ECTS Credits: 6   |   Semester: 1   |   Year: 1   |      |   Campus: University of L'Aquila   |   Language: English

Unit Coordinator: Corrado Lattanzio

Aims:

LEARNING OBJECTIVES.
The course aims at providing basic properties and main techniques to solve basic partial differential equations.
Those objectives contribute to the learning goals of the entire course of studies, as the inner coherence of the master degree in Mathematical Modelling was verified at the time of the planning of the master program.

LEARNING OUTCOMES.
At the end of the course, the student should:

1. know basic properties (existence, uniqueness, etc.) and main techniques (characteristics, separation of variables, Fourier methods, Green's functions, similarity solutions, etc.) to solve basic partial differential equations (semilinear first order PDEs, heat, Laplace, wave equations);
2. understand and be able to explain thesis and proofs in the field of basic partial differential equations;
3. have strengthened the logic and computational skills;
4. be able to read and understand other mathematical texts on related topics.

Content:

First order partial differential equations. Definition of characteristic vectors and characteristic surfaces. Characteristics for (semi)linear partial differential equations of first order in two independent variables. Existence and uniqueness to initial value problems for first order semilinear partial differential equations in two independent variables Duhamel’s principle for non homogeneous first order partial differential equations.

Second order partial differential equations. Classification of second order semilinear partial differential equations in two independent variables. Canonical form for second order semilinear partial differential equations in two independent variables. Classification for second order semilinear partial differential equations in many independent variables.

Heat equation. Derivation of heat equation and well–posed problems in one space dimension. Solution of Cauchy–Dirichlet problem for one dimensional heat equation by means of Fourier method of separation of variables. Energy method and uniqueness. Maximum principle. Fundamental solution. Solution of global Cauchy problem. Non homogeneous problem: Duhamel’s principle.

Laplace equation. Laplace and Poisson equation: well-posed problems; uniqueness by means of energy method. Mean value property and maximum principles. Laplace equation in a disk by means of separation of variables. Poisson’s formula. Harnack’s inequality and Liouville’s Theorem. Fundamental solution of Laplace operator. Solution of Poisson’s equation in the whole space. Green’s functions and Green’s representation formula.

Wave equation. Transversal vibrations of a string. Well–posed problems in one space dimension. D’Alembert formula. Characteristic parallelogram. Domain of dependence and range of influence. Fundamental solution for one dimensional wave equation. Duhamel’s principle for non homogeneous one dimensional wave equations. Special solutions of multi–d wave equation: planar and spherical waves. Well–posedness for initial, boundary value problems: uniqueness by means of energy estimates. Separation of variables. Domain of dependence and range of influence in several space variables. Fundamental solution for multi–dimensional wave equation. Solution of 3–d wave equation: Kirchhoff’s formula and strong Huygens’ principle. Wave equation in two dimensions: method of descent. Fundamental solution in 2–d. Duhamel’s principle for non homogeneous wave equation in 3–d: delayed potentials.

Pre-requisites:

Students must know the basic notions of mathematical analysis, including Fourier series and ordinary differential equations, and the basic notions of continuum mechanics.

Reading list:

- L.C. Evans. Partial Differential Equations. Graduate Studies in Mathematics, Vol. 19, AMS, 2010.

- S. Salsa. Partial Differential Equations in Actions: from Modelling to Theory. Springer–Verlag Italia, 2008.

- S. Salsa, G. Verzini. Equazioni a derivate parziali: complementi ed esercizi. Springer–Verlag Italia. 2005.

- W.A. Strauss. Partial Differential Equations: An Introduction. John Wiley & Sons Inc., 2008.

- E.C. Zachmanoglou, D.W. Thoe. lntroduction to Partial Differential Equations with Applications. Dover Publications, Inc., 1986.

Additional info:

http://people.disim.univaq.it/~corrado/

ECTS Credits: 6   |   Semester: 1   |   Year: 1   |      |   Campus: University of L'Aquila   |   Language: English

Unit Coordinator: Bruno Rubino

Aims:

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.

Content:

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.

Pre-requisites:

Ordinary differential equations

Reading list:

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

ECTS Credits: 9   |   Semester: 1   |   Year: 1   |      |   Campus: University of L'Aquila   |   Language: English

Unit Coordinator: Rosella Sampalmieri, Antonio Esposito

Aims:

The purpose of this introductory course is to give a uniform background
of basic knowledge in mathematical analysis along with a common
mathematical language. Indeed, due to the international nature of the
course, Students come from different countries, with different study
curricula and mathematical training. Furthermore, over the years it has
been noted that most of the Students have a training in analysis that is
more applied than theoretical (calculus-type courses), which prevents the
full understanding of the more theoretical topics addressed in subsequent
courses. Therefore numerous fundamental topics, traditionally present in
mathematical analysis courses will be resumed and deepened. The
Student who has successfully attended the course should be able to
easily understand the presentation of more advanced and sophisticated
mathematical arguments and proofs and be able to apply the theory to
the proposed problems.
On successful completion of this course, the Student should
1-have a clear understanding of the basic notions and concepts of the
real analysis
2-know and know how to apply the theoretical results learned to the
proposed problems, using appropriate mathematical language;
3-understand and consciously reproduce a demonstration;
4- be able to read and understand scientific texts on related topics.

