ECTS Credits: 9   |   Semester: 1   |   Year: 1   |   Campus: University of L'Aquila   |   Language: English   |   Code: DT0626

Unit Coordinator: Marco Di Francesco, Michele Palladino

Aims:

Introducing basic tools of advanced real analysis such as metric spaces, Banach spaces, Hilbert spaces, bounded operators, weak convergences, compact operators, weak and strong compactness in metric spaces, spectral theory, in order to allow the student to formulate and solve linear ordinary differential equations partial differential equations, classical variational problems, and numerical approximation problems in an "abstract" form. Provide a primer of abstract measure and integration to be used in advanced probability and analysis courses.

Content:

• Metric spaces, normed linear spaces. Topology in metric spaces. Compactness.
• Spaces of continuous functions. Convergence of function sequences. Approximation by polynomials. Compactness in spaces of continuous functions. Arzelà's theorem. Contraction mapping theorem.
• Crash course on abstract measure and integration. Measurable spaces and measurable functions. Borel and Lebesgue measures. Integrals on measure spaces. Limit exchange convergence theorems. Lp spaces. Product measures. Signed measures and Radon-Nicodym Theorem. Riesz representation theorem for measures.
• Introduction to the theory of linear bounded operators on Banach spaces. Bounded operators. Dual norm. Examples. Riesz' lemma. Norm convergence for bounded operators.
• Hilbert spaces. Elementary properties. Orthogonality. Orthogonal projections. Bessel's inequality. Orthonormal bases. Examples.
• Bounded operators on Hilbert spaces. Dual of a Hilbert space. Adjoin operator, self-adjoint operators, unitary operators. Applications. Weak convergence on Hilbert spaces. Banach-Alaoglu's theorem.
• Introduction to spectral theory. Compact operators. Spectral theorem for self-adjoint compact operators on Hilbert spaces. Hilbert-Schmidt operators. Functions of operators.

Pre-requisites:

Basic calculus and analysis in several variables, linear algebra.

Reading list:

• John K. Hunter, Bruno Nachtergaele, Applied Analysis. World Scientific.
• H. Brezis, Funtional Analysis, Sobolev Spaces, and partial differential equations. Springer.
• Piermarco Cannarsa, Teresa D’Aprile, Introduction to Measure Theory and Functional Analysis, Springer.

ECTS Credits: 6   |   Semester: 1   |   Year: 1   |   Campus: University of L'Aquila   |   Language: English   |   Code: I0183

Unit Coordinator: Corrado Lattanzio

Aims:

LEARNING OBJECTIVES.
The course aims at providing basic properties and main techniques to solve basic partial differential equations.
Those objectives contribute to the learning goals of the entire course of studies, as the inner coherence of the master degree in Mathematical Modelling was verified at the time of the planning of the master program.

LEARNING OUTCOMES.
At the end of the course, the student should:

1. know basic properties (existence, uniqueness, etc.) and main techniques (characteristics, separation of variables, Fourier methods, Green's functions, similarity solutions, etc.) to solve basic partial differential equations (semilinear first order PDEs, heat, Laplace, wave equations);
2. understand and be able to explain thesis and proofs in the field of basic partial differential equations;
3. have strengthened the logic and computational skills;
4. be able to read and understand other mathematical texts on related topics.

Content:

First order partial differential equations. Definition of characteristic vectors and characteristic surfaces. Characteristics for (semi)linear partial differential equations of first order in two independent variables. Existence and uniqueness to initial value problems for first order semilinear partial differential equations in two independent variables Duhamel’s principle for non homogeneous first order partial differential equations.

Second order partial differential equations. Classification of second order semilinear partial differential equations in two independent variables. Canonical form for second order semilinear partial differential equations in two independent variables. Classification for second order semilinear partial differential equations in many independent variables.

Heat equation. Derivation of heat equation and well–posed problems in one space dimension. Solution of Cauchy–Dirichlet problem for one dimensional heat equation by means of Fourier method of separation of variables. Energy method and uniqueness. Maximum principle. Fundamental solution. Solution of global Cauchy problem. Non homogeneous problem: Duhamel’s principle.

