At present UAQ counts over 20,000 students, around 650 teachers and researchers, and nearly 500 administrative and technical staff members. Officially established in 1952 (but its origins date back to the 16th century), UAQ has now 7 departments offering a wide range of Bachelor, Master and PhD programmes in biotechnologies, sciences, economics, engineering, education, humanities, medicine, psychology, and sport sciences. Internationalisation has played an increasingly important role at UAQ. The Engineering and the Sciences Faculties have a strong tradition of research in the area of Mathematical Modelling. The Dept of Pure and Applied Math has rich experience in managing International projects (starting in 1996 as coordinator of the FP4 "HCL" TMR , FMRX-CT96-0033). UAQ provides many services for its students, including Career Office, International Relations Office, Quality Assessment Office (of teaching, research and services), Centre for Students with Disabilities, Language Centre, Student Counseling Centre.
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Bruno Rubino
Department of Information Engineering, Computer Science and Mathematics
University of L'Aquila
via Vetoio (Coppito), 1 – 67100 L'Aquila (Italy)
Phone: +39 0862 434701
Fax: +39 0862 433180
* Students are required to earn 66 ECTS credits, at least, during their first year by successfully attending the following compulsory course units (Semester 1 and 2 amounting to 48 ECTS credits) and picking other 18 ECTS credits (minimum) from the elective ones listed below.
Students will know basic of properties (existence, uniqueness, etc.) and techniques (characteristics, separation of variables, Fourier methods, Green's functions, similarity solutions, etc.) to solve basic PDEs (conservation laws, heat, Laplace, wave equations).
Integral curves and surfaces of vector fields. First order partial differential equations. Linear and quasi linear partial differential equations (PDEs) of first order. Method of characteristics. The initial value problem: existence and uniqueness. Development of shocks.
The Cauchy-Kovalevsky theorem. Linear partial differential operators and their characteristic curves and surfaces. Methods for finding characteristic curves and surfaces. The initial value problem for linear first order equations in two independent variables. Holmgren's uniqueness theorem. Canonical form of first order equations. Classification and canonical forms of second order equations in two independent variables. Second order equations in two or more independent variables. The principle of superposition.
The divergence theorem and the Green's identities. Equations of Mathematical Physics.
LAPLACE'S EQUATION AND HARMONIC FUNCTIONS Elementary harmonic functions. Separation of variables. Inversion with respect to circles and spheres. Boundary value problems associated with Laplace's equation. Representation theorem. Mean value property. Maximum principle. Harnack’s inequality and Liouville’s theorem. Well-posedness of the Dirichlet problem. Solution of the Dirichlet problem for the unit disc. Fourier series and Poisson's integral. Analytic functions of a complex variable and Laplace's equation in two dimensions. The Neumann problem.
GREEN'S FUNCTIONS. Solution to the Dirichlet problem for a ball in three dimensions. Further properties of harmonic functions. The Dirichlet problem in unbounded domains. Method of electrostatic images.
THE WAVE EQUATION. Cauchy problem. Energy method and uniqueness. Domain of dependence and range of influence. Conservation of energy. One-dimensional wave equation. D’Alembert formula. Characteristic parallelogram. Non homogeneous equation and Duhamel’s method. Multi-dimensional wave equation. Well posed problems. Fundamental solution (n=3) and strong Huygens’ principle. Kirchhoff formula. Method of descent. Poisson?s formula (n=2). Wave propagation in regions with boundaries. Uniqueness of solution of the initial-boundary value problem. Separation of variables. Reflection of waves.
THE HEAT EQUATION. Heat conduction in a finite rod. Maximum principle and applications. Solution of the initial-boundary value problem for the one dimensional heat equation. Method of separation of variables. The initial value problem for the one dimensional heat equation. Fundamental solution. Non homogeneous case and Duhamel’s method. Heat conduction in more than one space dimension.
E. C. Zachmanoglou and Dale W. Thoe, lntroduction to Partial Differential Equations with Applications. Dover Publications, Inc.. 1986. ISBN 0-486-65251-3
L.C. Evans, Partial Differential Equations. American Mathematical Society. 2010. Second edition, ISBN-13: 978-0821849743
S. Salsa, Partial Differential Equations in Actions: from Modelling to Theory. Springer-Verlag Italia. 2008. ISBN 978-88-470-0751-2
W. A. Strauss, Partial Differential Equations, Student Solutions Manual: An Introduction. John Wiley & Sons, LTD. 2008. Second edition, ISBN-13: 978-0470260715
W. A. Strauss, Partial Differential Equations: an introduction. John Wiley & Sons, LTD. 2007. Second edition, ISBN-13 978-0470-05456-7
The course provides the basic methodologies for modeling, analysis and controller design for continuous-time linear time-invariant systems.
Frequency domain models of Linear Systems: Laplace Transform, Transfer Function, Block diagrams.
Time domain models of Linear Systems:State space representation. BIBO stability.
Control specifications for transient and steady-state responses. Polynomial and sinusoidal disturbances rejection.
The Routh-Hurwitz Criterion. PID controllers.
Analysis and controller design using the root locus.
Analysis and controller design using the eigenvalues assignment: controllability, observability, the separation principle.
Reference inputs in state space representations.
Controller design using MATLAB.
Advanced topics in control theory.
R.C. Dorf, R.H. Bishop, Modern Control Systems. Prentice Hall. 2008. Eleventh Edition
Ordinary Differential Equations
Linear systems of differential equations: uncoupled linear systems, diagonalization, exponentials of operators, the fundamental theorem for linear systems, planar linear systems, complex eigenvalues, multiple eigenvalues, stability theory, nonhomogeneous linear systems.