Content:

THE COURSE INCLUDES THE FOLLOWING TOPICS:
The set of real numbers and related axioms. Inferior and supreme.
The topology of real numbers: intervals, half lines, open sets, closed sets.
The topology of the Euclidean space R^n: spherical neighborhoods, open,
closed, connected, convex sets. Compact sets in Euclidean space.
Numerical sequences: limits of sequences and related theorems.
Real functions of real variable: limits of functions, continuous functions,
continuity and compactness, continuity and connection, discontinuity, monotonic functions.
Derivation in IR: the derivative of a real function, rules of derivation,
Taylor's theorem, study of the graph of a function.
The Riemann integral in IR: definition and existence of the integral,
properties of the integral, integration techniques.
Functions of several variables: limits of functions of several variables,
continuity, differentiability and differentiability, higher order derivatives,
necessary and sufficient conditions for the existence of local and global
extrema, constrained extrema, implicit function theorem (various
versions), functions vectors, transformations of coordinates.
Multiple integrals: double and triple integrals, reduction formulas, change
of variables in multiple integrals.
Curves and surfaces: curves, regular curves, length of a curve, rectifiable
curves, curvilinear integrals.
Surfaces: regular surfaces, regular surface area, surface integrals.
Vector fields: conservative vector fields, Green's formulas, Stokes's
theorem, Gauss's theorem.

Pre-requisites:

The prerequisites to this course are the following:
Propositional logic. Propositional calculus.
Sets, set operations, relations, functions. Cardinality of sets, countable sets, uncountable sets. Elementary number sets. Integers and rationals. Induction principle.
More on functions: injective and surjective functions, invertible functions, image and pre-image.
The set of real numbers. Separation axiom, Dedekind cuts. Infimum and supremum. Archimedean property. Complex numbers: cartesian and trigonetric form, basic properties, powers, complex roots, fundamental theorem of algebra.
Introduction to functions of real numbers. Elementary functions: exponential and logarithmic function, trigonometric functions, irrational functions. Monotone functions.
The topology of real numbers: intervals, half lines, open sets, closed sets. The topology of the Euclidean space Rn: balls, open and closed sets. Compact sets in the Euclidean space.
The prerequisites to this course can also be found in the section ''Real Analisys: Foundations ''of the Pre-Master's Foundation Programme (PMFP) in Applied Mathematics whose purpose is homogenising the competencies portfolios of prospective students of the two Master's Programmes in Mathematical Modelling and Mathematical Engineering, in particular in Mathematical Analisys.

Reading list:

-An Introduction to Real Analysis
John K. Hunter,
Department of Mathematics, University of California at Davis

- Calculus of several variables,
Serge Lang,
Yale University,
Addison-Wesley Publishing Company

-Principles of mathematical analysis -W.Rudin - McGraw-Hill 1976

Additional info:

Teaching methods

In addition to the classic lectures, the Students will be involved in informal discussions on the subjects presented, in order to stimulate a deeper understanding. Students will be asked to propose strategies to solve problems and exercises during the lessons. Moreover home-works will be assigned regularly and commented during the classes.
A support tutoring activity is also organized for the students.

Assessment methods

As far as summative assessment is concerned, there will be a written test followed by an oral examination.
There will also be 2/3 optional intermediate written tests, which aim to cover all the different parts of the program, whose passing will be equivalent to passing the written test.
The written tests will focus on both calculus and more theoretical exercises. A written test with a grade of 18/30 is considered sufficient. The oral tests will focus on definitions, statements of theorems and examples. The proof of 10 theorems presented in class will also be required, at the student's choice but within the whole program. The oral exam can add a maximum of 4 points to the mark of the written exam or decrease or cancel this mark if negative.

ECTS Credits: 6   |   Semester: 1   |   Year: 1   |      |   Campus: University of L'Aquila   |   Language: English

Unit Coordinator: Alessandro Borri

Aims:

The course provides basic and advanced methodologies for the modeling, analysis and design of control systems.

Content:

Frequency domain models of Linear Systems: Laplace Transform, Transfer Functions, 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.
Digital control: Z-Transform, discretization of continuous-time linear systems, finite-time response, deadbeat response.
Analysis and control design using the eigenvalues assignment: controllability, observability, the separation principle.
Controller design using MATLAB.

Pre-requisites:

Mathematical analysis.

