Optional group:
comune Orientamento unico DUE INSEGNAMENTI A SCELTA NEL GRUPPO 2 (CONSIGLIATI: AL310,AM310,FM310,GE310,AN420,CP410,AC310,IN410) - (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.
-
PAPPALARDI FRANCESCO
( syllabus)
Cardano equations for the solvability of third degree equations, rings and fields, the characteristic of a field, reminders of polynomial rings, field extensions, construction of some field extensions,
the subring generated by a subset, the subfield generated by a subset, algebraic and transcendental elements, algebraically closed fields.
Splitting fields. Simple extensions and maps between simple extensions, splitting fields, existence of the splitting field, uniqueness except for 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 automorphisms of a field, normal, separable and Galois extensions, characterisations of separable extensions, fundamental theorem of Galois correspondence, examples, Galois group of polynomials, root extensions, solvable groups and the Galois theorem on solving equations, Theorem of the existence of the primitive element.
The calculation of the 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,
Discriminant theory, Galois groups of polynomials of degree less than or equal to 4, examples of polynomials with Galois group S_p,
Dedekind theorem (only statement). Applications of Dedekind's Theorem, how to construct a polynomial with Galois group S_n.
Cyclotomic fields. Definitions, Galois group, under maximal real fields, quadratic sub-fields, Galois groups, cyclotomic polynomials and their properties, Galois inverse 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 plane points, constructible real numbers, characterization of constructible points in terms of fields, constructible subfields and construction of constructible numbers, cube duplication, angle trisection, circle quadrature and Gauss theorem for constructing regular polygons with ruler and compass.
( reference books)
J. S. Milne,Fields and Galois Theory.Course Notes, (2015).
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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.
-
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
5. Integration on product spaces
Measurability on cartesian products. Product measure. Fubini theorem.
6. 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.
-
PONTECORVO MASSIMILIANO
( syllabus)
1. Topological classification of curves and compact surfaces. Triangulations, Euler carachetristic.
2. Smooth and regular curves in Euclidean space. Immersions and imbeddings. Arc lenght. Curvature and the Fundamental Theorem of local geometry of plane curves.
3. Regular surfaces in R^3, local coordinates. Inverse image of a regular value. Maps and diffeomorphisms. Tangent plane and derivative of a map. Normal unit vector, orientation
and the Gauss map of a surface in R^3. Orientable surfaces, examples. The Moebius band is non-orientable.
4. Riemannian metric, examples. The shape operator is self-adjoint. Principal and asymptotic directions. The Mean and Gauss curvatures.
5. The geometry of the Gauss map. The sign of the Gauss curvature and position of the tangent plane. Theorems of Meusnieur and Olinde Rodrigues. Geometric properties of compact surfaces, Minimal surfaces and Ruled surfaces.
6. Isometries and local isometries. The Shape operator in isothermal coordinates. Proof of Gauss' Theorema Egregium. Examples, counter-examples and applications.
7. Homeworks.
8. 12 hours of lab for the visualization and computation on curves and surfaces.
( reference books)
[1] J.M. Lee, Introduction to topological manifolds. Springer, (2000). - – http://dx.doi.org/10.1007/b98853
[2] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976).
[3] E. Sernesi, Geometria 2. Boringhieri, (1994).
[4] M.Abate, F.Tovena, Curve e Superfici. Springer, (2006).
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OTIMAN ALEXANDRA IULIA
( syllabus)
Weekly meetings in the computer science laboratory for the use of Wolfram Mathematica software, the goal being the graphic representation of curves and surfaces
( reference books)
Alfred Gray, Elsa Abbena, Simon Salamon, "Modern Differential Geometry of Curves and Surfaces with Mathematica"
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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.
-
Derived from
20410414 CP410 - TEORIA DELLA PROBABILITÀ in Matematica LM-40 CAPUTO PIETRO, CANDELLERO ELISABETTA
( syllabus)
An introductory example: the branching process.
Introduction to measure theory: existence and uniqueness results for probability measures. First Borel–Cantelli lemma.
Random variables and their distriubutions.
Independence. Second Borel–Cantelli lemma. The 0-1 law.
Integration. Expected value. Monotone convergence theorem.
Dominated convergence theorem.
Jensen's inequality, Hoelder and Cauchy-Schwarz inequalities. Markov's inequality.
Examples of weak and strong laws of large numbers. Product spaces. Fubini's theorem. Joint laws.
Conditional expectation. Martingales and convergence theorems.
