Degree Course: Mathematics Syllabus for A.Y. 2016/2017

Italian version

FIRST YEAR

First semester

Course

Credits

Scientific Disciplinary Sector Code

Contact Hours

Exercise Hours

Laboratory Hours

Personal Study Hours

Type of Activity

Language

20401885 - AL110 - ALGEBRA 1 (objectives)

- TARTARONE FRANCESCA (syllabus) (reference books)

- CIGLIOLA ANTONIO

MAT/02

Basic compulsory activities

ITA

20401886 - AM110 - MATHEMATICAL ANALYSIS 1 (objectives)

- CHIERCHIA LUIGI

- MATALONI SILVIA

MAT/05

Basic compulsory activities

ITA

20401890 - IN110 - COMPUTER SCIENCE 1 (ALGORITHMS AND COMPUTER PROGRAMMING) (objectives)

- LIVERANI MARCO (syllabus) (reference books)

- MAIELI ROBERTO

INF/01

Basic compulsory activities

ITA

Optional group: Gruppo idoneità di lingue - (show)

Second semester

Course	Credits	Scientific Disciplinary Sector Code	Contact Hours	Exercise Hours	Laboratory Hours	Personal Study Hours	Type of Activity	Language
20401887 - AM120 - MATHEMATICAL ANALYSIS 2 (objectives) To obtain the fundamental notions about derivation and integration of real functions of one real variable and to develop abilities in calculus methods. To develop the fundamentals of the theory of real and complex series. - CHIERCHIA LUIGI - MATALONI SILVIA	10	MAT/05	60	36	-	-	Core compulsory activities	ITA
20401888 - GE110- GEOMETRY 1 (objectives) OBTAIN GOOD KNOWLEDGE OF THE CONCEPTS AND METHODS OF BASIC LINEAR ALGEBRA, WITH SPECIAL EMPHASIS ON THE STUDY OF LINEAR SYSTEMS, MATRICES AND DETERMINANTS, VECTOR SPACES AND LINEAR MAPS, AFFINE GEOMETRY. - LOPEZ ANGELO - CIGLIOLA ANTONIO	10	MAT/03	60	36	-	-	Basic compulsory activities	ITA
20401889 - CP110 - PROBABILITY 1 (objectives) INTRODUCTION TO THE MAIN ASPECTS OF PROBABILITY THEORY: PROBABILITY SPACES, REPEATED TRIALS, RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, SOME LIMIT THEOREMS AND A FIRST LOOK AT FINITE MARKOV CHAINS - CAPUTO PIETRO - Quattropani Matteo	10	MAT/06	60	36	-	-	Core compulsory activities	ITA

SECOND YEAR

First semester

Course

Credits

Scientific Disciplinary Sector Code

Contact Hours

Exercise Hours

Laboratory Hours

Personal Study Hours

Type of Activity

Language

20402075 - AL210 - ALGEBRA 2 (objectives)

- GABELLI STEFANIA

- SPIRITO Dario

MAT/02

Core compulsory activities

ITA

20402076 - AM210 - MATHEMATICAL ANALYSIS 3 (objectives)

- BIASCO LUCA

- PROCESI MICHELA

MAT/05

Core compulsory activities

ITA

20402079 - GE210 - GEOMETRY 2 (objectives)

- VERRA ALESSANDRO

- CIGLIOLA ANTONIO

MAT/03

Core compulsory activities

ITA

Optional group: comune Orientamento unico UN INSEGNAMENTO A SCELTA NEL GRUPPO 1 - (show)

20402088 - AN410 - NUMERICAL ANALYSIS 1 (objectives) Provide the basic elements (including implementation in a programming language) of elementary numerical approximation techniques, in particular those related to solution of linear systems and nonlinear scalar equations, interpolation and approximate integration - Derived from 20402088 AN410 - ANALISI NUMERICA 1 in MATEMATICA (DM 270) LM-40 N0 FERRETTI ROBERTO, (syllabus) 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. (reference books) [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.	7	MAT/08	60	12	-	-	Related or supplementary learning activities	ITA
20402090 - MC410 - COMPLEMENTARY MATHEMATICS 1 (objectives) Mathematics software, with particular attention to their use for teaching mathematics in school. 2. Analysis of the potential and criticality of the use of technological tools for teaching and learning mathematics - Derived from 20402090 MC410 - MATEMATICHE COMPLEMENTARI 1 in MATEMATICA (DM 270) LM-40 N0 BRUNO ANDREA	7	MAT/04	60	-	-	-	Related or supplementary learning activities	ITA

Second semester

Course

Credits

Scientific Disciplinary Sector Code

Contact Hours

Exercise Hours

Laboratory Hours

Personal Study Hours

Type of Activity

Language

20402077 - AM220 - MATHEMATICAL ANALYSIS 4 (objectives)

- BIASCO LUCA

- FELICI FABIO

MAT/05

Core compulsory activities

ITA

20402078 - FS210 - PHYSICS 1 (objectives)

- RIGGIO LORENZO

- GALLO PAOLA (syllabus) (reference books)

FIS/01

Basic compulsory activities

ITA

20402080 - GE220 - GEOMETRY 3 (objectives)

- MASCARENHAS MELO ANA MARGARIDA (syllabus)

Topological spaces. Examples and first properties. Metric spaces and continuous applications. Examples. The topological space associated with a metric space.
Metrizable and non metrizable spaces, topologically equivalent metric spaces. Closed subsets of a topological space and closing operators, neighborhood and border of a subset of a topological space.
Properties and examples of closures, interiors and borders of sub-spaces of a topological space. Limit points and derived set of a subset of a topological space. Basis for a topology. Characterization of a base.
Sub-bases for a topology. Separable topological spaces. Basic bases of surroundings for a point and axioms of numerability. Limit of a sequence. Continuity of applications between topological spaces.
Examples of continuous applications. Open and closed applications. Homeomormisms between topological spaces. Examples of homeomorphic and non-homeomorphic spaces. The product topology.
Examples and properties of product topologies. The topology produced on infinite collections of topological spaces. The quotient topology. Examples of identifications.
The quotient topology: examples and characterization through open saturated sets. Hausdorff or T2 spaces: definition, examples and first properties.
Properties of Hausdorff spaces. T0 and T1 spaces: examples and properties. Regular and normal spaces.
Normal spaces. The lemma of Uryshon and the theorem of the extension of Tietze.
Lindelof spaces: definitions, properties and examples. Compact spaces: definition, examples and first properties. Compacts in Hausdorff spaces.
Continuous and compact applications. The Heine-Borel theorem and compact subspaces of R.
Tychonoff's theorem. Generalizations of compactness: locally compact, paracompact, numerically compact and compact spaces for sequences.
Equivalence between compact and complete metric spaces and totally limited. Compactification of Alexandroff and Stone-Cech.
Stone-Cech compactification: existence and properties. Connected spaces: definition and first properties.
Properties of connected spaces and applications, connected components and connection by arcs.
Connected components for arcs and properties. Homotopy of maps.
Homotopic equivalence, examples. Fundamental group.
Functorial properties of the fundamental group. Hints on the theory of categories.
Properties of the fundamental group: invariance for arc connected components and properties of the fundamental group of contractible spaces. The fundamental group of S^1 is isomorphic to Z.
Coverings. Theorem of the lifting of the homotopies and some consequences. Morphism induced by a covering in the fundamental groups.
Coverings and fundamental group. Monodromy. Correspondence between subgroups of the fundamental group and isomorphism classes of coverings.

