Optional group:
CURRICULUM TEORICO SCEGLIERE QUATTRO INSEGNAMENTI (30 CFU) NEI SEGUENTI SSD MAT/01,02,03,05 TRA LE ATTIVITÀ CARATTERIZZANTI (B). - (show)
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30
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20410451 -
LM410 -THEOREMS IN LOGIC 1
(objectives)
To acquire a good knowledge of first order classical logic and its fundamental theorems.
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20410425 -
GE460- GRAPH THEORY
(objectives)
Provide tools and methods for graph theory.
-
Derived from
20410425 GE460 - TEORIA DEI GRAFI in Scienze Computazionali LM-40 SCHAFFLER LUCA
( 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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6
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MAT/03
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48
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12
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Core compulsory activities
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20410627 -
TN410 - INTRODUCTION TO NUMBER THEORY
(objectives)
Acquire a good knowledge of the concepts and methods of the elementary number theory, with particular reference to the study of the Diophantine equations and congruence equations. Provide prerequisites for more advanced courses of algebraic and analytical number theory.
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PAPPALARDI FRANCESCO
( syllabus)
Division, Factorization, Some Elementary Properties of Primes, Some Results and Problems Concerning Primes.
ARITHMETIC FUNCTIONS: The Divisor Function. The Moebius Function. The Euler Function. Dirichlet Convolution
CONGRUENCES: Sets of Residues,Some Interesting Congruences, Some Linear Congruences, Some Polynomial Congruences, Primitive Roots, the Theorem of Gauss.
QUADRATIC RESIDUES: The Legendre Symbol. Quadratic Reciprocity. The Jacobi Symbol.The Distribution of Quadratic Residues.
SUMS OF INTEGER SQUARES: Sums of Two Squares. Number of Representations. Sums of Four Squares. Sums of Three Squares.
ELEMENTARY PRIME NUMBER THEORY: Euclid's Theorem Revisited. The Von Mangoldt Function. Tchebycheff's Theorem. Some Results of Mertens
( reference books)
Chen, W; ELEMENTARY NUMBER THEORY. https://rutherglen.science.mq.edu.au/wchen/lnentfolder/lnent.html
Chowdhury, F.; Chowdhury, M. R. Essentials of Number Theory. Pi Publications, Dhaka, Bangladesh, 2005. ISBN 984-32-2836-7
Hardy, G. H.; Wright, E. M. An introduction to the theory of numbers. Fifth edition. The Clarendon Press, Oxford University Press, New York, 1979. xvi+426 pp. ISBN: 0-19-853170-2; 0-19-853171-0
Davenport, H. Aritmetica superiore. Un'introduzione alla teoria dei numeri. Editore: Zanichelli, 1994. 199 pp. ISBN: 8808091546
Gioia, A. A. The theory of numbers. An introduction. Reprint of the 1970 original. Dover Publications, Inc., Mineola, NY, 2001. xii+207 pp. ISBN: 0-486-41449-3
Rosen, K. H. Elementary number theory and its applications. Fourth edition. Addison-Wesley, Reading, MA, 2000. xviii+638 pp. ISBN: 0-201-87073-8
Tattersall, J. J. Elementary number theory in nine chapters. Cambridge University Press, Cambridge, 1999. viii+407 pp. ISBN: 0-521-58531-7
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6
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MAT/02
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48
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12
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Core compulsory activities
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ITA |
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20410637 -
AM450 - FUNCTIONAL ANALYSIS
(objectives)
To acquire a good knowledge of functional analysis: Banach and Hilbert spaces, weak topologies, linear and continuous operators, compact operators, spectral theory.
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9
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MAT/05
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48
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24
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Core compulsory activities
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ITA |
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20410746 -
AL440 – GROUP THEORY
(objectives)
Become familiar with the fundamental notions of group theory, particularly finite groups,
so as to be able to study and classify some important classes of finite groups.
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TARTARONE FRANCESCA
( syllabus)
Recalls on the elementary properties of groups. Direct and semi-direct products. Permutation groups and simplicity of alternate groups. Actions on groups. Sylow's theorems. Finitely generated abelian groups, free groups, nilpotent groups and solvable groups. Since this is an advanced course, other topics can also be included upon request of the class.
( reference books)
A. Machì, Gruppi. Una introduzione a idee e metodi della Teoria dei Gruppi, SPRINGER VERLAG (2007).
M. Artin, Algebra, BOLLATI BORINGHIERI (1997).
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6
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MAT/02
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48
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12
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Core compulsory activities
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ITA |
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20410757 -
AM410 - AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS
(objectives)
To acquire a good knowledge of general methods and basic techniques necessary to the study of classical and weak solutions for partial differential equations
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AM410- MODULE A - AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS
(objectives)
To acquire a good knowledge of general methods and basic techniques necessary to the study of classical and weak solutions for partial differential equations
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ESPOSITO PIERPAOLO
( syllabus)
Preliminaries: definition of hyper-surface, integration on hyper-surfaces, the divergence theorem; the Laplace equation: the mean value inequalities, the minimum and maximum principle, the Harnack inequality, the Green representation, the Poisson integral, convergence's theorems, interior estimates on the derivatives, the Perron method for the Dirichlet problem.
( reference books)
“Elliptic partial differential equations of second order. Reprint of the 1998 edition”, D. Gilbarg e N.S. Trudinger, Classics in Mathematics, Springer-Verlag
"Partial differential equations. Second edition", Lawrence C. Evans, Graduate Studies in Mathematics 19, American Mathematical Society
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3
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MAT/05
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24
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6
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Core compulsory activities
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ITA |
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AM410 - MODULE B - AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS
(objectives)
To acquire a good knowledge of general methods and basic techniques necessary to the study of classical and weak solutions for partial differential equations
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ESPOSITO PIERPAOLO
( syllabus)
Definition and basic properties of the Sobolev spaces W^{1,p} (Ω). Extension operators. Sobolev inequalities. The space W^{1,p}_0 (Ω). Variational formulation of some elliptic boundary value problems. Existence of weak solutions. Regularity of weak solutions.
( reference books)
"Analisi funzionale", H. Brézis, Liguori Editore
"Partial differential equations. Second edition", Lawrence C. Evans, Graduate Studies in Mathematics 19, American Mathematical Society
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3
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MAT/05
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24
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6
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Core compulsory activities
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ITA |
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20410758 -
AM410 - MODULE A - AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS
(objectives)
To acquire a good knowledge of general methods and basic techniques necessary to the study of classical and weak solutions for partial differential equations
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Derived from
20410757_1 AM410- MODULO A - INTRODUZIONE ALLE EQUAZIONI ALLE DERIVATE PARZIALI in Matematica LM-40 ESPOSITO PIERPAOLO
( syllabus)
Preliminaries: definition of hyper-surface, integration on hyper-surfaces, the divergence theorem; the Laplace equation: the mean value inequalities, the minimum and maximum principle, the Harnack inequality, the Green representation, the Poisson integral, convergence's theorems, interior estimates on the derivatives, the Perron method for the Dirichlet problem.