Discrete time stochastic processes. Gambilng. Optional stopping theorem and applications.
Convergence theorems for martingales. Examples and problems with martingales. Kolmogorov's strong law.
Convergence in distribution and characteristic functions. Central limit theorem. Modes of convergence of random variables.
( reference books)
D. Williams, Probability with martingales. Cambridge University Press, (1991).
R. Durrett, Probability: Theory and Examples. Thomson, (2000).
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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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20410137 -
IN410 - MODELLI DI CALCOLO
(objectives)
The course of Computer Institutions is devoted to the deepening of the mathematical aspects of the computation concept, to the study of the relationships between different computational models and computational complexity.
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Derived from
20410417 IN410-CALCOLABILITÀ E COMPLESSITÀ in Scienze Computazionali LM-40 PEDICINI MARCO
( syllabus)
1) Computability, complexity and representability:
- Introduction to decision problems, algorithmic and non-algorithmic procedures, deterministic computations, discrete procedures, notion of alphabet, of speech. Decidability and semi-decidability of a set. Deterministic, finitary and discrete computations. Formal algorithms: formal definition of algorithm, configurations of input, output, transition function. Example of formalization of an algorithm. Decidability for finished automata. Representation of the automata by matrices. Free Monoid of words. Formal semi-rings. Non-deterministic finite automata. Regular Languages. Equivalence between deterministic and non-deterministic automata.
- Turing machines: definition, decidability for Turing machine, stopping time, stopping space. Cost of computation. Complexity: worst case and average case. Independence of decision time from a finite number of input configurations. Complexity functions, complexity classes DTIME and DSPACE (deterministic time and space). Inclusion DTIME (T (n)) ⊂ DSPACE (T (n)) ⊂ DTIME (2 ^ {cT (n)}). Pumping Lemma. Simulation of algorithms, simulation of the half tape Turing machine, simulation of a multi-tape machine. Special Turing machines. Linear Speedup theorem for Turing machines with an extended alphabet. Evaluation of acceleration coefficient in relation to alphabets. Decisions of natural number sets. Independence from representation. Considerations concerning complexity.
- Turing computability: definition of Turing computable function, characteristic functions of Turing decidable sets, the class of Turing computable functions is closed by composition, concatenation, primitive recursion and minimization. Examples of Turing computable functions. Recursive Functions: equivalence between Turing computability and recursive functions. Ackermann function ([1] chapter 1,2,3,4,5 and [4] chapter 1).
- Time-constructible functions. Notion of T-clock. Examples of some time constructible function. Closure by composition.
- Non-deterministic Turing machines: characterization through the decidability of projection sets. Definition of the class of polynomial non-deterministic functions. NP-complete problems.
2) Lambda calculus and functional programming:
- Declarative programming: historical outline on the lambda calculus, basic definitions, the terms of the lambda calculus, the simple substitution. Relations on the lambda terms. Congruences, transition to the context. α-equivalence. alpha-equivalence passes to the context. Transitive closure of a relationship, owned by Church-Rosser. Listing of lambda-terms with respect to alpha-equivalence.
- Definition of beta-reduction and beta-equivalence. Church-Rosser's theorem for beta-reduction. Normal forms for beta-reduction. Beta-reduction strategies. Normalizing strategy: left reduction (left most-outer most). Head reduction. Soluble Terms. Head Normal Forms. Solvability characterization theorem.
- Representation of the recursive functions: lambda definability theorem. Existence of the fixed point for the lambda terms. Church Fixed Point and Curry fixed point.
- Representation of other data types in the lambda-calculus: pairs, lists, trees, the solution of recursive equations on lambda-terms ([2] chapters 1, 2, 5).
( 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).
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7
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MAT/01
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72
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Related or supplementary learning activities
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ITA |
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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.
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7
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MAT/02
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60
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-
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Related or supplementary learning activities
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ITA |
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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
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7
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MAT/05
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60
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-
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-
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-
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Related or supplementary learning activities
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ITA |
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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.
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7
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MAT/07
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60
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Related or supplementary learning 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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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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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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72
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Related or supplementary learning activities
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ITA |
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20410189 -
LM410 -THEOREMS IN LOGIC 1
(objectives)
To acquire a good knowledge of first order classical logic and its fundamental theorems
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7
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MAT/01
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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
( syllabus)
Cardano equations for the solvability of third degree equations, rings and fields, the characteristic of a field, reminders of polynomial rings, field extensions, construction of some field extensions,
the subring generated by a subset, the subfield generated by a subset, algebraic and transcendental elements, algebraically closed fields.