(reference books)

- Tasin Luca

MAT/03

Core compulsory activities

ITA

Optional group: comune Orientamento unico UN INSEGNAMENTO A SCELTA NEL GRUPPO 1 - (show)

20402089 - IN410 - COMPUTER SCIENCE 2 (objectives) FUNCTIONAL MODELS OF LAMBDA-CALCULUS. OBJECT-ORIENTED PROGRAMMING: CLASSES, FUNCTIONAL CLASSES. CLASS INHERITANCE. ABSTRACT CLASS DECLARATION. PUBLIC AND PRIVATE METHODS. METHOD LATE-BINDING. - Derived from 20402089 IN410 - INFORMATICA 2 in MATEMATICA (DM 270) LM-40 N0 PEDICINI MARCO, Pistone Paolo (syllabus) COMPLEXITY, COMPUTABILITY, REPRESENTABILITY: DECISION PROBLEMS, FINITE AUTOMATA AND ALGORITHMS. TURING-COMPUTABILITY. SPACE AND TIME COMPLEXITY OF ALGORITHMS. RAM MACHINES. COMPLEXITY FUNCTIONS. RECURSIVE FUNCTIONS. THE HALT PROBLEM FOR TURING MACHINES. FUNCTIONAL PROGRAMMING: LAMBDA CALCULUS. CHURCH-ROSSER THEOREM. STRATEGIES OF REDUCTION. SOLVABLE TERMS. BÖHM'S THEOREM. THEOREM FOR LAMBDA-DEFINABLE RECURSIVE FUNCTIONS. BETA-FUNCTIONAL MODELS OF LAMBDA-CALCULUS. OBJECT-ORIENTED PROGRAMMING: CLASSES, FUNCTIONAL CLASSES. CLASS INHERITANCE. ABSTRACT CLASS DECLARATION. PUBLIC AND PRIVATE METHODS. METHOD LATE-BINDING. (reference books) [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).	7	INF/01	60	12	-	-	Related or supplementary learning activities	ITA
20402091 - TN410 - INTRODUCTION TO NUMBER THEORY (objectives) TO ACQUIRE A GOOD KNOWLEDGE OF CONCEPTS AND METHODS OF ELEMENTARY NUMBER THEORY, WITH PARTICULAR RESPECT OF STUDY OF DIOPHANTINE EQUATIONS AND POLYNOMIAL CONGRUENCES. TO GIVE PREREQUISITES FOR ADVANCED COURSES OF ALGEBRAIC AND ANALYTIC NUMBER THEORY. - Derived from 20402091 TN410 - INTRODUZIONE ALLA TEORIA DEI NUMERI in MATEMATICA (DM 270) LM-40 N0 TARTARONE FRANCESCA (syllabus) CONGRUENCES AND POLYNOMIALS. LINEAR DIOPHANTINE EQUATIONS. SYSTEMS OF LINEAR CONGRUENCES. POLYNOMIAL CONGRUENCES. POLYNOMIAL CONGRUENCES MOD p: LAGRANGES’S THEOREM. p-ADIC APPROXIMATION. THE EXISTENCE OF PRIMITIVE ROOTS MOD p. QUADRATIC CONGRUENCES. QUADRATIC RESIDUES. THE LEGENDRE SYMBOL. GAUSS’S LEMMA AND THE LAW OF QUADRATIC RECIPROCITY.THE JACOBI SYMBOL. SUMS OF TWO, THREE, FOUR SQUARES. MULTIPLICATIVE FUNCTIONS. THE MÖBIUS INVERSION FORMULA. SOME NONLINEAR DIOPHANTINE EQUATIONS. CONTINUED FRACTIONS. (reference books) M. FONTANA, APPUNTIDEL CORSO TN1 (ARGOMENTI DELLA TEORIA CLASSICA DEI NUMERI), HTTP://WWW.MAT.UNIROMA3.IT/USERS/FONTANA/DIDATTICA/FONTANA_DIDATTICA.HTML DISPENSE. D.M. BURTON: ELEMENTARY NUMBER THEORY, MCGRAW-HILLINTERNATIONAL EDITION, 6TH EDITION (2007), 434 PP. G.A. JONES AND J.M. JONES, ELEMENTARY NUMBER THEORY, SPRINGER, 1ST EDITION (1998), 200 PP. H. DAVENPORT, ARITMETICA SUPERIORE.UN?INTRODUZIONE ALLA TEORIA DEI NUMERI, ZANICHELLI, (1994), 199 PP. G.H. HARDY AND E.M. WRIGHT, AN INTRODUCTION TO THE THEORY OH NUMBERS, THE CLARENDON PRESS, OXFORD UNIVERSITY, 5TH EDITION (1979), XVI+426 PP. W.J. LEVEQUE, FUNDAMENTALS OF NUMBER THEORY, DOVER PUBLICATIONS, (1996) K.H. ROSEN. ELEMENTARY NUMBER THEORY AND ITS APPLICATIONS, ADDISON-WESLEY, 6TH EDITION (2011), XV+752 PP.	7	MAT/02	60	-	-	-	Related or supplementary learning activities	ITA

THIRD YEAR

First semester

Course

Credits

Scientific Disciplinary Sector Code

Contact Hours

Exercise Hours

Laboratory Hours

Personal Study Hours

Type of Activity

Language

20402081 - FM210 - MATHEMATICAL PHYSICS 1 (objectives)

- GIULIANI ALESSANDRO (syllabus) (reference books)

- Vaia Faenia

MAT/07

Core compulsory activities

ITA

Optional group: comune Orientamento unico DUE INSEGNAMENTI A SCELTA NEL GRUPPO 2 - (show)