Local theory of nonlinear systems: initial value problem, hyperbolic equilibrium point, Stable Manifold Theorem. Hartman-Grobman Theorem. Stability and Liapunov functions. Saddles, nodes, foci and centers. Nonhyperbolic critical points. Center manifold theory.
Global theory of nonlinear systems: limit set, attractor, limit cycle, Poincaré map, stable manifold theorem for periodic orbits, Poincaré-Bendixson theory. Mathematical background: Fundaments of perturbation analysis. The Multiple Scale Method. Basic concepts of bifurcation analysis: Bifurcation points, Linear codimension of a bifurcation, Imperfections, Fundamental path, Center Manifold Theory.
Basic mechanisms of multiple bifurcations: divergence, Hopf, nonresonant or resonant double-Hopf, Divergence-Hopf, Double-zero bifurcation.
Lawrence Perko, Differential equations and dynamical systems, Springer-Verlag, 2001
Linear Algebra. Complex numbers. Differential and integral calculus of functions of real variables.
Basic functional analysis: normed and Banach spaces, Hilbert spaces, Lebesgue integral, linear operators, weak topologies, distribution theory, Sobolev spaces, fixed point theorems, calculus in Banach spaces, spectral theory.
Applications: ordinary differential equations, boundary value problems for partial differential equations, linear system theory, optimization theory.
Ruth F. Curtain, A.J. Pritchard, Functional Analysis in Modern Applied Mathematics, Academic Press, 1977
Linux/Unix OS and tools;
Basic Fortran (or C);
HPC architecture;
System Scheduler;
Message Passing Interface;
OpenMP;
GPU computing;
Applications: linear algebra, PDEs, ODEs.
Some knowledge of linear algebra and basic notions in elementary mechanics of a pointwise body could be helpful.
To get familiar with kinematics of continuum, a suitable notion of force distribution, a general method delivering balance equations in continuum mechanics, the formal way of describing material properties and energy balance mainly for solid matter.
Placements and motions. Rigid and affine motions. Deformation gradient, stretch and rotation. Stretching and spin. Test velocity fields and force distributions. Working and stress. Working balance principle. Balance equations. Frame indifference principle. Affine bodies. Cauchy continuum. Cauchy stress and Piola-Kirchhoff stress.
Material response. Material objectivity. Symmetry group and isotropy. Elastic and hyperelastic materials. Strain energy function. Constraints and reactive stress. Incompressibility. Mooney-Rivlin and neo-Hookean materials. Dissipative stress and dissipation principle. Fluids and solids. A general scheme for describing growth and relaxation via Kroner-Lee decomposition. Remodeling forces and stress. Eshelby tensor. Viscoelasticity.
Numerical simulations with Comsol Multiphysics.
C. Truesdell, A First Course in Rational Continuum Mechanics. Academic Press. 1977.
M. Gurtin, An Introduction to Continuum Mechanics. Academic Press. 1981.
P. Chadwick, Continuum Mechanics: Concise Theory and Problems. Dover Books on Physics. 1976.
The student will be requested to have a good preparation on the presented topics, and to be able to implement some of the algorithms in a programming language
Abstract: Basic cryptograpy and coding theory will be developed. Some protocols and algorithms will be discussed focusing an security and data integrity.
Programme: Elementary arithmetics: Integers, divisibility, prime numbers, Euclidean division and g.c.d., Congruence classes, Chinese remainder theorem, cyclic and abelian groups, Lagrange theorem, Euler theorem, the structure of invertible classes mod p^n, Fields with p elements, polynomials, Euclidean division and g.c.d., Congruence classes of polynomials, Finite fields, primitive elements and polynomials, Legendre/Jacoby symbols and quadratic reciprocity. Cryptography: Classical cryptosystems: Shift cyphers, Vigenère Chipher, Substitution Chiper, One time pads, LFSR Data Encryption Standard: Simplified DES and differential cryptanalysis, Attacks, password encryption RSA: the algorithm, Attacks, Primality testing, the public key concept. Discrete logarithms: Bit commitment, Diffie-Helman Key exchange, ELGAMAL Hash function: SHA, birthday attacks Digital signatures: RSA signatures, Hashing and signing, DSA Error correcting codes: Binary block codes, distance and correction of errors, classical bounds, linear codes, cyclic codes, Hamming codes, BCH and Reed-Solomon codes.
[1] Wade Trappe, Lawrence C. Washington, Introduction to cryptography: with coding theory 2nd ed.. Pearson Prentice Hall. 2006.
[2] https://www.disim.univaq.it/didattica/content.php?corso=424&pid=88&did=0&lid=en
Knowledge of all topics treated the Mathematical Analysis courses in the first and second year: real function of real variables, limits, differentiation, integration; sequences and series of funcions; ordinary differential equations.
Knowledge of basic topics of complex analysis: elementary functions of complex variable, differentiation, integration and main theorems on analytic functions.
Ability to use such knowledge in solving problems and exercises
Complex numbers. Sequences. Elementary functions of complex numbers. Limits, continuity. Differentiation. Analytic functions. Armonic functions.
Contour integrals. Cauchy's Theorem. Cauchy's integral formula. Maximum modulus theorem. Liuville's theorem.
Series representation of analytic functions. Taylor's theorem. Laurent's series and classification of singularities.
Calculus of residues. The residue theorem. Application in evaluation of integrals. Rouche's theorem.
Conformal mappings. Main theorems. Fractional linear transformations.