Laplace equation. Laplace and Poisson equation: well-posed problems; uniqueness by means of energy method. Mean value property and maximum principles. Laplace equation in a disk by means of separation of variables. Poisson’s formula. Harnack’s inequality and Liouville’s Theorem. Fundamental solution of Laplace operator. Solution of Poisson’s equation in the whole space. Green’s functions and Green’s representation formula.

Wave equation. Transversal vibrations of a string. Well–posed problems in one space dimension. D’Alembert formula. Characteristic parallelogram. Domain of dependence and range of influence. Fundamental solution for one dimensional wave equation. Duhamel’s principle for non homogeneous one dimensional wave equations. Special solutions of multi–d wave equation: planar and spherical waves. Well–posedness for initial, boundary value problems: uniqueness by means of energy estimates. Separation of variables. Domain of dependence and range of influence in several space variables. Fundamental solution for multi–dimensional wave equation. Solution of 3–d wave equation: Kirchhoff’s formula and strong Huygens’ principle. Wave equation in two dimensions: method of descent. Fundamental solution in 2–d. Duhamel’s principle for non homogeneous wave equation in 3–d: delayed potentials.

Pre-requisites:

Students must know the basic notions of mathematical analysis, including Fourier series and ordinary differential equations, and the basic notions of continuum mechanics.

Reading list:

- L.C. Evans. Partial Differential Equations. Graduate Studies in Mathematics, Vol. 19, AMS, 2010.

- S. Salsa. Partial Differential Equations in Actions: from Modelling to Theory. Springer–Verlag Italia, 2008.

- S. Salsa, G. Verzini. Equazioni a derivate parziali: complementi ed esercizi. Springer–Verlag Italia. 2005.

- W.A. Strauss. Partial Differential Equations: An Introduction. John Wiley & Sons Inc., 2008.

- E.C. Zachmanoglou, D.W. Thoe. lntroduction to Partial Differential Equations with Applications. Dover Publications, Inc., 1986.

Additional info:

http://people.disim.univaq.it/~corrado/

ECTS Credits: 6   |   Semester: 1   |   Year: 1   |   Campus: University of L'Aquila   |   Language: English   |   

Unit Coordinator: Michele Palladino, Cristina Pignotti

ECTS Credits: 6   |   Semester: 1   |   Year: 1   |   Campus: University of L'Aquila   |   Language: English   |   Code: I0459

Unit Coordinator: Bruno Rubino

Aims:

The course is intended to introduce and develop an understanding of the concepts in nonlinear dynamical systems and bifurcation theory, and an ability to analyze nonlinear dynamic models of physical systems. The emphasis is to be on understanding the underlying basis of local bifurcation analysis techniques and their applications to structural and mechanical systems.

Content:

Review of: first-order nonlinear ODE, first-order linear systems of autonomous ODE. Local theory for nonlinear dynamical systems: linearization, stable manifold theorem, stability and Liapunov functions, planar non-hyperbolic critical points, center manifold theory, normal form theory. Global theory for nonlinear systems: limit sets and attractors, limit cycles and separatrix cycles, Poincaré map. Hamiltonian systems. Poincaré-Bendixson theory. Bifurcation theory for nonlinear systems: structural stability, bifurcation at non-hyperbolic equilibrium points, Hopf bifurcations, bifurcation at non hyperbolic periodic orbits. Applications.

Pre-requisites:

Ordinary differential equations

Reading list:

Lawrence Perko, Differential equations and dynamical systems, Springer-Verlag, 2001

ECTS Credits: 3   |   Semester: 1   |   Year: 1   |   Campus: University of L'Aquila   |      |   Code: DT0807

Unit Coordinator: Roberta Antonetti

Aims:

The aim of this course is to provide students with knowledge of Italian language at level A1 (beginner). The course will concentrate on four aspects of communication: speaking, writing, listening and reading on a basic level. Attention is given to correct pronunciation of Italian.
Furthermore, the goals of the course are to enable students to:
- develop the language proficiency required to communicate effectively in Italian at level A1;
- develop awareness of the nature of language and language learning;
- develop transferable skills;
- form a sound base of the skills, language and attitudes required for progression to work or further study, either in Italian or another subject area.
On successful completion of this module, a student should be able to:
- understand familiar words and very basic phrases concerning himself/herself, his/her family and immediate concrete surroundings when people speak slowly and clearly;
- read familiar names, words and very simple sentences, for example on notices and posters or in catalogues;
- interact in a simple way provided the other person is prepared to repeat or rephrase things at a slower rate of speech;
- ask and answer simple questions in areas of immediate need or on very familiar topics;
- use simple phrases and sentences to describe where he/she lives and people he/she knows;
- write short messages, postcards, fill in forms with personal details or a hotel registration form.