20402083 - AL4 - ELEMENTS OF ADVANCED ALGEBRA (objectives) OBTAIN GOOD KNOWLEDGE OF THE CONCEPTS AND METHODS OF THE THEORY OF EQUATIONS IN ONE VARIABLE. UNDERSTAND AND BE ABLE TO APPLY THE “FUNDAMENTAL THEOREM OF CORRESPONDENCE OF GALOIS” IN ORDER STUDY THE “COMPLEXITY” OF A POLYNOMIAL. - PAPPALARDI FRANCESCO - TALAMANCA VALERIO	7	MAT/02	72	-	-	-	Related or supplementary learning activities	ITA
20402087 - GE310 - ELEMENTS OF ADVANCED GEOMETRY (objectives) A REFINED STUDY OF TOPOLOGY VIA ALGEBRAIC AND ANALYTIC TOOLS. - PONTECORVO MASSIMILIANO - TALAMANCA VALERIO	7	MAT/03	72	-	-	-	Related or supplementary learning activities	ITA
20402093 - CP410 - PROBABILITY 2 (objectives) To gain a solid knowledge of the basic aspects of probabilità theory: construction of probabilità measures on measurable spaces, 0-1 law, independence, conditional expectation, random variables, convergence of random variables, characteristic functions, central limit theorem, branching processes, discrete martingales. - Derived from 20402093 CP410 - PROBABILITA' 2 in MATEMATICA (DM 270) LM-40 N0 CAPUTO PIETRO (syllabus) 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 (reference books) 1] D.WILLIAMS, PROBABILITY WITH MARTINGALES. CAMBRIDGE MATHEMATICAL TEXTBOOKS, (1991).	7	MAT/06	60	-	-	-	Related or supplementary learning activities	ITA
20402279 - AC310 – COMPLEX ANALYSIS 1 (objectives) TO ACQUIRE A SOLID KNOWLEDGE OF HOLOMORPHIC AND MEROMORPHIC FUNCTIONS OF ONE COMPLEX VARIABLE AND THEIR MAIN PROPERTIES. TO DEVELOP PRACTICAL SKILLS IN THE USE OF COMPLEX FUNCTIONS, ESPECIALLY IN COMPLEX INTEGRATION AND IN COMPUTATION OF REAL DEFINITE INTEGRALS.	7	MAT/05	72	-	-	-	Related or supplementary learning activities	ITA
20410100 - AC310 - Complex analysis 1 - BESSI UGO (syllabus) Complex numbers; complex differentiability and Cauchy-Riemann formula; the Riemann sphere; linear fractionary maps; the line integral; Cauchy's theorem and Cauchy's formula; Lioville's theorem; the fundamental theorem of algebra; the identity principle; the Euler product for the sine; the mean value property, the maximum principle and Schwarz's lemma; the Moebius maps are the automorphisms of the disc; the hyperbolic metric on the disc and its geodesics; the contractions of the hyperbolic metric; Laurent series; theorem of the removable singularity; poles and essential singularities; the Casorati-Weierstrass theorem; the residues theorem; the argument principle; the theorem of Ropuche'; normal families and quasi-uniform convergence; harmonic functions are the real part of holomorphic functions; uniqueness for the Dirichlet problem; Poisson's kernel; functions which have the mean value property are harmonic; the Schwarz refelction principle; the Riemann mapping theorem. (reference books) W. Rudin, Real and complex analysis, Tata McGraw Hill. J. B. Conway, Functions of one complex variable, Springer. - FELICI FABIO	7	MAT/05	60	12	-	-	Related or supplementary learning activities	ITA

Optional group: comune Orientamento unico DUE INSEGNAMENTI A SCELTA AMPIA - (show)

