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.

Roberto Ferretti, "Appunti del corso di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Slides of the lessons, available from the course page: http://www.mat.uniroma3.it/users/ferretti/bacheca.html

CONGRUENCES AND POLYNOMIALS. LINEAR DIOPHANTINE EQUATIONS. SYSTEMS OF LINEAR CONGRUENCES. POLYNOMIAL CONGRUENCES. POLYNOMIAL CONGRUENCES MOD p: LAGRANGES’S THEOREM. p-ADIC APPROXIMATION. THE EXISTENCE OF PRIMITIVE ROOTS MOD p. QUADRATIC CONGRUENCES. QUADRATIC RESIDUES. THE LEGENDRE SYMBOL. GAUSS’S LEMMA AND THE LAW OF QUADRATIC RECIPROCITY.THE JACOBI SYMBOL. SUMS OF TWO, THREE, FOUR SQUARES. MULTIPLICATIVE FUNCTIONS. THE MÖBIUS INVERSION FORMULA. SOME NONLINEAR DIOPHANTINE EQUATIONS. CONTINUED FRACTIONS.

M. FONTANA, APPUNTIDEL CORSO TN1 (ARGOMENTI DELLA TEORIA CLASSICA DEI NUMERI), HTTP://WWW.MAT.UNIROMA3.IT/USERS/FONTANA/DIDATTICA/FONTANA_DIDATTICA.HTML DISPENSE.
D.M. BURTON: ELEMENTARY NUMBER THEORY, MCGRAW-HILLINTERNATIONAL EDITION, 6TH EDITION (2007), 434 PP.
G.A. JONES AND J.M. JONES, ELEMENTARY NUMBER THEORY, SPRINGER, 1ST EDITION (1998), 200 PP.
H. DAVENPORT, ARITMETICA SUPERIORE.UN?INTRODUZIONE ALLA TEORIA DEI NUMERI, ZANICHELLI, (1994), 199 PP.
G.H. HARDY AND E.M. WRIGHT, AN INTRODUCTION TO THE THEORY OH NUMBERS, THE CLARENDON PRESS, OXFORD UNIVERSITY, 5TH EDITION (1979), XVI+426 PP.
W.J. LEVEQUE, FUNDAMENTALS OF NUMBER THEORY, DOVER PUBLICATIONS, (1996)
K.H. ROSEN. ELEMENTARY NUMBER THEORY AND ITS APPLICATIONS, ADDISON-WESLEY, 6TH EDITION (2011), XV+752 PP.

Introduction: Cardano equations for the solubility of third-degree equations, rings and fields, the characteristics of a field, known facts about rings of polynomials, field extensions, construction of some field extensions, the sub-ring generated by a subset, the subfield generated by a subset, algebraic and transcendent elements, algebraically closed fields.

Splitting fields: Simple extensions and maps between simple extensions, splitting fields, existence of the splitting field, uniqueness up to isomorphisms of the splitting field, multiple roots, formal derivatives, separable polynomials and perfect fields, minimal polynomials and their characterizations.

The fundamental Theorem of Galois Theory: Group of the automorphisms of a field, normal, separable and Galois extensions, characterizations of separable extensions, Fundamental Theorem of Galois Correspondence, examples, Galois group of a polynomial, Radical extensions, solvable groups and Galois's Theorem on solving equations , Theorem of the existence of primitive elements.

The computation of Galois group: Galois groups as subgroups of $ S_n $, transitive subgroups of $ S_n $, characterization of irreducibility in terms of transitivity, polynomials with Galois groups in $ A_n $, Theory of discriminants, Galois groups of polynomials with degree up to $4$, examples of polynomials with Galois group $S_p$.

Cyclotomic fields: Definitions, Galois group, maximal real subfields, quadratic subfields, Galois groups, cyclotomic polynomials and their properties, Theorem of the inverse Galois theory for abelian groups.

Finite fields: Existence and uniqueness of finite fields, Galois group of a finite field, subfields of a finite field, enumeration of irreducible polynomials on finite fields. Construction of the algebraic closure of a finite field with $ p $ elements.

Constructions with ruler and compass: Definition of constructible points of the plane, constructible real numbers, characterization of constructible points in terms of fields, constructible subfields and construction of constructible numbers, cube duplication, trisection of angles, quadrature of the circle and Gauss's theorem for the construction of regular polygons with ruler and compass.

