METODI DIRETTI PER SISTEMI LINEARI: IL METODO DI GAUSS, LE FATTORIZZAZIONI LU, DI CHOLESKY E QR. METODI ITERATIVI PER SISTEMI LINEARI. METODI ITERATIVI PER EQUAZIONI SCALARI: METODI DI BISEZIONE, DI SOSTITUZIONI SUCCESSIVE, DI NEWTON E DERIVATI. APPROSSIMAZIONE DI FUNZIONI: INTERPOLAZIONE POLINOMIALE DI LAGRANGE E NEWTON, SEMPLICE E COMPOSITA. POLINOMIO DI HERMITE. APPROSSIMAZIONE DI ERRORE QUADRATICO MINIMO. TEORIA GENERALE DELLE FORMULE DI QUADRATURA INTERPOLATORIE. QUADRATURE DI NEWTON-COTES SEMPLICI E COMPOSITE. QUADRATURE GAUSSIANE.

[1] ALFIO QUARTERONI, RICCARDO SACCO, FAUSTO SALERI, MATEMATICA NUMERICA, SPRINGER, 1998;
[2] VALERIANO COMINCIOLI, ANALISI NUMERICA: METODI, MODELLI, APPLICAZIONI, APOGEO, 2005;
[3] ROBERTO FERRETTI, APPUNTI DEL CORSO DI ANALISI NUMERICA;
[5] ROBERTO FERRETTI, ESERCIZI D'ESAME DI ANALISI NUMERICA.

GEOMETRIA EUCLIDEA E GEOMETRIE NON-EUCLIDEE. GEOMETRIE LOCALMENTE EUCLIDEE. ISOMETRIE DEL PIANO. GRUPPI DISCRETI DI ISOMETRIE DEL PIANO. GEOMETRIA IPERBOLICA

R. TRUDEAU: “LA RIVOLUZIONE NON EUCLIDEA”- BORINGHIERI ED.
NIKULIN, SHAFAREVICH: “GEOMETRY AND GROUPS”; SPRINGER ED.

THE COURSE COMPUTER SCIENCE 2 (IN2 - MODELS OF COMPUTATION) IS DEVOTED TO THE INTRODUCTION OF MATHEMATICAL ASPECTS OF THE CONCEPT OF COMPUTATION, AND TO THE STUDY OF RELATIONS BETWEEN DIFFERENT MODELS OF COMPUTATION AND BETWEEN DIFFERENT PROGRAMMING STYLES. IN PARTICULAR, AN INTRODUCTION TO OBJECT ORIENTED PROGRAMMING IS GIVEN DURING THE COURSE.

[1] DEHORNOY, P., COMPLEXITE' ET DECIDABILITE'. SPRINGER-VERLAG, (1993).
[2] KRIVINE, J.-L., LAMBDA CALCULUS: TYPES AND MODELS. ‪ELLIS HORWOOD, (1993).
[3] SIPSER,M., INTRODUCTION TO THE THEORY OF COMPUTATION.THOMSON COURSE TECHNOLOGY, (2006).
[4] GABBRIELLI, M., MARTINI, S., LINGUAGGI DI PROGRAMMAZIONE: PRINCIPI E PARADIGMI. MCGRAW-HILL, (2011).

FURTHER TEXTBOOKS:

[4] AHO, HOPCROFT, ULLMAN, DESIGN AND ANALYSIS OF COMPUTER ALGORITHMS. ADDISON-WESLEY PUB. CO., (1974).
[5] AUSIELLO, G., GAMBOSI, G., D'AMORE F., LINGUAGGI, MODELLI, COMPLESSITA'. FRANCO ANGELI (2003).
[6] A. BERNASCONI, B. CODENOTTI, INTRODUZIONE ALLA COMPLESSITA' COMPUTAZIONALE. SPRINGER-VERLAG, (1998).
[7] SETHI, R., PROGRAMMING LANGUAGES: CONCEPTS AND CONSTRUCTS. ADDISON-WESLEY (ED. ITALIANA ZANICHELLI), (1996).
[8] HERMES, H., ENUMERABILITY, DECIDABILITY, COMPUTABILITY. DIE GRUNDLEHREN DER MATHEMATICHENWISSENSHAFTEN IN EINZELDARSTELLUNGEN, N. 127, SPRINGER-VERLAG, (1969).
[9] DARNELL, P. A. AND MARGOLIS, P. E., C A SOFTWARE ENGINEREEING APPROACH. SPRINGER-VERLAG, (1996).

Division, Factorization, Some Elementary Properties of Primes, Some Results and Problems Concerning Primes.
ARITHMETIC FUNCTIONS: The Divisor Function. The Moebius Function. The Euler Function. Dirichlet Convolution
CONGRUENCES: Sets of Residues,Some Interesting Congruences, Some Linear Congruences, Some Polynomial Congruences, Primitive Roots, the Theorem of Gauss.
QUADRATIC RESIDUES: The Legendre Symbol. Quadratic Reciprocity. The Jacobi Symbol.The Distribution of Quadratic Residues.
SUMS OF INTEGER SQUARES: Sums of Two Squares. Number of Representations. Sums of Four Squares. Sums of Three Squares.
ELEMENTARY PRIME NUMBER THEORY: Euclid's Theorem Revisited. The Von Mangoldt Function. Tchebycheff's Theorem. Some Results of Mertens

Chen, W; ELEMENTARY NUMBER THEORY. https://rutherglen.science.mq.edu.au/wchen/lnentfolder/lnent.html
Chowdhury, F.; Chowdhury, M. R. Essentials of Number Theory. Pi Publications, Dhaka, Bangladesh, 2005. ISBN 984-32-2836-7
Hardy, G. H.; Wright, E. M. An introduction to the theory of numbers. Fifth edition. The Clarendon Press, Oxford University Press, New York, 1979. xvi+426 pp. ISBN: 0-19-853170-2; 0-19-853171-0
Davenport, H. Aritmetica superiore. Un'introduzione alla teoria dei numeri. Editore: Zanichelli, 1994. 199 pp. ISBN: 8808091546
Gioia, A. A. The theory of numbers. An introduction. Reprint of the 1970 original. Dover Publications, Inc., Mineola, NY, 2001. xii+207 pp. ISBN: 0-486-41449-3
Rosen, K. H. Elementary number theory and its applications. Fourth edition. Addison-Wesley, Reading, MA, 2000. xviii+638 pp. ISBN: 0-201-87073-8
Tattersall, J. J. Elementary number theory in nine chapters. Cambridge University Press, Cambridge, 1999. viii+407 pp. ISBN: 0-521-58531-7

Fields extensions. Galois extensions and their Galois groups. Galois correspondence. Solvable groups and solvable polynomials. Constructibility.

