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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