J. S. Milne.Fields and Galois Theory. Course Notes v4.22 (March 30, 2011).
S. Gabelli. Teoria delle Equazioni e Teoria di Galois. Springer UNITEXT (La Matematica per il 3+2) 2008, XVII, 410 pagg., ISBN: 978-88-470-0618-8
E. Artin.Galois Theory. NOTRE DAME MATHEMATICAL LECTURES Number 2. 1942.
C. Procesi.Elementi di Teoria di Galois. Decibel, Zanichelli, (Seconda ristampa, 1991).

1. Abstract integration theory
Riemann integration theory. The concept of measurability. Step functions. Elementary properties of measures. Arithmetic in [0,∞]. Integration of positive functions. Integration of complex functions. Importance of sets
with null measure.
2. Positive Borel measures
Vector spaces. Topological preliminaries. Riesz representation theorem. Regularity properties of Borel measures. Lebesgue measure. Continuity properties of measurable functions.
3. L^p spaces
Inequalities and convex functions. L^p spaces. Approximation through continuous functions.
4. Basic theory of Hilbert spaces
Inner products and linear functionals. Dual space of L^2
4. Integration on product spaces
Measurability on cartesian products. Product measure. Fubini theorem.
4. Complex measures
Total variation. Absolute continuity. Radon-Nykodym theorem. Bounded linear functionals on L^p. The Riesz representation theorem.

"Analisi reale e complessa”, W. Rudin. Bollati Boringhieri.

1. Probability

Introductory example: The branching process.
Introduction to measure theory.
Measure spaces. Events. Uniqueness and extension of measure.
Probability measures. First Borel--Cantelli lemma.
Random variables, distribution function and law. Indipendence.
Second Borel--Cantelli lemma.
0--1 law for independent random variables.

2. Integration, expected value

Introduxction to integration theory. Monotone convergence theorem.
Expectation. Taking the limit under expectation.
Jensen's inequality. L_p norms.
H\"older inequality and Cauchy-Schwarz.
Markov's inequality.
Examples of weak and strong laws of large numbers.
Product measures. Fubini theorem.
Joint laws.

3. Conditional expectation, martingales and limit theorems

Conditional expectation with respect to a sub $\sigma$--algebra. Kolmogorov existence and uniqueness theorem.
Filtrations. Martingale. Gambilng. Stopping times.
Optional stopping.
Some applications to exit times from an interval.
Convergence theorem for martingales in L_1 and in L_2. Kolmogorov's strong law of large numbers.

4. Convergence ind distribution and the central limit theorem

Characteristic functions. Inversion theorem.
Equivalence between convergence in distributiona nd pointwise convergence of characteristic functions.
Various modes of convergence for random variables. Examples.

D. Williams.
Probability with martingales
Cambridge University Press, 1991


R. Durrett
Probability: Theory and Examples
Thomson, 2000

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 function: properties and examples of holomorphic and non-holomorphic functions. Cauchy-Riemann equations. The real and imaginary parts of holomorphic function 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 function: 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 place 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.
Demonstration 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.

L. V. Ahlfors: Complex Analysis, McGraw-Hill.
S. Lang: Complex analysis, GTM 103.
E. Freitag, R. Busam: Complex Analysis, Springer.

Introduction: Cardano equations for the solubility of third-degree equations, rings and fields, the characteristics of a field, known facts about rings of polynomials, field extensions, construction of some field extensions, the sub-ring generated by a subset, the subfield generated by a subset, algebraic and transcendent elements, algebraically closed fields.

Splitting fields: Simple extensions and maps between simple extensions, splitting fields, existence of the splitting field, uniqueness up to isomorphisms of the splitting field, multiple roots, formal derivatives, separable polynomials and perfect fields, minimal polynomials and their characterizations.

The fundamental Theorem of Galois Theory: Group of the automorphisms of a field, normal, separable and Galois extensions, characterizations of separable extensions, Fundamental Theorem of Galois Correspondence, examples, Galois group of a polynomial, Radical extensions, solvable groups and Galois's Theorem on solving equations , Theorem of the existence of primitive elements.

The computation of Galois group: Galois groups as subgroups of $ S_n $, transitive subgroups of $ S_n $, characterization of irreducibility in terms of transitivity, polynomials with Galois groups in $ A_n $, Theory of discriminants, Galois groups of polynomials with degree up to $4$, examples of polynomials with Galois group $S_p$.

Cyclotomic fields: Definitions, Galois group, maximal real subfields, quadratic subfields, Galois groups, cyclotomic polynomials and their properties, Theorem of the inverse Galois theory for abelian groups.

