Schmidt operators. Functions of operators.
      • Introduction to the theory of unbounded operators. Linear differential operators. Applications.
      • Introduction to infinite-dimensional differential calculus and variational methods.
    • Prerequisites

       

      Basic calculus and analysis in several variables, linear algebra.

    • Books

       

      • John K. Hunter, Bruno Nachtergaele, Applied Analysis. World Scientific.
      • H. Brezis, Funtional Analysis, Sobolev Spaces, and partial differential equations. Springer.

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  • Mathematical modelling of continuum media [3 credits]

    Mathematical modelling of continuum media

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

       

      The aim of the course is 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 understand constitutive models for fluids and to be able to solve simple boundary value problems for fluids and solids.

    • 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
      • Example of fluid models in various fields
    • Prerequisites

       

      Basic notions of functional analysis and multi variable calculus, standard properties of the heat equation, wave equation, Laplace and Poisson's equations.


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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) [3 credits]

    Italian Language and Culture for Foreigners (level A1)

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


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