Optional group:
comune Orientamento unico DUE INSEGNAMENTI A SCELTA NEL GRUPPO 2 - (show)
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14
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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.
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PAPPALARDI FRANCESCO
( syllabus)
Introduction: Cardano equations for the solubility of third-degree equations, rings and fields, the characteristics of a field, known facts about rings of polynomials, field extensions, construction of some field extensions, the sub-ring generated by a subset, the subfield generated by a subset, algebraic and transcendent elements, algebraically closed fields.
Splitting fields: Simple extensions and maps between simple extensions, splitting fields, existence of the splitting field, uniqueness up to isomorphisms of the splitting field, multiple roots, formal derivatives, separable polynomials and perfect fields, minimal polynomials and their characterizations.
The fundamental Theorem of Galois Theory: Group of the automorphisms of a field, normal, separable and Galois extensions, characterizations of separable extensions, Fundamental Theorem of Galois Correspondence, examples, Galois group of a polynomial, Radical extensions, solvable groups and Galois's Theorem on solving equations , Theorem of the existence of primitive elements.
The computation of Galois group: Galois groups as subgroups of $ S_n $, transitive subgroups of $ S_n $, characterization of irreducibility in terms of transitivity, polynomials with Galois groups in $ A_n $, Theory of discriminants, Galois groups of polynomials with degree up to $4$, examples of polynomials with Galois group $S_p$.
Cyclotomic fields: Definitions, Galois group, maximal real subfields, quadratic subfields, Galois groups, cyclotomic polynomials and their properties, Theorem of the inverse Galois theory for abelian groups.
Finite fields: Existence and uniqueness of finite fields, Galois group of a finite field, subfields of a finite field, enumeration of irreducible polynomials on finite fields. Construction of the algebraic closure of a finite field with $ p $ elements.
Constructions with ruler and compass: Definition of constructible points of the plane, constructible real numbers, characterization of constructible points in terms of fields, constructible subfields and construction of constructible numbers, cube duplication, trisection of angles, quadrature of the circle and Gauss's theorem for the construction of regular polygons with ruler and compass.
( reference books)
J. S. Milne.Fields and Galois Theory. Course Notes v4.22 (March 30, 2011).
S. Gabelli. Teoria delle Equazioni e Teoria di Galois. Springer UNITEXT (La Matematica per il 3+2) 2008, XVII, 410 pagg., ISBN: 978-88-470-0618-8
E. Artin.Galois Theory. NOTRE DAME MATHEMATICAL LECTURES Number 2. 1942.
C. Procesi.Elementi di Teoria di Galois. Decibel, Zanichelli, (Seconda ristampa, 1991).
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7
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MAT/02
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60
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12
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Related or supplementary learning activities
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ITA |
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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.
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ESPOSITO PIERPAOLO
( syllabus)
1. Abstract integration theory
Riemann integration theory. The concept of measurability. Step functions. Elementary properties of measures. Arithmetic in [0,∞]. Integration of positive functions. Integration of complex functions. Importance of sets
with null measure.
2. Positive Borel measures
Vector spaces. Topological preliminaries. Riesz representation theorem. Regularity properties of Borel measures. Lebesgue measure. Continuity properties of measurable functions.
3. L^p spaces
Inequalities and convex functions. L^p spaces. Approximation through continuous functions.
4. Basic theory of Hilbert spaces
Inner products and linear functionals. Dual space of L^2
4. Integration on product spaces
Measurability on cartesian products. Product measure. Fubini theorem.
4. Complex measures
Total variation. Absolute continuity. Radon-Nykodym theorem. Bounded linear functionals on L^p. The Riesz representation theorem.
( reference books)
"Analisi reale e complessa”, W. Rudin. Bollati Boringhieri.
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7
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MAT/05
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60
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12
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Related or supplementary learning activities
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ITA |
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20402087 -
GE310 - ELEMENTS OF ADVANCED GEOMETRY
(objectives)
A REFINED STUDY OF TOPOLOGY VIA ALGEBRAIC AND ANALYTIC TOOLS.