Finite fields: Existence and uniqueness of finite fields, Galois group of a finite field, subfields of a finite field, enumeration of irreducible polynomials on finite fields. Construction of the algebraic closure of a finite field with $ p $ elements.

Constructions with ruler and compass: Definition of constructible points of the plane, constructible real numbers, characterization of constructible points in terms of fields, constructible subfields and construction of constructible numbers, cube duplication, trisection of angles, quadrature of the circle and Gauss's theorem for the construction of regular polygons with ruler and compass.

J. S. Milne.Fields and Galois Theory. Course Notes v4.22 (March 30, 2011).
S. Gabelli. Teoria delle Equazioni e Teoria di Galois. Springer UNITEXT (La Matematica per il 3+2) 2008, XVII, 410 pagg., ISBN: 978-88-470-0618-8
E. Artin.Galois Theory. NOTRE DAME MATHEMATICAL LECTURES Number 2. 1942.
C. Procesi.Elementi di Teoria di Galois. Decibel, Zanichelli, (Seconda ristampa, 1991).

1. Abstract integration theory
Riemann integration theory. The concept of measurability. Step functions. Elementary properties of measures. Arithmetic in [0,∞]. Integration of positive functions. Integration of complex functions. Importance of sets
with null measure.
2. Positive Borel measures
Vector spaces. Topological preliminaries. Riesz representation theorem. Regularity properties of Borel measures. Lebesgue measure. Continuity properties of measurable functions.
3. L^p spaces
Inequalities and convex functions. L^p spaces. Approximation through continuous functions.
4. Basic theory of Hilbert spaces
Inner products and linear functionals. Dual space of L^2
4. Integration on product spaces
Measurability on cartesian products. Product measure. Fubini theorem.
4. Complex measures
Total variation. Absolute continuity. Radon-Nykodym theorem. Bounded linear functionals on L^p. The Riesz representation theorem.

"Analisi reale e complessa”, W. Rudin. Bollati Boringhieri.

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.

Roberto Ferretti, "Appunti del corso di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Slides of the lessons, available from the course page: http://www.mat.uniroma3.it/users/ferretti/bacheca.html

LAGRANGIAN AND HAMILTONIAN MECHANICS. VARIATIONAL PRINCIPLES. CONSTRAINTS. CYCLICI VARIABLES. CONSTANTS OF MOTION AND SYMMETRIES. HAMILTONIAN FLOWS. LIOUVILLE THEOREM AND POINCARE RECURRENCE THEOREM. CANONICAL TRANSFORMATIONS. GENERATING FUNCTIONS. HAMILTON-JACOBI METHOD AND ACTION-ANGLE VARIABLES. PERTURBATION THEORY. KAM THEOREM.

[1] G. GENTILE,INTRODUZIONE AI SISTEMI DINAMICI. 1.EQUAZIONI DIFFERENZIALI ORDINARIE, ANALISI QUALITATIVA E ALCUNE APPLICAZIONI.
AVAILABLE ON-LINE: http://www.mat.uniroma3.it/users/gentile/2016-2017/FM410/testo.html
[2] G. GENTILE,INTRODUZIONE AI SISTEMI DINAMICI. 2.FORMALISMO LAGRANGIANO E HAMILTONIANO.
AVAILABLE ON-LINE: http://www.mat.uniroma3.it/users/gentile/2016-2017/FM410/testo.html
[3] G. DELL'ANTONIO, ELEMENTI DI MECCANICA. LIGUORI EDITORE, (1996)
[4] V.I. ARNOLD, METODI MATEMATICI DELLA MECCANICA CLASSICA. EDITORI RIUNITI, (1979)
[5] G. GALLAVOTTI, MECCANICA ELEMENTARE. BOLLATI-BORINGHIERI, (1980)

Affine Spaces
Zariski topology. Affine closed subsets and radical ideals. Theorem of the zeros of Hilbert. Correspondence between closed subsets and radical ideals. Noetherian topological spaces. Irreducible closed subsets, irreducible components. Regular functions of affine closed subsets. Regular maps, isomorphisms. Principal open subsets. Examples. Projections are open. Finite morphisms.

Varieties
Projective spaces and their Zariski topology. Quasi-projective varieties. Rational and regular maps. Projective hypersurfaces. Birational equivalence. Principal open subsets and affine closed subsets. Affine varieties. Dimension of quasi-projective varieties. Finite and generically finite morphisms. Characterizations of birational equivalence. Characterization of generically finite morphisms. Costructible sets, Chevalley's theorem. Every variety is birational to a hypersurface.

