 1 Year
 Scientific Computing Pathway
 University of L'Aquila Place
 66 (min.*) ECTS Credits
 Read here Qualification
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
 Lecturer 1 Corrado Lattanzio

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 CauchyKovalevsky 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. Wellposedness 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. Onedimensional wave equation. D’Alembert formula. Characteristic parallelogram. Non homogeneous equation and Duhamel’s method. Multidimensional 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 initialboundary value problem. Separation of variables. Reflection of waves.
THE HEAT EQUATION. Heat conduction in a finite rod. Maximum principle and applications. Solution of the initialboundary 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 0486652513
L.C. Evans, Partial Differential Equations. American Mathematical Society. 2010. Second edition, ISBN13: 9780821849743
S. Salsa, Partial Differential Equations in Actions: from Modelling to Theory. SpringerVerlag Italia. 2008. ISBN 9788847007512
W. A. Strauss, Partial Differential Equations, Student Solutions Manual: An Introduction. John Wiley & Sons, LTD. 2008. Second edition, ISBN13: 9780470260715
W. A. Strauss, Partial Differential Equations: an introduction. John Wiley & Sons, LTD. 2007. Second edition, ISBN13 9780470054567
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Control Systems (6 credits)
 ECTS credits 6
 Semester 1
 University University of L'Aquila
 Lecturer 1 Alessandro D'Innocenzo

Objectives
The course provides the basic methodologies for modeling, analysis and controller design for continuoustime linear timeinvariant 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 steadystate responses. Polynomial and sinusoidal disturbances rejection.
The RouthHurwitz 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
 Lecturer 1 Bruno Rubino

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. HartmanGrobman 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, PoincaréBendixson 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 doubleHopf, DivergenceHopf, Doublezero bifurcation.

Books
Lawrence Perko, Differential equations and dynamical systems, SpringerVerlag, 2001
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Functional Analysis in Applied Mathematics and Engineering (9 credits)
 ECTS credits 9
 Semester 1
 University University of L'Aquila
 Lecturer 1 Marco Di Francesco

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
 Lecturer 1 Tommaso Ciotti
 Lecturer 2 Cinzia Di Martino

Prerequisites
None

Objectives
The aim of this course is to provide the student with knowledge of fundamental grammatical structures, vocabulary and comunicative structures of the Italian language. Notions of Italian culture will be given during the course.
On successful completion of this module, the student should be able to:
 recognise familiar words and simple expressions about himself, his family and his background;
 understand simple names and words, such as ads, catalogues, billboards;
 easily interact with an interlocutor, ask questions and give answers on familiar topics or immediate needs;
 describe the place where he lives and people he knows;
 write a short and simple text; fill forms with personal information (name, nationality, address, etc.) 
Topics
The aim of the course is to develop the following skills:
 AURAL COMPREHENSION: to understand a short speech, with long breaks and a slow pronunciation;
 WRITTEN COMPREHENSION: to understand simple and short texts, understanding names, wellknown words and expressions;
 ORAL EXPRESSION: to say easy and isolated phrases about people and places;
 WRITTEN EXPRESSION: to write easy and isolated phrases;
 ORAL INTERACTION: to interact in an easy way and slowly. Answer and ask easy questions, expressing immediate needs;
 WRITTEN INTERACTION: to ask and give personal informations.
During the course the following socialcommunicative actions will be analyzed and developed:
1. introduce himself;
2. ask and give personal informations;
3. greet and answer to greetings;
4. begin, maintain and finish a conversation;
5. give thanks and answer to thanks;
6. accept or refuse a invitation  invite someone;
7. search, ask and give information in everyday situations;
8. express desires;
9. introduce someone;
10. describe people, objects and places;
11. put events in a timeline;
12. ask and give a permission to do something;
13. give and understand easy instructions;
14. seek clarification and give an explanation.
The following grammar skills will be analyzed and developed:
Articoli determinativi e indeterminativi;
Aggettivi qualificativi di alta frequenza;
Aggettivi e pronomi possessivi e dimostrativi;
Pronomi personali soggetto;
Pronomi personali complemento in espressioni fisse;
Quantificatori;
Verbi di altissima frequenza;
Verbi servili;
Indicativo presente;
Passato prossimo (solo ricezione);
Condizionale in formule fisse di frequenza;
Congiunzioni (e additivo);
Principali preposizioni semplici in espressioni fisse (a casa; con le mani; di mio fratello);
Locuzioni avverbiali di alta frequenza (causa, tempo, luogo);
The following semantic fields will be analyzed:
 Family
 House
 Furniture
 Food and drinks
 Nationalities
 Job
 Free time
 Offices, shops, city 
Books
The following textbook will be used during the course:
"Nuovo Espresso 1", Alma Edizioni, Firenze 2014, lessons 16.
Further learning material will be provided during the lessons. 
More information
For further study and exercises:
http://italianoperstranieri.mondadorieducation.it
http://italianoperstranieri.loescher.it
https://www.almaedizioni.it/it/almatv/
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Semester 2