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7
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MAT/03
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60
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12
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Related or supplementary learning activities
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ITA |
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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.
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7
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MAT/06
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60
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Related or supplementary learning activities
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ITA |
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20410107 -
CR410 - CRITTOGRAFIA 1
(objectives)
Acquire a basic understanding of the notions and methods of public-key encryption theory, providing an overview of the models which are most widely used in this field.
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7
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MAT/03
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60
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Related or supplementary learning activities
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ITA |
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Optional group:
comune Orientamento unico DUE INSEGNAMENTI A SCELTA AMPIA - (show)
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14
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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 L-35 N0 PAPPALARDI FRANCESCO, TALAMANCA VALERIO
( syllabus)
Introduction: Cardano equations for the solubility of third-degree equations, rings and fields, the characteristics of a field, known facts about rings of polynomials, field extensions, construction of some field extensions, the sub-ring generated by a subset, the subfield generated by a subset, algebraic and transcendent elements, algebraically closed fields.
Splitting fields: Simple extensions and maps between simple extensions, splitting fields, existence of the splitting field, uniqueness up to isomorphisms of the splitting field, multiple roots, formal derivatives, separable polynomials and perfect fields, minimal polynomials and their characterizations.
The fundamental Theorem of Galois Theory: Group of the automorphisms of a field, normal, separable and Galois extensions, characterizations of separable extensions, Fundamental Theorem of Galois Correspondence, examples, Galois group of a polynomial, Radical extensions, solvable groups and Galois's Theorem on solving equations , Theorem of the existence of primitive elements.
The computation of Galois group: Galois groups as subgroups of $ S_n $, transitive subgroups of $ S_n $, characterization of irreducibility in terms of transitivity, polynomials with Galois groups in $ A_n $, Theory of discriminants, Galois groups of polynomials with degree up to $4$, examples of polynomials with Galois group $S_p$.
Cyclotomic fields: Definitions, Galois group, maximal real subfields, quadratic subfields, Galois groups, cyclotomic polynomials and their properties, Theorem of the inverse Galois theory for abelian groups.
Finite fields: Existence and uniqueness of finite fields, Galois group of a finite field, subfields of a finite field, enumeration of irreducible polynomials on finite fields. Construction of the algebraic closure of a finite field with $ p $ elements.
Constructions with ruler and compass: Definition of constructible points of the plane, constructible real numbers, characterization of constructible points in terms of fields, constructible subfields and construction of constructible numbers, cube duplication, trisection of angles, quadrature of the circle and Gauss's theorem for the construction of regular polygons with ruler and compass.
( reference books)
J. S. Milne.Fields and Galois Theory. Course Notes v4.22 (March 30, 2011).
S. Gabelli. Teoria delle Equazioni e Teoria di Galois. Springer UNITEXT (La Matematica per il 3+2) 2008, XVII, 410 pagg., ISBN: 978-88-470-0618-8
E. Artin.Galois Theory. NOTRE DAME MATHEMATICAL LECTURES Number 2. 1942.
C. Procesi.Elementi di Teoria di Galois. Decibel, Zanichelli, (Seconda ristampa, 1991).
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7
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MAT/02
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72
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Elective activities
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ITA |
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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 L-35 N0 ESPOSITO PIERPAOLO, BATTAGLIA LUCA
( syllabus)
1. Abstract integration theory
Riemann integration theory. The concept of measurability. Step functions. Elementary properties of measures. Arithmetic in [0,∞]. Integration of positive functions. Integration of complex functions. Importance of sets
with null measure.
2. Positive Borel measures
Vector spaces. Topological preliminaries. Riesz representation theorem. Regularity properties of Borel measures. Lebesgue measure. Continuity properties of measurable functions.
3. L^p spaces
Inequalities and convex functions. L^p spaces. Approximation through continuous functions.
4. Basic theory of Hilbert spaces
Inner products and linear functionals. Dual space of L^2
4. Integration on product spaces
Measurability on cartesian products. Product measure. Fubini theorem.
4. Complex measures
Total variation. Absolute continuity. Radon-Nykodym theorem. Bounded linear functionals on L^p. The Riesz representation theorem.