[1]. S. GABELLI, TEORIA DELLE EQUAZIONI E TEORIA DI GALOIS, UNITEXT 38, SPRINGER ITALIA, 2008
[2] J. S. MILNE.FIELDS AND GALOIS THEORY. COURSE NOTES HTTP://WWW.JMILNE.ORG/MATH/ 2003

TEORIA DEI RIVESTIMENTI. ESISTENZA DEL RIVESTIMENTO UNIVERSALE. OMOLOGIA SINGOLARE. INVARIANZA PER OMEOMORFISMO E PER OMOTOPIA. LA SUCCESSIONE DI MAYER-VIETORIS. APPLICAZIONI. ELEMENTI DI TOPOLOGIA DIFFERENZIALE. VARIETA' E APPLICAZIONI LISCE. CAMPI TANGENTI E CARATTERISTICA DI EULERO. ORIENTABILITA'.

E. SERNESI, GEOMETRIA 2, ZANICHELLI
J.M . LEE, INTRODUCTION TO TOPOLOGICAL MANIFOLDS, GRADUATE TEXTS IN MATHEMATICS N. 202, SPRINGER.
WILLIAM S. MASSEY ALGEBRAIC TOPOLOGY, AN INTRODUCTION SPRINGER GTM (1967)

ELEMENTI DI TEORIA DELLA MISURA. SPAZI DI PROBABILITÀ ASTRATTI. LEMMI DI BOREL-CANTELLI. VARIABILI ALEATORIE CONTINUE: LEGGI CONGIUNTE E MARGINALI, INDIPENDENZA, LEGGI CONDIZIONALI. MEDIA E MEDIA CONDIZIONALE. MOMENTI, VARIANZA E COVARIANZA. DISUGUAGLIANZE. CONVERGENZA QUASI CERTA E IN PROBABILITÀ. LEGGI DEI GRANDI NUMERI. CONVERGENZA IN DISTRIBUZIONE. FUNZIONI CARATTERISTICHE E TEOREMA DI LÉVY. TEOREMA LIMITE CENTRALE. MARTINGALE. PROCESSI DI RAMIFICAZIONE

1] D.WILLIAMS, PROBABILITY WITH MARTINGALES. CAMBRIDGE MATHEMATICAL TEXTBOOKS, (1991).

Reminders on complex numbers: algebraic and topological properties, the compactification to a point of the field of complex numbers and its identification with the sphere via stereographic projection, the polar representation of complex numbers.
Holomorphic functions: examples (polynomials, rational functions, exponential), non-examples (the complex conjugation), Cauchy-Riemann equations.
The algebra of the formal powers series: order of a power series, inverse, composition, derivative,
The quotient field of the domain of the formal series: the field of Laurent's series, residuals, inverse.
Convergence: punctual, absolute, uniform and uniform on the compact. The geometric series and its convergence properties.
Abel's Theorem for the convergence of power series: convergence radius, absolute and uniform convergence on compacts inside the convergence disk, divergence on the outside, holomorphicity of the limit function, convergence of the derived series.
Series of converging powers and holomorphic functions associated to them. Convergence of the derivative of a convergent series, of the sum and product of convergent series, of the convergent series composition, of the multiplicative inverse and of the inverse by composition.
Analytical functions: local series development of an analytical function; the analytic functions are infinitely derivable and all their derivatives are analytic; sum, product, inverse and composition of analytical functions are analytical; the analytic functions have a local analytic primitive, unique up to a constant, a convergent series defines an analytic function in its convergence disk, the inverse function theorem.
Normal form of an analytic function: each analytical function is locally, up tp translation and change of coordinates, the elevation to a power or is locally constant.
The open function theorem. Criterion of analytical isomorphism. Principle of the local and global maximum.
The zeros of an analytical function are discrete. Locally constant functions are constant in a connected open subset. Proof of the fundamental theorem of algebra (using the principle of the global maximum module).
The integral of a complex continuous function along a piecewise C^1-curve.
Criterion for the existence of a primitive: a continuous function admits a primitive if and only if its integral is zero along any closed curve. Example: the complex exponential as an analytic function.
The integral of uniformly convergent sequences and series.
A holomorphic function in a disk admits a primitive.
The integral of a holomorphic function along a continuous curve (not necessarily C^1).
The homotopy between continuous curves. The homotopic invariance theorem of the integral. Corollary: A holomorphic function on a simply connected open admits a primitive.
The integral (or local) formula of Cauchy (without proof). Cauchy formula for the series expansion of a holomorphic function (without proof). Corollaries: holomorphic functions are analytic, a power series is not holomorphic in at least one point of the boundary of the convergence disk, a whole function with an infinite polynomial expansion is a polynomial, a bounded entire function is constant (theorem of Liouville), integral formula of Cauchy for derivatives of a holomorphic function. Application: proof of the fundamental theorem of algebra using the Liouville theorem.
Proof of the integral (or local) formula of Cauchy. Proof of the Cauchy formula for the series expansion of a holomorphic fuction.
Winding number of a closed curve around a point: definition and examples.
Properties of the winding number of a closed curve: analytical definition, constancy in the connected components of the curve complement.
Homologous zero curves in an open subset. Relationship between homotopy and homology. The global formula of Cauchy: Dixon proof.
Chains and cycles. Integration along a chain. Winding number of a cycle. Zero homologous cycles. The first group of homology (with integral coefficients) of an open and its relation with the fundamental group. Examples.
The global Cauchy formula and Cauchy's Theorem on homological invariance: their equivalence.
The proof of Cauchy's Theorem on homological invariance (taken from the book by Alfhors).
Sequences of holomorphic functions uniformly converging on compacts: holomorphicity of the limit function; the derivative and the integral on a chain can be exchanged with the limit.
Laurent series (infinite) converging and holomorphic functions on a circular annulus.
Isolated singularities: removable singularities, poles, essential singularities. Examples. Rational functions. The theorems (by Riemann and Casorati-Weierstrass) of characterization of isolated singularities.
The residual theorem: local and global version.
Meromorphic functions and their properties. The logarithmic derivative of a meromorphic function and its properties. The principle of the argument. Corollary: number of zeros and poles within a simple closed curve. Application: proof of the fundamental Theorem of Algebra using the principle of the argument.
Roche's theorem. Application: proof of the fundamental Theorem of Algebra using Roche's theorem.
An approach to the problem of classification of domains through the map theorem and the uniformization theorem (without proof). Examples: the disk and the upper half-plane are biolomorphic (Cayley transform), the universal coating of C ^ * is the complex plane and the covering map is the exponential.
The complex projective line as a compactification of the complex plane. The complex projective line is homeomorphic to the sphere through stereographic projection. The linear projective group PGL2 acts on the complex projective line by means of linear (or Moebius) linear transformations.
The group of automorphisms of the complex plane.
Schwarz's Lemma. The group of automorphisms of the unitary disk.
The classification of the subgroups of the automorphisms of the complex plane that act in a free and properly discontinuous manner.
Coating quotients of the complex plane.
Elliptic (or doubly periodic) functions with respect to a lattice. The only holomorphic elliptic functions are constants. Properties of zeros and poles of elliptic functions inside a fundamental domain.
The Weiertrass P function and its derivative P ': definition and convergence properties.
Properties of the P function and its derivative P ': ellipticity, parity / disparity, poles and zeros.
The polynomial relationship between P and P '. All elliptic functions are expressed as rational functions in P and P '. Corollary: the field of elliptic functions.