Splitting fields. Simple extensions and maps between simple extensions, splitting fields, existence of the splitting field, uniqueness except for 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 automorphisms of a field, normal, separable and Galois extensions, characterisations of separable extensions, fundamental theorem of Galois correspondence, examples, Galois group of polynomials, root extensions, solvable groups and the Galois theorem on solving equations, Theorem of the existence of the primitive element.
The calculation of the 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,
Discriminant theory, Galois groups of polynomials of degree less than or equal to 4, examples of polynomials with Galois group S_p,
Dedekind theorem (only statement). Applications of Dedekind's Theorem, how to construct a polynomial with Galois group S_n.
Cyclotomic fields. Definitions, Galois group, under maximal real fields, quadratic sub-fields, Galois groups, cyclotomic polynomials and their properties, Galois inverse 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 plane points, constructible real numbers, characterization of constructible points in terms of fields, constructible subfields and construction of constructible numbers, cube duplication, angle trisection, circle quadrature and Gauss theorem for constructing regular polygons with ruler and compass.
( reference books)
J. S. Milne,Fields and Galois Theory.Course Notes, (2015).
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7
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MAT/02
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72
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-
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-
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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
( 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
5. Integration on product spaces
Measurability on cartesian products. Product measure. Fubini theorem.
6. 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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-
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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.
-
Derived from
20402087 GE310 - ISTITUZIONI DI GEOMETRIA SUPERIORE in Matematica L-35 N0 PONTECORVO MASSIMILIANO, OTIMAN ALEXANDRA IULIA
( syllabus)
1. Topological classification of curves and compact surfaces. Triangulations, Euler carachetristic.
2. Smooth and regular curves in Euclidean space. Immersions and imbeddings. Arc lenght. Curvature and the Fundamental Theorem of local geometry of plane curves.
3. Regular surfaces in R^3, local coordinates. Inverse image of a regular value. Maps and diffeomorphisms. Tangent plane and derivative of a map. Normal unit vector, orientation
and the Gauss map of a surface in R^3. Orientable surfaces, examples. The Moebius band is non-orientable.
4. Riemannian metric, examples. The shape operator is self-adjoint. Principal and asymptotic directions. The Mean and Gauss curvatures.
5. The geometry of the Gauss map. The sign of the Gauss curvature and position of the tangent plane. Theorems of Meusnieur and Olinde Rodrigues. Geometric properties of compact surfaces, Minimal surfaces and Ruled surfaces.
6. Isometries and local isometries. The Shape operator in isothermal coordinates. Proof of Gauss' Theorema Egregium. Examples, counter-examples and applications.
7. Homeworks.
8. 12 hours of lab for the visualization and computation on curves and surfaces.
( reference books)
[1] J.M. Lee, Introduction to topological manifolds. Springer, (2000). - – http://dx.doi.org/10.1007/b98853
[2] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976).
[3] E. Sernesi, Geometria 2. Boringhieri, (1994).
[4] M.Abate, F.Tovena, Curve e Superfici. Springer, (2006).
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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.
-
Derived from
20410413 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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-
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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
-
Derived from
20410412 MC310 - ISTITUZIONI DI MATEMATICHE COMPLEMENTARI in Matematica LM-40 BRUNO ANDREA
( syllabus)
1.Euclidean Geometry; rudiments of history of Greek Mathematics. Ruler and compass constructions. The classical problems. Euclid's Elements.
2. The V Postulate's problem: Posidonius attempt. Equivalent Postulates: Playfair, Wallis, transitivity of parallelism. Saccheri's work. Saccheri's Quadrilaterals. The three hyposes. Saccheri-Lagrange's Theorem and the exclusion of obtuse angle hypothesis. The birth of Non-Euclidean Geometry in Bolyai and Lobatchewaki.
3. Symmetries of the plane: Isometries of the plane and their type. Characterization of an isometry by the image of three points not on a line, Chasles' Theorem. Discrete groups of isometries. Rosettes, Friezes and Wallpaper groups. The Theorem of addiction of the angles. Leonardo's Theorem and the classification of finite groups of isometries. Sketch of the classification of the Frieze groups. Crystallographic restriction's Theorem and the classification of wallpaper groups.