Content:

Students will be required to show knowledge and understanding of the broad topic areas listed below. These provide contexts for the acquisition of vocabulary and the study of grammar and structures. 
VOCABULARY AND TOPIC AREAS 
Clothes and accessories; colors; countries, nationalities and languages; culture, customs and celebrations; education e.g. learning institutions, education and training, learning tools, subjects; family and friends; feelings and emotions; food and drink; interests, sports and activities; jobs; measurements e.g. size, shape, weight; numbers (cardinal, ordinal) and money; rooms and furniture; shops and places; street directions; the human body and health e.g. parts of the body, health and illness; time expressions e.g. telling the time, dates, days of the week, months, seasons; travel and transport; weather.

FUNCTIONAL SYLLABUS
Accepting/ thanking; applying for a job; asking about personal information; asking permission; buying and asking prices; describing people and objects; giving opinions; giving instructions; greeting and introducing; inviting, refusing; making suggestions; requesting, offering; talking about likes and dislikes; talking about routines; telling the time.

GRAMMAR SYLLABUS
Sounds and spelling; nouns: gender, number; irregular plural nouns; definite article; indefinite article; adjectives, first group adjectives and second group adjectives, position of adjectives; formal address; pronouns, subject pronouns and direct and indirect object pronouns; possessive adjectives; prepositions simple and prepositions with articles; interrogatives; adverbs; present tense of regular and irregular verbs; modal verbs and the verb sapere; present perfect and some verbs with an irregular past participle.

Pre-requisites:

There are no prerequisites for attending the course.

Reading list:

"Dieci A1" - Alma edizioni, Firenze 2019.
“New Italian grammar in practice”, Alma edizioni, Firenze 2015.

ECTS Credits: 9   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English   |   Code: DT0113

Unit Coordinator: Corrado Lattanzio

Aims:

LEARNING OBJECTIVES.
The course aims at providing advanced mathematical notions used in the field of (applied) mathematical analysis and their applications to a variety of topics, including the basic equations of mathematical physics and some current research topics about linear and nonlinear partial differential equations.
Those objectives contribute to the learning goals of the entire course of studies, as the inner coherence of the master degree in Mathematics was verified at the time of the planning of the master program.
LEARNING OUTCOMES.
At the end of the course, the student should:
1. know the advanced mathematical notions used in the field of (applied) mathematical analysis, as measure theory, Sobolev Spaces, distributions, and their applications to the theory of linear and non-linear partial differential equations;
2. understand and be able to explain thesis and proofs in advanced mathematical analysis;
3. have strengthened the logic and computational skills;
4. be able to read and understand other mathematical texts on related topics.

Content:

Distributions. Locally integrable functions. The space of test function D(Ω). 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. Fourier transform and tempered distributions. Convolution between distributions and regularization of distributions. Denseness of D(Ω) in D′(Ω).

Sobolev spaces. Definition of weak derivatives and his motivation. Sobolev spaces Wk,p(Ω) and their properties. Interior and global approximation by smooth functions. Extensions. Traces. Embeddings theorems: Gagliardo–Nirenberg–Sobolev inequality and 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. Embedding theorem for p = n. Characterization of the dual space H-1.

Second order parabolic equations. Definition of parabolic operator. Weak solutions for linear parabolic equations. Existence of weak solutions: Galerkin approximation, construction of approximating solutions, energy estimates, existence and uniqueness of solutions.

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.

Measures. System of sets, Positive Measures, Outer Measures, Construction of Measures, Signed Measures, Borel and Radon Measures
Integration. Measurable Functions, Simple Functions, Convergence Almost Everywhere, Integral of Measurable Functions, Convergences of Integrals, Fubini-Tonelli Theorems.
Differentiation. The Radon-Nikodym Theorem, Differentiation on Euclidean space, Differentiation of the real line.
Radon measures and continuous functions. Spaces of continuous functions, Riesz Theorem.