( reference books)
"Analisi reale e complessa”, W. Rudin. Bollati Boringhieri.
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7
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MAT/05
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72
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-
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Elective activities
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ITA |
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20402087 -
GE310 - ELEMENTS OF ADVANCED GEOMETRY
(objectives)
A REFINED STUDY OF TOPOLOGY VIA ALGEBRAIC AND ANALYTIC TOOLS.
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7
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MAT/03
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72
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Elective activities
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ITA |
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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.
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Derived from
20402088 AN410 - ANALISI NUMERICA 1 in Scienze Computazionali LM-40 FERRETTI ROBERTO, CACACE SIMONE
( syllabus)
Linear Systems
Direst methods: Gaussian elimination. Pivoting strategies. Gaussian elimination as a factorization. Doolittle and Choleshy factorizations. Iterative methods: Jacobi, Gauss-Seidel, SOR, Richardson, and related convergence results. Comparison of direct vs iterative solvers. Stability of algorithms for the solution of linear systems.
Iterative Methods for Scalar Nonlinear Equations
The intermediate zero theorem. The algorithms of bisection, Newton, secants, chords, and related convergence results. (Reference: Chapter 1 excluding Section 1.2.3, and Appendices A.1, A.2)
Approximation of Functions
General approximation strategies. Interpolating polynomial in Lagrange and Newton form. Representation of the interpolation error. Convergence of the interpolating polynomial for analytic functions. Refinement strategies in interpolation: Chebyshev nodes, composite approximations. Error estimates. Hermite polynomial, construction and representation of the error. Least Squares approximations. (Reference: Chapter 5 excluding Section 5.2, and Appendix A.4)
Numerical Integration
General principles of numerical integration. Polya's Theorem on the convergence of interpolatory quadrature formulae. Closed and open Newton-Cotes formulae. Stability results and error estimation. Generalized Newton-Cotes formulae and their convergence. Gaussian quadratures and their convergence. (Reference: Chapter 6)
Laboratory Activity
C language coding of some of the major algorithms, and in particular: Gaussian elimination, iterative methods for linear systems and scalar equations, Lagrange/Newton interpolation with a refinement strategy.
N.B.: References are provided with respect to the course notes.
( reference books)
Roberto Ferretti, "Appunti del corso di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/corso.pdf
Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf
Slides of the lessons, available from the course page: http://www.mat.uniroma3.it/users/ferretti/bacheca.html
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7
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MAT/08
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72
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Elective activities
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ITA |
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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.
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7
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INF/01
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60
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12
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Elective activities
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ITA |
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20402090 -
MC410 - COMPLEMENTARY MATHEMATICS 1
(objectives)
To acquire deep understanding of the principal geometry arguments treated in high-school mathematics
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Derived from
20410343 MC310 - ISTITUZIONI DI MATEMATICHE COMPLEMENTARI in Matematica LM-40 BRUNO ANDREA, Savarese Michele
( syllabus)
1. Euclidean Geometry
Rudiments of Greek mathematics history. Ruler and compass constructions. Classical problems. The Elements. Axioms, definitions and postulates of Book I. Theorems I-XXVIII with proofs. Theorems XXIX, XXX, XXXI, XXXII: the role of V Postulate.
2. The question of V Postulate
The attempt by Posidonio. Equivalent propositions: Playfair, Wallis, transitivity of parallelism. Saccheri's quadrilateral. Quadrilateri di Saccheri. Saccheri-Lagrange theorem and the exclusion of the obtuse angle hypothesis. The non-euclidean geometries of Bolyai and Lobachevski.
3. Isometries of the plane
Even and odd isometries. Characterisation of an isometry by the image of three points not on a line. Chasles' Theorem. Products of reflections. Discrete groups of isometries. Finite groups, friezes, crystals. The theorem of addition of the angle. Leonardo's Theorem and the characterisation of finite groups.
Sketch of proof of the theorem of classification of frieze groups. Crystallographic restriction Theorem and the classification of wallpaper groups.