Big data models and algorithms (3 credits)
 ECTS credits 3
 Semester 2
 University University of L'Aquila
 Lecturer 1 Mattia D'Emidio

Prerequisites
Basic courses on design and analysis of algorithms and data structures. Mathematical and programming maturity. Fundamentals of data analysis. 
Objectives
Upon completion of this course the student will have reliably demonstrated the ability to design, analyze and implement algorithms for massive data sets using stateoftheart algorithmic techniques in the area. Furthermore, the student will be able to understand: i) storage strategies that are suited for largescale datasets (e.g. distributed, unstructured); ii) alternative processing models that are relevant to big data; iii) fundamentals of largescale data mining. Finally, the student will acquire basic knowledge of experimental algorithmic techniques and data analysis. 
Topics
LargeScale Data Mining Models, Algorithms, Storage Techniques for Massive Datasets 
Books
J. Leskovec, A. Rajaraman, J. D. Ullman. Mining of Massive Datasets. 2nd Edition.
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Data analytics and Data mining (6 credits)
 ECTS credits 6
 Semester 2
 University University of L'Aquila
 Lecturer 1 Mattia D'Emidio
 Lecturer 2 Giovanni Felici
 Lecturer 3 Fabrizio Rossi

Prerequisites
Basic programming skills, introductory statistic.

Objectives
Learn fundamental techniques to examine raw data with the purpose of drawing datadriven decisions. The course deals with the main methods for supervised and nonsupervised learning. Particular attention will be given to the statistical foundations of learning. The most established techniques to extract information from data to orient decisions will be treated both in their theoretical motivations and in their practical details. Open source tools will support the course step by step, providing continuous verification of the material.

Topics
Introduction to analytics. Data collection, cleaning and preprocessing. Exploratory Data Analysis and Visualization. Statistical inference and regression models. Optimization formulations of data analysis and learning problems. Statistical foundations of learning. Clustering and Principal Component Analysis. Decision trees  Logic methods. Support vector machines  Feature selection and extraction. Methods and tools for supervised learning.

Books
Python Data Science Handbook. Essential Tools for Working with Data
Jake VanderPlas
O'Reilly Media (2016)
An Introduction to Statistical Learning
Gareth James, Daniela Witten, Trevor Hastie, Robert Tibshirani Springer Texts in Statistics (2015)
An Introduction to R
Version 3.4.1 (2017)
W. N. Venables, D. M. Smith and the R Core Team
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Parallel computing (6 credits)
 ECTS credits 6
 Semester 2
 University University of L'Aquila
 Lecturer 1 Adriano Festa
 Lecturer 2 Protasov Vladimir

Topics
Linux/Unix OS and tools;
Basic Fortran (or C);
HPC architecture;
System Scheduler;
Message Passing Interface;
OpenMP;
GPU computing;
Applications: linear algebra, PDEs, ODEs.
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Italian Language and Culture for foreigners (level A2) (3 credits)
 ECTS credits 3
 Semester 2
 University University of L'Aquila
 Lecturer 1 Tommaso Ciotti
 Lecturer 2 Cinzia Di Martino

