Università degli Studi di Roma "Tor Vergata"

The Lebesgue measure and the Lebesgue integral; Hilbert spaces; Lp spaces.

To be decided with the students:
possible books are Royden "Real Analysis", Brezis "Functional Analysis", Bressan "Lecture notes on Functional
Analysis"

Synthetic program
Part I. [1]
Introduction to finite differences and finite elements.
Methods for partial differential equations in zero dimensions: the BVP of ordinary differential equations.
Finite difference methods for elliptic equations

Part II [1]
Initial value problems for partial differential equations in zero spatial dimensions, the IVPs of ODEs
Zero-Stability and convergence for initial value problems
A-and L- stability and methods for stiff problems

Part III [1]
Classification of linear partial differential equations of second order elliptic, parabolic, hyperbolic
Derivation of the PDE conservation laws and transport, diffusion, reaction-diffusion, transport-diffusion, transport-reaction-diffusion
Fourier analysis of linear PDEs
Diffusion equation
Transport equation and outline methods for hyperbolic systems
Methods of high order
Nonlinear conservation laws

Part IV. [2]
Solution linear systems large and sparse generated from time to time by discrete and semidiscrete models. Notes on the efficient solution of some linear structured systems.

Part V. [*]
Finite element methods and weak formulation. Application to linear elliptic and parabolic 1D, nods to the 2D case.
Part VI. [1]
Application to model linear and nonlinear problems.

[1] R. J. LeVeque -- Finite Difference Methods for ODEs and PDEs, Steady State and Time Dependent Problems. SIAM, Philadelphia, 2007
[2] Bertaccini, Di Fiore, Zellini, Complessita' e iterazione, Boringhieri, 2013

Preliminaries: measure spaces, measurable functions, Lebesgue integral, classical inequalities, L^pspaces of functions.
Probability spaces, independent events, conditioning on events
Random variables, distribution and distribution function of a random variable, density of an absolute continuous random variable, main probability distributions and their properties, applications to modeling.
Expectation and variance of a random variables, classical inequalities, L^pspaces of random variables, characteristic functions, covariance of random variables, the Capital Asset Pricing Model.
Random vectors, joint and marginal distribution, distribution function and density of random vectors, multivariate Gaussian distributions.
Modes of convergence of sequence of random variables: almost sure, in probability, in ‎L^pspaces, in distribution, laws of large numbers, Borel-Cantelli’s lemma, the Central Limit theorem, applications to confidence intervals and the χ^2 test.
Independent random variables, information generated by a collection of random variables, conditioning random variables on available information, existence, properties and computation of conditional expectation, conditional expectation in L^2.
Stochastic processes, stopping times, discrete martingales and their main properties, arrested martingales, maximal inequalities convergence of martingales.
Cox-Ross-Rubinstein market model, Fundamental Theorem of Asset Pricing for a discrete market model.

D. Williams: Probability with martingales. Cambridge University Press, 1991.
P. Billingsley: Probability and measure. 2nd edition. Wiley series in Probability and Mathematical Statistics, 1986.
Paolo Baldi: Stochastic Differential Equations, to appear, (pre-print available upon request from the instructor)
Instructor’s notes will be provided for many topics of the course.

1) Elements of Classical Mechanics: Hamilton’s principle; Lagrangian for a free particle and for a system of non-interacting and interacting particles; Conservation laws (Energy, Momentum, Angular Momentum); Integration of the equations of motion; One-dimensional motions; two-body problem; motion in a central field; free, forced, and forced and damped oscillators; Legendre transformations, Hamiltonian, and Hamilton's equations; Poisson brackets.



2) Quantum Mechanics:

2.1 Basic concepts of quantum mechanics: uncertainty principle; the configurations space and the wave function; the superposition principle; operators, eigenfunctions and eigenvalues; commutator of two operators; continuous spectrum: coordinates operator; the concept of measurement in quantum mechanics.

2.2: Classical limit of the wave function; wave equation and Hamiltonian operator; conservative physical quantities; energy and wave function of the stationary states, finite and infinite motion.

2.3: Matrices: matrix elements of an operator, Hermitian matrices, the product of matrices;

2.4: Momentum operator: translation operator; conservation of momentum; eigenvalues and eigenfunctions.

2.5: Eigenvalue equations for a physical quantity in the representation of energy (stationary states); matrices in diagonal form, complete set of common eigenfunctions; Transformations of matrices; uncertainty relations.

2.6: Hamiltonian for a system of free particles, of interacting particles, and of particles in an external field

2.6: Schrödinger equation: basic properties; Schrödinger equation for a free particle and eigenfunctions; classical limit of the Schrödinger equation; current density operator and continuity equation for the probability density.