Fourier transform for L^1 functions. Applications. Fourier transform for L^2 functions. Plancherel theorem.
Laplace transform and applications.
J.E. Marsden, M.J. Hoffman, Basic complex analysis. Freeman New York.
W. Rudin, Real and complex analysis. Mc Graw Hill.
The course aims to give an introduction to the theory of stochastic processes with special emphasis on applications and examples. On successful completion of this module the students should become familiar with some of the most known classes of stochastic processes (such as martingales, markov processes, diffusion processes) and to acquire both the mathematical tools and intuition for being able to describe systems with randomness evolving in time in terms of a probability model and to analyze it charcterizing some of its properties.
Stochastic Processes: definition, finite dimensional distributions, stationarity, sample path spaces, construction, examples.
Filtrations, stopping times, conditional expectation.
Markov processes: definition, main properties and examples. Birth and death processes.
Poisson process with applications on queueing models.
Martingales: definition, main properties and examples. Branching processes.
Brownian motion: definition, construction and main properties.
Brownian Bridge, Geometric Brownian Motion, Ornstein-Uhlenbeck process.
Ito integral and stochastic differential equations. Applications and examples.
P. Billingsley, Probability and measure. John Wiley and Sons.
G. Grimmett, D. Stirzaker, Probability and random Processes. Oxford University Press.
B. Oksendal, Stochastic Differential Equations. Springer-Verlag.
Ability to recognize and model network optimization problems as Integer Linear Programming problems. Knowledge of fundamental algorithmic techniques for solving large scale Integer Linear Programming problems. Knowledge of commercial and open source Integer Linear Programming solvers.
1. Formulations of Integer and Binary Programs: The Assignment Problem; The Stable Set Problem; Set Covering, Packing and Partitioning; Minimum Spanning Tree; Traveling Salesperson Problem (TSP); Formulations of logical conditions.
2. Mixed Integer Formulations: Modeling Fixed Costs; Uncapacitated Facility Location; Uncapacitated Lot Sizing; Discrete Alternatives; Disjunctive Formulations.
3. Optimality, Relaxation and Bounds. Geometry of R^n: Linear and affine spaces; Polyhedra: dimension, representations, valid inequalities, faces, vertices and facets; Alternative (extended) formulations; Good and Ideal formulations.
4. LP based branch-and-bound algorithm: Preprocessing, Branching strategies, Node and variable selection strategies, Primal heuristics.
5. Cutting Planes algorithms. Valid inequalities. Automatic Reformulation: Gomory's Fractional Cutting Plane Algorithm. Strong valid inequalities: Cover inequalities, lifted cover inequalities; Clique inequalities; Subtour inequalities. Branch-and-cut algorithm.
6. Software tools for Mixed Integer Programming.
7. Lagrangian Duality: Lagrangian relaxation; Lagrangian heuristics.
8. Network Problems: formulations and algorithms. Constrained Spanning Tree Problems; Constrained Shortest Path Problem; Multicommodity Flows; Symmetric and Asymmetric Traveling Salesman Problem; Vehicle Routing Problem Steiner Tree Problem; Network Design. Local Search Tabu search and Simulated Annealing MIP based heuristics.
9. Heuristics for network problems: local search, tabu search, simulated annealing, MIP based heuristics.
L.A. Wolsey, Integer Programming. Wiley. 1998.
*Students are required to earn 60 ECTS credits, at least, during their second year by successfully attending the following compulsory course units (Semester 1 and 2 amounting to 48 ECTS credits) and picking other 12 ECTS credits (minimum) from the elective ones listed below.
Basic notions of functional analysis, functions of complex values, standard properties of classical solutions of semilinear first order equations, heat equation, wave equation, Laplace and Poisson's equations.
Knowledge of mathematical methods that are widely used by researchers in the area of Applied Mathematics, as Sobolev Spaces, distributions. Application of this knowledge to a variety of topics, including the basic equations of mathematical physics and some current research topics about linear and nonlinear partial differential equations.
Distributions. Locally integrable functions. The space of test function D(U). Distributions. Distributions associated to Locally integrable functions. Singular distributions. Examples. Operations on distributions: sum, products times functions, change of variables, restrictions, tensor product. Differentiation and his properties; comparison with classical derivatives. Differentiation of jump functions. Partition of unity. Support of a distribution; compactly supported distributions.
Convolution. Convolution in Lp spaces. Regularity of the convolution. Regularizing sequences and smoothing by means of convolutions. Convolution between distributions and regularization of distributions. Denseness of D(U) in D'(U).
Sobolev spaces. Definition of weak derivatives and his motivation. Sobolev spaces Wk,p(U) and their properties. Interior and global approximation by smooth functions. Extensions. Traces. Embeddings theorems: Gagliardo-Nirenberg-Sobolev inequality and Embedding theorem for p < n. Embedding theorem for p = n. Hölder spaces. Morrey inequality. Embedding theorem for p > n. Sobolev inequalities in the general case. Compact embeddings: Rellich-Kondrachov theorem, Poincaré inequalities. Characterization of the dual space H-1.
Second order parabolic equations. Definition of parabolici operator. Weak solutions for linear parabolici equations. existence of weak solutions: Galerkin approximation, construction of approximating solutions, energy estimates, existence and uniqueness of solutions. Existence of solutions of viscous scalar conservation laws.