Prerequisites
Italian language and culture  level A1

Objectives
The aim of this course is to provide the student with knowledge of preintermediate grammatical structures, vocabulary and comunicative structures of the Italian language. Many notions of Italian culture will be given during the course.
On successful completion of this module, the student should be able to:
 recognize words and expression of common usage relating to context concerning himself (for instance basic informations concerning himself and his family, shopping, local geography and job). Catch the essence of short, easy and clear messages and ads.
 read short and easy texts finding specific informations in materials of everyday use such as ads, plans, menus and timetables. Understand short and easy personal correspondence;
comunicate in simple tasks requiring only an exchange of information concerning usual activities and usual topics. Take part to short conversations, even if usually he doesn't understand what he needs to carry on the conversation;
use expressions and phrases to describe his family and other people, his living conditions and his current job;
write simple notes and short messages on topics concerning immediate needs.
Write a very simple personal letter (for instance to thank somebody). 
Topics
The aim of the course is to develop the following skills:
 AURAL COMPREHENSION: to understand everything necessary aimed at the satisfaction of needs of a concrete type, provided that the speaker speaks slowly and clearly.
 WRITTEN COMPREHENSION: to understand simple and short texts of familiar content and concrete type, formulated in a common vocabulary of everyday life and job;
 ORAL EXPRESSION: to describe and introduce in an easy way people, living conditions, daily tasks, to say what he likes or dislikes etc. using expressions and phrases linked together in order to create a list;
 WRITTEN EXPRESSION: to write expressions and phrases linked by easy connective as "e", "ma" and "perché";
 ORAL INTERACTION: to interact with ease in structured situations and short conversations with the collaboration of the interlocutor. To take part in easy routine conversations; to ask and answer to simple questions; to share ideas and information about familiar topics in everyday situations;
 WRITTEN INTERACTION: to write short and simple notes about immediate needs using conventional formulae.
The following socialcommunicative actions will be analyzed and developed:
1. introduce himself;
2. ask and give personal informations;
3. greet and answer to greetings;
4. begin, maintain and finish a conversation;
5. give thanks and answer to thanks;
6. accept or refuse a invitation  invite someone;
7. search, ask and give information in everyday situations;
8. express desires;
9. introduce someone;
10. describe people, objects and places;
11. describe a place and put an element in a place;
12. speak about himself and ask questions about past events;
13. put events in a timeline;
14. express and ask questions about time and dates;
15. put events in a sequence;
16. express moods, feelings and emotions;
17. express the wish to do something;
18. ask and give the permission to do something;
19. order or ban somebody to do something;
20. give and understand simple instructions;
21.give an explanation;
22. express judgments and personal opinions;
23. make simple assumptions.
The following grammar skills will be analyzed and developed:
opposizione articolo determinativo e indeterminativo;
aggettivi qualificativi di alta frequenza;
aggettivi numerali, cardinali e ordinali;
aggettivi e pronomi possessivi e dimostrativi;
pronomi personali soggetto;
pronomi personali complemento;
uso appropriato del che;
quantificatori;
verbi di alta frequenza;
verbi servili;
indicativo presente;
passato prossimo;
futuro semplice;
imperfetto (ricezione);
condizionale in formule fisse di richiesta;
congiunzioni (e additivo, ma avversativo, o disgiuntivo);
principali preposizioni semplici in espressioni fisse (a casa; con le mani; di mio fratello);
locuzioni avverbiali di alta frequenza (causa, tempo, luogo);
frasi impersonali;
verbi riflessivi;
verbi zerovalenti;
subordinate causali e temporali.
The following semantic fields will be analyzed:
family
house
forniture
food and drinks
nationalities
job
free time
offices, shops, city
natural events
university
body and health 
Books
The following textbook will be used during the course:
"Nuovo Espresso 1", Alma Edizioni, Firenze 2014, lessons 710.
Further learning material will be provided during the lessons.
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Electives

Combinatorics and cryptography (6 credits)
 ECTS credits 6
 Semester 2
 University University of L'Aquila
 Lecturer 1 Riccardo Aragona