4. Gauss' Geometry; The geometry on the Sphere. Locally euclidean geometries. Uniformly discontinuous groups of isometries. The Torus, Mobius band, Klein bottle. Classification of uniforly discontinuous groups of isometries. Sketch of the proof of the Theorem of classification of locally euclidean geometries
5. Moduli of geometries on the torus and hyperbolic geometry: Similar geometries. Similar geometries on the Torus. The modular figure. Poincaré upper plane model.Lines and distance. What was repugnant to Saccheri and that was not for Aristotle.
( reference books)
R. Trudeau: "La rivoluzione non Euclidea" Bollati Boringhieri
V, Nikulin, I. Shafarevich "Geometries and groups" Springer ed.
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7
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MAT/04
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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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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
20410414 CP410 - TEORIA DELLA PROBABILITÀ in Matematica LM-40 CAPUTO PIETRO, CANDELLERO ELISABETTA
( syllabus)
An introductory example: the branching process.
Introduction to measure theory: existence and uniqueness results for probability measures. First Borel–Cantelli lemma.
Random variables and their distriubutions.
Independence. Second Borel–Cantelli lemma. The 0-1 law.
Integration. Expected value. Monotone convergence theorem.
Dominated convergence theorem.
Jensen's inequality, Hoelder and Cauchy-Schwarz inequalities. Markov's inequality.
Examples of weak and strong laws of large numbers. Product spaces. Fubini's theorem. Joint laws.
Conditional expectation. Martingales and convergence theorems.
Discrete time stochastic processes. Gambilng. Optional stopping theorem and applications.
Convergence theorems for martingales. Examples and problems with martingales. Kolmogorov's strong law.
Convergence in distribution and characteristic functions. Central limit theorem. Modes of convergence of random variables.
( reference books)
D. Williams, Probability with martingales. Cambridge University Press, (1991).
R. Durrett, Probability: Theory and Examples. Thomson, (2000).
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7
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MAT/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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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.
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7
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MAT/02
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60
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-
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-
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-
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Elective activities
|
ITA |
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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
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7
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MAT/05
|
60
|
-
|
-
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-
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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.
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7
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MAT/07
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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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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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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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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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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.
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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)
Lecture notes available on the course website
J.J. Sakurai, Jim Napolitano - Meccanica Quantistica Moderna - Seconda Edizione [Zanichelli, Bologna, 2014]
An english version of the book is also available:
Sakurai J.J., Modern Quantum Mechanics (Revised Edition) [Addison-Wesley, 1994]
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7
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FIS/02
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60
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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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Derived from
20401116 ELEMENTI DI CHIMICA in Fisica L-30 N0 IUCCI GIOVANNA
( syllabus)
1. ATOMIC THEORY AND ATOMIC STRUCTURE. Atoms, molecules, moles; atomic and molecular weight. Atomic models: Rutheford, Bohr. Quantum theory, quantum numbers and energy levels. Polyelectronic atoms; periodic system.
2. CHEMICAL BONDS. Ionic bond. Covalent bond: and bonds. Polyatomic molecules: molecular structure. Hybridization and resonance. Molecular orbital. Metallic bond. Intermolecular forces.
3. NOMENCLATURE AND CHEMICAL REACTIONS. Oxides, hydroxides, acids, salts, ions. Balancing chemical reactions: redox reactions.
4. STATES OF AGGREGATION. Gas state, ideal gas law. Solid state: ionic, covalent, molecular and metallic solids. Conductors, semiconductors, insulators. Liquid and amorphous states. Phase transitions and phase diagrams.
5. SOLUTIONS. Concentration, colligative properties; electrolyte solutions.
6. THERMODYNAMICS. Matter, energy, heat, first and second principles; enthalpy, entropy, free energy.
7. CHEMICAL EQUILIBRIUM. Equilibrium constant and free energy. Gas-phase and heterogeneous equilibria. Le Chatelier’s principle. Van’t Hoff equation.
8. EQUILIBRIA IN SOLUTION. Acid-base equilibria: acids and bases, pH, dissociation constant, polyprotic acids, hydrolysis, buffers. Acid-base titrations and pH indicators. Solubility equilibria: solubility product, common ion effect.
9. ELECTROCHEMISTRY. Batteries, electrode potentials, Nernst’s equation. Electrolysis.
10. CHEMICAL KINETICS. Reaction speed, speed constant. Influence of the temperature on the reaction speed: Arrhenius equation. Catalysts.
Numerical exercises on all the listed subjects.