S. Lang: Complex Analysis. Fourth edition. Graduate Texts in Mathematics, 103. Springer-Verlag, New York, 1999.
L. V. Ahlfors: Complex Analysis. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York, 1978.
R. Shakarchi: Problems and solutions for complex analysis. Springer-Verlag, New York, 1999.

Fields extensions. Galois extensions and their Galois groups. Galois correspondence. Solvable groups and solvable polynomials. Constructibility.

[1]. S. GABELLI, TEORIA DELLE EQUAZIONI E TEORIA DI GALOIS, UNITEXT 38, SPRINGER ITALIA, 2008
[2] J. S. MILNE.FIELDS AND GALOIS THEORY. COURSE NOTES HTTP://WWW.JMILNE.ORG/MATH/ 2003

TEORIA DEI RIVESTIMENTI. ESISTENZA DEL RIVESTIMENTO UNIVERSALE. OMOLOGIA SINGOLARE. INVARIANZA PER OMEOMORFISMO E PER OMOTOPIA. LA SUCCESSIONE DI MAYER-VIETORIS. APPLICAZIONI. ELEMENTI DI TOPOLOGIA DIFFERENZIALE. VARIETA' E APPLICAZIONI LISCE. CAMPI TANGENTI E CARATTERISTICA DI EULERO. ORIENTABILITA'.

E. SERNESI, GEOMETRIA 2, ZANICHELLI
J.M . LEE, INTRODUCTION TO TOPOLOGICAL MANIFOLDS, GRADUATE TEXTS IN MATHEMATICS N. 202, SPRINGER.
WILLIAM S. MASSEY ALGEBRAIC TOPOLOGY, AN INTRODUCTION SPRINGER GTM (1967)

GEOMETRIA EUCLIDEA E GEOMETRIE NON-EUCLIDEE. GEOMETRIE LOCALMENTE EUCLIDEE. ISOMETRIE DEL PIANO. GRUPPI DISCRETI DI ISOMETRIE DEL PIANO. GEOMETRIA IPERBOLICA

R. TRUDEAU: “LA RIVOLUZIONE NON EUCLIDEA”- BORINGHIERI ED.
NIKULIN, SHAFAREVICH: “GEOMETRY AND GROUPS”; SPRINGER ED.

ELEMENTI DI TEORIA DELLA MISURA. SPAZI DI PROBABILITÀ ASTRATTI. LEMMI DI BOREL-CANTELLI. VARIABILI ALEATORIE CONTINUE: LEGGI CONGIUNTE E MARGINALI, INDIPENDENZA, LEGGI CONDIZIONALI. MEDIA E MEDIA CONDIZIONALE. MOMENTI, VARIANZA E COVARIANZA. DISUGUAGLIANZE. CONVERGENZA QUASI CERTA E IN PROBABILITÀ. LEGGI DEI GRANDI NUMERI. CONVERGENZA IN DISTRIBUZIONE. FUNZIONI CARATTERISTICHE E TEOREMA DI LÉVY. TEOREMA LIMITE CENTRALE. MARTINGALE. PROCESSI DI RAMIFICAZIONE

1] D.WILLIAMS, PROBABILITY WITH MARTINGALES. CAMBRIDGE MATHEMATICAL TEXTBOOKS, (1991).

Modules and Algebras; modules and rings of fractions; Integral extensions of rings; Noetherian and Artinian rings; discrete valuation domains and Dedekind domains.

1. M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969.
2. I. Kaplanski, Commutative Rings, The University of Chicago Press, 1974.
3. R. Y. Sharp, Steps in Commutative Algebra, London Mathematical Society Student Texts, 51, Cambridge University Press, 2000.