( reference books)
“Elliptic partial differential equations of second order. Reprint of the 1998 edition”, D. Gilbarg e N.S. Trudinger, Classics in Mathematics, Springer-Verlag
"Partial differential equations. Second edition", Lawrence C. Evans, Graduate Studies in Mathematics 19, American Mathematical Society
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3
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MAT/05
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24
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6
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Core compulsory activities
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ITA |
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20410759 -
AM410 - MODULO B - - AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS
(objectives)
To acquire a good knowledge of general methods and basic techniques necessary to the study of classical and weak solutions for partial differential equations
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3
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MAT/05
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24
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6
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Core compulsory activities
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ITA |
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20410455 -
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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6
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MAT/01
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48
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12
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Core compulsory activities
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ITA |
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20410882 -
AC310 - Complex analysis
(objectives)
To acquire a broad knowledge of holomorphic and meromorphic functions of one complex variable and of their main properties. To acquire good dexterity in complex integration and in the calculation of real definite integrals.
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MASCARENHAS MELO ANA MARGARIDA
( syllabus)
Complex numbers: algebraic and topological properties. Geometric representation of complex numbers: polar coordinates and the complex exponential.
Complex functions with complex variables: continuity and properties, differentiability and first properties. Holomorphic functions: properties and examples of holomorphic and non-holomorphic functions. Cauchy-Riemann equations. The real and imaginary parts of holomorphic functions are harmonic conjugated.
Equations of Cauchy-Riemann: proof. Examples. Sequences and complex series. Properties. Power series with complex values. Abel's theorem and Hadamard's formula.
Proof of Abel's Theorem. Taylor's formula for series of complex powers. The exponential and the trigonometric functions as analytical functions. Basic properties.
Periodicity of the complex exponential function. The complex logarithm: first considerations. The ring of formal powers series with complex coefficients: basic properties. Analytical functions: definition and first properties.
Series of converging powers are analytical within the convergence region. Composition of analytical functions. Theorem of the inverse function.
Inverse by composition of a formal series and its convergence. Complex powers and properties. The binomial series and properties. Consequences of the inverse theorem: the canonical form of an analytic function.
Local properties of analytical functions: open function theorem, invertibility criterion, principle of the maximum local module. The fundamental theorem of algebra. Parameterized curves. A holomorphic function with zero derivative is constant.
The locus of the zeros of a non-constant analytical function is discrete. Analytical continuation of functions defined on open connected sets. Principle of the maximum global module. Integrals in paths: definition and first properties. Examples.
A continuous function in a connected open admits a primitive if and only if its integral along a closed curve is zero. Integration of uniformly converging series of functions. Examples. Local primitive of a holomorphic function.
Local primitive of a holomorphic function. The Goursat theorem. Integral of a holomorphic function along a continuous path.
The homotopical form of the Cauchy Theorem. Global primitive of a holomorphic function in a simply connected domain. Applications to the study of the logarithm.
The integral formula of Cauchy. Cauchy formula for development in series and applications: a holomorphic and analytical function; the theorem of Liouville and the fundamental theorem of algebra.
Integral formula for derivatives. The number of windings of a curve with respect to a point. Curves homologous to 0. The global formula of Cauchy.
Proof of the global Cauchy formula. Examples.
The first homology group of an open set with values in integers. The Cauchy formula for homological invariance. Examples.
Applications of the Cauchy theorem: uniform limit on holomorphic function compacts is holomorphic. Examples. Laurent series.
Series expansion of a holomorphic function in a circular crown in the Laurent series. Isolated singularities and the field of meromorphic functions. Examples. Statement of the classification theorem of isolated singularities and residual theorem: local and global versions.
Proof of the classification theorem of isolated singularities and proof of the residues theorem. The logarithmic derivative and the principle of the argument. Calculation of residues.
Classification of the connected open of C. The Riemann map theorem and the uniformization theorem (without proof). The Riemann sphere as a compactification of the complex plane. The group of linear transformations of the projective line and the linear transformations produced by them. The group of automorphisms of the complex plane.
The lemma of Schwarz and the group of automorphisms of the unitary disc. Elements of global analytical functions and function. The logarithm as a global analytical function.
The n-th rooty as a global analytical function. The bundle of germs of analytical functions and its properties. The Riemann surface associated with a global analytical function.
Examples and properties of Riemann surface. The Riemann surface associated with an algebraic function and properties. Summary and considerations on the course program.
( reference books)
L. V. Ahlfors: Complex Analysis, McGraw-Hill.
S. Lang: Complex analysis, GTM 103.
E. Freitag, R. Busam: Complex Analysis, Springer.
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MAT/03
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22
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10
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Core compulsory activities
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| 5
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MAT/05
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26
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14
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Core compulsory activities
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ITA |
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Optional group:
CURRICULUM TEORICO: SCEGLIERE 1-2 INSEGNAMENTI (per un totale di 9 CFU) NEI SEGUENTI SSD MAT/06,07,08,09 TRA LE ATTIVITÀ CARATTERIZZANTI (B) - (show)
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9
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20410419 -
MS410-Statistical Mechanics
(objectives)
To acquire the mathematical basic techniques of statistical mechanics for interacting particle or spin systems, including the study of Gibbs measures and phase transition phenomena, and apply them to some concrete models, such as the Ising model in dimension d = 1,2 and in the mean field approximation.
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Derived from
20410419 MS410-MECCANICA STATISTICA in Scienze Computazionali LM-40 GIULIANI ALESSANDRO
( syllabus)
INTRODUCTION TO STATISTICAL MECHANICS AND GIBBS STATES
– The goals of statistical mechanics
– Review of thermodynamics. Convex functions and Legendre transform
– Models of statistical mechanics: canonical ensemble, grand canonical and Gibbs states.
– The Ising model and the lattice gas models. Existence of the thermodynamic limit for the free energy of the Ising or lattice gas model.
– The general structure of Gibbs states. Extremal states and mixtures. The notion of phase transition: loss of analyticity and non-uniqueness of the Gibbs state.
THE ISING MODEL
– Review of known results on the Ising model in one or more dimensions.
– The solution of the one-dimensional Ising model via the transfer matrix method.
– The mean field Ising model: exact solution. Phase transition and loss of equivalence between statistical ensembles
– Ising with long-range interactions (Kac potentials) in the mean-field limit. The Maxwell construction.
– FKG and Griffiths inequalities. Existence of the infinite volume correlation functions of states with + and − conditions in the ferromagnetic Ising model.
- The geometric representation of the Ising model: high and low temperature contours.
- Existence of a phase transition in the low temperature Ising model: the Peierls's argument.
– Absence of a phase transition at high temperature and exponential decay of boundary effects boundary conditions.
– Lee-Yang theorem and analyticity of the pressure at non-zero magnetic field.
– Existence of a phase transition in the one dimensional Ising model with power law interaction |x − y|^{−p}, 1
( reference books)
S. Friedli, Y. Velenik: Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction, Cambridge University Press, 2017.
G. Gallavotti: Statistical Mechanics. A short treatise, ed. Springer-Verlag, 1999.
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9
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MAT/07
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48
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24
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Core compulsory activities
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ITA |
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20410420 -
AN420 - NUMERICAL ANALYSIS 2
(objectives)
Introduce to the study and implementation of more advanced numerical approximation techniques, in particular related to approximate solution of ordinary differential equations, and to a further advanced topic to be chosen between the optimization and the fundamentals of approximation of partial differential equations.
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Derived from
20410420 AN420 - ANALISI NUMERICA 2 in Scienze Computazionali LM-40 FERRETTI ROBERTO
( syllabus)
Ordinary Differential Equations
Finite difference approximation for ordinary differential equations: Euler's method. Consistency, stability, absolute stability. Second order Runge-Kutta methods.