Local geometry
Local ring of a variety in a point. Zariski cotangent space. Zariski tangent space. Singular and non singular points.

L. Caporaso
Introduction to algebraic geometry
Notes of the course

I. Shafarevich
Basic Algebraic geometry
Springer-Verlag, Berlin, 1994

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

J.J. SAKURAI, J. NAPOLITANO. MECCANICA QUANTISTICA MODERNA. SECONDA EDIZIONE, ZANICHELLI, BOLOGNA, 2014

An english version of the book is also available:
J.J. SAKURAI, J. NAPOLITANO. MODERN QUANTUM MECHANICS. SECOND EDITION, PEARSON EDUCATION LIMITED 2014

1) Introduction to information theory

- Error correction codes. Symmetrical binary channel.
Block codes. Hamming codes. $ (7.4) $ - Hamming.
The generating matrix of the code in the linear case.
Decoding in the case of the $ (7.4) $ - Hamming code. Syndromes.
Decoding by syndromes. The effectiveness of a given code.
The channel capacity. Symmetrical codes. Graphic representation
associated with a code.

- Probability. Probability spaces. Discrete probability.
Probability in retrospect. Maximal likelihood principle. Definition of entropy. Information content according to Shannon. Redundancy. Joint Entropy. Decomposition rule for the entropy computation. Gibbs inequality. Jensen inequality.

- Inference.

([1] chapters 1, 2, 3)

2) Data Compression

- Source coding theorem. The measure of information content
of a random variable. Raw information content. Information content
and compression with the loss. Essential information content. Shannon's theorem. Sets of typicality. Asymptotic equipartition principle.

- Symbolic Codes. Encoding without loss of information. Prefix codes. Univocal decoding. Kraft inequality. Optimal codes. Huffman codes.

- Flow Codes. Arithmetic codes. Bayesian model. Huffman codes with a header. Arithmetic codes with Laplace predictive modelling of the distribution. Arithmetic codes with Dirichlet predictive model. Lempel-Ziv coding

([1] chap 4,5,6)

3) Channel coding in the presence of noise

- dependent random variables. Joint Entropy.
Conditional Entropy. Mutual information. Conditioned mutual information.

- Communication on channels in the presence of noise. Discrete channel
without memory (DMC). Examples: symmetrical binary channel (BSC), channel
binary with cancellation (BEC), the teletype with noise (NT),
the zeta-channel (Z). Information carried by a channel.

- The coding theorem of the source in the case with noise.
Optimal decoding. Probability of error on the block and on average on the single bit. Typical sequences and sets of joint typicality. Decoding by means of typicality sets. Information rate in the case of use of a channel beyond
capacity.

- Error correction codes and applications.

([1] chap 8,9,10,11)

4) Further codes

- Hash codes.
- Linear Codes.

([1] chapt 12,14)

[1] David J. C. MacKay, Information Theory, Inference and Learning Algorithms. (2004) Cambridge University Press;
[2] R. Blahut, Principles and Practice of Information Theory, (1987) Addison Wesley.
[3] Thomas and Cover, Elements of Information Theory, (1991) Wiley;