4. The geometry after Gauss
The geometry on the Sphere. Locally euclidean geometries. Uniformly discontinuous groups of isometries. The Torus, Moebius strip and the Klein bottle. Classification of uniformly discontinuous groups Sketch of the proof of the Theorem of Classification of locally euclidean geometries.
5. Geometries on the Torus and the Hyperbolic geometry
Similar geometries. Similar geometries on the Torus. The modular figure. Poincaré Half plane model. Lines and distance. What is repugnant for Saccheri, but not for Aristotle
( reference books)
R. Trudeau: La Rivoluzione non euclidea. Bollati Boringhieri ed, 1991
V. Nikulin, I. Shafarevich: Geometries and groups. Springer ed, 1987
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7
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MAT/04
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60
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Elective activities
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ITA |
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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.
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7
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MAT/06
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60
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-
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Elective activities
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ITA |
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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.
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Derived from
20402104 GE410 - GEOMETRIA ALGEBRICA 1 in Matematica LM-40 N0 MASCARENHAS MELO ANA MARGARIDA
( syllabus)
- Classical theory of algebraic varieties in affine spaces over algebraically closed fields.
- Local geometry, normalization.
- Divisors, linear systems and morphisms of projective varieties.
( reference books)
Algebraic Geometry texts:
R. Hartshorne, Algebraic geometry, Graduate Texts in Math. No. 52. Springer-Verlag, New York-Heidelberg, 1977.
I. Shafarevich, Basic algebraic geometry vol. 1, Springer-Verlag, New York-Heidelberg, 1994.
J. Harris, Algebraic geometry (a first course), Graduate Texts in Math. No. 133. Springer-Verlag, New York-Heidelberg, 1992.
Algebra texts:
* M. Artin, Algebra, Bollati Boringhieri 1997.
* M.F. Atiyah, I.G. Mac Donald, Introduzione all'algebra commutativa, Feltrinelli 1991.
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7
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MAT/03
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60
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-
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Elective activities
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ITA |
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20402115 -
ST410 - STATISTICS 1
(objectives)
Acquire a good understanding of the basic statistical mathematical methodologies for inference problems and statistical modeling. Develop a knowledge of some specific statistical packages for the practical application of acquired theoretical tools
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7
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SECS-S/01
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60
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12
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Elective activities
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ITA |
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20402122 -
FS420 - QUANTUM MECHANICS
(objectives)
THE COURSE AIMS TO PROVIDE A BASIC KNOWLEDGE OF QUANTUM MECHANICS, DISCUSSING THE MAIN EXPERIMENTAL EVIDENCES AND THE FOLLOWING THEORETICAL INTERPRETATIONS WHICH LED TO THE CRISIS OF CLASSICAL PHYSICS AND ILLUSTRATING ITS FUNDAMENTAL PRINCIPLES: THE CONCEPT OF PROBABILITY, THE WAVE-PARTICLE DUALITY, THE UNCERTAINTY PRINCIPLE. THE QUANTUM DYNAMICS, THE SCHRODINGER EQUATION AND ITS RESOLUTION FOR SOME PHYSICAL RELEVANT SYSTEMS WILL BE THEN DESCRIBED.
-
Derived from
20410015 MECCANICA QUANTISTICA in Fisica L-30 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
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7
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FIS/02
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60
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-
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-
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Elective activities
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ITA |
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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.
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7
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CHIM/03
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60
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Elective activities
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ITA |
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20410038 -
GRAPH THEORY
(objectives)
Provide tools and methods of graph theory
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7
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MAT/03
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60
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Elective activities
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ITA |
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20410040 -
COMPUTATIONAL METHODS IN SYSTEMS BIOLOGY
(objectives)
Acquire the basic knowledge of biological systems and problems related to their understanding, also in relation to deviations from normal functioning and thus to the insurgence of pathologies. Take care of the modeling aspect as well as of numerical simulation, especially for problems formulated by means of equations and discrete systems. Acquire the knowledge of the major bio-informatics algorithms useful to analyze biological data
-
Derived from
20410147 IN470 - METODI COMPUTAZIONALI PER LA BIOLOGIA in Scienze Computazionali LM-40 CASTIGLIONE Filippo
( syllabus)
Outline of the course; Introduction and generality; Bioinformatics and algorithms; Computational biology in the clinic and in the pharmaceutical industry; Pharmacokinetics and pharmacodynamics;
Introduction to Systems Biology: what is computational biology; The roles of mathematical modeling and bioinformatics; what is he aiming for; what are the problems; Theoretical tools used in bio-mathematics and bioinformatics.