20402083 - AL4 - ELEMENTS OF ADVANCED ALGEBRA (objectives) OBTAIN GOOD KNOWLEDGE OF THE CONCEPTS AND METHODS OF THE THEORY OF EQUATIONS IN ONE VARIABLE. UNDERSTAND AND BE ABLE TO APPLY THE “FUNDAMENTAL THEOREM OF CORRESPONDENCE OF GALOIS” IN ORDER STUDY THE “COMPLEXITY” OF A POLYNOMIAL. - Derived from 20402083 AL310 - ISTITUZIONI DI ALGEBRA SUPERIORE in MATEMATICA (DM 270) L-35 N0 PAPPALARDI FRANCESCO, TALAMANCA VALERIO	7	MAT/02	72	-	-	-	Elective activities	ITA
20402087 - GE310 - ELEMENTS OF ADVANCED GEOMETRY (objectives) A REFINED STUDY OF TOPOLOGY VIA ALGEBRAIC AND ANALYTIC TOOLS. - Derived from 20402087 GE310 - ISTITUZIONI DI GEOMETRIA SUPERIORE in MATEMATICA (DM 270) L-35 N0 PONTECORVO MASSIMILIANO, TALAMANCA VALERIO	7	MAT/03	72	-	-	-	Elective activities	ITA
20402088 - AN410 - NUMERICAL ANALYSIS 1 (objectives) THE COURSE IS INTENDED TO GIVE THE FUNDAMENTALS OF NUMERICAL APPROXIMATION TECHNIQUES, WITH A SPECIAL EMPHASIS ON THE SOLUTION OF LINEAR SYSTEMS AND NONLINEAR SCALAR EQUATIONS, POLYNOMIAL INTERPOLATION AND APPROXIMATE INTEGRATION FORMULAE. BESIDES BEING INTRODUCTORY, SUCH TECHNIQUES WILL BE USED IN THE SEQUEL AS BUILDING BLOCKS FOR MORE COMPLEX SCHEMES. - Derived from 20402088 AN410 - ANALISI NUMERICA 1 in MATEMATICA (DM 270) LM-40 N0 CACACE Simone, FERRETTI ROBERTO	7	MAT/08	72	-	-	-	Elective activities	ITA
20402090 - MC410 - COMPLEMENTARY MATHEMATICS 1 (objectives) To acquire deep understanding of the principal geometry arguments treated in high-school mathematics - Derived from 20402090 MC410 - MATEMATICHE COMPLEMENTARI 1 in MATEMATICA (DM 270) LM-40 N0 BRUNO ANDREA (syllabus) GEOMETRIA EUCLIDEA E GEOMETRIE NON-EUCLIDEE. GEOMETRIE LOCALMENTE EUCLIDEE. ISOMETRIE DEL PIANO. GRUPPI DISCRETI DI ISOMETRIE DEL PIANO. GEOMETRIA IPERBOLICA (reference books) R. TRUDEAU: “LA RIVOLUZIONE NON EUCLIDEA”- BORINGHIERI ED. NIKULIN, SHAFAREVICH: “GEOMETRY AND GROUPS”; SPRINGER ED.	7	MAT/04	60	-	-	-	Elective activities	ITA
20402093 - CP410 - PROBABILITY 2 (objectives) To gain a solid knowledge of the basic aspects of probabilità theory: construction of probabilità measures on measurable spaces, 0-1 law, independence, conditional expectation, random variables, convergence of random variables, characteristic functions, central limit theorem, branching processes, discrete martingales. - Derived from 20402093 CP410 - PROBABILITA' 2 in MATEMATICA (DM 270) LM-40 N0 CAPUTO PIETRO (syllabus) 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 (reference books) 1] D.WILLIAMS, PROBABILITY WITH MARTINGALES. CAMBRIDGE MATHEMATICAL TEXTBOOKS, (1991).	7	MAT/06	60	-	-	-	Elective activities	ITA
20402094 - AL410 - COMMUTATIVE ALGEBRA (objectives) THE PURPOSE OF THIS COURSE IS TO DEEPEN THE KNOWLEDGE OF SOME TOOLS AND FUNDAMENTAL PROPERTIES OF COMMUTATIVE RINGS AND THEIR MODULES, WITH PARTICULAR EMPHASIS TO THE CASE OF RINGS ARISING IN ALGEBRAIC NUMBER THEORY AND ALGEBRAIC GEOMETRY. - Derived from 20402094 AL410 - ALGEBRA COMMUTATIVA in MATEMATICA (DM 270) LM-40 N0 FONTANA MARCO	7	MAT/02	60	-	-	-	Elective activities	ITA
20402097 - AM410 - ELLITTIC PARTIAL DIFFERENTIAL EQUATIONS (objectives) To develop a good knowledge of the general methods and the classical techniques useful in the study of partial differential equations - Derived from 20402097 AM410 - EQUAZIONI ALLE DERIVATE PARZIALI DI TIPO ELLITTICO in MATEMATICA (DM 270) LM-40 N0 ESPOSITO PIERPAOLO (syllabus) 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 (reference books) "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	7	MAT/05	60	-	-	-	Elective activities	ITA
20402103 - FM410 - MATHEMATICAL PHYSICS 3 (objectives) CONTINUING THE STUDY, BEGAN DURING FM210, OF DYNAMIC SYSTEMS OF PHYSICAL INTEREST WITH MOST STYLISH AND POWERFUL TECHNIQUES, SUCH AS THE LAGRANGIAN AND HAMITONIAN FORMALISM, THAT ARE IN THE VAST RANGE OF APPLICATIONS OF ANALYSIS AND MATHEMATICAL PHYSICS. - Derived from 20402103 FM410 - FISICA MATEMATICA 3 in MATEMATICA (DM 270) LM-40 N0 GENTILE GUIDO (syllabus) 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. (reference books) 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).	7	MAT/07	60	-	-	-	Elective activities	ITA
20402104 - GE410 - ALGEBRAIC GEOMETRY 1 (objectives) Introduction to the study of topological and geometrical structures defined using algebraic methods. Refinement of the algebraic knowledge using applications to the study of algebraic varieties in affine and projective spaces. - Derived from 20402104 GE410 - GEOMETRIA ALGEBRICA 1 in MATEMATICA (DM 270) LM-40 N0 LOPEZ ANGELO (syllabus) Classical theory of algebraic varieties, affine and projective, over algebraically closed fields. Local geometry, normalization. Divisors, linear systems and associated morfisms. (reference books) 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.	7	MAT/03	60	-	-	-	Elective activities	ITA
20402113 - MC430 - LABORATORY: DIDACTICS FOR MATHEMATICS (objectives) Problem solving with examples from secondary school mathematical curricula, with the help of a computer and a direct use of numerical and symbolic calculus and dynamical geometry software (MATHEMATICA, introduction on CABRI and GEOGEBRA). All examples, in an interactive and “laboratorial” lessons, point at experience limits and opportunities of using computers at school on selected arguments like numerical approximation or visualization in geometry as well as analysis. - Derived from 20402113 MC430 - LABORATORIO DI DIDATTICA DELLA MATEMATICA in MATEMATICA (DM 270) LM-40 N0 FALCOLINI CORRADO (syllabus) 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. (reference books) 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)	7	MAT/04	60	-	-	-	Elective activities	ITA
20402114 - ME410 - ELEMENTARY MATHEMATICS FROM AN ADVANCED POINT OF VIEW (objectives) To acquire deep understanding of some of the principal arguments treated in high-school mathematics - Derived from 20402114 ME410 - MATEMATICHE ELEMENTARI DA UN PUNTO DI VISTA SUPERIORE in MATEMATICA (DM 270) LM-40 N0 FONTANA MARCO (syllabus) -- 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. (reference books) -- 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)	7	MAT/02	60	-	-	-	Elective activities	ITA
20402115 - ST410 - STATISTICS 1 - Derived from 20402115 ST410 - STATISTICA 1 in MATEMATICA (DM 270) LM-40 N0 DI BIAGIO LORENZO, Fasulo Andrea	7	SECS-S/01	72	-	-	-	Elective activities	ITA