Time structure of the exchange of amounts, capital and interest: exchange of amounts, time, price, price of time, convention for measuring time, deferred contracts and rights, transactions with a fixed schedule, regular investment / indebtedness transactions, laws financial, the law of simple interests, the law of compound interests, fundamental definitions based on the value function, factors rates and intensity, instantaneous intensity.
Contracts, exchange, prices: prices on the primary and secondary market, some types are bond contracts, zero coupon securities (ZCB), fixed coupon bonds (CB), accruals, and tel. Risks: time, uncertainty, risk, credit risk for mortgages and bonds,
Evaluation in conditions of certainty, the exponential law: the exponential function as a law of financial equivalence, rates and equivalent intensity in exponential and linear law, evaluation of a financial transaction based on exponential law, equity, functional properties of exponential law, breakdown of financial transactions;
annuities and amortization schedules: definitions, present value of annuities in constant installments, deferred immediate annuity of duration m, deferred perpetual annuity, immediate anticipated annuity of duration m, anticipated perpetual annuity, deferred annuities of n years, deferred (deferred) annuity, perpetual deferred payment in advance. Earnings in appearance, deferred income in constant installments, early repayments at constant installments, deferred annuities with constant capital, the repayment schedule, single repayment amortization;
internal rate of return: the case of periodic payments, the Descartes theorem, the case of a ZCB, the case of an investment transaction, the case of a financial transaction consisting of three amounts, the case of a CB quoted on par, the Newton method, the APR (Gross annual percentage rate);
theory of financial equivalence laws: the value function in a spot contract, the value function in a forward contract, the ownership of uniformity over time, discount and capitalization factors, hypothesis of consistency between spot contracts and forward contracts , the property of divisibility, interest rates and intensity, equivalent rates, the instantaneous intensity of interest, integral form of the discount factor, uniform laws, divisible laws, Cantelli's theorem, yield maturity intensity (yield to maturity) , linear and hyperbolic capitalization, linearity of present value.
Financial transactions in the market: value function and market prices: the characteristic assumptions of the market, the perfect market, the principle of non-arbitrage, the absence of arbitrage, the single price law, zero coupon coupons, decryption theorem with respect to maturity, coupon bonds nothing not unitary, amount independence theorem, ZCB portfolios with different maturity, price linearity theorem, forward contracts, implicit price theorems, implicit rates, considerations on tax effects, case of Treasury Obligation Bonds (BOT);
interest rate maturity structure: spot maturity structures, implicit maturity structures, dominance relationship between implied interest rate structure and spot rate structure, discrete time tables, discrete time schedules with a continuous pattern, parity, risky structures and credit spreads; time index and variability indices: payment flow timescales, maturity and maturity life, duration, the case of constant-rate annuities, second-order moments, duration and dispersion of portfolios, indices of variability of a payment flow , analysis of price sensitivity, semi-elasticity, elasticity, convexity, "thumb rule";
measurement of the interest rate maturity structure: methods based on the internal rate of return, methods based on linear algebra, methods based on the parity rate, swap rate as parity rate, methods based on the estimation of a model, Masera model , Nelson -Siegel-Svensson model;
arbitrage valuation of variable rate plans: random financial transactions, variable rate coupons, effects of perfect indexing of interest shares, reinvestment security, valuation of the stochastic ZCB, logic of the replicating portfolio, valuation of the individual indexed coupon, valuation of the flow of indexed coupons, valuation of the coupon at a variable rate at issue and in place, equivalence with a roll-over strategy;
interest rate sensitive contracts (outlines): the valuation of contracts dependent on nominal interest rates, recalls on the theory of the structure by maturity in certainty conditions, models of the structure by maturity in certainty conditions, examples of IRS contracts, stochastic models for contracts IRS, a class of uni models that varied over time, the basic assumptions, the dynamics of IRS contracts, the hedging argument, risk measures, the Vasicek model.

Castellani, G., De Felice, M., Moriconi, F. Manuale di finanza. Tassi d’interesse. Mutui e obbligazioni. Il Mulino, 2005
Allevi, E., Bosi, G., Riccardi, R., Zuanon, M., Matematica finanziaria e attuariale, Pearson, 2017
Luenberger, D., G., Introduzione alla matematica finanziaria, Apogeo, 2015
Cesari, R., Introduzione alla Finanza Matematica. Mercati azionari, rischi e portafogli, McGraw-Hill, 2012
Castellani, G., De Felice, M., Moriconi, F. Manuale di finanza. Tassi d’interesse. Mutui e obbligazioni. Il Mulino, 2005
Castellani, G., De Felice, M., Moriconi, F. Modelli stocastici e contratti derivati. Il Mulino, 2005
Naccarato, A., Pierini, A., “BEKK element-by-element estimation of a volatility matrix, A portfolio simulation”, in Mathematical and Statistical Methods for Actuarial Sciences and Finance, (editors Perna, C., Sibillo, M.), Springer, 2014
Lecture notes (delivered to class)

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.

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.

Outline of the course; Introduction and generality; Bioinformatics and algorithms; Computational biology in the clinic and in the pharmaceutical industry; Pharmacokinetics and pharmacodynamics;

Introduction to Systems Biology: what is computational biology; The roles of mathematical modeling and bioinformatics; what is he aiming for; what are the problems; Theoretical tools used in bio-mathematics and bioinformatics.