First order nonlinear hyperbolic equations. Scalar conservation laws: derivation, examples. Weak solutions, Rankine-Hugoniot conditions, entropy conditions. L1 stability, uniqueness and comparison for weak entropy solutions. Convergence of the vanishing viscosity and existence of the weak, entropy solution. Riemann problem. Definition of hyperbolic system. Quasilinear hyperbolic systems, symmetric and symmetrizable systems. Existence of solutions: approximations, a priori estimate, local existence of classical solutions.
V.S. Vladimirov, Equations of Mathematical Physics. Marcel Dekker, Inc..
C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics. Springer.
L.C. Evans, Partial Differential Equations. AMS.
M.E. Taylor, Partial Differential Equations, Nonlinear equations. Springer.
H. Brezis, Sobolev Spaces and Partial Differential Equations. Springer.
Basic notions of functional analysis, functions of complex values, standard properties of the heat equation, wave equation, Laplace and Poisson's equations.
This course is designed to give an overview of fluid dynamics from a mathematical viewpoint and to introduce students to the mathematical modeling of fluid dynamic type. At the end of the course students will be able to perform a qualitative and quantitative analysis of solutions for particular fluid dynamics problems and to use concepts and mathematical techniques learned from this course for analysis of other partial differential equations.
Derivation of the governing equations: Euler and Navier-Stokes.
Eulerian and Lagrangian description of fluid motion; examples of fluid flows.
Vorticity equation in 2D and 3D.
Dimensional analysis: Reynolds number, Mach Number, Frohde number.
From compressible to incompressible models.
Fluid dynamic modeling in various fields: biofluids, atmosphere and ocean, astrophysics.
Existence of solutions for viscid and inviscid fluids.
Linux/Unix OS and tools;
Basic Fortran (or C);
HPC architecture and libraries;
Application (ex ODEs, PDEs, elastodynamics).
A good knowledge of the basic arguments of a course of Functional Analysis, in particular, a good knowledge of the theory of Lebesgue's integral and the L^p spaces.
The first module of the course, in particular a good knowledge of the theory of distributions and Sobolev spaces.
Aim of the course is the knowledge of advanced techniques of mathematical analysis and in particular the basic techniques of the modern theory of the partial differential equations.
Abstract Measure theory.
AC and BV functions.
Fourier transforms.
Second order elliptic equations.
Variational methods.
The student will be requested to have a good preparation on the presented topics, and to be able to implement some of the algorithms in a programming language
Abstract: Basic cryptograpy and coding theory will be developed. Some protocols and algorithms will be discussed focusing an security and data integrity.
Programme: Elementary arithmetics: Integers, divisibility, prime numbers, Euclidean division and g.c.d., Congruence classes, Chinese remainder theorem, cyclic and abelian groups, Lagrange theorem, Euler theorem, the structure of invertible classes mod p^n, Fields with p elements, polynomials, Euclidean division and g.c.d., Congruence classes of polynomials, Finite fields, primitive elements and polynomials, Legendre/Jacoby symbols and quadratic reciprocity. Cryptography: Classical cryptosystems: Shift cyphers, Vigenère Chipher, Substitution Chiper, One time pads, LFSR Data Encryption Standard: Simplified DES and differential cryptanalysis, Attacks, password encryption RSA: the algorithm, Attacks, Primality testing, the public key concept. Discrete logarithms: Bit commitment, Diffie-Helman Key exchange, ELGAMAL Hash function: SHA, birthday attacks Digital signatures: RSA signatures, Hashing and signing, DSA Error correcting codes: Binary block codes, distance and correction of errors, classical bounds, linear codes, cyclic codes, Hamming codes, BCH and Reed-Solomon codes.
[1] Wade Trappe, Lawrence C. Washington, Introduction to cryptography: with coding theory 2nd ed.. Pearson Prentice Hall. 2006.
[2] https://www.disim.univaq.it/didattica/content.php?corso=424&pid=88&did=0&lid=en
The goal of this course is to provide the motivations, definitions and techniques for the translation of topological problems into algebraic ones, hopefully easier to deal with. On successful completion of this module, the student should understand the fundamental concepts of algebraic geometry and should be aware of potential applications of algebraic topological invariants in other fields as theoretical physics , including the computational fluid mechanics and electrodynamics.
General topology: topological spaces and continuous maps, induced, quotient and product topology, metric spaces, Hausdorff spaces, compact spaces, connected spaces, paths and path connected spaces
Manifolds and surfaces: the pancake problems, n-dimensional manifolds, surfaces and classification of surfaces.
Homotopy: Retracts and contractible spaces, paths and multiplication, the fundamental group, the fundamental group of the circle.
Covering spaces: the fundamental group of a covering space, the fundamental group of a orbit space, lifting theory and existence theorems, the Borsuk-Ulam theorem, the Seifert-Van Kampen theorem, the fundamental group of a surface.
Introduction to singular homology: standard and simplicial simplexes.
Czes Kosniowski, A first course in algebraic topology. Cambridge University Press. 1980.
Aim of the course is to present some mathematical models currently used in the analysis of collective phenomena, such as vehicular and pedestrian traffic, and flocking phenomena. Emphasis will be given to the mathematical treatment of specific problems coming from real world applications.
Macroscopic traffic models. LWR model, its derivation. Fundamental diagrams. The Riemann problem, examples. Second order models for traffic flow: Payne-Whitham model, description, drawbacks; Aw-Rascle model, shocks description, domains of invariance, instabilities near vacuum.
Theory: systems of conservation laws, strict hyperbolicity, Rankine-Hugoniot conditions; Lax admissibility condition. The Riemann problem for systems: the linear case; GNL and LD fields; rarefactions and contact discontinuities. BV functions, examples and properties. A compactness theorem.