Single step implicit methods: backward Euler and Crank-Nicolson methods. Convergence of single step methods. Multi-step methods: general structure, complexity, absolute stability. Stability and consistency of multi-step methods. Adams methods, BDF methods, Predictor-Corrector methods. (Reference: Chapter 7 of curse notes "Appunti del corso di Analisi Numerica")
Partial Differential Equations
Finite difference approximation for partial differential equations. Semi-discrete approximations and convergence. The Lax-Richtmeyer theorem. Transport equation: the method of characteristics. The "Upwind" (semi-discrete and fully-discrete) scheme, consistency and stability. Heat equation: Fourier approximation. Finite difference scheme, consistency and stability. Poisson equation: Fourier approximation. Finite difference scheme, convergence. (Reference: notes by R. LeVeque, "Finite Difference methods for differential equations", selected chapters 1, 2, 3, 12, 13)
( reference books)
Roberto Ferretti, "Appunti del corso di Analisi Numerica", in pdf on the course page
Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", in pdf on the course page
Lecture slides in pdf on the course page
Additional notes provided by the teacher
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9
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MAT/08
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48
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24
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Core compulsory activities
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ITA |
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20410416 -
FM410-Complements of Analytical Mechanics
(objectives)
To deepen the study of dynamical systems, with more advanced methods, in the context of Lagrangian and Hamiltonian theory.
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20410416-1 -
FM410-Complements of Analytical Mechanics - MODULE A
(objectives)
To deepen the study of dynamical systems, with more advanced methods, in the context of Lagrangian and Hamiltonian theory.
-
Derived from
20410084 COMPLEMENTI DI MECCANICA ANALITICA - MOD A in Fisica L-30 REUVERS Robin Johannes Petrus
( syllabus)
Linear dynamical systems. Forced harmonic oscillator with or without
friction. Stability theorems. Parametric resonance. Chain of
coupled harmonic oscillators: continuum limit and equations of
vibrating rope. Classic elastic diffusion. Hidden prime integrals
in the two-body problem and the harmonic oscillator problem
( reference books)
V.I. Arnol'd, Mathematical Methods of Classical Mechanics, Editors
Riuniti, Rome, 1979 G. Gallavotti, Meccanica Elementare, ed. P.
Boringhieri, Turin, 1986 G. Gentile, Introduction to systems
dynamics, 1 (Ordinary differential equations, qualitative analysis and
some applications) and
2 (Lagrangian and Hamiltonian mechanics) L.D. Landau, E.M. Lifshitz,
Meccanica, Editori Riuniti, Rome, 1976
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3
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MAT/07
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30
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Core compulsory activities
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ITA |
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20410416-2 -
FM410-Complements of Analytical Mechanics - Module B
(objectives)
To deepen the study of dynamical systems, with more advanced methods, in the context of Lagrangian and Hamiltonian theory.
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3
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MAT/07
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30
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Core compulsory activities
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ITA |
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20410769 -
FM410-Complements of Analytical Mechanics – Moldule A
(objectives)
To deepen the study of dynamical systems, with more advanced methods, in the context of Lagrangian and Hamiltonian theory.
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Derived from
20410084 COMPLEMENTI DI MECCANICA ANALITICA - MOD A in Fisica L-30 REUVERS Robin Johannes Petrus
( syllabus)
Linear dynamical systems. Forced harmonic oscillator with or without
friction. Stability theorems. Parametric resonance. Chain of
coupled harmonic oscillators: continuum limit and equations of
vibrating rope. Classic elastic diffusion. Hidden prime integrals
in the two-body problem and the harmonic oscillator problem
( reference books)
V.I. Arnol'd, Mathematical Methods of Classical Mechanics, Editors
Riuniti, Rome, 1979 G. Gallavotti, Meccanica Elementare, ed. P.
Boringhieri, Turin, 1986 G. Gentile, Introduction to systems
dynamics, 1 (Ordinary differential equations, qualitative analysis and
some applications) and
2 (Lagrangian and Hamiltonian mechanics) L.D. Landau, E.M. Lifshitz,
Meccanica, Editori Riuniti, Rome, 1976
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3
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MAT/07
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24
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6
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Core compulsory activities
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ITA |
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20410770 -
FM410-Complements of Analytical Mechanics – Moldule B
(objectives)
To deepen the study of dynamical systems, with more advanced methods, in the context of Lagrangian and Hamiltonian theory.
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3
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MAT/07
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24
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6
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Core compulsory activities
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ITA |
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20410441 -
CP420-Introduction to Stochastic Processes
(objectives)
Introduction to the theory of stochastic processes. Markov chains: ergodic theory, coupling, mixing times, with applications to random walks, card shuffling, and the Monte Carlo method. The Poisson process, continuous time Markov chains, convergence to equilibrium for some simple interacting particle systems.
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6
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MAT/06
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48
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12
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Core compulsory activities
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ITA |
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20410626 -
IN440 - COMBINATORIAL OPTIMISATION
(objectives)
Acquire skills on key solution techniques for combinatorial optimization problems; improve the skills on graph theory; acquire advanced technical skills for designing, analyzing and implementing algorithms aimed to solve optimization problems on graphs, trees and flow networks.
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Derived from
20410626 IN440 - OTTIMIZZAZIONE COMBINATORIA in Scienze Computazionali LM-40 LIVERANI MARCO, Onofri Elia
( syllabus)
Notes on graph theory: graph, directed graph, tree, free and rooted tree, connection, strong connection, acyclicality; isomorphisms between graphs, planarity, Kuratowski's theorem, Euler's formula; coloring of graphs, Eulerian paths, Hamiltonian circuits.
Review of algorithm theory and computational complexity: complexity classes, the class of NP, NP-complete, NP-hard problems. Problems of decision, search, enumeration and optimization; problems of nonlinear programming, convex programming, linear programming and integer linear programming; combinatorial optimization problems. Recalls on the elements of combinatorics.
Optimization problems on graphs: visit of graphs, verification of fundamental properties, connection, presence of cycles, strongly connected components, topological ordering of a graph. Minimum spanning tree (Prim and Kruskal algorithms). Paths of minimum cost (Dijkstra and Bellman-Ford algorithms, dynamic programming technique, Floyd-Warshall algorithm, computation of the transitive closure of a graph). Networks and calculation of the maximum flow on a network, maximum flow / minimum cut theorem, Ford-Fulkerson algorithm, Edmonds-Karp algorithm, preflow algorithms, "push-relabel" algorithms.
Partitioning problems of graphs, trees and paths into connected components, objective functions and algorithmic techniques.
Stable marriage problem, generalizations and applications of the problem, Gale and Shapley algorithm.
Huffman codes.
Programming laboratory for the implementation of algorithms in Python language and with the help of Mathematica software.
( reference books)
T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, "Introduction to algorithms"
Lecture notes and other teaching material in Italian provided by the teacher and made available on the course website (http://www.mat.uniroma3.it/users/liverani/IN440) and on the Microsoft Teams platform
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9
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MAT/09
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48
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24
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Core compulsory activities
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ITA |
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20410693 -
FM420 - Dynamic Systems
(objectives)
To acquire a solid knowledge on some advanced problems of interest in the theory of Dynamical Systems
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6
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MAT/07
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48
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12
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Core compulsory activities
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ITA |
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Optional group:
COMUNE AI 2 CURRICULA TEORICO E MODELLISTICO: SCEGLIERE QUATTRO INSEGNAMENTI (30 CFU) TRA LE ATTIVITÀ AFFINI INTEGRATIVE (C). - (show)
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30
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20410416 -
FM410-Complements of Analytical Mechanics
(objectives)
To deepen the study of dynamical systems, with more advanced methods, in the context of Lagrangian and Hamiltonian theory.