20402119 - LM410 - MATHEMATICAL LOGIC 1 (objectives) Understanding of main theorems (together with the proofs) about classical first-order logic. - Derived from 20710016 TEOREMI SULLA LOGICA 1 in FILOSOFIA (DM 270) L-5 ABRUSCI VITO MICHELE (syllabus) • 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 (reference books) V. Michele Abrusci, Lorenzo Tortora de Falco: LOGICA Volume 1 Dimostrazioni e modelli al primo ordine. Springer, 2014	7	MAT/01	60	-	-	-	Elective activities	ITA
20402122 - FS420 - QUANTUM MECHANICS (objectives) GAIN KNOWLEDGE OF THE BASIC PRINCIPLES OF QUANTUM MECHANICS APPLIED TO SIMPLE PHYSICAL SYSTEMS - Derived from 20401808 ISTITUZIONI DI FISICA TEORICA in FISICA (DM 270) L-30 N0 LUBICZ VITTORIO, TARANTINO CECILIA (syllabus) 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. (reference books) J.J. SAKURAI, J. NAPOLITANO. MECCANICA QUANTISTICA MODERNA. SECONDA EDIZIONE, ZANICHELLI, BOLOGNA, 2014 An english version of the book is also available: J.J. SAKURAI, J. NAPOLITANO. MODERN QUANTUM MECHANICS. SECOND EDITION, PEARSON EDUCATION LIMITED 2014	7	FIS/02	60	-	-	-	Elective activities	ITA
20402279 - AC310 – COMPLEX ANALYSIS 1 (objectives) TO ACQUIRE A SOLID KNOWLEDGE OF HOLOMORPHIC AND MEROMORPHIC FUNCTIONS OF ONE COMPLEX VARIABLE AND THEIR MAIN PROPERTIES. TO DEVELOP PRACTICAL SKILLS IN THE USE OF COMPLEX FUNCTIONS, ESPECIALLY IN COMPLEX INTEGRATION AND IN COMPUTATION OF REAL DEFINITE INTEGRALS.	7	MAT/05	72	-	-	-	Elective activities	ITA
20402328 - MC440 - FIRST ORDER CLASSICAL LOGIC	7	MAT/01	60	-	-	-	Elective activities	ITA
20410100 - AC310 - Complex analysis 1 - Derived from 20410100 AC310 - ANALISI COMPLESSA 1 in MATEMATICA (DM 270) L-35 BESSI UGO, FELICI FABIO (syllabus) Complex numbers; complex differentiability and Cauchy-Riemann formula; the Riemann sphere; linear fractionary maps; the line integral; Cauchy's theorem and Cauchy's formula; Lioville's theorem; the fundamental theorem of algebra; the identity principle; the Euler product for the sine; the mean value property, the maximum principle and Schwarz's lemma; the Moebius maps are the automorphisms of the disc; the hyperbolic metric on the disc and its geodesics; the contractions of the hyperbolic metric; Laurent series; theorem of the removable singularity; poles and essential singularities; the Casorati-Weierstrass theorem; the residues theorem; the argument principle; the theorem of Ropuche'; normal families and quasi-uniform convergence; harmonic functions are the real part of holomorphic functions; uniqueness for the Dirichlet problem; Poisson's kernel; functions which have the mean value property are harmonic; the Schwarz refelction principle; the Riemann mapping theorem. (reference books) W. Rudin, Real and complex analysis, Tata McGraw Hill. J. B. Conway, Functions of one complex variable, Springer.	7	MAT/05	60	12	-	-	Elective activities	ITA
20410068 - MATHEMATICS OF FINANCIAL MARKETS - Derived from 20410068 MF410 - MODELLI MATEMATICI PER I MERCATI FINANZIARI in MATEMATICA (DM 270) LM-40 PIERINI ANDREA	7	SECS-S/06	60	-	-	-	Elective activities	ITA
20410070 - LOGIC AND ARITHMETIC - Derived from 20710122 TEOREMI SULLA LOGICA, 2 in SCIENZE FILOSOFICHE (DM 270) LM-78 TORTORA DE FALCO LORENZO	7	MAT/01	60	-	-	-	Elective activities	ITA
20402291 - ST420 – MULTIVARIATE STATISTICAL ANALYSIS, MATHEMATICAL STATISTICS - Derived from 20402291 ST420- STATISTICA 2, STATISTICA MATEMATICA in MATEMATICA (DM 270) LM-40 NACCARATO ALESSIA, PIERINI ANDREA	7	SECS-S/01	60	12	-	-	Elective activities	ITA
20402280 - AN430 – NUMERICAL ANALYSIS 3 - Derived from 20402280 AN430-ANALISI NUMERICA 3 in MATEMATICA (DM 270) LM-40 TERESI LUCIANO (syllabus) Selected examples of partial differential equations The weak form of boundary value problems Galerkin method The Finite Elements Method (FEM) Convergence of MEF solutions Data structure and FEM implementation Numerical methods for the solution of algebraic systems (reference books) Mark S. Gockenbach, Understanding and Implementing the Finite Elements Method, SIAM 2006	7	MAT/08	60	-	-	-	Elective activities	ITA
20402110 - IN450 - INFORMATICS 6: ALGORITHMS FOR CRYPTOGRAPHY - Derived from 20402110 IN450 - INFORMATICA 6, ALGORITMI PER LA CRITTOGRAFIA in MATEMATICA (DM 270) LM-40 PEDICINI MARCO (syllabus) Introduction to modern cryptography: definition of security in encryption, Distinguisher, integrity, digital signature, authentication, abstract cryptographic primitives. Basic concepts of the theory of numbers: MCD, modular arithmetic and basic algorithms, polynomials and rational polynomials, finite fields, solution of equations. Vector spaces and linear maps. Algorithms: products of matrices in GF (2), products of dense arrays, Strassen algorithm, Gaussian elimination, matrix inversion, linear algebra on sparse matrices, iterative algorithms. Groebner bases: Buchberger algorithm. Brute-force cryptanalysis: attacks by dictionaries, block ciphers, substitution-permutation networks, Feistel type systems, DES, brute force on DES, AES. Hash functions, the family of hash functions SHA, linear model for SHA-0, looking for collisions, brute force and parallelism, efficient brute force. Birthday paradox: operating modes ECB, CBC, CBC-MAC, sorting algorithms, hash tables, binary trees, analysis of pseudo-random functions. Security of block ciphers. Time memory trade-off. Walsh-Hadamard transform: linear cryptanalysis, differential cryptanalysis, the study of the S-box, Walsh transform, differential characteristics, normal algebraic form, generalization of the Walsh transform in the case of finite fields GF (p). Analysis of complexity. Algebraic attacks, stream ciphers, the key stream generators based on LFSRs, correlation attacks, decoding methods, fast correlation attacks, algorithmic aspects of correlation based attacks. (reference books) [1] Antoine Joux, Algorithmic Cryptanalysis, (2010) CRC Press; [2] Douglas Stinson, Cryptography: Theory and Practice, 3rd edition, (2006) Chapman and Hall/CRC.	7	INF/01	60	-	-	-	Elective activities	ITA
20410038 - GRAPH THEORY - Derived from 20410038 GE460 - TEORIA DEI GRAFI in MATEMATICA (DM 270) LM-40 CAPORASO LUCIA	7	MAT/03	60	-	-	-	Elective activities	ITA
20410040 - COMPUTATIONAL METHODS IN SYSTEMS BIOLOGY - Derived from 20410040 IN470 - INFORMATICA 10 - METODI COMPUTAZIONALI PER LA BIOLOGIA in MATEMATICA (DM 270) LM-40 CASTIGLIONE Filippo	7	INF/01	60	-	-	-	Elective activities	ITA