Introduction to molecular and cellular biology (first part): basic knowledge of genetics, proteomics and cellular processes; Ecology and evolution; the basic molecule; molecular bonds; the chromosomes; DNA and its replication;

Introduction to molecular and cellular biology (second part); genomics; The central dogma of biology; The genome project; the structure of the human genome Analysis of genes; transcription of DNA; the viruses;

Laboratory: generation of random numbers; the functions srand48 and drand48; random generation of arbitrary length nucleotide strings (program1.c); random generation of amino acid strings of arbitrary length (program2.c);

Introduction to information theory; Shannon Entropy; Conditional Entropy; Mutual Information; Indices of biological diversity; Shannon Index; True diversity; Reny index;

Laboratory: the genetic code; C program of transcription DNA sequence and translation into proteins;

Introduction to stochastic processes; basic definition; examples; model of queues; Bernoulli and Poisson process; Markov processes; stochastic processes in bioinformatics and bio-mathematics; the autocorrelation; Outline of the Random Walks and the BLAST algorithm of sequence alignment as a stochastic process and principal algorithm for the consultation of biological sequence databases;

Laboratory: development of an algorithm in C for the calculation of the Shannon Entropy of a text in English (or in Italian) any (e.g., http://www.textfiles.com/etext/)

Random walks. The BLAST algorithm for aligning sequences as a random path; Laboratory: C implementation of different algorithms for the generation of a random walk in 1D and 2D on the lattice and in R or R ^ 2 signal and calculation of the mean square displacement;

Compare sequences: similarity and homology; pairwise alignment; editing distance; scoring matrices PAM and BLOSUM; Needleman-Wunsch's algorithm; local alignment; Smith-Waterman's algorithm; BLAST algorithm;

Laboratory: C implementation of an algorithm for the generation of a signal with noise and calculation of the correlogram in the presence or absence of a true signal;

Multiple Sequence Alignment; consensus sequence; star alignment algorithms; ClustalW; entropy and circular sum scoring functions;

Biological data banks; reasons; data format; taxonomy; Primary DBs; Secondary DBs; NCBI, EMBL, DDBJ; NCBI EBI-Entrez; Exact matching / string searching: general; the agony of Knuth-Morris-Pratt;

Exact matching / string searching: the Boyer-Moore agoritm;

Exercise on an implementation of the Knuth-Morris-Pratt exact matching algorithm. Exercise on biological databases; primary databases; secondary databases; NCBI, EMBL, DDBJ; NCBI EBI-Entrez; Use of the BLAST algorithm

Phylogenetic Analysis; phylogenetic trees; dimension of the research space of phylogenetic algorithms; Methods of construction of phylogenetic trees; Data used for phylogenetic analysis; The Unweighted Pair Method Method with Arithmetic mean (UPGMA) algorithm; the Neighbor Joining Method algorithm; Hidden Markov Models; decoding; the Viterbi Algorithm; Evaluation;

Laboratory: completion of the exercise on mutation, selection and evolution of nucleotide strings (genotype) translated into amino acid strings (phenotype); Selection is made based on the presence of certain substrings in the phenotype that determines the fitness value; Implemented details, display of the convergence criterion and results, discussion, etc .;

Machine Learning; generality'; supervised and unsupervised learning; model selection; undefitting; overfitting; Polynomial curve fitting; machine learning as an estimate of the parameters and the problem of overfitting; subdivision of the training set into testing and testing; concept of bias and variance trade-off; Artificial Neural Networks; definizone; the percussion of Rosenblatt; the percettrone learning algorithm; the multi-layer perceptron;

Laboratory: completion of the implementation in ANSI C of the evolutionary algorithm of nucleotide strings (genotype) translated, through the use of the genetic code, into amino acid strings (phenotype);

Hidden Markov Models; The Forward Algorithm; The Backward Algorithm; Posterior Decoding; Learning; Baum-Welch Algorithm; Use of Hidden Markov Models for the analysis of bio-sequences; gene finding;

Artificial Neural Networks; the error-back propagation algorithm for learning MLP; types of neural networks; convolution networks; reinforcement networks; unsupervised learning and self-organizing maps; Introduction to graph theory; representation, terminology, concepts; paths; cycles; connettivita '; distance; connected components; distance;

Introduction to graph theory; visit breadth-first search; depth-first search; Dijkstra's algorithm; six-degree of separation; small world networks; centrality measures; degree centrality; eigenvector centrality; betweennes centrality; closeness centrality; The network biology; generality'; concepts; types of biological data used to build networks; network biology and network medicine; problems and algorithms used; centrality measures; random networks; scale-free networks; preferential attachment; scale-free network in biology;

Laboratory: completion of the exercise on the evolutionary algorithm; Implemented details, display of the convergence criterion and results, discussion, etc .;