1. Laplace's equation

Mean-value inequalities, maximum and minimum principle, the Harnack inequality, the Green representation, the Poisson integral, convergence theorems, interior estimates of derivatives, the Dirichlet problem and the method of sub-harmonic functions

2. The classical maximum principle

Weak maximum principle, strong maximum principle, a-priori bounds, symmetry properties and the method of moving planes

3. Poisson's equation and the Newtonian potential

Hölder continuity, the Dirichlet problem for Poisson's equation, Hölder estimates for second derivatives, estimates at the boundary, Hölder estimates for the first derivatives

4. Classical solutions: the Schauder approach

Schauder interior estimates, boundary and global estimates, the Dirichlet problem, interior and boundary regularity

"Elliptic Partial Differential Equations of Second Order", by David Gilbarg and Neil S. Trudinger, Classics in Mathematics,volume 224, Springer-Verlag Berlin Heidelberg, 2nd Edition, 2001

"Elliptic Partial Differential Equations: Second Edition", by Qing Han and Fanghua Lin, Courant Lecture Notes, volume 1, AMS American Mathematical Society, 2nd Edition, 2011

"Partial Differential Equations: Second Edition", by Lawrence C. Evans, Graduate Studies in Mathematics, volume 19, AMS American Mathematical Society, 2010

MECCANICA LAGRANGIANA E SISTEMI VINCOLATI. VARIABILI CICLICHE. COSTANTI DEL MOTO E SIMMETRIE. SISTEMI DI OSCILLATORI LINEARI E PICCOLE OSCILLAZIONI. MECCANICA HAMILTONIANA. FLUSSI HAMILTONIANI. TEOREMA DI LIOUVILLE E DEL RITORNO. TRASFORMAZIONI CANONICHE. FUNZIONI GENERATRICI. METODO DI HAMILTON-JACOBI E VARIABILI AZIONE-ANGOLO. INTRODUZIONE ALLA TEORIA DELLE PERTURBAZIONI.

1] G. GENTILE,INTRODUZIONE AI SISTEMI DINAMICI. 1.EQUAZIONI DIFFERENZIALI ORDINARIE, ANALISI QUALITATIVA E ALCUNE APPLICAZIONI.
DISPONIBILE IN RETE:HTTP://WWW.MAT.UNIROMA3.IT/USERS/GENTILE/2011/TESTO/TESTO.HTML.
2] G. GENTILE,INTRODUZIONE AI SISTEMI DINAMICI. 2.FORMALISMO LAGRANGIANO E HAMILTONIANO.DISPONIBILE IN RETE:HTTP://WWW.MAT.UNIROMA3.IT/USERS/GENTILE/2011/TESTO/TESTO.HTML.
3] G. DELL'ANTONIO, ELEMENTI DI MECCANICA. LIGUORI EDITORE, (1996).
4] V.I. ARNOLD, METODI MATEMATICI DELLA MECCANICA CLASSICA. EDITORI RIUNITI, (1979).
5] G. GALLAVOTTI, MECCANICA ELEMENTARE. BOLLATI-BORINGHIERI, (1980).

Classical theory of algebraic varieties, affine and projective, over algebraically closed fields. Local geometry, normalization. Divisors, linear systems and associated morfisms.

I. SHAFAREVICH BASIC ALGEBRAIC GEOMETRY VOL. 1 SPRINGER-VERLAG, NEW YORK-HEIDELBERG, 1977.
J. HARRIS ALGEBRAIC GEOMETRY (A FIRST COURSE) GRADUATE TEXTS IN MATH. NO. 133. SPRINGER-VERLAG, NEW YORK-HEIDELBERG, 1977.
R. HARTSHORNE ALGEBRAIC GEOMETRY GRADUATE TEXTS IN MATH. NO. 52. SPRINGER-VERLAG, NEW YORK-HEIDELBERG, 1977.

USO DI PROGRAMMI DIDATTICI NELL'INSEGNAMENTO DELLA MATEMATICA: I SOFTWARE CABRI, GEOGEBRA E MATHEMATICA.
COMANDI PER IL CALCOLO SIMBOLICO E NUMERICO, LA VISUALIZZAZIONE DI GRAFICI, CURVE E SUPERFICI E LA LORO ANIMAZIONE AL VARIARE DI PARAMETRI.
ESEMPI DI PROBLEMI: PROPRIETÀ DEI TRIANGOLI NELLA GEOMETRIA EUCLIDEA ED ESEMPI DI GEOMETRIE NON EUCLIDEE, APPROSSIMAZIONE DI PI GRECO E DI ALTRI NUMERI IRRAZIONALI, SOLUZIONI DI EQUAZIONI E DISEQUAZIONI, SOLUZIONI DI SISTEMI, DETERMINAZIONE E VISUALIZZAZIONE DI PARTICOLARI LUOGHI GEOMETRICI, DERIVATA DI UNA FUNZIONE, CALCOLO APPROSSIMATO DI AREE.

DISPENSE DEL DOCENTE SU UN ELENCO DI PROBLEMI DA VISUALIZZARE E RISOLVERE (SIMULANDO UN LABORATORIO SCOLASTICO) CON L'AIUTO DEL SOFTWARE MATHEMATICA.
PER APPROFONDIMENTI SULLA VISUALIZZAZIONE CON MATHEMATICA DI CURVE E SUPERFICI:
RENZO CADDEO, ALFRED GRAY LEZIONI DI GEOMETRIA DIFFERENZIALE - CURVE E SUPERFICI VOL. 1,
ED. CUEC (COOPERATIVA UNIVERSITARIA EDITRICE CAGLIARITANA)

-- TEORIA DELLA CARDINALITÀ. ALCUNI PARADOSSI CLASSICI. INSIEMI NUMERABILI. INSIEMI INFINITI NON NUMERABILI. TEOREMI DI CANTOR. TEOREMA DI CANTOR-BERNSTEIN.
-- ANELLI BOOLEANI. ALGEBRE DI BOOLE, CAMPI DI INSIEMI, SPAZI BOOLEANI E RETICOLI. TEOREMI DI RAPPRESENTAZIONE. APPLICAZIONI ALLA LOGICA SIMBOLICA ED AI CIRCUITI ELETTRICI
-- TEORIA DELLA DIVISIBILITÀ IN DOMINI (ANELLI COMMUTATIVI UNITARI, PRIVI DI DIVISORI DELLO ZERO). FATTORIZZAZIONI DI ELEMENTI, ESISTENZA DI MCD, MCM, DOMINI DI BÉZOUT. FATTORIZZAZIONI DI IDEALI. DOMINI DI NUMERI ALGEBRICI.
-- NUMERI DI FIBONACCI. PRINCIPALI PROPRIETÀ. IL RAPPORTO FN / FN-1, OSSIA TRA UN TERMINE E IL SUO PRECEDENTE NELLA SUCCESSIONE DEI NUMERI DI FIBONACCI, AL TENDERE DI N ALL'INFINITO TENDE AL NUMERO ALGEBRICO AUREO. RELAZIONI CON IL TRIANGOLO DI TARTAGLIA ED I COEFFICIENTI BINOMIALI. RELAZIONI CON IL MASSIMO COMUN DIVISORE E LA DIVISIBILITÀ.
-- TERNE PITAGORICHE. TERNE PITAGORICHE PRIMITIVE E TEOREMA DI CLASSIFICAZIONE. PROPRIETÀ GEOMETRICHE ED ARITMETICHE.