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|
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20410416-1 -
FM410-Complements of Analytical Mechanics - MODULE A
(objectives)
To deepen the study of dynamical systems, with more advanced methods, in the context of Lagrangian and Hamiltonian theory.
-
Derived from
20410084 COMPLEMENTI DI MECCANICA ANALITICA - MOD A in Fisica L-30 REUVERS Robin Johannes Petrus
( syllabus)
Linear dynamical systems. Forced harmonic oscillator with or without friction. Stability theorems. Parametric resonance. Chain of coupled harmonic oscillators: continuum limit and equations of vibrating rope. Classic elastic diffusion. Hidden prime integrals in the two-body problem and the harmonic oscillator problem
( reference books)
V.I. Arnol'd, Mathematical Methods of Classical Mechanics, Editors Riuniti, Rome, 1979 G. Gallavotti, Meccanica Elementare, ed. P. Boringhieri, Turin, 1986 G. Gentile, Introduction to systems dynamics, 1 (Ordinary differential equations, qualitative analysis and some applications) and 2 (Lagrangian and Hamiltonian mechanics) L.D. Landau, E.M. Lifshitz, Meccanica, Editori Riuniti, Rome, 1976
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3
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MAT/07
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30
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Related or supplementary learning activities
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ITA |
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20410416-2 -
FM410-Complements of Analytical Mechanics - Module B
(objectives)
To deepen the study of dynamical systems, with more advanced methods, in the context of Lagrangian and Hamiltonian theory.
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3
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MAT/07
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30
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20410451 -
LM410 -THEOREMS IN LOGIC 1
(objectives)
To acquire a good knowledge of first order classical logic and its fundamental theorems.
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20410438 -
MF410 - Computational Finance
(objectives)
Basic knowledge of financial markets, introduction to computational and theoretical models for quantitative finance, portoflio optimization, risk analysis. The computational aspects are mostly developed within the Matlab environment.
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9
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SECS-S/06
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60
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20410419 -
MS410-Statistical Mechanics
(objectives)
To acquire the mathematical basic techniques of statistical mechanics for interacting particle or spin systems, including the study of Gibbs measures and phase transition phenomena, and apply them to some concrete models, such as the Ising model in dimension d = 1,2 and in the mean field approximation.
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Derived from
20410419 MS410-MECCANICA STATISTICA in Scienze Computazionali LM-40 GIULIANI ALESSANDRO
( syllabus)
INTRODUCTION TO STATISTICAL MECHANICS AND GIBBS STATES
– The goals of statistical mechanics
– Review of thermodynamics. Convex functions and Legendre transform
– Models of statistical mechanics: canonical ensemble, grand canonical and Gibbs states.
– The Ising model and the lattice gas models. Existence of the thermodynamic limit for the free energy of the Ising or lattice gas model.
– The general structure of Gibbs states. Extremal states and mixtures. The notion of phase transition: loss of analyticity and non-uniqueness of the Gibbs state.
THE ISING MODEL
– Review of known results on the Ising model in one or more dimensions.
– The solution of the one-dimensional Ising model via the transfer matrix method.
– The mean field Ising model: exact solution. Phase transition and loss of equivalence between statistical ensembles
– Ising with long-range interactions (Kac potentials) in the mean-field limit. The Maxwell construction.
– FKG and Griffiths inequalities. Existence of the infinite volume correlation functions of states with + and − conditions in the ferromagnetic Ising model.
- The geometric representation of the Ising model: high and low temperature contours.
- Existence of a phase transition in the low temperature Ising model: the Peierls's argument.
– Absence of a phase transition at high temperature and exponential decay of boundary effects boundary conditions.
– Lee-Yang theorem and analyticity of the pressure at non-zero magnetic field.
– Existence of a phase transition in the one dimensional Ising model with power law interaction |x − y|^{−p}, 1
( reference books)
S. Friedli, Y. Velenik: Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction, Cambridge University Press, 2017.
G. Gallavotti: Statistical Mechanics. A short treatise, ed. Springer-Verlag, 1999.
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9
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MAT/07
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20410420 -
AN420 - NUMERICAL ANALYSIS 2
(objectives)
Introduce to the study and implementation of more advanced numerical approximation techniques, in particular related to approximate solution of ordinary differential equations, and to a further advanced topic to be chosen between the optimization and the fundamentals of approximation of partial differential equations.
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Derived from
20410420 AN420 - ANALISI NUMERICA 2 in Scienze Computazionali LM-40 FERRETTI ROBERTO
( syllabus)
Ordinary Differential Equations
Finite difference approximation for ordinary differential equations: Euler's method. Consistency, stability, absolute stability. Second order Runge-Kutta methods.
Single step implicit methods: backward Euler and Crank-Nicolson methods. Convergence of single step methods. Multi-step methods: general structure, complexity, absolute stability. Stability and consistency of multi-step methods. Adams methods, BDF methods, Predictor-Corrector methods. (Reference: Chapter 7 of curse notes "Appunti del corso di Analisi Numerica")
Partial Differential Equations
Finite difference approximation for partial differential equations. Semi-discrete approximations and convergence. The Lax-Richtmeyer theorem. Transport equation: the method of characteristics. The "Upwind" (semi-discrete and fully-discrete) scheme, consistency and stability. Heat equation: Fourier approximation. Finite difference scheme, consistency and stability. Poisson equation: Fourier approximation. Finite difference scheme, convergence. (Reference: notes by R. LeVeque, "Finite Difference methods for differential equations", selected chapters 1, 2, 3, 12, 13)
( reference books)
Roberto Ferretti, "Appunti del corso di Analisi Numerica", in pdf on the course page
Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", in pdf on the course page
Lecture slides in pdf on the course page
Additional notes provided by the teacher
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MAT/08
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20410442 -
IN420 - Information Theory
(objectives)
Introduce key questions in the theory of signal transmission and quantitative analysis of signals, such as the notions of entropy and mutual information. Show the underlying algebraic structure. Apply the fundamental concepts to code theory, data compression and cryptography.
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Derived from
20410442 IN420 - TEORIA DELL'INFORMAZIONE in Scienze Computazionali LM-40 BONIFACI VINCENZO
( syllabus)
1. Introduction to information theory.
Reliable transmission of information. Shannon's information content. Measures of information. Entropy, mutual information, informational divergence. Data compression. Error correction. Data processing theorems. Fundamental inequalities. Information diagrams. Informational divergence and maximum likelihood.
2. Source coding and data compression
Typical sequences. Typicality in probability. Asymptotic equipartitioning property. Block codes and variable length codes. Coding rate. Source coding theorem. Lossless data compression. Huffman code. Universal codes. Ziv-Lempel compression.
3. Channel coding
Channel capacity. Discrete memoryless channels. Information transmitted over a channel. Decoding criteria. Noisy channel coding theorem.
4. Further codes and applications
Hamming space. Linear codes. Generating matrix and check matrix. Cyclic codes. Hash codes.
( reference books)
T.M. Cover, J.A. Thomas. Elements of Information Theory. Wiley, 1991.
R.E. Blahut. Algebraic Codes for Data Transmission. Cambridge University Press, 2003.