( reference books)
P.W.Atkins, L. Jones; CHEMISTRY: MOLECULES, MATTER, AND CHANGE
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7
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CHIM/03
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60
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20410038 -
GRAPH THEORY
(objectives)
Provide tools and methods of graph theory
-
Derived from
20410425 GE460 - TEORIA DEI GRAFI in Scienze Computazionali LM-40 MASCARENHAS MELO ANA MARGARIDA
( syllabus)
Graphs: basic definitions. Simple and non simple graphs, planarity, connectivity, degree, regularity, incidence and adjacency matrices. Examples of families of graphs. The '' handshaking lemma ''.
Graphs obtained from others: complement, subgraph, cancellation and contraction. Isomorphisms and automorphisms of graphs. Connectivity: paths, cycles. A graph is bipartite if and only if each cycle has equal length.
Connetivity and connected component. Connectivity for sides and vertices. Eulerian and semi-Eulerian graphs.
Euler's theorem: a connected graph is Eulerian if and only if every vertex has an even degree. Hamiltonian graphs. Sufficient conditions to guarantee that a graph is Hamiltonian: the theorems of Ore and Dirac.
the Ore theorem. Trees and forests. The cyclomatic number and the "cutset" rank of a graph.
Fundamental system of cycles and cuts associated with a generating forest. Enumeration of generating forests. The Cayley theorem.
Generating trees: the "greedy" algorithm for the "connector problem". Planar graphs. K3.3 and K5 are not planar. Statement of the theorem of Kuratovski and variations.
Euler's formula for planar graphs. The dual of a planar graph.
Correspondence between cycles and cuts for planar graphs and their dual. Dual abstract. A graph that admits a dual abstract is planar. Colorings: initial considerations and some properties.
Colorings: the 5 colors theorem. Graphs on surfaces: classification of topological surfaces.
Coloring of faces and duality between this problem and the coloring of vertices. Reduction of the proof of the 4-color theorem to the coloring of cubic graph faces. '' The marriage problem ": Hall's theorem.
Hall's theorem in the language of transversals. Criteria of existence of transversal and partial transversivities. Application to the construction of Latin squares. Direct graphs: basic notions and orientability.
The Max-Flow Min-Cut theorem and Menger's theorem.
Complexity of algorithms and applications to the theory of graphs.
Introduction to the theory of matroids: definitions using bases and independent elements. Graphical and cographic matroids, vector matroids and the problem of representability.
Definition of matroid using the cycles and the rank function. Minors of a matroid. Transverse matroids and the Rado Theorem for matroids.
Union of matroids and applications: existence of disjoint bases in a matroid. Duality for matroids and applications to graphic and cographic matroids. Planar matroids and the generalization of Kuratovski's theorem for matroids.
Elements of algebraic graph theory: the incidence matrix and the Laplacian matrix of an oriented graph. The vertex space and the space of edges of a graph. Subspaces of the cycles and subspace of the cuts of a defined oriented graph of the incidence matrix.
Basis for the space of the cycles and for the space of the cuts of a graph. The Riemann-Roch theorem for graphs.
Proof of the "generalized Matrix Tree theorem". The contraction / restriction algorithm for matroids. Examples. The number of acyclical orientations of a graph.
Graph polynomials: the chromatic polynomial, the "reliabiliaty" polynomial. Examples. The polynomial rank of a matroid. Properties and first applications.
Proof of the structure theorem for functions on matroids that satisfy contraction/restriction properties. Their incarnation through the polynomial rank.
Whitey's moves and two isomorphisms for graphs. Isomorphism between graphical matroids implies isomorphism between graphs in case the graphs are 3 connected.
Whitney's Theorem for graphic matroids: sketch of the demonstration. Characterization for minors excluded from binary and regular matroids. The theorem of Seymour.
( reference books)
R. Diestel: Graph theory, Spriger GTM 173.
R. Wilson: Introduction to Graph theory, Prentice Hall.
B. Bollobas: Modern Graph theory, Springer GTM 184.
J. A. Bondy, U.S.R. Murty: Graph theory, Springer GTM 244.
N. Biggs: Algebraic graph theory, Cambridge University Press.
C. D. Godsil, G. Royle: Algebraic Graph theory, Springer GTM 207.
J. G. Oxley: Matroid theory. Oxford graduate texts in mathematics, 3.
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7
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MAT/03
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60
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20402291 -
ST420 – MULTIVARIATE STATISTICAL ANALYSIS, MATHEMATICAL STATISTICS
(objectives)
Provide statistical models and parameter estimates. Studying asymptotic theory of estimators
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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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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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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
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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 (2020), Computational Finance. MATLAB oriented modeling, Routledge-Giappichelli Studies in Business and Management, ISBN 978-0-367-49303-5
https://www.giappichelli.it/computational-finance
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7
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SECS-S/06
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60
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20410137 -
IN410 - MODELLI DI CALCOLO
(objectives)
The course of Computer Institutions is devoted to the deepening of the mathematical aspects of the computation concept, to the study of the relationships between different computational models and computational complexity.