-- STEVEN GIVANT - PAUL HALMOS, INTRODUCTION TO BOOLEAN ALGEBRAS. UNDERGRADUATE TEXTS IN MATHEMATICS.SPRINGER, NEW YORK, 2009. XIV+574.

-- PAUL R. HALMOS, LECTURES ON BOOLEAN ALGEBRAS. VAN NOSTRAND MATHEMATICAL STUDIES, NO. 1, D. VAN NOSTRAND CO., INC., PRINCETON, N.J. 1963 V+147 PP.

-- IRA J. PAPICK, ALGEBRA CONNECTIONS: MATHEMATICS FOR MIDDLE SCHOOL TEACHERS, PRENTICE HALL, 2005

-- HANS RADEMACHER, HIGHER MATHEMATICS FROM AN ELEMENTARY POINT OF VIEW. EDITED BY D. GOLDFELD. WITH NOTES BY G. CRANE. BIRKHÄUSER, BOSTON, MASS., 1983 II+138 PP.

-- DAVID SHARPE, RINGS AND FACTORIZATION. CAMBRIDGE UNIVERSITY PRESS, CAMBRIDGE, 1987. X+111 PP.

-- J. ELDON WHITESITT, BOOLEAN ALGEBRA AND IST APPLICATIONS, DOVER PUBLICATIONS INC., NEW YORK, 1995 (PREVIOUSLY PUBLISHED BY ADDISON-WESLEY, READING MA, 1961)

Random variables, Theory of Estimation, Hypothesis testing (classical and Bayesian setting), Testing based on likelihood ratio, Statistical sampling

Statistical Inference, Casella G., Berger R. L., Duxbury Advanced Series, Second Edition

• PROVABILITY AND SATISFIABILITY, TRANSFORMATION (REDUCTION) OF PROOFS
• COMPACTNESS THEOREM FOR FIRST-ORDER LOGIC
• COMPLETENESS THEOREM FOR FIRST-ORDER LOGIC
• CUT-ELIMINATION THEOREM FOR FIRST-ORDER LOGIC

V. Michele Abrusci, Lorenzo Tortora de Falco: LOGICA Volume 1 Dimostrazioni e modelli al primo ordine. Springer, 2014

• PROVABILITY AND SATISFIABILITY, TRANSFORMATION (REDUCTION) OF PROOFS
• COMPACTNESS THEOREM FOR FIRST-ORDER LOGIC
• COMPLETENESS THEOREM FOR FIRST-ORDER LOGIC
• CUT-ELIMINATION THEOREM FOR FIRST-ORDER LOGIC

V. Michele Abrusci, Lorenzo Tortora de Falco: LOGICA Volume 1 Dimostrazioni e modelli al primo ordine. Springer, 2014

QUANTUM MECHANICS: THE CRISIS OF CLASSICAL PHYSICS. WAVES AND PARTICLES. STATE VECTORS AND OPERATORS. MEASUREMENTS AND OBSERVABLES. THE POSITION OPERATOR. TRANSLATIONS AND MOMENTUM. TIME EVOLUTION AND THE SCHRODINGER EQUATION. PARITY. ONE-DIMENSIONAL PROBLEMS. HARMONIC OSCILLATOR. SYMMETRIES AND CONSERVATION LAWS. TIME INDEPENDENT PERTURBATION THEORY. TIME DEPENDENT PERTURBATION THEORY.

J.J. SAKURAI, JIM NAPOLITANO - MECCANICA QUANTISTICA MODERNA - SECONDA EDIZIONE [ZANICHELLI, BOLOGNA, 2014]

An english version of the book is also available:
SAKURAI J.J., MODERN QUANTUM MECHANICS (REVISED EDITION) [ADDISON-WESLEY, 1994]