2.7: One-dimensional motions: Schrödinger equation and general principles; potential well with infinite walls; potential step; coefficients of transmission and reflection; potential well with finite walls; potential barrier and tunnel effect; Dirac notation: definition and properties; position observable and wave functions; measurement process; Harmonic oscillator;

2.8: Angular momentum: rotation operation; conservation of angular momentum; commutation relations; angular momentum in polar coordinates; eigenvalues and eigenfunctions of the angular momentum along z; eigenvalues and eigenfuncions of the square modulus of the angular momentum; spherical harmonics; matrix elements of the components of the angular momentum.

2.7: Motion in a central field: the two-body problem: the reduced mass; Schrödinger equation for the radial part of the wave function and effective potential; motion in Coulomb field: hydrogen atom.
2.8: Spin: basic concepts; commutation relations; eigenvalues of Sz and S2; integer spin or half- integer; spinors; matrix elements of the components of the spin: Pauli matrices for the spin operator in two dimensions; total angular momentum (spin and orbital angular momentum); composition of angular momenta.

2.9: Identical particles: indistinguishable particles in quantum mechanics; two identical particles of spin 1/2; symmetric and antisymmetric states (exchange operator); Pauli principle; fermions and bosons; state of N fermions and N bosons; the wave function for fermions and bosons.

Part 1): L. D. Landau & E. M. Lifshitz, “Mechanics”; H. Goldstein: “Classical Mechanics”
Part 2): L. D. Landau & E. M. Lifshitz, “Quantum Mechanics: non relativistic theory”; S. Gasiorowicz, “Quantum Physics”; J. J. Sakurai, “Modern Quantum Mechanics”

1. The main principles of Statistical Physics
Statistical distribution, Liuville theorem, statistical matrix and statistical distributions in quantum mechanics.

2. Gibbs distribution
Ideal gas: Boltzmann and Maxwell distributions. Quantum gases: Fermi-Dirac and Bose-Einstein distributions. Fermi surface of the ideal Fermi gas. Heat capacity of the Fermi gas. Phenomenon of the Bose-condensation. Black radiation.

3. Solids
Lattice heat capacity. Thermal expansion. Classical treatment of the lattice vibrations. Phonons.

4. Phase transitions of the II order
Landau theory of phase transitions. Order parameter fluctuations.

5. Superconductivity
General description and historical review. Ginzburg-Landau phenomenology. Flux quantization. Superconductivity of the second type. Evaluation of the lower and upper critical fields (8 hours).

L. D. Landau & E. M. Lifshitz, “Statistical Physics”

Free and constrained systems, constraints and their classification, possible and virtual displacements, ideal constraints, D'Alembert's Principle.

- Truss structures, beams and frames, the force method, the deformation method, stability of beams and frames.

- The geometry of deformation, transmission of force, elastic constitutive theory, boundary-value problems in Elasticity, yield conditions.

- Extension, bending, torsion, and shear of prismatic bars; thin-walled beams.

- Energetic principles, the Ritz method, the Galerkin method, the Finite Element Method.

S. Krenk and J. Høgsberg: Statics and Mechanics of Structures, Springer 2013

K. D. Hjelmstad: Fundamentals of Structural Mechanics, Springer 2005

A. Jhennane: Introduction to Finite Element Analysis Using MATLAB and Abaqus, CRC Press 2013


Other reference books:

F. Gantmacher: Lectures in Analytical Mechanics, MIR Publishers 1975

T.J. Oden and E.A. Ripperger: Mechanics of Elastic Structures – Second Edition, McGraw-Hill 1981

M.S. Gockenbach: Understanding and Implementing the Finite Element Method, SIAM 2006

Continuum Mechanics


Main Topic: Principles and
methods of continuum thermodynamics (global, partwise and pointwise
versions of energy balance and entropy imbalance; energy and entropy
inflows; free-energy growth inequality and its constitutive consequences;
heat propagation, standard and not; heat waves; thermomechanics in
virtual working format; equilibrium notions in continuum and statistical
mechanics; energy- and entropy-driven processes)
If time allows: Transport
structures in continuum mechanics, theory (diffusion,
convection/advection, reaction/diffusion, gradient flows,
creation/annihilation mechanisms) and applications (Fourier, Fick,
Darcy, defect dynamics induced by pulsed-laser illumination)

Notes from the teacher.

Program
Brownian motion, definitions and general facts; main properties, asymptotics, invariance, stopping rules.

Continuous time martingales (discrete time martingales and conditional expectations are assumed to be already known and will be briefly recalled)

Markov processes, Diffusion processes.

Stochastic integral: definitions, main properties. Ito formula and applications.

Differenziali Stochastic differential equations: existence and uniqueness, Markov property, diffusions, stochastic representation of the solutions of partial differential problems

Notes and e lists of exercises (in english) will be distributed.