Reading list:

[1] K. J. Astrom, R. M. Murray, "Feedback Systems: An Introduction for Scientists and Engineers". Princeton Univ. Pr., 2nd Edition, 2021
[2] R. C. Dorf, R. H. Bishop. "Modern Control Systems". Prentice Hall, 12th Edition, 2008 
[3] G. F. Franklin, J. D. Powell, A. Emami-Naeini. "Feedback control of dynamic systems". Prentice Hall, 4th Edition, 2002
[4] G. F. Franklin, J. D. Powell, M. L. Workman. "Digital Control of Dynamic Systems". Addison-Wesley, 3rd Edition, 1998
[5] K. Ogata. "Discrete-Time Control Systems". Prentice Hall, 2nd Edition, 1995
[6] A. Isidori. "Sistemi di controllo". Siderea, 1996
[7] Lectures slides

ECTS Credits: 3   |   Semester: 1   |   Year: 1   |      |   Campus: University of L'Aquila   |   

Unit Coordinator: Roberta Antonetti

Aims:

The aim of this course is to provide students with knowledge of Italian language at level A1 (beginner). The course will concentrate on four aspects of communication: speaking, writing, listening and reading on a basic level. Attention is given to correct pronunciation of Italian.
Furthermore, the goals of the course are to enable students to:
- develop the language proficiency required to communicate effectively in Italian at level A1;
- develop awareness of the nature of language and language learning;
- develop transferable skills;
- form a sound base of the skills, language and attitudes required for progression to work or further study, either in Italian or another subject area.
On successful completion of this module, a student should be able to:
- understand familiar words and very basic phrases concerning himself/herself, his/her family and immediate concrete surroundings when people speak slowly and clearly;
- read familiar names, words and very simple sentences, for example on notices and posters or in catalogues;
- interact in a simple way provided the other person is prepared to repeat or rephrase things at a slower rate of speech;
- ask and answer simple questions in areas of immediate need or on very familiar topics;
- use simple phrases and sentences to describe where he/she lives and people he/she knows;
- write short messages, postcards, fill in forms with personal details or a hotel registration form.

Content:

Students will be required to show knowledge and understanding of the broad topic areas listed below. These provide contexts for the acquisition of vocabulary and the study of grammar and structures. 
VOCABULARY AND TOPIC AREAS 
Clothes and accessories; colors; countries, nationalities and languages; culture, customs and celebrations; education e.g. learning institutions, education and training, learning tools, subjects; family and friends; feelings and emotions; food and drink; interests, sports and activities; jobs; measurements e.g. size, shape, weight; numbers (cardinal, ordinal) and money; rooms and furniture; shops and places; street directions; the human body and health e.g. parts of the body, health and illness; time expressions e.g. telling the time, dates, days of the week, months, seasons; travel and transport; weather.

FUNCTIONAL SYLLABUS
Accepting/ thanking; applying for a job; asking about personal information; asking permission; buying and asking prices; describing people and objects; giving opinions; giving instructions; greeting and introducing; inviting, refusing; making suggestions; requesting, offering; talking about likes and dislikes; talking about routines; telling the time.

GRAMMAR SYLLABUS
Sounds and spelling; nouns: gender, number; irregular plural nouns; definite article; indefinite article; adjectives, first group adjectives and second group adjectives, position of adjectives; formal address; pronouns, subject pronouns and direct and indirect object pronouns; possessive adjectives; prepositions simple and prepositions with articles; interrogatives; adverbs; present tense of regular and irregular verbs; modal verbs and the verb sapere; present perfect and some verbs with an irregular past participle.

Pre-requisites:

There are no prerequisites for attending the course.

Reading list:

"Dieci A1" - Alma edizioni, Firenze 2019.
“New Italian grammar in practice”, Alma edizioni, Firenze 2015.

ECTS Credits: 9   |   Semester: 2   |   Year: 1   |      |   Campus: University of L'Aquila   |   Language: English

Unit Coordinator: Mariapia Palombaro, Gennaro Ciampa

Aims:

Knowledge of basic topics of Functional Analysis, functional spaces and Lebesgue integral.
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.

Content:

• Metric spaces, normed linear spaces.
• 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 measure and integration. Limit exchange theorems. Lp spaces. Hilbert spaces.
• Introduction to the theory of linear bounded operators on Banach spaces. Bounded operators.
• Complex numbers. Sequences. Elementary functions of complex numbers. Limits, continuity. Differentiation. Analytic functions. Harmonic functions.
• Contour integrals. Cauchy's Theorem. Cauchy's integral formula. Maximum modulus theorem. Liouville's theorem. Morera's theorem.
• Series representation of analytic functions. Taylor's theorem. Laurent's series and classiScation of singularities.
• Calculus of residues. The residue theorem. Application in evaluation of integrals on the real line and Principal Value. The logarithmic residue, Rouche's theorem.
• Fourier transform for L^1 functions. Applications. Fourier transform for L^2 functions. Plancherel theorem.
• Laplace transform and applications.