Bio-mathematical models; prediction using theoretical models; the itertative paradigm of mathematical modeling; data-driven models; limited and non-population growth models; analytical derivation and examples; logistics growth; ecological models limited by density; The Lotka-Volterra model; the experiment by Huffaker and Kenneth; the SIR epidemic model and some of its variants; Perelson's model for HAART; the Java Populus application for the solution of continuous models of population dynamics; hints to the numerical resolution methods of differential equation systems;

Discrete models; spin models (Ising models); Cellular automata; Boolean networks; Agent-based models; data fitting and parameter estimation; software tools available; Cellular automata; introduction and history; definition; the 1-dimensional automaton; Wolfram classification; the 2-dimensional automaton; Conway's Game of Life; Software available for CA simulation; dedicated hardware (CA-Machine); the prey-predator model as a two-dimensional cellular automaton; relationship with the system of ordinary derivation equations; stochastic models; Stochastic CAs as discrete stochastic dynamic systems and stochastic processes; example of CA: Belousov-Zabotonsky reactions;

[-] E.S. Allman, J.A. Rhodes. Mathematical Models in Biology: An Introduction (2004) Cambridge University Press.
[-] W.J. Ewens, G.R. Grant. Statistical Methods in Bioinformatics, An Introduction (2005) Springer Verlag.
[-] R. Durbin, S. Eddy, A. Krogh, G. Mitchison. Biological sequence analysis - Probabilistic models of proteins and nucleic acids (1998) Cambridge University Press.

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 function: properties and examples of holomorphic and non-holomorphic functions. Cauchy-Riemann equations. The real and imaginary parts of holomorphic function 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 function: 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 place 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.
Demonstration 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.

L. V. Ahlfors: Complex Analysis, McGraw-Hill.
S. Lang: Complex analysis, GTM 103.
E. Freitag, R. Busam: Complex Analysis, Springer.

1. Probability

Introductory example: The branching process.
Introduction to measure theory.
Measure spaces. Events. Uniqueness and extension of measure.
Probability measures. First Borel--Cantelli lemma.
Random variables, distribution function and law. Indipendence.
Second Borel--Cantelli lemma.
0--1 law for independent random variables.

2. Integration, expected value

Introduxction to integration theory. Monotone convergence theorem.
Expectation. Taking the limit under expectation.
Jensen's inequality. L_p norms.
H\"older inequality and Cauchy-Schwarz.
Markov's inequality.
Examples of weak and strong laws of large numbers.
Product measures. Fubini theorem.
Joint laws.

3. Conditional expectation, martingales and limit theorems

Conditional expectation with respect to a sub $\sigma$--algebra. Kolmogorov existence and uniqueness theorem.
Filtrations. Martingale. Gambilng. Stopping times.
Optional stopping.
Some applications to exit times from an interval.
Convergence theorem for martingales in L_1 and in L_2. Kolmogorov's strong law of large numbers.

4. Convergence ind distribution and the central limit theorem

Characteristic functions. Inversion theorem.
Equivalence between convergence in distributiona nd pointwise convergence of characteristic functions.
Various modes of convergence for random variables. Examples.

D. Williams.
Probability with martingales
Cambridge University Press, 1991


R. Durrett
Probability: Theory and Examples
Thomson, 2000

Introduction to cryptography. Classic ciphers. Introduction to cryptanalysis.
Introduction to public-key cryptography.
The RSA cryptosystem.
Primality tests.
Factorization algorithms.
Some attacks on the RSA.
The discrete logarithm problem. Diffie-Hellman key exchange.
Elgamal cryptosystem.
Massey-Omura cryptosystem.
Digital signatures.
Overview of some cryptographic protocols.

Baldoni, Ciliberto, Piacentini: Aritmetica, crittografia e codici
D. Stinson: Cryptography - theory and practice

CONGRUENCES AND POLYNOMIALS. LINEAR DIOPHANTINE EQUATIONS. SYSTEMS OF LINEAR CONGRUENCES. POLYNOMIAL CONGRUENCES. POLYNOMIAL CONGRUENCES MOD p: LAGRANGES’S THEOREM. p-ADIC APPROXIMATION. THE EXISTENCE OF PRIMITIVE ROOTS MOD p. QUADRATIC CONGRUENCES. QUADRATIC RESIDUES. THE LEGENDRE SYMBOL. GAUSS’S LEMMA AND THE LAW OF QUADRATIC RECIPROCITY.THE JACOBI SYMBOL. SUMS OF TWO, THREE, FOUR SQUARES. MULTIPLICATIVE FUNCTIONS. THE MÖBIUS INVERSION FORMULA. SOME NONLINEAR DIOPHANTINE EQUATIONS. CONTINUED FRACTIONS.