Reminders on complex numbers: algebraic and topological properties, the compactification to a point of the field of complex numbers and its identification with the sphere via stereographic projection, the polar representation of complex numbers.
Holomorphic functions: examples (polynomials, rational functions, exponential), non-examples (the complex conjugation), Cauchy-Riemann equations.
The algebra of the formal powers series: order of a power series, inverse, composition, derivative,
The quotient field of the domain of the formal series: the field of Laurent's series, residuals, inverse.
Convergence: punctual, absolute, uniform and uniform on the compact. The geometric series and its convergence properties.
Abel's Theorem for the convergence of power series: convergence radius, absolute and uniform convergence on compacts inside the convergence disk, divergence on the outside, holomorphicity of the limit function, convergence of the derived series.
Series of converging powers and holomorphic functions associated to them. Convergence of the derivative of a convergent series, of the sum and product of convergent series, of the convergent series composition, of the multiplicative inverse and of the inverse by composition.
Analytical functions: local series development of an analytical function; the analytic functions are infinitely derivable and all their derivatives are analytic; sum, product, inverse and composition of analytical functions are analytical; the analytic functions have a local analytic primitive, unique up to a constant, a convergent series defines an analytic function in its convergence disk, the inverse function theorem.
Normal form of an analytic function: each analytical function is locally, up tp translation and change of coordinates, the elevation to a power or is locally constant.
The open function theorem. Criterion of analytical isomorphism. Principle of the local and global maximum.
The zeros of an analytical function are discrete. Locally constant functions are constant in a connected open subset. Proof of the fundamental theorem of algebra (using the principle of the global maximum module).
The integral of a complex continuous function along a piecewise C^1-curve.
Criterion for the existence of a primitive: a continuous function admits a primitive if and only if its integral is zero along any closed curve. Example: the complex exponential as an analytic function.
The integral of uniformly convergent sequences and series.
A holomorphic function in a disk admits a primitive.
The integral of a holomorphic function along a continuous curve (not necessarily C^1).
The homotopy between continuous curves. The homotopic invariance theorem of the integral. Corollary: A holomorphic function on a simply connected open admits a primitive.
The integral (or local) formula of Cauchy (without proof). Cauchy formula for the series expansion of a holomorphic function (without proof). Corollaries: holomorphic functions are analytic, a power series is not holomorphic in at least one point of the boundary of the convergence disk, a whole function with an infinite polynomial expansion is a polynomial, a bounded entire function is constant (theorem of Liouville), integral formula of Cauchy for derivatives of a holomorphic function. Application: proof of the fundamental theorem of algebra using the Liouville theorem.
Proof of the integral (or local) formula of Cauchy. Proof of the Cauchy formula for the series expansion of a holomorphic fuction.
Winding number of a closed curve around a point: definition and examples.
Properties of the winding number of a closed curve: analytical definition, constancy in the connected components of the curve complement.
Homologous zero curves in an open subset. Relationship between homotopy and homology. The global formula of Cauchy: Dixon proof.
Chains and cycles. Integration along a chain. Winding number of a cycle. Zero homologous cycles. The first group of homology (with integral coefficients) of an open and its relation with the fundamental group. Examples.
The global Cauchy formula and Cauchy's Theorem on homological invariance: their equivalence.
The proof of Cauchy's Theorem on homological invariance (taken from the book by Alfhors).
Sequences of holomorphic functions uniformly converging on compacts: holomorphicity of the limit function; the derivative and the integral on a chain can be exchanged with the limit.
Laurent series (infinite) converging and holomorphic functions on a circular annulus.
Isolated singularities: removable singularities, poles, essential singularities. Examples. Rational functions. The theorems (by Riemann and Casorati-Weierstrass) of characterization of isolated singularities.
The residual theorem: local and global version.
Meromorphic functions and their properties. The logarithmic derivative of a meromorphic function and its properties. The principle of the argument. Corollary: number of zeros and poles within a simple closed curve. Application: proof of the fundamental Theorem of Algebra using the principle of the argument.
Roche's theorem. Application: proof of the fundamental Theorem of Algebra using Roche's theorem.
An approach to the problem of classification of domains through the map theorem and the uniformization theorem (without proof). Examples: the disk and the upper half-plane are biolomorphic (Cayley transform), the universal coating of C ^ * is the complex plane and the covering map is the exponential.
The complex projective line as a compactification of the complex plane. The complex projective line is homeomorphic to the sphere through stereographic projection. The linear projective group PGL2 acts on the complex projective line by means of linear (or Moebius) linear transformations.
The group of automorphisms of the complex plane.
Schwarz's Lemma. The group of automorphisms of the unitary disk.
The classification of the subgroups of the automorphisms of the complex plane that act in a free and properly discontinuous manner.
Coating quotients of the complex plane.
Elliptic (or doubly periodic) functions with respect to a lattice. The only holomorphic elliptic functions are constants. Properties of zeros and poles of elliptic functions inside a fundamental domain.
The Weiertrass P function and its derivative P ': definition and convergence properties.
Properties of the P function and its derivative P ': ellipticity, parity / disparity, poles and zeros.
The polynomial relationship between P and P '. All elliptic functions are expressed as rational functions in P and P '. Corollary: the field of elliptic functions.

S. Lang: Complex Analysis. Fourth edition. Graduate Texts in Mathematics, 103. Springer-Verlag, New York, 1999.
L. V. Ahlfors: Complex Analysis. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York, 1978.
R. Shakarchi: Problems and solutions for complex analysis. Springer-Verlag, New York, 1999.

SPAZI MISURA E MISURE. L’INTEGRALE DI LEBESGUE E I TEOREMI DI CONVERGENZA. GLI SPAZI L^PIL TEOREMA DI RICOPRIMENTO DI BESICOVICH E LA DIFFERENZIAZIONE DELLE MISURE.

RUDIN, W.: ANALISI REALE E COMPLESSA, MILANO 1991
ROYDEN, REAL ANALYSIS, CHINA MACHINE PRESS, 2004.

CLASSIFICAZIONE DELLE EQUAZIONI ALLE DERIVATE PARZIALI SEMILINEARI E LORO FORMA CANONICA. STUDIO DI PROBLEMI CONCRETI RELATIVI ALL'EQUAZIONE DELLE ONDE, DEL CALORE E DI LAPLACE.

A.N.TICHONOV; A.A.SAMARSKIJ: EQUAZIONI DELLA FISICA MATEMATICA EDIZIONI MIR. ZAEMANOGLOU THOE : INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS WITH APPLICATIONS. DOVER

EQUAZIONI DIFFERENZIALI ORDINARIE E SISTEMI: METODI AD UN PASSO E A PIU' PASSI PER IL PROBLEMA DI CAUCHY, IN PARTICOLARE METODI DI RUNGE-KUTTA, METODI DI ADAMS, METODI BDF. APPLICAZIONI.
TEORIA GENERALE DEI METODI PER EQUAZIONI A DERIVATE PARZIALI. PRINCIPALI METODI ALLE DIFFERENZE E VARIAZIONALI PER EQUAZIONI ELLITTICHE, PARABOLICHE ED IPERBOLICHE LINEARI.

[1] ALFIO QUARTERONI, RICCARDO SACCO, FAUSTO SALERI, MATEMATICA NUMERICA, SPRINGER, 1998;
[2] VALERIANO COMINCIOLI, ANALISI NUMERICA: METODI, MODELLI, APPLICAZIONI, APOGEO, 2005;
[3] ROBERTO FERRETTI, APPUNTI DEL CORSO DI ANALISI NUMERICA;
[5] ROBERTO FERRETTI, ESERCIZI D'ESAME DI ANALISI NUMERICA.