Introduction to molecular and cellular biology (first part): basic knowledge of genetics, proteomics and cellular processes; Ecology and evolution; the basic molecule; molecular bonds; the chromosomes; DNA and its replication;
Introduction to molecular and cellular biology (second part); genomics; The central dogma of biology; The genome project; the structure of the human genome Analysis of genes; transcription of DNA; the viruses;
Laboratory: generation of random numbers; the functions srand48 and drand48; random generation of arbitrary length nucleotide strings (program1.c); random generation of amino acid strings of arbitrary length (program2.c);
Introduction to information theory; Shannon Entropy; Conditional Entropy; Mutual Information; Indices of biological diversity; Shannon Index; True diversity; Reny index;
Laboratory: the genetic code; C program of transcription DNA sequence and translation into proteins;
Introduction to stochastic processes; basic definition; examples; model of queues; Bernoulli and Poisson process; Markov processes; stochastic processes in bioinformatics and bio-mathematics; the autocorrelation; Outline of the Random Walks and the BLAST algorithm of sequence alignment as a stochastic process and principal algorithm for the consultation of biological sequence databases;
Laboratory: development of an algorithm in C for the calculation of the Shannon Entropy of a text in English (or in Italian) any (e.g., http://www.textfiles.com/etext/)
Random walks. The BLAST algorithm for aligning sequences as a random path; Laboratory: C implementation of different algorithms for the generation of a random walk in 1D and 2D on the lattice and in R or R ^ 2 signal and calculation of the mean square displacement;
Compare sequences: similarity and homology; pairwise alignment; editing distance; scoring matrices PAM and BLOSUM; Needleman-Wunsch's algorithm; local alignment; Smith-Waterman's algorithm; BLAST algorithm;
Laboratory: C implementation of an algorithm for the generation of a signal with noise and calculation of the correlogram in the presence or absence of a true signal;
Multiple Sequence Alignment; consensus sequence; star alignment algorithms; ClustalW; entropy and circular sum scoring functions;
Biological data banks; reasons; data format; taxonomy; Primary DBs; Secondary DBs; NCBI, EMBL, DDBJ; NCBI EBI-Entrez; Exact matching / string searching: general; the agony of Knuth-Morris-Pratt;
Exact matching / string searching: the Boyer-Moore agoritm;
Exercise on an implementation of the Knuth-Morris-Pratt exact matching algorithm. Exercise on biological databases; primary databases; secondary databases; NCBI, EMBL, DDBJ; NCBI EBI-Entrez; Use of the BLAST algorithm
Phylogenetic Analysis; phylogenetic trees; dimension of the research space of phylogenetic algorithms; Methods of construction of phylogenetic trees; Data used for phylogenetic analysis; The Unweighted Pair Method Method with Arithmetic mean (UPGMA) algorithm; the Neighbor Joining Method algorithm; Hidden Markov Models; decoding; the Viterbi Algorithm; Evaluation;
Laboratory: completion of the exercise on mutation, selection and evolution of nucleotide strings (genotype) translated into amino acid strings (phenotype); Selection is made based on the presence of certain substrings in the phenotype that determines the fitness value; Implemented details, display of the convergence criterion and results, discussion, etc .;
Machine Learning; generality'; supervised and unsupervised learning; model selection; undefitting; overfitting; Polynomial curve fitting; machine learning as an estimate of the parameters and the problem of overfitting; subdivision of the training set into testing and testing; concept of bias and variance trade-off; Artificial Neural Networks; definizone; the percussion of Rosenblatt; the percettrone learning algorithm; the multi-layer perceptron;
Laboratory: completion of the implementation in ANSI C of the evolutionary algorithm of nucleotide strings (genotype) translated, through the use of the genetic code, into amino acid strings (phenotype);
Hidden Markov Models; The Forward Algorithm; The Backward Algorithm; Posterior Decoding; Learning; Baum-Welch Algorithm; Use of Hidden Markov Models for the analysis of bio-sequences; gene finding;