Wave front tracking algorithm: approximate rarefactions, possible types of interactions. Bounds on number of waves and on total variation. Compactness of approximate solutions. The initial-boundary value problem on the half line: boundary Riemann problem, interactions with the boundary, control of the total variation by means of a Lyapunov-type functional. The Toll gate problem.
Networks, basic definitions, conservation of the flux. Examples. Distributions along the roads, maximization of the flux. Riemann problem on a junction composed by 2 incoming roads and 2 outgoing roads. The case of 2 incoming roads and 1 outgoing road: the "right of way" rule. Junction between one incoming and one outgoing road, different fluxes.
Pedestrian flow: normal and panic situation. Macroscopic models for evacuation, conservation of "mass", eikonal equation. The Hughes model for pedestrian flow. The eikonal equation: non uniqueness, viscosity solutions, relation with vanishing viscosity approximation. The Hughes model in one space dimension. Curve of turning points, Rankine-Hugoniot conditions. The case of constant initial density and of symmetric initial data; conservation of the left and right mass; an example with mass exchange across the turning point. Macroscopic models for pedestrian flow that include: knowledge of a preferred path, discomfort from walking along walls, tendency of avoiding high densities of pedestrian in a neighborhood (nonlocal term of convolution type), angle of vision, obstacle in the domain. Linearized stability around a smooth solution.
Introduction to the theory of flocking. Examples: Krause model for opinion dynamics, Cucker-Smale model, model for attraction-repulsion phenomena. The Cucker-Smale flocking model: basic properties, estimates on the kinetic energy. A "flocking theorem": proof by bootstrapping method (Ha and Tadmor). Some drawbacks of the model. Introduction to the kinetic limit for flocking: the N-particle distribution function, Liouville equation, marginal distribution, continuity equation. The formal mean-field limit: a Vlasov-type equation.
M.D. Rosini, Macroscopic models for vehicular flows and crowd dynamics: theory and applications. Springer. 2013. http://link.springer.com/book/10.1007/978-3-319-00155-5/page/1
M. Garavello, B. Piccoli, Traffic flow on networks. Conservation laws models. AIMS Series on Applied Mathematics. 2006. http://www.aimsciences.org/books/am/AMVol1.html
Basic calculus and analysis (differential and integral calculus with functions of many variables).
Ordinary differential equations.
Basics in finite dimensional dynamical systems.
Elementary methods for the solution of linear partial differential equations (separation of the variables).
1) To learn the basics in the mathematical modelling of population dynamics.
2) To provide a mathematical description of ODE models in population dynamics and the intepretation of the qualitative behaviour of the solutions.
3) To get the basic notions in mathematical models in epidemiology and reaction kinetics.
4) To learn the mathematical modelling of population models in heterogeneous environment, described by partial differential equations.
5) To deal with advanced models in biology such as chemotaxis models and structured dynamics equations.
6) To get a sound background in reaction diffusion phenomena, Turing instability, and pattern formation.
Continuous Population Models for Single Species. Continuous Growth Models. Delay models. Linear Analysis of Delay Population Models: Periodic Solutions.
Continuous models for Interacting Populations. PredatorPrey Models: Lotka-Volterra Systems. Realistic Predator–Prey Models. Competition Models: Principle of Competitive Exclusion. Mutualism or Symbiosis.
Reaction kinetics. Enzyme Kinetics: Basic Enzyme Reaction. Transient Time Estimates and Nondimensionalisation. Michaelis-Menten Quasi-Steady State Analysis.
Dynamics of Infectious Diseases: Epidemic Models and AIDS. Simple Epidemic Models (SIR, SI) and Practical Applications. Modelling Venereal Diseases. AIDS: Modelling the Transmission Dynamics of the Human Immunodeficiency Virus (HIV).
Time-space dependent models: PDEs in biology. Diffusion equations. Diffusion and Random walk. The gaussian distribution. Smoothing and decay properties of the diffusion operator. Nonlinear diffusion.
Reaction–diffusion models for one single species. Diffusive Malthus equation and critical patch size. Travelling waves. Fisher–Kolmogoroff equation.
Reaction–diffusion systems. Multi species waves in Predator-Prey Systems. Turing instability and spatial patterns.
Chemotaxis modelling. Diffusion vs. Chemotaxis: stability vs. instability. Diffusion vs. Chemotaxis: stability and blow–up. Chemotaxis with nonlinear diffusion. Models with maximal density.
Nonlocal interaction models in biology. Mathematical models of swarms. Approximation with interacting particle systems. Asymptotic behaviour.
Structured population dynamics. An example in ecology: competition for resources. Continuous traits. Evolutionary stable strategy in a continuous model.
James D. Murray, Mathematical Biology I: an introduction. Springer.
James D. Murray, Mathematical Biology II: Spatial models and biomedical applications . Springer.
Benoit Perthame, Transport equations in biology. Birkaeuser.
The course aims to give an introduction to the theory of stochastic processes with special emphasis on applications and examples. On successful completion of this module the students should become familiar with some of the most known classes of stochastic processes (such as martingales, markov processes, diffusion processes) and to acquire both the mathematical tools and intuition for being able to describe systems with randomness evolving in time in terms of a probability model and to analyze it charcterizing some of its properties.
Stochastic Processes: definition, finite dimensional distributions, stationarity, sample path spaces, construction, examples.
Filtrations, stopping times, conditional expectation.
Markov processes: definition, main properties and examples. Birth and death processes.
Poisson process with applications on queueing models.
Martingales: definition, main properties and examples. Branching processes.
Brownian motion: definition, construction and main properties.