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INF/01
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20410459 -
MC430 - LABORATORY: DIDACTICS FOR MATHEMATICS
(objectives)
1. Mathematics software, with particular attention to their use for teaching mathematics in school. 2. Analysis of the potential and criticality of the use of technological tools for teaching and learning mathematics.
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Derived from
20410459 MC430 - LABORATORIO DI DIDATTICA DELLA MATEMATICA in Matematica LM-40 FALCOLINI CORRADO
( syllabus)
Teaching mathematics with the help of a computer: GeoGebra and Mathematica softwares.
Commands for numerical and symbolic calculus, graphics visualization, parametric surfaces and curves with animations in changing parameters.
Solving problems: triangle's properties in Euclidean and non-Euclidean geometry with examples, approximation of pi and other irrational numbers, solutions of equations and inequalities, systems of equations, defining and visualizing geometrical loci, function integral and derivatives, approximation of surface area.
( reference books)
List of problems given in class concerning visualization and solutions with the help of software Mathematica or GeoGebra.
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MAT/04
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20410424 -
IN450 - 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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Derived from
20410424 IN450- ALGORITMI PER LA CRITTOGRAFIA in Scienze Computazionali LM-40 PEDICINI MARCO
( syllabus)
1. Classic Cryptography
- Basic cryptosystems: encryption by substitution, by translation, by permutation, affine cryptosystem, by Vigenère, by Hill. Stream encryption (synchronous and asynchronous), Linear feedback shift registers (LFSR) on finite fields, Autokey cypher. Product cyphers. Basic cryptanalysis: classification of attacks; cryptoanalysis for affine cyphers, for substitution cypher (frequency analysis), for Vigenere cypher: Kasiski test, coincidence index; cryptoanalysis of Hill's cypher and LFSR: algebraic attacks, cube attack.
2. Application of Shannon theory to cryptography
- Security of cyphers: computational security, provable security, unconditional security. Basics of probability: discrete random variables, joint probability, conditional probability, independent random variables, Bayes' theorem. Random variables associated with cryptosystems. Perfect secrecy for encryption systems. Vernam cryptosystem. Entropy. Huffman codes. Spurious Keys and Unicity distance.
3. Block cyphers
- iterative encryption schemes; Substitution-Permutation Networks (SPN); Linear cryptanalysis for SPN: Piling-Up Lemma, linear approximation of S-boxes, linear attacks on S-boxes; Differential cryptanalysis for SPN; Feistel cyphers; DES: description and analysis; AES: description; Notes on finite fields: operations on finite fields, Euclid's generalized algorithm for the computation of the GCD and inverse; Operating modes for block cyphers.
4. Hash functions and codes for message authentication
- Hash functions and data integrity. Safe hash functions: resistance to the pre-image, resistance to the second pre-image, collision resistance. The random oracle model: ideal hash functions, properties of independence. Randomized algorithms, collision on the problem of the second pre-image, collision on the problem of the pre-image. Iterated hash functions; the construction of Merkle-Damgard. Safe Hash Algorithm (SHA-1). Authentication Codes (MAC): nested authentication codes (HMAC).
( reference books)
[1] Antoine Joux, Algorithmic Cryptanalysis, (2010) CRC Press.
[2] Douglas Stinson, Cryptography: Theory and Practice, 3rd edition, (2006) Chapman and Hall/CRC.
[3] Delfs H., Knebl H., Introduction to Cryptography, (2007) Springer Verlag.
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20410425 -
GE460- GRAPH THEORY
(objectives)
Provide tools and methods for graph theory.
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Derived from
20410425 GE460 - TEORIA DEI GRAFI in Scienze Computazionali LM-40 SCHAFFLER LUCA
( 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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MAT/03
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20410426 -
IN480 - PARALLEL AND DISTRIBUTED COMPUTING
(objectives)
Acquire parallel and distributed programming techniques, and know modern hardware and software architectures for high-performance scientific computing. Parallelization paradigms, parallelization on CPU and GPU, distributed memory systems. Data-intensive, Memory Intensive and Compute Intensive applications. Performance analysis in HPC systems.
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9
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INF/01
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20410428 -
CR510 – ELLIPTIC CRYPTOSYSTEMS
(objectives)
Acquire a basic knowledge of the concepts and methods related to the theory of public key cryptography using the group of points of an elliptic curve on a finite field. Apply the theory of elliptic curves to classical problems of computational number theory such as factorization and primality testing.
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Derived from
20410428 CR510 – CRITTOSISTEMI ELLITTICI in Scienze Computazionali LM-40 CAPUANO LAURA
( syllabus)
1. Theory of Elliptic Curves
Weierstrass Equation, The structure of the group on rational points, formulas for the addition and duplication. Generalities on the intersections between lines and curves in P2(K)
Preparatory results for the proof of the associativity of points on elliptic curves. Proof of the associativity of the sum for the points of an elliptic curve. Other equations for elliptic curves, Legendre's equation, Cubic equations, Quartic equations, intersections of two cubic surfaces. The j-invariant, elliptic curve in characteristic 2, Endomorphisms, singular curves, elliptic curves module n.
2. Torsion points
Torsion points, Division polynomials. Weil's pairing
3. Elliptic curve on finite fields
Frobenius endomorphism. The problem of determining the order of the group. Curves on subfields, Legendre's symbols, Point orders, Shanks's "Baby Step, Giant Step" algorithm. Particular families of elliptic curves. Schoof's algorithm.
4. Cryptosystems on Elliptical Curves.
The Discrete Logarithm Problem. Algorithms for calculating the discrete logarithm: Baby-Step Giant-Step and Polig-Hellman. MOV attack. Attack on anomalous curves. Diffie-Hellman Key Exchange. Cryptosystems by Massey Omura and ElGamal. El Gamal Signature Scheme. Cryptosystems on elliptic curves analysis on the factorization problem. A cryptosystem based on Weil coupling. Factorization of internal numbers using elliptic curves. Using Pari.
( reference books)
Lawrence C. Washington, Elliptic Curves: Number Theory and Crptography. Chapman & Hall (CRC) 2003.
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MAT/02
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20410462 -
GE510 - ALGEBRAIC GEOMETRY 2
(objectives)
Introduce to the study of algebraic geometry, with particular emphasis on beams, schemes and cohomology.
-
TURCHET AMOS
( syllabus)
- Introduction and motivation
- Affine and projective schemes, morphisms and properties
- Diviors, Sheaves of modules and invertible sheaves, differentials
- Sheaf cohomology
- Applications to the study of algebraic curves
( reference books)
R. Hartshorne, Algebraic geometry, Graduate Texts in Math. No. 52. Springer-Verlag, New York-Heidelberg, 1977
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MAT/03
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20410434 -
FS450 - Elements of Statistical Mechanics
(objectives)
Gain knowledge of fundamental principles of statistical 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)
e-Learning Moodle Platform with Supplementary Material
( reference books)
Suggested textbook:
1) C. Di Castro and R. Raimondi, Statistical Mechanics and Applications in Condensed Matter, Cambridge University Press, 2015.
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FIS/02
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60
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20410560 -
IN400- Python and MATLAB programming
(objectives)
Acquire the ability to implement high-level programs in the interpreted languages Python and MATLAB. Understand the main constructs used in Python and MATLAB and their application to scientific computing and data processing scenarios.