Artificial Neural Networks; the error-back propagation algorithm for learning MLP; types of neural networks; convolution networks; reinforcement networks; unsupervised learning and self-organizing maps; Introduction to graph theory; representation, terminology, concepts; paths; cycles; connettivita '; distance; connected components; distance;
Introduction to graph theory; visit breadth-first search; depth-first search; Dijkstra's algorithm; six-degree of separation; small world networks; centrality measures; degree centrality; eigenvector centrality; betweennes centrality; closeness centrality; The network biology; generality'; concepts; types of biological data used to build networks; network biology and network medicine; problems and algorithms used; centrality measures; random networks; scale-free networks; preferential attachment; scale-free network in biology;
Laboratory: completion of the exercise on the evolutionary algorithm; Implemented details, display of the convergence criterion and results, discussion, etc .;
Bio-mathematical models; prediction using theoretical models; the itertative paradigm of mathematical modeling; data-driven models; limited and non-population growth models; analytical derivation and examples; logistics growth; ecological models limited by density; The Lotka-Volterra model; the experiment by Huffaker and Kenneth; the SIR epidemic model and some of its variants; Perelson's model for HAART; the Java Populus application for the solution of continuous models of population dynamics; hints to the numerical resolution methods of differential equation systems;
Discrete models; spin models (Ising models); Cellular automata; Boolean networks; Agent-based models; data fitting and parameter estimation; software tools available; Cellular automata; introduction and history; definition; the 1-dimensional automaton; Wolfram classification; the 2-dimensional automaton; Conway's Game of Life; Software available for CA simulation; dedicated hardware (CA-Machine); the prey-predator model as a two-dimensional cellular automaton; relationship with the system of ordinary derivation equations; stochastic models; Stochastic CAs as discrete stochastic dynamic systems and stochastic processes; example of CA: Belousov-Zabotonsky reactions;
( reference books)
[-] E.S. Allman, J.A. Rhodes. Mathematical Models in Biology: An Introduction (2004) Cambridge University Press.
[-] W.J. Ewens, G.R. Grant. Statistical Methods in Bioinformatics, An Introduction (2005) Springer Verlag.
[-] R. Durbin, S. Eddy, A. Krogh, G. Mitchison. Biological sequence analysis - Probabilistic models of proteins and nucleic acids (1998) Cambridge University Press.
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7
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INF/01
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60
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Elective activities
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ITA |
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20410043 -
FIRST-ORDER CLASSICAL LOGIC
(objectives)
Obtain a good understanding of the principles of the classical logic of the first order and the sequencing of sequences for it, as well as of the main results that concern it.
-
Derived from
20710016 TEOREMI SULLA LOGICA 1 in Filosofia L-5 TORTORA DE FALCO LORENZO
( syllabus)
Part 1: Some preliminary notions.
Order relations and trees, inductive definitions, proofs by induction, axiom of choice and Kőnig's lemma.
Part 2: Provability and satisyability.
First order formal language: alphabet, terms, formulas, sequents. Structures for first order languages: structures, terms and formulas with parameters in a structure, value of terms, formulas and sequents. The calculus of sequents for first order logic: Gentzen's LK. Derivable sequents and derivations. Correctness of the rules of LK. Canonical analysis and fundamental theorem: construction of the canonical analysis (with and without cuts) and proof of the fundamental theorem of the canonical analysis. Consequences of the fundamental theorem: completeness theorem, compactness theorem, eliminability of cuts, L"owenheim-Skolem's theorem.
Part 3: Towards proof-theory: the cut-elimination theorem.
The cut-elimination procedure. Definition of the elementary steps of cut-elimination. First proof strategy (big reduction steps). Second proof strategy (reversion of derivations). The complexity of the cut-elimination procedure (sketch). Some immediate consequences of the cut-elimination theorem.