Brownian Bridge, Geometric Brownian Motion, Ornstein-Uhlenbeck process.
Ito integral and stochastic differential equations. Applications and examples.
P. Billingsley, Probability and measure. John Wiley and Sons.
G. Grimmett, D. Stirzaker, Probability and random Processes. Oxford University Press.
B. Oksendal, Stochastic Differential Equations. Springer-Verlag.
Mathematical Analysis, Fourier transform.
This course provides an introduction to the classical kinetic theory of gases and the principles of kinetic modeling.
A special focus is given to the derivation of hydrodynamic equations from kinetic models by means of non-perturbative techniques and to the analysis of numerical schemes for the simulation of fluid flows.
On successful completion of this module the student has the knowledge on the basic principles and the simulation strategies of kinetic models.
Boltzmann equation and the principles of kinetic description.
Kinetic models: BGK,Maxwell molecules, Vlasov equation and Fokker-Planck equation.
The closure problem and methods of reduced description: Chapman-Enskog expansion, Grad's Moment method.
Non-perturbative techniques in kinetic theory: the method of the slow invariant manifold.
Overview on Lattice Boltzmann models.
Monte Carlo simulations of lattice gas models.
The course is an introduction to Time Series Analysis and Forecasting. The level is the first-year graduate in Mathematics with a prerequisite knowledge of basic inferential statistical methods.
The aim of the course is to present important concepts of time series analysis (stationarity of stochastic processes, ARIMA models, forecasting etc.). At the end of the course, the student should be able to select an appropriate ARIMA model for a given time series.
Stochastic processes (some basic concepts)
Stationary stochastic processes
Autocovariance and autocorrelation functions
Ergodicity of a stationary stochastic process
Estimation of moment functions of a stationary process
ARIMA models
Estimatiom of ARIMA models
Building ARIMA models
Forecasting from ARIMA models
[1]Time Series Analysis Univariate and Multivariate Methods, 2nd Edition, W. W. Wei, 2006, Addison Wesley.
[2] Time Series Analysis, J. Hamilton, 1994, Princeton University Press.
[3] Time Series Analysis and Its Applications with R Examples, Shumway, R. and Stoffer, D., 2006, Springer.
[4]Introduction to Time Series and Forecasting. Second Edition, P. Brockwell and R. Davis, 2002, Springer.
I assume familiarity with vector and topological spaces, and with the standard model of the real numbers. I assume that you know the basic facts about metric spaces, normed and seminormerd spaces, Banach and Hilbert spaces.
On successful completion of this course, the student should:
- Know the fundamental fixed point theorems for set-valued maps and the basic existence results for equilibrium problems and variational inequalities.
- Explain some interconnections among these various results.
- Apply this analysis to game and economic theory
Sperner’s lemma
The Knaster-Kuratowski-Mazurkiewicz lemma
Brouwer's fixed point theorem
Variational inequalities and equilibrium problems
Generalized monotonicity and convexity
Brézis-Nirenberg-Stampacchia theorem and Fan's minimax principle
Continuity of correspondences
Browder, Kakutani and Fan-Glicksberg fixed point theorems
Gale-Nikaido-Debreu theorem
Nash equilibrium of games and abstract economies
Walrasian equilibrium of an economy
An application to traffic network
Basic notions of functional analysis, functions of complex values, standard properties of classical solutions of semilinear first order equations, heat equation, wave equation, Laplace and Poisson's equations.
Knowledge of mathematical methods that are widely used by researchers in the area of Applied Mathematics, as Sobolev Spaces, distributions. Application of this knowledge to a variety of topics, including the basic equations of mathematical physics and some current research topics about linear and nonlinear partial differential equations.
Distributions. Locally integrable functions. The space of test function D(U). Distributions. Distributions associated to Locally integrable functions. Singular distributions. Examples. Operations on distributions: sum, products times functions, change of variables, restrictions, tensor product. Differentiation and his properties; comparison with classical derivatives. Differentiation of jump functions. Partition of unity. Support of a distribution; compactly supported distributions.
Convolution. Convolution in Lp spaces. Regularity of the convolution. Regularizing sequences and smoothing by means of convolutions. Convolution between distributions and regularization of distributions. Denseness of D(U) in D'(U).
Sobolev spaces. Definition of weak derivatives and his motivation. Sobolev spaces Wk,p(U) and their properties. Interior and global approximation by smooth functions. Extensions. Traces. Embeddings theorems: Gagliardo-Nirenberg-Sobolev inequality and Embedding theorem for p < n. Embedding theorem for p = n. Hölder spaces. Morrey inequality. Embedding theorem for p > n. Sobolev inequalities in the general case. Compact embeddings: Rellich-Kondrachov theorem, Poincaré inequalities. Characterization of the dual space H-1.
Second order parabolic equations. Definition of parabolici operator. Weak solutions for linear parabolici equations. existence of weak solutions: Galerkin approximation, construction of approximating solutions, energy estimates, existence and uniqueness of solutions. Existence of solutions of viscous scalar conservation laws.
First order nonlinear hyperbolic equations. Scalar conservation laws: derivation, examples. Weak solutions, Rankine-Hugoniot conditions, entropy conditions. L1 stability, uniqueness and comparison for weak entropy solutions. Convergence of the vanishing viscosity and existence of the weak, entropy solution. Riemann problem. Definition of hyperbolic system. Quasilinear hyperbolic systems, symmetric and symmetrizable systems. Existence of solutions: approximations, a priori estimate, local existence of classical solutions.
V.S. Vladimirov, Equations of Mathematical Physics. Marcel Dekker, Inc..