The concept of bitwise operation. Estimation of the number of bitwise operations (machine time) to execute fundamental operatons.
Polynomial and exponential algorithms. Divisibility. Euclid algorithm and bezout identity executed through bitwise operations. Congruences in Z. Chinese
remainder and successive squaring algorithm.

RSA
The Adleman, Shamir e Rivest algorithm . Formulation and analysis of the algorithm. RSA attacks.
Recalls on squaring congruences; Legendre and Jacobi symbols. A plynomial algorithm to calculate Jacobi symbol. Rabin cryptosystem: description, security analysis and examples. Prime numbers distribution. Primality tests: Fermat, Pocklington algorithms. Euler and strong pseudo-primality condition. Carmichael numbers. Montecarlo algorithms. Solovay-Strassen and Miller-Rabin tests. Pollard factorization methods: p-1and $\rho$.

FINITE FIELDS

N. Koblitz, A Course in Number Theory and Cryptography, GTM-Springer.

HANDBOOK OF APPLIED CRYPTOGRAPHY (Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone)

SPAZI MISURA E MISURE. L’INTEGRALE DI LEBESGUE E I TEOREMI DI CONVERGENZA. GLI SPAZI L^PIL TEOREMA DI RICOPRIMENTO DI BESICOVICH E LA DIFFERENZIAZIONE DELLE MISURE.

RUDIN, W.: ANALISI REALE E COMPLESSA, MILANO 1991
ROYDEN, REAL ANALYSIS, CHINA MACHINE PRESS, 2004.

CLASSIFICAZIONE DELLE EQUAZIONI ALLE DERIVATE PARZIALI SEMILINEARI E LORO FORMA CANONICA. STUDIO DI PROBLEMI CONCRETI RELATIVI ALL'EQUAZIONE DELLE ONDE, DEL CALORE E DI LAPLACE.

A.N.TICHONOV; A.A.SAMARSKIJ: EQUAZIONI DELLA FISICA MATEMATICA EDIZIONI MIR. ZAEMANOGLOU THOE : INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS WITH APPLICATIONS. DOVER

THE COURSE COMPUTER SCIENCE 2 (IN2 - MODELS OF COMPUTATION) IS DEVOTED TO THE INTRODUCTION OF MATHEMATICAL ASPECTS OF THE CONCEPT OF COMPUTATION, AND TO THE STUDY OF RELATIONS BETWEEN DIFFERENT MODELS OF COMPUTATION AND BETWEEN DIFFERENT PROGRAMMING STYLES. IN PARTICULAR, AN INTRODUCTION TO OBJECT ORIENTED PROGRAMMING IS GIVEN DURING THE COURSE.

[1] DEHORNOY, P., COMPLEXITE' ET DECIDABILITE'. SPRINGER-VERLAG, (1993).
[2] KRIVINE, J.-L., LAMBDA CALCULUS: TYPES AND MODELS. ‪ELLIS HORWOOD, (1993).
[3] SIPSER,M., INTRODUCTION TO THE THEORY OF COMPUTATION.THOMSON COURSE TECHNOLOGY, (2006).
[4] GABBRIELLI, M., MARTINI, S., LINGUAGGI DI PROGRAMMAZIONE: PRINCIPI E PARADIGMI. MCGRAW-HILL, (2011).

FURTHER TEXTBOOKS:

[4] AHO, HOPCROFT, ULLMAN, DESIGN AND ANALYSIS OF COMPUTER ALGORITHMS. ADDISON-WESLEY PUB. CO., (1974).
[5] AUSIELLO, G., GAMBOSI, G., D'AMORE F., LINGUAGGI, MODELLI, COMPLESSITA'. FRANCO ANGELI (2003).
[6] A. BERNASCONI, B. CODENOTTI, INTRODUZIONE ALLA COMPLESSITA' COMPUTAZIONALE. SPRINGER-VERLAG, (1998).
[7] SETHI, R., PROGRAMMING LANGUAGES: CONCEPTS AND CONSTRUCTS. ADDISON-WESLEY (ED. ITALIANA ZANICHELLI), (1996).
[8] HERMES, H., ENUMERABILITY, DECIDABILITY, COMPUTABILITY. DIE GRUNDLEHREN DER MATHEMATICHENWISSENSHAFTEN IN EINZELDARSTELLUNGEN, N. 127, SPRINGER-VERLAG, (1969).
[9] DARNELL, P. A. AND MARGOLIS, P. E., C A SOFTWARE ENGINEREEING APPROACH. SPRINGER-VERLAG, (1996).

Division, Factorization, Some Elementary Properties of Primes, Some Results and Problems Concerning Primes.
ARITHMETIC FUNCTIONS: The Divisor Function. The Moebius Function. The Euler Function. Dirichlet Convolution
CONGRUENCES: Sets of Residues,Some Interesting Congruences, Some Linear Congruences, Some Polynomial Congruences, Primitive Roots, the Theorem of Gauss.
QUADRATIC RESIDUES: The Legendre Symbol. Quadratic Reciprocity. The Jacobi Symbol.The Distribution of Quadratic Residues.
SUMS OF INTEGER SQUARES: Sums of Two Squares. Number of Representations. Sums of Four Squares. Sums of Three Squares.
ELEMENTARY PRIME NUMBER THEORY: Euclid's Theorem Revisited. The Von Mangoldt Function. Tchebycheff's Theorem. Some Results of Mertens

Chen, W; ELEMENTARY NUMBER THEORY. https://rutherglen.science.mq.edu.au/wchen/lnentfolder/lnent.html
Chowdhury, F.; Chowdhury, M. R. Essentials of Number Theory. Pi Publications, Dhaka, Bangladesh, 2005. ISBN 984-32-2836-7
Hardy, G. H.; Wright, E. M. An introduction to the theory of numbers. Fifth edition. The Clarendon Press, Oxford University Press, New York, 1979. xvi+426 pp. ISBN: 0-19-853170-2; 0-19-853171-0
Davenport, H. Aritmetica superiore. Un'introduzione alla teoria dei numeri. Editore: Zanichelli, 1994. 199 pp. ISBN: 8808091546
Gioia, A. A. The theory of numbers. An introduction. Reprint of the 1970 original. Dover Publications, Inc., Mineola, NY, 2001. xii+207 pp. ISBN: 0-486-41449-3
Rosen, K. H. Elementary number theory and its applications. Fourth edition. Addison-Wesley, Reading, MA, 2000. xviii+638 pp. ISBN: 0-201-87073-8
Tattersall, J. J. Elementary number theory in nine chapters. Cambridge University Press, Cambridge, 1999. viii+407 pp. ISBN: 0-521-58531-7