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20410566 -
FS470 - Principles of astrophysics
(objectives)
Provide the student with a first view of some of the fundamental topics of Astrophysics and Cosmology using the mathematical and physical knowledge acquired in the first two years
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Derived from
20410499 Principi di Astrofisica in Fisica L-30 LA FRANCA FABIO, MATT GIORGIO
( syllabus)
Topics Part A
• Coordinates and Telescopes
• Elements of Spectroscopy
• Stars and Stellar Evolution • Galaxies
• Active Galactic Nuclei
Program Part A
• Overview
• Celestial coordinates (1.3)
• Telescopes and resolving power (6.1)
• Parallax distance (3.1)
• Flux, brightness, apparent and absolute magnitudes, colors (3.2, 3.3, 3.6)
• The black body (3.4, 3.5)
• Hertzsprung-Russel diagram (8.2)
• Open and globular clusters: position, stellar populations and HR diagram (13.3)
• White dwarfs, Novae and SuperNovae (notes and partly in 15 and 16)
• The classification of galaxies (24.1)
• The rotation curve of galaxies and dark matter (25.3)
• The center of the Galaxy and its Black Hole (25.4)
• Hubble's law and expansion of the Universe (27.2)
• Probability of collision between stars and galaxies (handouts)
• Black Holes: outline of General Relativity (outline 17)
• Active Galactic Nuclei (28.1, 28.2, 28.3)
Topics Part B
• Structure and stellar evolution
• Elements of Spectroscopy
• Distances and expansion of the Universe • Galaxies
• GRB and gravitational waves
Program Part B
• Acretion disks and X-ray emission in Active Galactic Nuclei (28.2)
• Stars of Neutrons and Pulsars (16.6, 16.7)
• Gamma Ray Bursts (handouts)
• Gravitational Waves (lecture notes)
• Spectroscopy: eq. Boltzmann-excitation and Saha-ionization (8.1)
• Spectroscopy: speed, temperature and density measurements (handouts)
• Eq. of star structure, time and Kelvin-Helmholtz instability (11.1-4)
• Nuclear reactions of hydrogen (11.3)
• Jeans mass of gravitational collapse, free-fall time and Initial Mass Function (12.2, 12.3)
• The Milky Way and the local group (25.1, 25.2)
• Metallicity (25.2)
• Transit of Venus and measurement of the Earth-Sun distance (handouts)
• Distance scale (27.1)
• Hubble's law and expansion of the Universe (27.2)
• Local Group, Clusters of Galaxies, large scale structure of the Universe (27.3)
• The Big Bang and the background radiation (29.2 brief notes and lecture notes)
( reference books)
A copy of the lecture notes can be downloaded from the course website.
In brackets, the paragraphs from “An Introduction to Modern Astrophysics, II ed. - B.W. Carrol, D.A. Ostlie - Ed. Pearson, Addison Wesley ”(copies available in the library). The discussion in the course has been simplified compared to what is reported in the text. Alternative text in Italian: Attilio Ferrari, Stars, Galaxies, Universe - Fundamentals of Astrophysics - Ed. Springer
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FIS/05
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20410592 -
LM400 - INTRODUZIONE ALLA LOGICA
(objectives)
To Introduce students to themes, concepts, methods and results of logic that are at the basis of every discipline, in order to provide students - having any kind of background- with a deep interdisciplinary approach and an appropriate training for school teaching
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M-FIL/02
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20410529 -
LM510 - LOGICAL THEORIES 1
(objectives)
Address some questions of the theory of the proof of the twentieth century, in connection with the themes of contemporary research
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MAT/01
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36
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20410627 -
TN410 - INTRODUCTION TO NUMBER THEORY
(objectives)
Acquire a good knowledge of the concepts and methods of the elementary number theory, with particular reference to the study of the Diophantine equations and congruence equations. Provide prerequisites for more advanced courses of algebraic and analytical number theory.
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Derived from
20410627 TN410 - INTRODUZIONE ALLA TEORIA DEI NUMERI in Matematica LM-40 PAPPALARDI FRANCESCO
( syllabus)
Division, Factorization, Some Elementary Properties of Primes, Some Results and Problems Concerning Primes.
ARITHMETIC FUNCTIONS: The Divisor Function. The Moebius Function. The Euler Function. Dirichlet Convolution
CONGRUENCES: Sets of Residues,Some Interesting Congruences, Some Linear Congruences, Some Polynomial Congruences, Primitive Roots, the Theorem of Gauss.
QUADRATIC RESIDUES: The Legendre Symbol. Quadratic Reciprocity. The Jacobi Symbol.The Distribution of Quadratic Residues.
SUMS OF INTEGER SQUARES: Sums of Two Squares. Number of Representations. Sums of Four Squares. Sums of Three Squares.
ELEMENTARY PRIME NUMBER THEORY: Euclid's Theorem Revisited. The Von Mangoldt Function. Tchebycheff's Theorem. Some Results of Mertens
( reference books)
Chen, W; ELEMENTARY NUMBER THEORY. https://rutherglen.science.mq.edu.au/wchen/lnentfolder/lnent.html
Chowdhury, F.; Chowdhury, M. R. Essentials of Number Theory. Pi Publications, Dhaka, Bangladesh, 2005. ISBN 984-32-2836-7
Hardy, G. H.; Wright, E. M. An introduction to the theory of numbers. Fifth edition. The Clarendon Press, Oxford University Press, New York, 1979. xvi+426 pp. ISBN: 0-19-853170-2; 0-19-853171-0
Davenport, H. Aritmetica superiore. Un'introduzione alla teoria dei numeri. Editore: Zanichelli, 1994. 199 pp. ISBN: 8808091546
Gioia, A. A. The theory of numbers. An introduction. Reprint of the 1970 original. Dover Publications, Inc., Mineola, NY, 2001. xii+207 pp. ISBN: 0-486-41449-3
Rosen, K. H. Elementary number theory and its applications. Fourth edition. Addison-Wesley, Reading, MA, 2000. xviii+638 pp. ISBN: 0-201-87073-8
Tattersall, J. J. Elementary number theory in nine chapters. Cambridge University Press, Cambridge, 1999. viii+407 pp. ISBN: 0-521-58531-7
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20410626 -
IN440 - COMBINATORIAL OPTIMISATION
(objectives)
Acquire skills on key solution techniques for combinatorial optimization problems; improve the skills on graph theory; acquire advanced technical skills for designing, analyzing and implementing algorithms aimed to solve optimization problems on graphs, trees and flow networks.
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Derived from
20410626 IN440 - OTTIMIZZAZIONE COMBINATORIA in Scienze Computazionali LM-40 LIVERANI MARCO, Onofri Elia
( syllabus)
Notes on graph theory: graph, directed graph, tree, free and rooted tree, connection, strong connection, acyclicality; isomorphisms between graphs, planarity, Kuratowski's theorem, Euler's formula; coloring of graphs, Eulerian paths, Hamiltonian circuits.
Review of algorithm theory and computational complexity: complexity classes, the class of NP, NP-complete, NP-hard problems. Problems of decision, search, enumeration and optimization; problems of nonlinear programming, convex programming, linear programming and integer linear programming; combinatorial optimization problems. Recalls on the elements of combinatorics.
Optimization problems on graphs: visit of graphs, verification of fundamental properties, connection, presence of cycles, strongly connected components, topological ordering of a graph. Minimum spanning tree (Prim and Kruskal algorithms). Paths of minimum cost (Dijkstra and Bellman-Ford algorithms, dynamic programming technique, Floyd-Warshall algorithm, computation of the transitive closure of a graph). Networks and calculation of the maximum flow on a network, maximum flow / minimum cut theorem, Ford-Fulkerson algorithm, Edmonds-Karp algorithm, preflow algorithms, "push-relabel" algorithms.