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Derived from
20410417 IN410-CALCOLABILITÀ E COMPLESSITÀ in Scienze Computazionali LM-40 PEDICINI MARCO
( syllabus)
1) Computability, complexity and representability:
- Introduction to decision problems, algorithmic and non-algorithmic procedures, deterministic computations, discrete procedures, notion of alphabet, of speech. Decidability and semi-decidability of a set. Deterministic, finitary and discrete computations. Formal algorithms: formal definition of algorithm, configurations of input, output, transition function. Example of formalization of an algorithm. Decidability for finished automata. Representation of the automata by matrices. Free Monoid of words. Formal semi-rings. Non-deterministic finite automata. Regular Languages. Equivalence between deterministic and non-deterministic automata.
- Turing machines: definition, decidability for Turing machine, stopping time, stopping space. Cost of computation. Complexity: worst case and average case. Independence of decision time from a finite number of input configurations. Complexity functions, complexity classes DTIME and DSPACE (deterministic time and space). Inclusion DTIME (T (n)) ⊂ DSPACE (T (n)) ⊂ DTIME (2 ^ {cT (n)}). Pumping Lemma. Simulation of algorithms, simulation of the half tape Turing machine, simulation of a multi-tape machine. Special Turing machines. Linear Speedup theorem for Turing machines with an extended alphabet. Evaluation of acceleration coefficient in relation to alphabets. Decisions of natural number sets. Independence from representation. Considerations concerning complexity.
- Turing computability: definition of Turing computable function, characteristic functions of Turing decidable sets, the class of Turing computable functions is closed by composition, concatenation, primitive recursion and minimization. Examples of Turing computable functions. Recursive Functions: equivalence between Turing computability and recursive functions. Ackermann function ([1] chapter 1,2,3,4,5 and [4] chapter 1).
- Time-constructible functions. Notion of T-clock. Examples of some time constructible function. Closure by composition.
- Non-deterministic Turing machines: characterization through the decidability of projection sets. Definition of the class of polynomial non-deterministic functions. NP-complete problems.
2) Lambda calculus and functional programming:
- Declarative programming: historical outline on the lambda calculus, basic definitions, the terms of the lambda calculus, the simple substitution. Relations on the lambda terms. Congruences, transition to the context. α-equivalence. alpha-equivalence passes to the context. Transitive closure of a relationship, owned by Church-Rosser. Listing of lambda-terms with respect to alpha-equivalence.
- Definition of beta-reduction and beta-equivalence. Church-Rosser's theorem for beta-reduction. Normal forms for beta-reduction. Beta-reduction strategies. Normalizing strategy: left reduction (left most-outer most). Head reduction. Soluble Terms. Head Normal Forms. Solvability characterization theorem.
- Representation of the recursive functions: lambda definability theorem. Existence of the fixed point for the lambda terms. Church Fixed Point and Curry fixed point.
- Representation of other data types in the lambda-calculus: pairs, lists, trees, the solution of recursive equations on lambda-terms ([2] chapters 1, 2, 5).
( 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).
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7
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MAT/01
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72
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20410149 -
IN490 - PROGRAMMING LANGUAGES
(objectives)
Present the main concepts of the theory of formal languages and their application to the classification of programming languages. Introduce the main techniques for syntactic analysis of programming languages. Learn to recognize the structure of a programming language and techniques to implement its abstract machine. Knowing the object-oriented paradigm and another non-imperative paradigm.
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Derived from
20410427 IN490 - LINGUAGGI DI PROGRAMMAZIONE in Scienze Computazionali LM-40 LOMBARDI FLAVIO
( syllabus)
The objective of Linguaggi di Programmazione course is to introduce main formal language theory concepts and results as well as their application for programming language classification.
Most relevant approaches for syntactic analysis of programming languages are introduced. Learning how to recognize the structure of a programming language and the implementation techniques for the abrstract machine.