Second semester

Course

Credits

Scientific Disciplinary Sector Code

Contact Hours

Exercise Hours

Laboratory Hours

Personal Study Hours

Type of Activity

Language

20402082 - FS220 - PHYSICS 2 (objectives)

- BIANCHI Stefano

- PLASTINO WOLFANGO

FIS/01

Related or supplementary learning activities

ITA

20402131 - SCIENTIFIC ENGLISH (objectives)

- SERNESI EDOARDO

Other activities

ITA

Optional group: comune Orientamento unico DUE INSEGNAMENTI A SCELTA NEL GRUPPO 2 - (show)

20402085 - AM310 - ELEMENTS OF ADVANCED ANALYSIS (objectives) The student is going to learn the basics of the Lebesgue integration theory: measure spaces, measurability, Lebesgue integral, L^p spaces, differentiation. - BESSI UGO (syllabus) Definition of a sigma-algebra; measures on a sigma-algebra; definition of the integral; monotone and dominated convergence theorems; continuity and derivation under the integral sign; the L^p spaces; Holder and Minkowski inequalities; L^p is complete; the Borel sets and Lusin's theorem; the compactly supported, C^\infty functions are dense in L^p(R^n); convergence in measure and almost-uniform convergence; Egorov's theorem; Caratheodory's extension theorem and the construction of Lebesgue's measure; product measures; the theorems of Fubini and Tonelli; complex measures; the Radon=Nykodim theorem and differentiation of measures. (reference books) W. Rudin, Real and complex analysis, Tata-McGraw Hill. H. L. Royden, Real analysis, China Machine Press. R. L. Wheeden, A. Zygmund, Measure and integral, Dekker. - CODOGNI GIULIO	7	MAT/05	72	-	-	-	Related or supplementary learning activities	ITA
20402086 - FM310 - MATHEMATICAL PHYSICS 2 (objectives) The aim of the course is to develop a good knowledge of fundamental methods in the solution of elementary problems in partial differential equations - PELLEGRINOTTI ALESSANDRO - SIGNORINO MARIA CRISTINA	7	MAT/07	72	-	-	-	Related or supplementary learning activities	ITA
20402092 - AN420 - NUMERICAL ANALYSIS 2 (objectives) THE COURSE PRESENTS A REVIEW OF NUMERICAL METHODS OF INCREASING IMPACT FOR APPLICATION. IN THIS LINE OF WORK, THE ELEMENTARY SCHEMES INTRODUCED IN THE FIRST COURSE ARE USED AS BUILDING BLOCKS FOR MORE COMPLEX METHODS, WITH THE FINAL GOAL OF INTRODUCING THE STUDENT (IN A SOMEWHAT SIMPLIFIED FRAMEWORK) TO THE GENERAL ASPECTS OF THE APPROXIMATE SOLUTION OF OPTIMIZATION PROBLEMS AND SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS. ALL TECHNIQUES WILL BE TESTED ON SOME BENCHMARK PROBLEMS OF INTEREST FOR APPLICATION. - Derived from 20402092 AN420 - ANALISI NUMERICA 2 in MATEMATICA (DM 270) LM-40 N0 FERRETTI ROBERTO (syllabus) 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. (reference books) [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.	7	MAT/08	60	-	-	-	Related or supplementary learning activities	ITA
20402102 - CR410 - CRYPTOGRAPHY 1 (objectives) ACQUIRE GOOD KNOWLEDGE OF THE CONCEPTS AND MATHEMATICAL METHODS OF PUBLIC KEY CRYPTOGRAPHY PROVIDING AN OVERVIEW OF THOSE WHICH ARE CURRENTLY IN USE.	7	MAT/02	60	-	-	-	Related or supplementary learning activities	ITA
20410107 - CR410 - CRITTOGRAFIA 1 - Derived from 20410107 CR410 - CRITTOGRAFIA 1 in MATEMATICA (DM 270) LM-40 MEROLA FRANCESCA (syllabus) Introduction to cryptography. Historical background. Definition of cryptosystem. Classical ciphers. Introduction to cryptoanalysis. Introduction to public key cryptography. Computational complexity. The knapsack problem. The Merkle-Hellman cryptosystem. The RSA cryptosystem. Primality testing. Factorization algorithms. Some RSA attacks. the Rabin cipher. The discrete logarithm problem. Diffie-Hellman key exchange. Elgamal cryptosystem. Digital signature. Signature schemes. The RSA scheme. The Elgamal scheme. Notes on some cryptographic protocols. (reference books) Stinson - Cryptography, theory and practice. Chapman and Hall. Baldoni, Ciliberto, Piacentini-Cattaneo - Aritmetica, crittografia e codici. Springer.	7	MAT/03	60	-	-	-	Related or supplementary learning activities	ITA

Optional group: comune Orientamento unico DUE INSEGNAMENTI A SCELTA AMPIA - (show)