Prerequisites
Basics of Algebra

Objectives
The course aims to provide the arithmetical and algebraic background and the basic techniques for symmetric cryptography, publickey cryptography and error correction coding.
At the end of the course the student should be able to understand the fundamental concepts of modular arithmetic and finite fields and to be able to apply them to the study of basic cryptographic techniques and basic error correcting codes described during the course.
On successful completion of this course, the student should
1) have knowledge of the basic techniques of cryptography and error correction codes introduced;
2) understand the fundamental concepts of arithmetic and algebra and their interactions and be aware of their applications in cryptography and coding theory;
3) have knowledge of how to apply the notions of arithmetic and algebra to the study of cryptographic techniques and error correction codes;
4) understand and analyze the mathematical and application problems underlying the cryptographic schemes studied;
5) demonstrate skill in reasoning and arithmetic calculation and ability to understand the proofs of the theoretical and cryptographic results studied;
6) demonstrate ability to read and understand other scientific texts on related subjects. 
Topics
Topics of the module include:
Overview of Cryptography and attack scenarios.
Elementary arithmetics: Integers, divisibility, prime numbers, Euclidean division and g.c.d., Bezout's Identity, Eucledian Algorithm, Extended Eucledian Algorithm, Congruence classes, Chinese remainder theorem, cyclic and abelian
groups, Lagrange theorem, Fermat's Little Theorem, Euler theorem, the structure of invertible classes mod N, Fields with p elements, Primitive Roots, polynomials, Euclidean division and g.c.d., Congruence classes of polynomials, Finite fields, primitive elements and polynomials.
Introduction to Probability. Probability and Ciphers, Introduction to Shannon Theory, Perfect secrecy, Shannon Theorem, one time pad, Substitution Ciphers.
Symmetric Cryptography, Feistel Networks, Substitution Permutation Networks, Advanced Encryption Standard  Rijandel.
Group generated by a round functions and Imprimitive attack.
Differential cryptanalysis, example of differential cryptanalysis on a small variant of PRESENT.
Publickey Cryptography, Discrete logarithms problem (DLP), Computational DiffieHellmann Problem (DHP), between DLP and DHP, DiffieHelman Key exchange.
RSA Algorithm, Trial Division, Fermat's test, Miller Rabin Test, AKS primality test, Factoring and factoringrelated problems (SQRROOT and RSA Problem), Security of RSA, Coppersmith Theorem, Hastad Attack, Wiener Attack.
Hash function, Digital signatures, RSA signatures, Hashing and signing, DSA.
Error correcting codes, Binary block codes, distance and correction of errors, singleton bound, Hamming bound, GilbertVarshamov bound, linear codes, Syndrome decoding, dual codes, Hamming codes, Simplex codes,cyclic codes, ReedSolomon codes. 
Books
1) Trappe and Washington, "Introduction to Cryptography with Coding Theory", second edition, Pearson Pretince Hall, 2006;
2) Smart, "Cryptography made simple", Information Security and Cryptography, Springer, 2016;
3) Heys, "A Tutorial on Linear and Differential Cryptanalysis",
https://www.engr.mun.ca/~howard/PAPERS/ldc_tutorial.pdf  Link https://www.disim.univaq.it/didattica/content
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Complex Analysis (6 credits)
 ECTS credits 6
 Semester 2
 University University of L'Aquila
 Lecturer 1 Corrado Lattanzio

Prerequisites
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

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. 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 classification 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. 
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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Discrete and continuum mechanics with applications (9 credits)
 ECTS credits 9
 Semester 2
 University University of L'Aquila
 Lecturer 1 Francesco Dell'Isola

Prerequisites
The student must know the basics of algebra and mathematical analysis. He must also have a basic knowledge of the mechanics of the material point and of the rigid body.

Objectives

The goal of this course is to provide the concepts of solid Mechanics using the notion of continuum models as well as of discrete models. On successful completion of this module, the student should know the fundaments of the Analytic Mechanics and how to apply it to solve the problem of an elastically deformable body.


Topics
Space of configurations: the finite dimensional case. Space of configurations: the infinite dimensional case. The configuration space for the EulerBernoulli beam. Elements of model theory. The problem of the determination of the motion for finite dimensional systems. The principle of minimum action, Lagrangians: the finite dimensional case. Deduction of EulerLagrange conditions for the finite dimensional case. Numerical methods for solving ordinary differential equations. The principle of minimum action for infinite dimensional systems. Space of threedimensional Continuous configurations. Euler theory of the deformable beam. Stable equilibrium as a minimum of energy. The concept of constraint for finite dimensional systems: the Dini theorem. Discretization of infinitedimensional models. Timoshenko beam. Wave propagation: applications.