Understanding the Object Oriented paradigm together with other non imperative approaches.
( reference books)
[1] Maurizio Gabbrielli, Simone Martini,Programming Languages - Principles and paradigms, 2/ed. McGraw-Hill, (2011).
[2] Dean Wampler, Alex Payne, Programming Scala: Scalability = Functional Programming + Objects, 2 edizione. O’Reilly Media, (2014).
[3] David Parsons, Foundational Java Key Elements and Practical Programming. Springer- Verlag, (2012).
Course Slides
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7
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INF/01
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60
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20410157 -
AL430 - COMMUTATIVE RING AND IDEAL
(objectives)
To provide the students with the technical and theoretical basis for addressing recent literature and current issues in the multiplicative theory of ideals, developing the themes that originated from the works of L. Kronecker, W. Krull, E. Noether, P. Samuel, P. Jaffard, R. Gilmer
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7
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MAT/02
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60
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20410139 -
AN430 - FINITE ELEMENTS METHOD
(objectives)
Introduce the Finite Element Method for the Numeric Solution of Partial Derivative Equations; in particular: computational fluid dynamics, transport problems; mechanics of computational solids.
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Derived from
20410421 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)
1) Integral Form at a Glance,
classroom notes
2) When functions have no value(s): Delta functions and distributions
Steven G. Johnson, MIT course 18.303 notes, 2011
3) Understanding and Implementing the Finite Elements Method
Mark S. Gockenbach, SIAM, 2006 Cap. 1 Some model PDE’s
Cap. 2 The weak formo of a BVP Cap. 3 The Galerkin method
Cap. 4 Piecewise polynomials and the finite element method (sections 4.1, 4.2) Cap. 5 Convergence of the finite element method (sections 5.1 ~ 5.4)
4) Computational Science and Engineering
Gilbert Strang, Wellesley-Cambridge Press, 2007 Cap 3.1, page 236~241;
Cap. 7.2 Iterative methods, page 563~567
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7
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MAT/08
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60
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20410142 -
FS440 - DATA ACQUISITION
(objectives)
To provide the student with basic knowledge of how to build a nuclear physics experiment based on data collection by detector, equipment control and experiment, monitoring of the good functioning of the equipment and the quality of the Acquired data.
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7
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FIS/04
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60
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20410143 -
IN440 - COMBINATORIAL OPTIMISATION
(objectives)
Acquire skills on key resolution techniques for combinatorial optimization issues; To deepen the skills on graph theory; Acquire advanced technical skills for designing, analyzing and deploying algorithms for troubleshooting graphics, trees, and flow networks.
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7
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INF/01
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60
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20410145 -
FS450 - STATISTICAL MECHANICS
(objectives)
Acquire the basic knowldege of the fundamental principles of statistiscal mechanics for classical and quantum systems.
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Derived from
20401806 ELEMENTI DI MECCANICA STATISTICA in Fisica L-30 N0 RAIMONDI ROBERTO
( syllabus)
CONTENTS OF THE LECTURES: the numbers in round brackets refer to the chapter and section of the textbook adopted.
Kinetic theory of gases. Boltzmann equation and H theorem. (1, Par.2.1,2.2,2.3,2.4)
Maxwell-Boltzmann distribution. (1, Par. 2.5)
Phase space and Liouville theorem. (1, Par. 3.1,3.2)
Gibbs ensembles. Micro canonical ensemble. Definition of entropy. (1, Par. 3.3,3.4)
The ideal gas in the micro canonical ensemble. (1, Par. 3.6)
The equipartition theorem. (1, Par. 3.5)
The canonical ensemble. (1, Par.4.1).
The partition function and the free energy. Fluctuations of energy in the canonical ensemble. (1 Par. 4.4)
The grand canonical ensemble. The grand potential. The ideal gas in the grand canonical ensemble. (1 Par. 4.3).
Fluctuations of the particle number. (1 Par. 4.4)
Classical theory of the linear response and fluctuation-dissipation theorem. (1, Par. 8.4).
Einstein and Langevin theories of the Brownian motion. (Par. 1 par. 11.1,11.2).
Johnson-Nyqvist theory of thermal noise. (1 Par. 11.3).