C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics. Springer.
L.C. Evans, Partial Differential Equations. AMS.
M.E. Taylor, Partial Differential Equations, Nonlinear equations. Springer.
H. Brezis, Sobolev Spaces and Partial Differential Equations. Springer.
Basic notions of functional analysis, functions of complex values, standard properties of the heat equation, wave equation, Laplace and Poisson's equations.
This course is designed to give an overview of fluid dynamics from a mathematical viewpoint and to introduce students to the mathematical modeling of fluid dynamic type. At the end of the course students will be able to perform a qualitative and quantitative analysis of solutions for particular fluid dynamics problems and to use concepts and mathematical techniques learned from this course for analysis of other partial differential equations.
Derivation of the governing equations: Euler and Navier-Stokes.
Eulerian and Lagrangian description of fluid motion; examples of fluid flows.
Vorticity equation in 2D and 3D.
Dimensional analysis: Reynolds number, Mach Number, Frohde number.
From compressible to incompressible models.
Fluid dynamic modeling in various fields: biofluids, atmosphere and ocean, astrophysics.
Existence of solutions for viscid and inviscid fluids.
Linux/Unix OS and tools;
Basic Fortran (or C);
HPC architecture and libraries;
Application (ex ODEs, PDEs, elastodynamics).
Mathematical Analysis, Fourier transform.
This course provides an introduction to the classical kinetic theory of gases and the principles of kinetic modeling.
A special focus is given to the derivation of hydrodynamic equations from kinetic models by means of non-perturbative techniques and to the analysis of numerical schemes for the simulation of fluid flows.
On successful completion of this module the student has the knowledge on the basic principles and the simulation strategies of kinetic models.
Boltzmann equation and the principles of kinetic description.
Kinetic models: BGK,Maxwell molecules, Vlasov equation and Fokker-Planck equation.
The closure problem and methods of reduced description: Chapman-Enskog expansion, Grad's Moment method.
Non-perturbative techniques in kinetic theory: the method of the slow invariant manifold.
Overview on Lattice Boltzmann models.
Monte Carlo simulations of lattice gas models.
Browse the tabs below to get useful information about L'Aquila
L'Aquila is an Italian city of about 70,000 inhabitants and nearly 30,000 university students. It is the capital of the Abruzzo region and it is conveniently located 100 km (62 miles) to the east of Rome. The city is on a hill at 720 m (2365 ft) above sea level and is surrounded by mountains, most notably to the north by the Gran Sasso range, which includes the highest peaks (up to 2900 m) of the Apennines, with a number of small lakes, trails and mountain climbing routes as well as deep caves. Within the province of L’Aquila there are also two national parks (Parco Nazionale Gran Sasso Monti della Laga and Parco Nazionale della Majella).
The city itself is full of history, traditions, beautiful buildings (like the Spanish Fortress) and churches (like the Basilica of Collemaggio). There are also a lot of good restaurants, pubs and places where students get together every night. The city is also home to L'Aquila Rugby: this team won the Italian championship five times!
For further practical and historical information on L'Aquila, click here.
The InterMaths organizing committee will take care of finding accommodation for all the InterMaths students. Information about rent, utility bills and possible reduction will be emailed to selected students. In general, the cost per person may range from 300 to 400 euros per month (utility bills included). All selected students will receive detailed information by email about accommodation in good time before leaving their own country (usually in June), along with contracts and instructions on how to pay for the deposit, if applicable.
Please visit our partner website or read below: www.mathmods.eu
as for the Trenitalia, check the article for possible regional trains to and from L'Aquila.
There are three bus companies operating in L'Aquila that you will find particulary usefull in your everyday life
Besides these busses, there are also
The public transport in L'Aquila is covered by the AMA bus company. Depending on how busy the trip can get thet run orange/blue/green/black busses "l'autobus" or blue vans called "pulmino".
The final destination of the bus is always written on the front side of the bus and together with the bus number can be seen from far. You need to get used to the way the timetable is written, as you can either search for number, or the bus stop (not for the time or "from-to destination trip"). To search for the time online, please check the name of your bus stop and then search for this destination in Linee e Orari to find the bus number you need to take. The other way is to search for the number and then check where it goes. Moreover, there is nowadays nice and clear time table on every bus stop. You can find the map of lines here.
You are obliged to enter a bus with a ticket and validate it in the yellow machine inside the bus. The complete list of ticket selling places can be found here. The closest one to university Coppito is Self copy SaS di Epifano - just across of the road from University. Please mind, that not every place sells all types of tickets.
Here are the most useful types of tickets with prices:
The controllers occasionaly get on the bus and wear the dark blue company clothes. They don't speak english and the fine for not having a validated ticket can get up to 160 Euros.
* the same system as for Arpa holds here as well: in order to be allowed to buy the montly ticket you need to register and buy a card at Terminal Bus Station or Sangritana Viaggi e vacanze Fontana Luminosa. You need to fill in a form and bring ID or passport and 2 passport size fotos. The card is valid until you lose it and costs around 15 Euros.
Regional public bus transport is run by the Arpa company - blue or white with blue stipes busses.
It is the bus company you will use daily for reaching the University. For more information about this daily routine see this article about transfer between L'Aquila and Pizzoli, or how to get to Rome from L'Aquila.
Timetables for other Arpa rides can be found here.
Arpa is the intercity Abruzzo regional bus company that offers transport within the region and also runs to Rome. It is the best option for Rome trips (11 Euros one way). Their offices can be found in several places, moreover, it is possible to buy a ticket at every SISAL place - for example at the Bar in Motel Amiternum.