20402085 - AM310 - ELEMENTS OF ADVANCED ANALYSIS (objectives) The student is going to learn the basics of the Lebesgue integration theory: measure spaces, measurability, Lebesgue integral, L^p spaces, differentiation. - Derived from 20402085 AM310 - ISTITUZIONI DI ANALISI SUPERIORE in MATEMATICA (DM 270) L-35 N0 BESSI UGO, CODOGNI GIULIO (syllabus) Definition of a sigma-algebra; measures on a sigma-algebra; definition of the integral; monotone and dominated convergence theorems; continuity and derivation under the integral sign; the L^p spaces; Holder and Minkowski inequalities; L^p is complete; the Borel sets and Lusin's theorem; the compactly supported, C^\infty functions are dense in L^p(R^n); convergence in measure and almost-uniform convergence; Egorov's theorem; Caratheodory's extension theorem and the construction of Lebesgue's measure; product measures; the theorems of Fubini and Tonelli; complex measures; the Radon=Nykodim theorem and differentiation of measures. (reference books) W. Rudin, Real and complex analysis, Tata-McGraw Hill. H. L. Royden, Real analysis, China Machine Press. R. L. Wheeden, A. Zygmund, Measure and integral, Dekker.	7	MAT/05	72	-	-	-	Elective activities	ITA
20402086 - FM310 - MATHEMATICAL PHYSICS 2 (objectives) The aim of the course is to develop a good knowledge of fundamental methods in the solution of elementary problems in partial differential equations - Derived from 20402086 FM310 - FISICA MATEMATICA 2 in MATEMATICA (DM 270) L-35 N0 PELLEGRINOTTI ALESSANDRO, SIGNORINO MARIA CRISTINA	7	MAT/07	72	-	-	-	Elective activities	ITA
20402089 - IN410 - COMPUTER SCIENCE 2 (objectives) The course Theory of Computation and Interaction provides a in-deep view of theoretical aspects related to the concept of computation and the study of relations between different models of computation. The basic knowledge on information technology is here extended with new concepts and theoretical viewpoints. The course is divided into two units of 6 CFU. At choice, the student can decide to pass the first unit or both (12 CFU). More specifically, the course provides a formal presentation of the concepts of algorithm and computability. After the introduction of the classical concept of computability as formalized by Alan M. Turing, we address the basic concepts of algorithmic complexity and problem decidability, functional models and functional programming. In the second unit, we focus on interactive paradigms in the theory of computation which allow the description of additional complexity classes and their use in the semantics of programming languages. - Derived from 20402089 IN410 - INFORMATICA 2 in MATEMATICA (DM 270) LM-40 N0 PEDICINI MARCO (syllabus) 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. (reference books) [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).	7	INF/01	72	-	-	-	Elective activities	ITA
20402091 - TN410 - INTRODUCTION TO NUMBER THEORY (objectives) TO ACQUIRE A GOOD KNOWLEDGE OF CONCEPTS AND METHODS OF ELEMENTARY NUMBER THEORY, WITH PARTICULAR RESPECT OF STUDY OF DIOPHANTINE EQUATIONS AND POLYNOMIAL CONGRUENCES. TO GIVE PREREQUISITES FOR ADVANCED COURSES OF ALGEBRAIC AND ANALYTIC NUMBER THEORY. - Derived from 20402091 TN410 - INTRODUZIONE ALLA TEORIA DEI NUMERI in MATEMATICA (DM 270) LM-40 N0 TARTARONE FRANCESCA (syllabus) CONGRUENCES AND POLYNOMIALS. LINEAR DIOPHANTINE EQUATIONS. SYSTEMS OF LINEAR CONGRUENCES. POLYNOMIAL CONGRUENCES. POLYNOMIAL CONGRUENCES MOD p: LAGRANGES’S THEOREM. p-ADIC APPROXIMATION. THE EXISTENCE OF PRIMITIVE ROOTS MOD p. QUADRATIC CONGRUENCES. QUADRATIC RESIDUES. THE LEGENDRE SYMBOL. GAUSS’S LEMMA AND THE LAW OF QUADRATIC RECIPROCITY.THE JACOBI SYMBOL. SUMS OF TWO, THREE, FOUR SQUARES. MULTIPLICATIVE FUNCTIONS. THE MÖBIUS INVERSION FORMULA. SOME NONLINEAR DIOPHANTINE EQUATIONS. CONTINUED FRACTIONS. (reference books) M. FONTANA, APPUNTIDEL CORSO TN1 (ARGOMENTI DELLA TEORIA CLASSICA DEI NUMERI), HTTP://WWW.MAT.UNIROMA3.IT/USERS/FONTANA/DIDATTICA/FONTANA_DIDATTICA.HTML DISPENSE. D.M. BURTON: ELEMENTARY NUMBER THEORY, MCGRAW-HILLINTERNATIONAL EDITION, 6TH EDITION (2007), 434 PP. G.A. JONES AND J.M. JONES, ELEMENTARY NUMBER THEORY, SPRINGER, 1ST EDITION (1998), 200 PP. H. DAVENPORT, ARITMETICA SUPERIORE.UN?INTRODUZIONE ALLA TEORIA DEI NUMERI, ZANICHELLI, (1994), 199 PP. G.H. HARDY AND E.M. WRIGHT, AN INTRODUCTION TO THE THEORY OH NUMBERS, THE CLARENDON PRESS, OXFORD UNIVERSITY, 5TH EDITION (1979), XVI+426 PP. W.J. LEVEQUE, FUNDAMENTALS OF NUMBER THEORY, DOVER PUBLICATIONS, (1996) K.H. ROSEN. ELEMENTARY NUMBER THEORY AND ITS APPLICATIONS, ADDISON-WESLEY, 6TH EDITION (2011), XV+752 PP.	7	MAT/02	60	-	-	-	Elective activities	ITA
20402092 - AN420 - NUMERICAL ANALYSIS 2 (objectives) THE COURSE PRESENTS A REVIEW OF NUMERICAL METHODS OF INCREASING IMPACT FOR APPLICATION. IN THIS LINE OF WORK, THE ELEMENTARY SCHEMES INTRODUCED IN THE FIRST COURSE ARE USED AS BUILDING BLOCKS FOR MORE COMPLEX METHODS, WITH THE FINAL GOAL OF INTRODUCING THE STUDENT (IN A SOMEWHAT SIMPLIFIED FRAMEWORK) TO THE GENERAL ASPECTS OF THE APPROXIMATE SOLUTION OF OPTIMIZATION PROBLEMS AND SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS. ALL TECHNIQUES WILL BE TESTED ON SOME BENCHMARK PROBLEMS OF INTEREST FOR APPLICATION. - Derived from 20402092 AN420 - ANALISI NUMERICA 2 in MATEMATICA (DM 270) LM-40 N0 FERRETTI ROBERTO (syllabus) 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. (reference books) [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.	7	MAT/08	60	-	-	-	Elective activities	ITA
20402102 - CR410 - CRYPTOGRAPHY 1 (objectives) ACQUIRE GOOD KNOWLEDGE OF THE CONCEPTS AND MATHEMATICAL METHODS OF PUBLIC KEY CRYPTOGRAPHY PROVIDING AN OVERVIEW OF THOSE WHICH ARE CURRENTLY IN USE. - Derived from 20410107 CR410 - CRITTOGRAFIA 1 in MATEMATICA (DM 270) LM-40 MEROLA FRANCESCA (syllabus) Introduction to cryptography. Historical background. Definition of cryptosystem. Classical ciphers. Introduction to cryptoanalysis. Introduction to public key cryptography. Computational complexity. The knapsack problem. The Merkle-Hellman cryptosystem. The RSA cryptosystem. Primality testing. Factorization algorithms. Some RSA attacks. the Rabin cipher. The discrete logarithm problem. Diffie-Hellman key exchange. Elgamal cryptosystem. Digital signature. Signature schemes. The RSA scheme. The Elgamal scheme. Notes on some cryptographic protocols. (reference books) Stinson - Cryptography, theory and practice. Chapman and Hall. Baldoni, Ciliberto, Piacentini-Cattaneo - Aritmetica, crittografia e codici. Springer.	7	INF/01	72	-	-	-	Elective activities	ITA
20402106 - GE430 - DIFFERENTIAL GEOMETRY 2 (objectives) We present an introduction to Riemannian Geometry based on the study of geodesics. We prove selected results showing some relations between curvature and topology. - Derived from 20402106 GE430 - GEOMETRIA DIFFERENZIALE 2 in MATEMATICA (DM 270) LM-40 PONTECORVO MASSIMILIANO	7	MAT/03	60	-	-	-	Elective activities	ITA