Quantum statistical mechanics and the density matrix. (1, Par. 6.2,6.3,6.4)
Fermi-Dirac and Bose-einstein quantum statistics. ( 1, Par. 7.1)
The Fermi gas. The Sommerfeld expansion and the electron specific heat. (1, Par. 7.2)
The Bose gas. The Bose-Einstein condensation. (1, Par. 7.3)
Quantum theory of black-body radiation. (1, Par. 7.5)
Web page with additional material about the lecture course
https://sites.google.com/a/personale.uniroma3.it/robertoraimondi/home/teaching/elementi
( reference books)
Suggested textbook:
1) C. Di Castro and R. Raimondi, Statistical Mechanics and Applications in Condensed Matter, Cambridge University Press, 2015.
Further reading:
2) K. Huang, Meccanica Statistica, Zanichelli, 1997.
3) L. Peliti, Appunti di Meccanica Statistica, Bollati Boringhieri, 2003.
4) Joel L. Lebowitz, Statistical mechanics: A selective review of two central issues, Reviews of Modern Physics, 71, S346 (1999).
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7
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FIS/02
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60
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20410147 -
IN470- COMPUTATIONAL METHODS IN SYSTEMS BIOLOGY
(objectives)
Acquire the basic knowledge of biological systems and the issues related to their understanding also in relation to deviations from normal functioning and thus on the onset of pathologies. Maintain the modeling aspect as well as that of numerical simulation, especially problems formulated by equations and discrete systems. Acquire the knowledge of the major bio-informatics algorithms useful for analyzing biological data.
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7
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INF/01
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60
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20410148 -
IN480 - PARALLEL AND DISTRIBUTED COMPUTING
(objectives)
Acquire parallel and distributed programming techniques and knowledge of modern hardware and software architectures for high performance scientific computing. Introduce distributed iterative methods for the simulation of numeric problems. Acquire the knowledge of newly conceived languages for dynamic programming in scientific computation, such as Julia's language.
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7
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ING-INF/05
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60
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20410187 -
IN460 - Laboratory of geometric and graphic programming
(objectives)
Acquire conceptual and programming tools for the geometric modeling of curves, surfaces and solids,
and computer-aided design. Acquire the knowledge of the main computer graphics techniques, also on the web platform,
with python and javascript languages.
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7
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ING-INF/05
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60
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20410189 -
LM410 -THEOREMS IN LOGIC 1
(objectives)
To acquire a good knowledge of first order classical logic and its fundamental theorems
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Derived from
20410451-2 LM410 -TEOREMI SULLA LOGICA 1 - MODULO B in Matematica LM-40 TORTORA DE FALCO LORENZO
( syllabus)
Modulo A:
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.
Modulo B: 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.
( reference books)
V.M. Abrusci, L. Tortora de Falco, Logica Volume 1- Dimostrazioni e modelli al primo
ordine. Springer, (2014).
-
Derived from
20410451-1 LM410 -TEOREMI SULLA LOGICA 1 - MODULO A in Matematica LM-40 TORTORA DE FALCO LORENZO, MAIELI ROBERTO, Acclavio Matteo
( syllabus)
Modulo A:
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.
Modulo B: 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.
( reference books)
V.M. Abrusci, L. Tortora de Falco, Logica Volume 1- Dimostrazioni e modelli al primo
ordine. Springer, (2014).
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MAT/01
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60
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20410190 -
LM420 - THEOREMS IN LOGIC 2
(objectives)
To support the students into an in-depth analysis of the main results of first order classical logic and to study some of their remarkable consequences
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Derived from
20710122 TEOREMI SULLA LOGICA, 2 in Scienze filosofiche LM-78 TORTORA DE FALCO LORENZO
( syllabus)
Logic and Arithmetic: incompleteness
Part 1: Decidability and fundamental results of recursion theory. Primitive recursive functions and elementary functions: definitions and examples, elementary coding of the finite sequences of natural numbers, an alternative definition of the set of elementary functions. Ackermann's function and the (partial) recursive functions. Arithmetical hierarchy and representation (in N) of recursive functions. Arithmetization of syntax: coding of terms and formulas, satisfiability in N of Delta formulas is elementary, coding of sequence and derivations. The fundamental theorems of recursion theory. Decidability, semi-decidability, undecidability.
Part 2: Peano arithmetic. Peano's axioms and first order Peano’s axioms. The models of (first order) Peano arithmetic. The representable functions in (first order) Peano arithmetic. Incompleteness and undecidability: Church’s undecidability theorem, fixed point, Gödel’s first incompleteness theorem, Gödel’s second incompleteness theorem, final remarks on incompleteness, hints on incompleteness and second order logic.
( reference books)
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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Elective activities
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ITA |
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Università Roma Tre