( reference books)
V.M. Abrusci, L. Tortora de Falco, Logica Volume 1- Dimostrazioni e modelli al primo
ordine. Springer, (2014).
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7
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MAT/01
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60
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-
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-
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-
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Elective activities
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ITA |
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20402110 -
IN450 - INFORMATICS 6: ALGORITHMS FOR CRYPTOGRAPHY
(objectives)
Acquire the knowledge of the main encryption algorithms. Deepen the mathematical skills necessary for the description of the algorithms. Acquire the cryptanalysis techniques used in the assessment of the security level provided by the encryption systems
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7
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INF/01
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60
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-
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-
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-
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Elective activities
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ITA |
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20402280 -
AN430 – NUMERICAL ANALYSIS 3
(objectives)
Introduce to the main techniques used for Numerical Analysis applications to partial differential equations, such as the finite difference and variational methods
-
Derived from
20410355 AN430 - METODO DEGLI ELEMENTI FINITI in Scienze Computazionali LM-40 TERESI LUCIANO
( syllabus)
The purpose of this course is to give a brief introduction to the Finite Elements Method (FEM), a gold standard for the numerical solution of PDEs systems. Oddly enough, the widespread use of the FEM is not accompanied by an adequate knowledge of the mathematical framework underlying the method.
This course, starting from the weak formulation of balance equations, will give an overview of the techniques used to reduce a differential problem into an algebraic one. During the course, some selected problems in mechanics and physics will be solved, covering the three main types of equations: elliptic, parabolic and hyperbolic.
The course will cover the following topics: - Applied Linear Algebra. - Boundary Value Problems. - Initial Value Problems Moreover, the students will be introduced to the use COMSOL Multiphysics, a scientific software for numerical simulations based on the Finite Element Method.
( reference books)
Mark S. Gockenbach,
Understanding and Implementing the Finite Elements Method,
SIAM, 2006
Gilbert Strang
Computational Science and Engineering
Wellesley-Cambridge Press, 2007
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7
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MAT/08
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60
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-
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-
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-
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Elective activities
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ITA |
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20410107 -
CR410 - CRITTOGRAFIA 1
(objectives)
Acquire a basic understanding of the notions and methods of public-key encryption theory, providing an overview of the models which are most widely used in this field.
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7
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MAT/03
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60
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-
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-
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-
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Elective activities
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ITA |
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20410068 -
MATHEMATICS OF FINANCIAL MARKETS
(objectives)
Acquire knowledge of the basic notions of financial mathematics. Deepen the valuation of financial assets and bonds, the forward rate structure of interest rates. Studying CAPM and APT Templates for portfolio choices, utility functions, stock price dynamics in discrete and continuous time, valuation of derivatives
-
Derived from
21201730 FINANZA COMPUTAZIONALE in Finanza e impresa LM-16 CESARONE FRANCESCO
( syllabus)
MODULE 1
1 A rapid introduction to MATLAB
1.1 MATLAB basics: Preliminary elements; Variable assignment; Workspace; Arithmetic operations; Vectors and matrices; Standard operations of linear algebra; Element-by-element multiplication and division; Colon (:) operator; Predefined function; inline Function; Anonymous Function.