20402108 - IN430 - INFORMATICS 4: ADVANCED COMPUTING TECHNIQUES (objectives) THE COURSE IN430 – COMPUTER SCIENCE 4, ADVANCED COMPUTATIONAL TECHNIQUES IS FOCUSED ON THE ACQUISITION OF OBJECT ORIENTED PROGRAMMING LANGUAGES AND APPLICATION OF CONCEPTUAL TOOLS FOR ANALYSIS AND DEVELOPMENT IN OBJECT ORIENTEND PROGRAMMING. THIS COURSE INCLUDES AN INTRODUCTION TO MODELING AND DESIGN OF CLASSES THROUGH UML DIAGRAMS, AND THE STUDY OF SPECIFICATION AND IMPLEMENTATION OF ALGORITHMS FOR GRAPH ANALYSIS. - Derived from 20402108 IN430 - INFORMATICA 4, TECNICHE INFORMATICHE AVANZATE in MATEMATICA (DM 270) LM-40 N0 LOMBARDI FLAVIO	7	INF/01	60	-	-	-	Elective activities	ITA
20402109 - IN440 - COMPUTER SCIENCE 5: COMBINATORIAL OPTIMISATION (objectives) The aim of the course is to acquire skills on resolution techniques for combinatorial optimization problems, deepening the skills on graph theory, advanced technical skills for design, analysis and computer implementation of algorithms for solving optimization problems on graphs, trees and networks.	7	INF/01	60	-	-	-	Elective activities	ITA
20402117 - FS410 - PHYSICS 3: RELATIVITY AND RELATIVISTIC THEORIES (objectives) FAR ACQUISIRE ALLO STUDENTE I FONDAMENTI CONCETTUALI E OSSERVATIVI DELLA TEORIA DELLA RELATIVITÀ GENERALE, COME TEORIA GEOMETRICA DELLA GRAVITAZIONE, E LE SUE IMPLICAZIONI A LIVELLO FENOMENOLOGICO E FILOSOFICO; INSEGNARGLI A PADRONEGGIARE IL FORMALISMO QUADRIDIMENSIONALE DELLO SPAZIO-TEMPO EINSTEINIANO, QUALE VARIETÀ RIEMANNIANA, NEI SUOI ASPETTI DI CALCOLO TENSORIALE E DI GEOMETRIA DIFFERENZIALE. - Derived from 20410010 ESPERIMENTAZIONI DI FISICA I in FISICA (DM 270) L-30 DE VINCENZI MARIO, BARTIROMO ROSARIO, Verducci Monica	7	FIS/02	60	-	-	-	Elective activities	ITA
20402123 - MA410 - APPLIED AND INDUSTRIAL MATHEMATICS (objectives) CALCULUS AND, IN PARTICULAR, DIFFERENTIAL EQUATIONS ARE IMPORTANT IN THE RESEARCH AND DEVELOPMENT OF APPLIED AND INDUSTRIAL MATEMATICS. THESE MATHEMATICAL TOOLS ARE NECESSARY TO UNDERSTAND A NUMBER OF PHYSICAL, CHEMICAL, BIOLOGICAL AND FINANCIAL PHENOMENA, AND TO IMPROVE THE QUALITY OF INDUSTRIAL PRODUCTS AND PROCESSES. MODELING AND SIMULATION COULD BE THE BEST TERMS TO DESCRIBE THE SPIRIT OF THIS COURSE. CONCRETE PROBLEMS WILL BE CONSIDERED AS EXAMPLES. - Derived from 20402123 MA410 - MATEMATICA APPLICATA E INDUSTRIALE in MATEMATICA (DM 270) LM-40 N0 SPIGLER RENATO, CONCEZZI MORENO	7	MAT/08	60	-	-	-	Elective activities	ITA
20402187 - AL440 – GROUP THEORY (objectives) ADVANCED COURSE IN GROUP THEORY. THE INTENT IS TO DEEPEN THE MAIN GROUP THEORY CONCEPTS STUDIED IN AL210, ALSO THROUGH SEMINARS GIVEN BY STUDENTS ABOUT TOPICS IN FINITE GROUP AND FREE GROUP THEORY.	7	MAT/02	60	-	-	-	Elective activities	ITA
20402249 - CH410 - ELEMENTS OF CHEMISTRY (objectives) Acquiring knowledge of the basic principles of General Chemistry and the ability to apply the acquired knowledge to simple problems of Chemistry. - Derived from 20401116 ELEMENTI DI CHIMICA in FISICA (DM 270) L-30 N0 IUCCI GIOVANNA (syllabus) Atomic Structure Chemical bonds and Molecular structure Chemical compounds and reactions Aggregation states Thermodynamics Solutions Kinetics Chemical equilibrium Reactions in solution Electrochemistry (reference books) M. Schiavello, L. Palmisano; “Fondamenti di Chimica”, Edises P. Michelin Lausarot, G. A. Vaglio; “Stechiometria per la Chimica Generale” Piccin - Derived from 20401116 ELEMENTI DI CHIMICA in FISICA (DM 270) L-30 N0 IUCCI GIOVANNA (syllabus) Atomic Structure Chemical bonds and Molecular structure Chemical compounds and reactions Aggregation states Thermodynamics Solutions Kinetics Chemical equilibrium Reactions in solution Electrochemistry (reference books) M. Schiavello, L. Palmisano; “Fondamenti di Chimica”, Edises P. Michelin Lausarot, G. A. Vaglio; “Stechiometria per la Chimica Generale” Piccin	7	CHIM/03	60	-	-	-	Elective activities	ITA
20402290 - IN420 - COMPUTER SCIENCE 3, INFORMATION THEORY - Derived from 20402290 IN420-INFORMATICA 3, TEORIA DELL'INFORMAZIONE in MATEMATICA (DM 270) LM-40 PEDICINI MARCO (syllabus) 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. (reference books) [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).	7	INF/01	60	-	-	-	Elective activities	ITA
20410107 - CR410 - CRITTOGRAFIA 1 - Derived from 20410107 CR410 - CRITTOGRAFIA 1 in MATEMATICA (DM 270) LM-40 MEROLA FRANCESCA (syllabus) Introduction to cryptography. Historical background. Definition of cryptosystem. Classical ciphers. Introduction to cryptoanalysis. Introduction to public key cryptography. Computational complexity. The knapsack problem. The Merkle-Hellman cryptosystem. The RSA cryptosystem. Primality testing. Factorization algorithms. Some RSA attacks. the Rabin cipher. The discrete logarithm problem. Diffie-Hellman key exchange. Elgamal cryptosystem. Digital signature. Signature schemes. The RSA scheme. The Elgamal scheme. Notes on some cryptographic protocols. (reference books) Stinson - Cryptography, theory and practice. Chapman and Hall. Baldoni, Ciliberto, Piacentini-Cattaneo - Aritmetica, crittografia e codici. Springer.	7	MAT/03	60	-	-	-	Elective activities	ITA
20410069 - FS410 - LABORATORIO DI DIDATTICA DELLA FISICA - Derived from 20410010 ESPERIMENTAZIONI DI FISICA I in FISICA (DM 270) L-30 DE VINCENZI MARIO, BARTIROMO ROSARIO, Verducci Monica	7	FIS/08	60	-	-	-	Elective activities	ITA
20410074 - LM430 - TEORIA ASSIOMATICA DEGLI INSIEMI - Derived from 20710092 TEORIE LOGICHE 2 - LM in SCIENZE FILOSOFICHE (DM 270) LM-78 TORTORA DE FALCO LORENZO	7	MAT/01	60	-	-	-	Elective activities	ITA
20402095 - AL420 - ALGEBRAIC THEORY OF NUMBERS - Derived from 20402095 AL420 - TEORIA ALGEBRICA DEI NUMERI in MATEMATICA (DM 270) LM-40 GABELLI STEFANIA	7	MAT/02	60	-	-	-	Elective activities	ITA
20402100 - CP420 - STOCHASTIC PROCESSES - Derived from 20402100 CP420 - PROCESSI STOCASTICI in MATEMATICA (DM 270) LM-40 MARTINELLI FABIO	7	MAT/06	60	-	-	-	Elective activities	ITA
20402327 - FM450 - MATHEMATICAL FOUNDATIONS OF QUANTUM MECHANICS - Derived from 20402327 FM450- ASPETTI MATEMATICI DELLA MECCANICA QUANTISTICA in MATEMATICA (DM 270) LM-40 N0 CORREGGI MICHELE	7	MAT/07	175	-	-	-	Core compulsory activities	ITA
20410094 - GE470 - Riemann surfaces - Derived from 20410094 GE470 - SUPERFICI DI RIEMANN in MATEMATICA (DM 270) LM-40 SERNESI EDOARDO	7	MAT/03	60	-	-	-	Elective activities	ITA
20410073 - IN460 - INFORMATICA 7, GEOMETRIA COMPUTAZIONALE - Derived from 20410073 IN460 - INFORMATICA 7, GEOMETRIA COMPUTAZIONALE in MATEMATICA (DM 270) LM-40 (docente da definire)	7	INF/01	60	-	-	-	Elective activities	ITA

20401599 - FINAL EXAM

225

Final examination and foreign language test

ITA