As for Pizzoli - you can get the tickets in every bar in Pizzoli. For Pizzoli - L'Aquila trip the ticket Tariffa 2 is needed. You are obliged to validate it in the machine inside the bus.
You'll find all Gaspari timetables on their webpage. There are 2 lines that might be on your interest:
Gaspari tickets can be bought both, on the bus and online. Their page hasn't been translated into english yet but this article describing how to buy tickets on gaspari webpage in english might be useful. We recommend to buy your ticket online, so you assure yourself with the seat on the bus in case it gets busy (which has recently been happening). When entering the bus, pasangers with online tickets go first.
As for car transport possibilities in&around L'Aquila you can consider following options
If you wish to bring your own car to L'Aquila there is no problem. There has so far been at least one person bringing their car from each generation. Parking next to our student residences in Pizzoli is possible just next to your flat. When arriving to L'Aquila by car please mind that in Italy we pay for highway everytime we enter.
L'Aquila can be reached either via motorway or road:
There is only one taxi company in L'Aquila, Radio Taxi. They can be reached by phone +39086225165 or in front of Motel Amiternum.
Erasmus Mundus students are meant to get the special price of 15 Euros for the trip Pizzoli-L'Aquila and vise versa. The usual trip Motel Amiternum-city center is aroun 4 Euros.
Radio Taxi are available every day until 12am; on Thursdays and Saturdays until 3am. Aside from these times it is possible to make a pick up agreement in advance, however, if you have a early flight (therefore early gaspari bus) we reccomend to stay at friends place in L'Aquila for this special occasion.
Students often use auto rental services for weekend or daily trips. There are several car rental company, just search for 'car rental in L'Aquila' in your browser or ask older students.
In 2014 there were students renting a car from Europcar, na Via della Croce Rossa. For approximately 2-4 days it was around 30 Euros a day but drivers under 25 have to pay an extra fee.
Aquilasmus is a student association, part of ESN (Erasmus Student Network). Aquilasmus offers several services to Erasmus students, like organizing parties, trips, international dinners, cineforums and more. Take a look at their website and join their Facebook group to get to know other international students and be involved in their activities.
Check this article to see more information about cinemas, theathers, music, bars, restaurants,pubs & clubs, discos.
Free time and nightlife
Aquilasmus is a student association, part of ESN (Erasmus Student Network). Aquilasmus offers several services to Erasmus students, like organizing parties, trips, international dinners, cineforums and more. Take a look at their website and join their Facebook group to get to know other international students and be involved in their activities.
Cinemas
Theatres
Music
Before the 2009 earthquake most people and students used to gather at the many cafes and bars in L'Aquila city centre. Now, while most buildings there are still to be reconstructed and great part of the area is not yet accessible to people, a bunch of bars have proudly reopened their doors. You will find lots of students hanging out mostly on Thursday nights (typically, university night) and Saturday nights. Just ask the taxi/bus driver to drop you at "Fontana Luminosa" (the big fountain near the castle) and walk into the main road "Corso Federico II". You'll see that most people gather in that small square or head right into via Garibaldi. As a consequence of the earthquake, several other good pubs and clubs have had to move to other areas of the city. So, take a look at the rest of list, too.
City centre
Viale della Croce Rossa
After the earthquake several pubs moved from the city centre, which was off-limits for several months, to this road which connects the "Fontana Luminosa" to Viale Corrado IV
Between the train station and Viale Corrado IV
Viale Corrado IV (main road connecting Hotel Amiternum to the city centre)
Other areas
For informantion on popular excursions you can make around L'Aquila (ski resorts, beaches and more), read here.
Check these articles for more info about shoping in L'Aquila or shops in Pizzoli.
As for fashion or souvenirs there are many shops in Roma or Pescara on various price level.
Many collective sports like rugby, football as well as tennis can be played in student organisations. Check the map below for University Sports Centre (CUS - Centi Colella), where students have reduced fee. It should be the bus stop s.s.17
Please mind that in Italy it is necessary to have a health check certificate from a 'family doctor' ( = the general doctor) before joining any sport facility (f.e. a gym). Feel free to e-mail us in case you have difficulties getting one.
Thanks to the great geographical location, both, L'Aquila and Pizzoli offer great oportunities to hike or just go for a nice walk into the mountains or woods.
All Pizzoli hikes start at the 'Pizzoli castle' when you continue up the road and after you pass few houses you will find yourself in the nice Abruzzo woodland.
As for L'Aquila hikes, you may pick any hill you see from the center
It is quite likely to meet horses, cheep, cows or even some wild animals on your way - all being more scared than yourself and therefore harmless. Sometime you may stumble across cheep dogs that might look angry but if you don't show any signs of agression and simply ignore them they will gladly return the favore.
Dispite having no cycle paths in L'Aquila, it is possible to cycle on the road, however, this is on everyones own responsibility. You can get propper cycling stufff in Decathlon in Laquilone or other cycling shops in L'Aquila. There is also an active cycling club that occasionally organises bike events in Gran Sasso .
there is a big stable in Paganica, check AMA busses to see. One lecture costs 10 Euros and it is possible to prepay 10 lectures. Mind, the transportation from L'Aquila can take a while.
there are few gyms in L'Aquila and in Pizzoli as well. Please see the map above for precise location and this article about sporting facilities in L'Aquila for more information.
If you are interested in Yoga classes or similar, there used to be some in Asilo Occupato but we recommend to ask Laquilasmus.
»Year1 L'Aquila: SciCom
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