Partitioning problems of graphs, trees and paths into connected components, objective functions and algorithmic techniques.
Stable marriage problem, generalizations and applications of the problem, Gale and Shapley algorithm.
Huffman codes.
Programming laboratory for the implementation of algorithms in Python language and with the help of Mathematica software.
( reference books)
T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, "Introduction to algorithms"
Lecture notes and other teaching material in Italian provided by the teacher and made available on the course website (http://www.mat.uniroma3.it/users/liverani/IN440) and on the Microsoft Teams platform
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20410637 -
AM450 - FUNCTIONAL ANALYSIS
(objectives)
To acquire a good knowledge of functional analysis: Banach and Hilbert spaces, weak topologies, linear and continuous operators, compact operators, spectral theory.
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MAT/05
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20410441 -
CP420-Introduction to Stochastic Processes
(objectives)
Introduction to the theory of stochastic processes. Markov chains: ergodic theory, coupling, mixing times, with applications to random walks, card shuffling, and the Monte Carlo method. The Poisson process, continuous time Markov chains, convergence to equilibrium for some simple interacting particle systems.
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6
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MAT/06
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48
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12
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ITA |
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20410746 -
AL440 – GROUP THEORY
(objectives)
Become familiar with the fundamental notions of group theory, particularly finite groups,
so as to be able to study and classify some important classes of finite groups.
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Derived from
20410746 AL440 - TEORIA DEI GRUPPI in Matematica LM-40 TARTARONE FRANCESCA
( syllabus)
Recalls on the elementary properties of groups. Direct and semi-direct products. Permutation groups and simplicity of alternate groups. Actions on groups. Sylow's theorems. Finitely generated abelian groups, free groups, nilpotent groups and solvable groups. Since this is an advanced course, other topics can also be included upon request of the class.
( reference books)
A. Machì, Gruppi. Una introduzione a idee e metodi della Teoria dei Gruppi, SPRINGER VERLAG (2007).
M. Artin, Algebra, BOLLATI BORINGHIERI (1997).
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6
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MAT/02
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48
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12
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-
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ITA |
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20410757 -
AM410 - AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS
(objectives)
To acquire a good knowledge of general methods and basic techniques necessary to the study of classical and weak solutions for partial differential equations
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AM410- MODULE A - AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS
(objectives)
To acquire a good knowledge of general methods and basic techniques necessary to the study of classical and weak solutions for partial differential equations
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Derived from
20410757_1 AM410- MODULO A - INTRODUZIONE ALLE EQUAZIONI ALLE DERIVATE PARZIALI in Matematica LM-40 ESPOSITO PIERPAOLO
( syllabus)
Preliminaries: definition of hyper-surface, integration on hyper-surfaces, the divergence theorem; the Laplace equation: the mean value inequalities, the minimum and maximum principle, the Harnack inequality, the Green representation, the Poisson integral, convergence's theorems, interior estimates on the derivatives, the Perron method for the Dirichlet problem.
( reference books)
“Elliptic partial differential equations of second order. Reprint of the 1998 edition”, D. Gilbarg e N.S. Trudinger, Classics in Mathematics, Springer-Verlag
"Partial differential equations. Second edition", Lawrence C. Evans, Graduate Studies in Mathematics 19, American Mathematical Society
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3
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MAT/05
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24
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6
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-
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ITA |
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AM410 - MODULE B - AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS
(objectives)
To acquire a good knowledge of general methods and basic techniques necessary to the study of classical and weak solutions for partial differential equations
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3
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MAT/05
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24
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6
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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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20410758 -
AM410 - MODULE A - AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS
(objectives)
To acquire a good knowledge of general methods and basic techniques necessary to the study of classical and weak solutions for partial differential equations
-
Derived from
20410757_1 AM410- MODULO A - INTRODUZIONE ALLE EQUAZIONI ALLE DERIVATE PARZIALI in Matematica LM-40 ESPOSITO PIERPAOLO
( syllabus)
Preliminaries: definition of hyper-surface, integration on hyper-surfaces, the divergence theorem; the Laplace equation: the mean value inequalities, the minimum and maximum principle, the Harnack inequality, the Green representation, the Poisson integral, convergence's theorems, interior estimates on the derivatives, the Perron method for the Dirichlet problem.
( reference books)
“Elliptic partial differential equations of second order. Reprint of the 1998 edition”, D. Gilbarg e N.S. Trudinger, Classics in Mathematics, Springer-Verlag
"Partial differential equations. Second edition", Lawrence C. Evans, Graduate Studies in Mathematics 19, American Mathematical Society
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3
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MAT/05
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24
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6
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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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20410759 -
AM410 - MODULO B - - AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS
(objectives)
To acquire a good knowledge of general methods and basic techniques necessary to the study of classical and weak solutions for partial differential equations
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3
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MAT/05
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24
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6
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-
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Related or supplementary learning activities
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ITA |
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20410568 -
IN470- 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
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Derived from
20410568 IN470 - METODI COMPUTAZIONALI PER LA BIOLOGIA in Scienze Computazionali LM-40 Mastrostefano Enrico
( syllabus)
The course consists of online and laboratory lessons, focusing primarily on gene sequence alignment techniques. We will also cover biological networks and agent-based models in the biological domain.
Regarding alignment techniques, we will explore DNA/RNA sequencing and alignment algorithms, as well as phylogenetic tree reconstruction methods. Specifically, we will study techniques for whole genome reconstruction (Whole Genome Sequencing) using de Bruijn graphs and the alignment of pairwise and multiple subsequences, which are employed to identify the biological functions associated with different genes.
We will examine some of the main genetic databases and the online tools available for performing alignments (NCBI, BLAST, Clustal)..
Next, we will study some characteristics and methodologies used in the analysis of complex networks, such as centrality measures and clustering, applied to protein networks and gene regulatory networks.
Finally, we will introduce some agent-based modeling techniques used in clinical and immunological applications.
- Topics:
- Introduction to Biology
- Sequencing techniques, genome sequencing, de Bruijn graphs
- Sequence alignment: Knuth-Morris-Pratt algorithm, online databases, pairwise alignment, scoring matrices, Needleman-Wunsch algorithm, BLAST tool
- Multiple sequence alignment: Markov chains, phylogenetic trees, Clustal, UPGMA, Neighbor-Joining algorithm
- Biological networks: protein networks and gene regulatory networks
- Cellular automata and agent-based models
( reference books)
Python:https://github.com/steguar/DAIL/blob/main/Lecture_1/Lecture_1_Python_crash_ course.ipynb
• Understanding Bioinformatics, Marketa Zvelebil & Jeremy O. Baum
• Biological sequence analysis, R. Durbin et al. (CAP 1,2,6,7)
• Bioinformatics Algorithms: an Active Learning Approach, Pavel A. Pevzner and Phillip Compeau
• Bioinformatics - an Introduction, Jeremy Ramsden
• Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids
• Statistical Methods in Bioinformatics, An Introduction, Warren J. Ewens , Gregory Grant
• Networks, M. Newman (ER and CM random graphs, Epidemics on Networks)
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6
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INF/01
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48
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12
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ITA |
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20410455 -
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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6
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MAT/01
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48
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12
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ITA |
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20410875 -
FM530 - Mathematical Methods for Machine Learning
(objectives)
Linear algebra concepts are key for understanding and creating machine learning algorithms, especially as applied to deep learning and neural networks. This course reviews linear algebra with applications to statistics, image processing and optimization–and above all a full explanation of the structure of Neural Networks.