M. FONTANA, APPUNTIDEL CORSO TN1 (ARGOMENTI DELLA TEORIA CLASSICA DEI NUMERI), HTTP://WWW.MAT.UNIROMA3.IT/USERS/FONTANA/DIDATTICA/FONTANA_DIDATTICA.HTML DISPENSE.
D.M. BURTON: ELEMENTARY NUMBER THEORY, MCGRAW-HILLINTERNATIONAL EDITION, 6TH EDITION (2007), 434 PP.
G.A. JONES AND J.M. JONES, ELEMENTARY NUMBER THEORY, SPRINGER, 1ST EDITION (1998), 200 PP.
H. DAVENPORT, ARITMETICA SUPERIORE.UN?INTRODUZIONE ALLA TEORIA DEI NUMERI, ZANICHELLI, (1994), 199 PP.
G.H. HARDY AND E.M. WRIGHT, AN INTRODUCTION TO THE THEORY OH NUMBERS, THE CLARENDON PRESS, OXFORD UNIVERSITY, 5TH EDITION (1979), XVI+426 PP.
W.J. LEVEQUE, FUNDAMENTALS OF NUMBER THEORY, DOVER PUBLICATIONS, (1996)
K.H. ROSEN. ELEMENTARY NUMBER THEORY AND ITS APPLICATIONS, ADDISON-WESLEY, 6TH EDITION (2011), XV+752 PP.

Principles of Object Oriented Design
Abstraction, Polimorphism, Inheritance, Aggregation
Object Oriented Programming Models and UML
UML Use Case, Sequence, Class e Object, Deployment diagrams
Software Analysis and Developmenmt for Java Virtual Machine: I/O, Stream, Networking
(Scientific, Real-time,...) Efficient Distributed Computing, Multithreading and Concurrency in Cloud e Mobile

Manuale di Java 9 De Sio Cesari Claudio Hoepli Informatica

Under the general title of this Course, we consider both, asymptotic and
numerical methods mostly to handle ordinary differential equations and systems,
and possibly some partial differential equations, and applications. Among the
asymptotic methods, the so-called ``singular perturbations'' are addressed.
These two subjects, which represent basic techniques of Applied Mathematics,
are usually taught separately. In this course we will use both, at the same
time, in the same problems.

Asymptotic Methods:

Introduction to Asymptotic Analysis. Asymptotic series. Regular and singular
perturbations of ordinary differential equations and systems.

Numerical Methods:

Numerical solution, using several algorithms, of ordinary differential
equations and systems. This will be mostly done in the computing laboratory.

Robert E. O'malley,
Singular Perturbation Methods for Ordinary Differential Equations,
Springer-Verlag, Applied Mathematical Sciences, 89, New York, 1991.

John K. Hunter,
Asymptotic Analysis and Singular Perturbation Theory,
Lecture Notes, https://www.math.ucdavis.edu/~hunter/notes/asy.pdf,
2004

Teacher's Lectures notes available on web.

Introduction to cryptography. Classic ciphers. Introduction to cryptanalysis.
Introduction to public-key cryptography.
The RSA cryptosystem.
Primality tests.
Factorization algorithms.
Some attacks on the RSA.
The discrete logarithm problem. Diffie-Hellman key exchange.
Elgamal cryptosystem.
Massey-Omura cryptosystem.
Digital signatures.
Overview of some cryptographic protocols.

Baldoni, Ciliberto, Piacentini: Aritmetica, crittografia e codici
D. Stinson: Cryptography - theory and practice

Introduction to Python and/or Julia programming

1. Python or Julia? Getting started
2. Language syntax by examples

Polyhedral geometry

3. Linear and affine spaces
4. Convex sets, affine and convex coordinates
5. Simplicial, cuboidal and cellular complexes

Basic computer graphics

6. Affine transformations
7. hierarchical structures and scene graphs

Introduction to Geometric Computing

8. Parametric representation
9. Curves, surfaces, solids
10. Rational and polynomial maps
11. tensor product patches
12. Solid modeling. Motion modeling

A. Paoluzzi, Geometric Programming for Computer-Aided Design, Wiley, 2003. (free download from uniroma3.it domain)
https://github.com/cvdlab-courses/ggpl
https://github.com/plasm-language/pyplasm
https://github.com/cvdlab/lar-cc