1.2 M-file: Script and Function
1.3 Programming fundamentals: if, else, and elseif scheme; for loops; while loops
1.4 Matlab graphics
1.5 Preliminary exercises on programming
1.6 Exercises on the financial evaluation basics
MODULE 2
2 Preliminary elements on Probability Theory and Statistics
2.1 Random variables
2.2 Probability distributions
2.3 Continuous random variable
2.4 Higher-order moments and synthetic indices of a distribution
2.5 Some probability distributions: Uniform, Normal, Log-normal, Chi-square, Student-t
3 Linear and Non-linear Programming
3.1 Some Matlab built-in functions for optimization problems
3.2 Multi-objective optimization: Determining the efficient frontier
4 Portfolio Optimization
4.1 Portfolio of equities: Prices and returns
4.2 Risk-return analysis: Mean-Variance; Effects of the diversification in an Equally Weighted portfolio; Mean-MAD; Mean-MinMax; VaR; Mean-CVaR; Mean-Gini portfolios
4.3 Bond portfolio immunization
MODULE 3
5 Further elements on Probability Theory and Statistics
5.1 Introduction to the Monte Carlo simulation
5.2 Stochastic processes: Brownian motion; Ito’s Lemma; Geometrical Brownian motion
6 Pricing of derivatives with an underlying security
6.1 Binomial model (CRR): A replicating portfolio of stocks and bonds; Calibration of the model; Multi-period case
6.2 Black-Scholes model: Assumptions of the model; Pricing of a European call; Pricing equation for a call; Implied Volatility
6.3 Option Pricing with Monte Carlo Method: Solution in integral form; Path Dependent Derivatives
( reference books)
F. Cesarone, Computational Finance: a MATLAB oriented modeling, draft
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7
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SECS-S/06
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60
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-
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-
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-
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Elective activities
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ITA |
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20410070 -
LOGIC AND ARITHMETIC
(objectives)
To deepen the knowledge of the main results of the classical logic of the first order and study some of their significant consequences.
-
Derived from
20710122 TEOREMI SULLA LOGICA, 2 in Scienze filosofiche LM-78 TORTORA DE FALCO LORENZO
( syllabus)
Towards model theory: some consequences of the compactness theorem.
Proof of the compactness theorem for languages of any cardinality. Languages with equality. The compactness theorem for languages with equality. Correctness and completeness for languages with equality. L"owenheim-Skolem's theorem for (denumerable) languages with equality. The limits of the expressive power of first order languages. Elementary equivalence, substructures, elementary substructures. Isomorphsims and elementary equivalence. The notion of substructure. Elementary substructures and diagrams. The preservation theorems. Generalisations of the L"owenheim-Skolem's theorem. Completeness of a theory.
Logic and Arithmetic: incompleteness
Decidability and fundamental results of recursion theory: primitive recursive functions and elementary functions, Ackermann's function and the (partial) recursive functions, arithmetical hierarchy and representation (in N) of recursive functions, arithmetization of syntax, fundamental theorems of recursion theory, decidability, semi-decidability, undecidability.
Peano arithmetic: Peano's axioms, the models of (first order) Peano arithmetic, the representable functions in (first order) Peano arithmetic, incompleteness and undecidability.
( reference books)
V.M. Abrusci, L. Tortora de Falco, Logica Volume 1- Dimostrazioni e modelli al primo
ordine. Springer, (2014).
V.M. Abrusci, L. Tortora de Falco, Logica Volume 2- Incompletezza, teoria assiomatica degli insiemi. Springer, (2018).
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7
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MAT/01
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60
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-
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-
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-
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Elective activities
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ITA |
|
20410133 -
IN480 - CALCOLO PARALLELO E DISTRIBUITO
(objectives)
Acquire techniques in parallel and distributed programming, and the knowledge of modern hardware and software architectures for high-performance scientific computing. Learn distributed iterative methods for simulating numerical problems. Acquire the knowledge of the newly developed languages for dynamic programming in scientific computing, such as the Julia language
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Derived from
20410148 IN480 - CALCOLO PARALLELO E DISTRIBUITO in Scienze Computazionali LM-40 PAOLUZZI ALBERTO, D'AUTILIA ROBERTO
( syllabus)
This course introduces to techniques of parallel and distributed computing, and to hardware and software architectures for high-performance scientific and technical computing. Some space will be given to iterative distributed methods for simulation of numerical problems, and to methods for assessment of very large geometric models and meshes. The programming language used is Julia, novel dynamic language for scientific computing. Specific learning goals are:
Solve compute-intensive problems faster;
Solve larger problems in the same amount of time;
Solve same size problems with higher accuracy in the same amount of time.
( reference books)
Blaise N. Barney, HPC Training Materials, gentle concession of Lawrence Livermore National Laboratory's Computational Training Center
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7
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INF/01
|
48
|
12
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-
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-
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Elective activities
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ITA |
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Università Roma Tre