EQUAZIONI DIFFERENZIALI ORDINARIE E SISTEMI: METODI AD UN PASSO E A PIU' PASSI PER IL PROBLEMA DI CAUCHY, IN PARTICOLARE METODI DI RUNGE-KUTTA, METODI DI ADAMS, METODI BDF. APPLICAZIONI.
TEORIA GENERALE DEI METODI PER EQUAZIONI A DERIVATE PARZIALI. PRINCIPALI METODI ALLE DIFFERENZE E VARIAZIONALI PER EQUAZIONI ELLITTICHE, PARABOLICHE ED IPERBOLICHE LINEARI.

[1] ALFIO QUARTERONI, RICCARDO SACCO, FAUSTO SALERI, MATEMATICA NUMERICA, SPRINGER, 1998;
[2] VALERIANO COMINCIOLI, ANALISI NUMERICA: METODI, MODELLI, APPLICAZIONI, APOGEO, 2005;
[3] ROBERTO FERRETTI, APPUNTI DEL CORSO DI ANALISI NUMERICA;
[5] ROBERTO FERRETTI, ESERCIZI D'ESAME DI ANALISI NUMERICA.

COMPILAZIONE ED ESECUZIONE DI PROGRAMMI JAVA. TIPI DI DATO, ARITMETICA E ARRAYS.STRUTTURE DI CONTROLLO. CREAZIONE DI OGGETTI. CREAZIONE DI DOMINI DI CLASSI. UTILIZZO COORDINATO DI MOLTEPLICI CLASSI: ASSOCIAZIONE, AGGREGAZIONE E COMPOSIZIONE DI CLASSI. EREDITARIETA', POLIMORFISMO E INTERFACCE. GESTIONE DELLE ECCEZIONI. LIBRERIE JAVA. PROGRAMMAZIONE GENERICA IN JAVA.STREAM DI INPUT/OUTPUT. COMPILAZIONE AUTOMATICA CON ANT. JAVA E I DATABASE (JDBC). IL MULTITHREADING IN JAVA. APPLICAZIONI.

[1] GABBRIELLI, M., MARTINI, S., LINGUAGGI DI PROGRAMMAZIONE: PRINCIPI E PARADIGMI. MCGRAW-HILL, (2011).
[2] PARSONS, D., FUNDATIONAL JAVA: KEY ELEMENTS AND PRACTICAL PROGRAMMING, SPRINGER-VERLAG (2012).
[3] SEDGEWICK, R., WAYNE, K., AN INTRODUCTION TO PROGRAMMING IN JAVA: AN INTERDISCIPLINARY APPROACH. HTTP://INTROCS.CS.PRINCETON.EDU/JAVA/HOME. ADDISON-WESLEY (2012).

TESTI DI APPROFONDIMENTO:

[4] RAMNATH, S., DATHAN, B., OBJECT-ORIENTED ANALYSIS AND DESIGN, SPRINGER-VERLAG, (2010).

SARANNO SVILUPPATI E ANALIZZATI MODELLI MATEMATICI DI PROBLEMI APPLICATIVI, ANCHE DI INTERESSE INDUSTRIALE, BASATI SOPRATTUTTO SU EQUAZIONI DIFFERENZIALI ORDINARIE O ALLE DERIVATE PARZIALI.
IL CORSO HA UNA COMPONENTE MODELLISTICA ED UNA NUMERICA. E SARÀ ORGANIZZATO, ALMENO IN PARTE, "PER PROBLEMI" PIUTTOSTO CHE "PER METODI", OSSIA PARTENDO DA UN CERTO NUMERO DI PROBLEMI APPLICATIVI E CERCANDONE LA SOLUZIONE, INTRODUCENDO VIA VIA GLI STRUMENTI NECESSARI, QUALI I METODI NUMERICI PIÙ OPPORTUNI.
SARANNO INVITATI A TENERE CONFERENZE SU ARGOMENTI SPECIFICI DEI MATEMATICI APPLICATI OPERANTI IN ALTRE SEDI UNIVERSITARIE O IN LABORATORI DI RICERCA , ATTIVI NEL CAMPO DELLA MATEMATICA APPLICATA E/O INDUSTRIALE.

1] R. KNOBEL, ''AN INTRODUCTION TO THE MATHEMATICAL THEORY OF WAVES'', AMER. MATH. SOC., PROVIDENCE, 2000.

2] K.W. MORTON AND D.F. MAYERS, ''NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS'', CAMBRIDGE UNIV. PRESS, CAMBRIDGE, 2005.

3] APPUNTI E MATERIALE DALLE LEZIONI.

PROBABILITY, ENTROPY, INFERENCE. DATA COMPRESSION: SOURCE CODING THEOREM, SYMBOLIC CODES, STREAM CODES.

INFORMATION THEORY. NOISY CHANNEL CODING: DEPENDENT RANDOM VARIABLES, COMMUNICATION ON A NOISY CHANNEL,
PERFECT SECRECY. ERROR CORRECTION CODES. LINEAR CODES, HAMMING CODES, GOLAY CODES, CYCLIC CODES, BCH CODE,
REED-SOLOMON CODES. MCELIECE CIPHER.

[1] MacKay, D.J.C., Information Theory, Inference and Learning Algorithms, CUP (2004).
[2] Blahut, R., Principles and Practice of Information Theory, Addison-Wesley (1985).
[3] Cover, T.M. and Thomas, J.A., Elements of Information Theory, Wiley (1991).
[4] W. Trappe, l.C. Washington, Introduction to Cryptography with Coding Theory, 2nd edition, Prentice-Hall (2005).