Books
Gurtin, Morton E. An introduction to continuum mechanics. Vol. 158. Academic press, 1982.
Sanjay Govindjee. A First Course on Variational Methods in Structural Mechanics and Engineering. University of California, Berkeley.
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Network optimization (6 credits)
 ECTS credits 6
 Semester 2
 University University of L'Aquila
 Lecturer 1 Fabrizio Rossi

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 branchandbound 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. Branchandcut 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.
 Link http://www.di.univaq.it/rossi/networkdesign.html
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Numerical methods for linear algebra and optimisation (6 credits)
 ECTS credits 6
 Semester 2
 University University of L'Aquila
 Lecturer 1 Raffaele D'Ambrosio

Prerequisites
Basic Numerical Analysis and Linear Algebra.

Objectives
The Aim of this course is to provide the student with knowledge of Numerical Linear Algebra and Numerical Optimisation and ability to analyze theoretical properties and design mathematical software based on the proposed schemes.
On successful completion of this module, the student should
 have profound knowledge and understanding of the most relevant numerical methods for Numerical Linear Algebra and Numerical Optimisation and the design of accurate and efficient mathematical software;
 demonstrate skills in choosing the most suitable method in relation to the problem to be solved and ability to provide theoretical analysis and mathematical software based on the proposed schemes;
 demonstrate capacity to read and understand other texts on the related topics. 
Topics
MATRIX FACTORIZATIONS
LU decomposition, Cholesky decomposition. Singular value decomposition and applications (image processing, recommender systems). QR decomposition and least squares. Householder triangularization. Conditioning and stability in the case of linear systems.
EIGENVALUE PROBLEMS
Approximation of the spectral radius. Power method and its variants. Reduction to Hessemberg form. Rayleigh quotient, inverse iteration. QR algorithm with and without shift. Jacobi method. GivensHouseholder algorithm. Google PageRank.
ITERATIVE METHODS FOR LINEAR SYSTEMS
Overview of iterative methods. Arnold iterations, Krylov iterations. GMRES. Lanczos method. Conjugate gradient. Preconditioners. Preconditioned conjugate gradient.
NUMERICAL OPTIMISATION
Continuous versus discrete optimization. Constrained and unconstrained optimization. Global and local optimization. Overview of optimization algorithms. Convexity.
Line search methods. Convergence of line search methods. Rate of convergence. Steepest descent, quasiNewton methods. Steplength selection algorithms. Trust region methods. Cauchy point and related algorithms. Dogleg method. Global convergence. Algorithms based on nearly exact solutions. Conjugate gradient methods. Basic properties. Rate of convergence. Preconditioning. Nonlinear conjugate gradient methods: FletcherReeves method, PolakRibiere method. 
Books
 J. Stoer, R. Bulirsch, Introduction to numerical analysis , Springer. 2002.
 J. Nocedal, S. J. Wright, Numerical optimization , Springer. 1999.
 A. Quarteroni, R. Sacco, F. Saleri, P. Gervasio, Numerical Mathematics, Springer (2014).
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Stochastic processes (6 credits)
 ECTS credits 6
 Semester 2
 University University of L'Aquila
 Lecturer 1 DIMITRIOS TSAGKAROGIANNIS

Prerequisites
Probability theory and Real Analysis

Objectives
Students should:
1. Develop the skills to model simple real problems and propose a solution;
2. Solve theoretical problems, using the appropriate mathematical tools;
3. Read the related texts and gain access to more advanced courses;
4. Get a first flavour of the relevant research problems. 
Topics
1. Discrete time processes: Markov chains in finite and countable space, limiting distribution;
2. Continuous time processes: density and distribution of intoevent time for Poisson process, applications and extensions: e.g. birthanddeath processes, queues, epidemics;
3. Renewal processes: ordinary renewal process, renewal theorem, equilibrium
renewal process, application to queues;
4. Wiener processes and basic stochastic calculus: basic definitions and properties, It\^o's formula, Stochastic Differential Equations. 
Books
1. Markov Chains, J.R. Norris, Cambridge University Press;
2. Introduction to Stochastic Processes, G. Lawler, Chapman & Hall;
3. Basic Stochastic Processes, A Course Through Exercises, Z. Brzezniak and T. Zastawniak, Springer;
4. Probability and Random Processes, G. Grimmett and D. Stirzaker, 3rd Edition, Oxford University Press;
5. A first look at Rigorous Probability Theory, J. Rosenthal, World Scientific.
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