Pre-requisites:

Knowledge of all topics treated the Mathematical Analysis courses in the first and second year: real functions of real variables, limits, differentiation, integration; sequences and series of funcions; ordinary differential equations

Reading list:

Additional info:

Teaching methods

Lectures and tutorials

Assessment methods

Written and possibly oral exam

ECTS Credits: 3   |   Semester: 2   |   Year: 1   |      |   Campus: University of L'Aquila   |   Language: English

Unit Coordinator: Antonio Cicone

Aims:

Learning Objectives

The aim of this course is to provide the student with knowledge of
Parallel Computing and the ability to analyze theoretical properties and
design mathematical software for high-performance computation.

Learning Outcomes

On successful completion of this module, the student should:
- have profound knowledge and understanding of the most relevant
numerical methods for numerical computation and the design of accurate
and highly performant mathematical software;
- demonstrate skills in choosing the most suitable numerical method to
be implemented depending on the problem to be solved. Furthermore,
they should demonstrate ability in developing mathematical software and
in providing its theoretical analysis;
- demonstrate the ability to read and understand other texts on related
topics.

Content:

NUMERICAL INTEGRATION
Quadrature Formulae, The Midpoint or Rectangle Formula, The
Trapezoidal Formula, The Cavalieri-Simpson Formula.
FOURIER TRANSFORM
Orthogonal Polynomials in Approximation Theory, Fourier Trigonometric
Polynomials, The Fast Fourier Transform.
NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS
The Cauchy Problem, One-Step Numerical Methods, Analysis of One-Step Methods, The Zero-Stability, Convergence Analysis, The Absolute
Stability, Difference Equations, Runge-Kutta Methods, Derivation of an
Explicit RK Method, Systems of ODEs.

Pre-requisites:

Basic Numerical Analysis and Linear Algebra.
Contents

Reading list:

A. Quarteroni, R. Sacco, F. Saleri, P. Gervasio, Numerical Mathematics,
Springer (2014).
The book in pdf is available for all students of the University at
https://link-springer-com.univaq.clas.cineca.it/book/10.1007/978-0-387-22750-4

ECTS Credits: 3   |   Semester: 2   |   Year: 1   |      |   Campus: University of L'Aquila   |   Language: English

Unit Coordinator: Antonio Cicone

Aims:

Learning Objectives

The aim of this course is to provide the student with knowledge of
Parallel Computing and the ability to analyze theoretical properties and
design mathematical software for high-performance computation.

Learning Outcomes

On successful completion of this module, the student should:
- have profound knowledge and understanding of the most relevant
numerical methods for numerical computation and the design of accurate
and highly performant mathematical software;
- demonstrate skills in choosing the most suitable numerical method to
be implemented depending on the problem to be solved. Furthermore,
they should demonstrate ability in developing mathematical software and
in providing its theoretical analysis;
- demonstrate the ability to read and understand other texts on related
topics.

Content:

Introduction to Parallel Computing
Motivation and Need for Parallel Computing
Types of Parallelism (Data, Task, and Instruction)
Parallel Architectures and Models
Parallel Programming Paradigms (e.g., Shared Memory, Distributed Memory)
Parallel Algorithms and Analysis
Parallel Performance Metrics and Analysis
Parallel Computing Platforms (e.g., Multicore CPUs, GPUs, Clusters)
Parallel Programming Languages and Libraries (e.g., OpenMP, MPI, CUDA)
Parallel Software Development Tools and Environments
Synchronization and Communication in Parallel Computing
Load Balancing Techniques in Parallel Computing
Parallel I/O and File Systems
Parallelism in Scientific Computing and Engineering Applications
Parallelism in Data Science and Big Data Analytics
Parallelism in Web and Cloud Computing
Parallelism in High-Performance Computing (HPC) and Supercomputing
Parallelism and Energy Efficiency
Parallelism and Scalability
Case Studies of Parallel Computing Applications

Pre-requisites:

Basic Numerical Analysis and Linear Algebra.

Reading list:

A. Quarteroni, R. Sacco, F. Saleri, P. Gervasio, Numerical Mathematics,
Springer (2014).

The book in pdf is available for all students of the University at
https://link-springer-com.univaq.clas.cineca.it/book/10.1007/978-0-387-22750-4

ECTS Credits: 3   |   Semester: 2   |   Year: 1   |      |   Campus: University of L'Aquila   |   

Unit Coordinator: Roberta Antonetti

Aims:

Pre-requisites:

Reading list:

"Dieci A2" - Alma edizioni, Firenze 2019. 
“New Italian grammar in practice”, Alma edizioni, Firenze 2015.