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Derived from
20410875 FM530 - METODI MATEMATICI PER IL MACHINE LEARNING in Scienze Computazionali LM-40 TERESI LUCIANO, REUVERS Robin Johannes Petrus
( syllabus)
Highlights of Linear Algebra:
Matrix-matrix multiplication; column & row space; rank
The four fundamental subspaces of linear algebra
Fundamentals of Matrix factorizations:
A=LU rows & columns point of view
A=LU elimination & factorization; permutations
A=RU=VU; Orthogonal matrices
Eigensystems and Linear ODE
Intro to PSym; the energy function
Gradient and Hessian
Singular Value Decomposition
Eckart-Young; derivative of a matrix norm
Principal Component Analysis
Generalized evectors;
Norms
Least Squares
Convexity & Newton’s method
Newton & L-M method; Recap of non-linear regression
Lagrange multipliers
Machine Learning:
Gradient Descend; exact line search; GD in action; GD with Matlab
Learning & Loss; Intro to Deep Neural Network; DNN with Matlab
Loss functions: Quadratic VS Cross entropy
Stocastics Gradient Descend (SGD) & Kaczmarcz; SGD convergence rates & ADAM
Matlab interface for DNN
Construction of DNN: the key steps
Backpropagation and the Chain Rule
Machine Learning examples with Wolfram Mathematica
Convolutional NN + Mathematica examples of 1D convolution
Convolution and 2D filters + Mathematica examples of 2D convolution
Matlab Live Script, Network Designer, Pretrained Net
( reference books)
Lecture notes
G. Strang,
Linear Algebra and Learning from Data,
Wellesley-Cambridge Press
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9
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MAT/07
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48
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24
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ITA |
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20410882 -
AC310 - Complex analysis
(objectives)
To acquire a broad knowledge of holomorphic and meromorphic functions of one complex variable and of their main properties. To acquire good dexterity in complex integration and in the calculation of real definite integrals.
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Derived from
20410882 AC310 - ANALISI COMPLESSA in Matematica LM-40 MASCARENHAS MELO ANA MARGARIDA
( syllabus)
Complex numbers: algebraic and topological properties. Geometric representation of complex numbers: polar coordinates and the complex exponential.
Complex functions with complex variables: continuity and properties, differentiability and first properties. Holomorphic functions: properties and examples of holomorphic and non-holomorphic functions. Cauchy-Riemann equations. The real and imaginary parts of holomorphic functions are harmonic conjugated.
Equations of Cauchy-Riemann: proof. Examples. Sequences and complex series. Properties. Power series with complex values. Abel's theorem and Hadamard's formula.
Proof of Abel's Theorem. Taylor's formula for series of complex powers. The exponential and the trigonometric functions as analytical functions. Basic properties.
Periodicity of the complex exponential function. The complex logarithm: first considerations. The ring of formal powers series with complex coefficients: basic properties. Analytical functions: definition and first properties.
Series of converging powers are analytical within the convergence region. Composition of analytical functions. Theorem of the inverse function.
Inverse by composition of a formal series and its convergence. Complex powers and properties. The binomial series and properties. Consequences of the inverse theorem: the canonical form of an analytic function.
Local properties of analytical functions: open function theorem, invertibility criterion, principle of the maximum local module. The fundamental theorem of algebra. Parameterized curves. A holomorphic function with zero derivative is constant.
The locus of the zeros of a non-constant analytical function is discrete. Analytical continuation of functions defined on open connected sets. Principle of the maximum global module. Integrals in paths: definition and first properties. Examples.
A continuous function in a connected open admits a primitive if and only if its integral along a closed curve is zero. Integration of uniformly converging series of functions. Examples. Local primitive of a holomorphic function.
Local primitive of a holomorphic function. The Goursat theorem. Integral of a holomorphic function along a continuous path.
The homotopical form of the Cauchy Theorem. Global primitive of a holomorphic function in a simply connected domain. Applications to the study of the logarithm.
The integral formula of Cauchy. Cauchy formula for development in series and applications: a holomorphic and analytical function; the theorem of Liouville and the fundamental theorem of algebra.
Integral formula for derivatives. The number of windings of a curve with respect to a point. Curves homologous to 0. The global formula of Cauchy.
Proof of the global Cauchy formula. Examples.
The first homology group of an open set with values in integers. The Cauchy formula for homological invariance. Examples.
Applications of the Cauchy theorem: uniform limit on holomorphic function compacts is holomorphic. Examples. Laurent series.
Series expansion of a holomorphic function in a circular crown in the Laurent series. Isolated singularities and the field of meromorphic functions. Examples. Statement of the classification theorem of isolated singularities and residual theorem: local and global versions.
Proof of the classification theorem of isolated singularities and proof of the residues theorem. The logarithmic derivative and the principle of the argument. Calculation of residues.
Classification of the connected open of C. The Riemann map theorem and the uniformization theorem (without proof). The Riemann sphere as a compactification of the complex plane. The group of linear transformations of the projective line and the linear transformations produced by them. The group of automorphisms of the complex plane.
The lemma of Schwarz and the group of automorphisms of the unitary disc. Elements of global analytical functions and function. The logarithm as a global analytical function.
The n-th rooty as a global analytical function. The bundle of germs of analytical functions and its properties. The Riemann surface associated with a global analytical function.
Examples and properties of Riemann surface. The Riemann surface associated with an algebraic function and properties. Summary and considerations on the course program.
( reference books)
L. V. Ahlfors: Complex Analysis, McGraw-Hill.
S. Lang: Complex analysis, GTM 103.
E. Freitag, R. Busam: Complex Analysis, Springer.
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| 4
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MAT/03
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22
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10
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Attività formative affini ed integrative
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| 5
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MAT/05
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26
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14
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Attività formative affini ed integrative
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ITA |
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20410693 -
FM420 - Dynamic Systems
(objectives)
To acquire a solid knowledge on some advanced problems of interest in the theory of Dynamical Systems
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6
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MAT/07
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48
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12
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ITA |
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20411002 -
IN510 – QUANTUM COMPUTING
(objectives)
Module A
PPresent the computational paradigm of Quantum Computing. By the end of the course, students should be able to understand even complex Quantum algorithms and to analyze and write simple quantum algorithms.
Module B
Study of the quantum circuit model and its universality, in-depth exploration of key quantum techniques for algorithm design and analysis, and the introduction to some quantum programming languages and software platforms for the specification of quantum computations.
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20411056 -
FM500 - NON-LINEAR MODELS IN MATHEMATICAL PHYSICS
(objectives)
Acquire a solid knowledge of some advanced problems in the study of some nonlinear equations of mathematical physics
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6
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MAT/07
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-
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Related or supplementary learning activities
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ITA |
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20410952 -
AM560 - Geometric analysis
(objectives)
To acquire a good knowledge of general methods and techniques necessary to the study of partial differential equations in relation with geometric-differential.
problems on surfaces and manifolds
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BATTAGLIA LUCA
( syllabus)
Introduction to Riemannian geometry.
The problem of prescribing Gaussian curvature on compact surfaces.
The problem of prescribing scalar curvature on manifolds of dimension greater or equal to three.
( reference books)
"Introduction à l'analyse non linéaire sur les variétés", E. Hebey, Diderot Editeur
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6
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MAT/05
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48
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12
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ITA |
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Università Roma Tre