Pre-requisites:

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.

Reading list:

- L. Ambrosio, G. Da Prato, A. Mennucci. Introduction to measure theory and integration. Edizioni della Normale.

- L. Ambrosio, N. Fusco, D. Pallara. Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs.

- V.I. Bogachev. Measure theory, Volume I, Springer.

- H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, Springer.

- P. Cannarsa, T. D’Aprile. Introduction to Measure Theory and Functional Analysis. Springer.

- C.M. Dafermos. Hyperbolic Conservation Laws in Continuum Physics, Springer.

- L.C. Evans. Partial Differential Equations. Graduate Studies in Mathematics, Vol. 19, AMS.

- L. Evans, R. Gariepy. Measure Theory and Fine Properties of Functions, CRC Press. Revised Edition.

- G.B. Folland. Real analysis: Modern techniques and their applications. New York Wiley

- G. Gilardi. Analisi 3. McGraw–Hill.

- L. Grafakos, Classical Fourier Analysis. Springer.

- V.S. Vladimirov. Equations of Mathematical Physics. Marcel Dekker, Inc

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English   |   

Unit Coordinator: Donatella Donatelli

Aims:

Content:

- Derivation of the governing equations: Euler and Navier-Stokes
- Eulerian and Lagrangian description of fluid motion; examples of fluid flows
- Poiseulle and Couette fluid types
- Vorticity equation in 2D and 3D
- Dimensional analysis: Reynolds number, Mach Number, Frohde number.
- From compressible to incompressible models
- Existence of solutions for viscid and inviscid fluids
- Fluid dynamic modeling in various fields: mixture of fluids, combustion, astrophysics, geophysical fluids (atmosphere, ocean)
- Modeling for biofluids: hemodynamics, cerebrospinal fluids, cancer modelling, animal locomotion, bioconvection for swimming microorganisms.

Pre-requisites:

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English   |   

Unit Coordinator: Donato Pera

Aims:

The aim of the course is the study of analytical, numerical and computational methods (on parallel computing structures), for the solution of partial differential equations considered as basic elements for the construction of mathematical models for natural disasters. During the course will be introduced basic concepts related to the analytical and numerical solution for wave equations, elastodynamics equations and advection-reaction-diffusion systems.
In addition to the classical analytical and numerical approach, some basic elements for the simulation of the studied problems on parallel computing structures will be introduced, with reference to the programming of Shared Memory, Distributed Memory and/or GPU computing architectures.
The course activities are consistent with the professional profiles proposed by the Mathematical Engineering master course in relation to the acquisition of programming skills for complex computing structures and the solution of theoretical models.

Learning outcomes

At the end of the course the student should be able to:

1) Know the basic aspects related to the analytical and numerical solutions of the proposed models.
2) Use parallel computing codes related to the models proposed during the course.
3) Propose solutions related to problems similar to the models proposed during the course.
4) Understand technical-scientific texts on related topics.

Content:

Wave equations analytical and numerical methods, elastodynamic equations analytical and numerical methods.
Diffusion equations analytical and numerical methods.
Advection-reaction-diffusion systems analytical and numerical methods.
Introduction to parallel computing architectures:
Shared memory systems, distributed memory systems and GPU computing.
Models for performance evaluation Speedup, Efficiency and Amdahl's law.
Introduction to Linux/Unix operating systems and scheduling for HPC applications.
Message passing interface (MPI) programming
MPI basic notions, point-to-point communications, collective communications

Pre-requisites:

Basic knowledge of mathematical analysis, numerical analysis and scientific programming

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English   |   

Unit Coordinator: Federica Di Michele

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English   |   

Unit Coordinator: TBD

Aims:

TBD

Content:

TBD

Pre-requisites:

TBD

ECTS Credits: 9   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English   |   Code: DT0113

Unit Coordinator: Corrado Lattanzio

Aims:

LEARNING OBJECTIVES.
The course aims at providing advanced mathematical notions used in the field of (applied) mathematical analysis and their applications to a variety of topics, including the basic equations of mathematical physics and some current research topics about linear and nonlinear partial differential equations.
Those objectives contribute to the learning goals of the entire course of studies, as the inner coherence of the master degree in Mathematics was verified at the time of the planning of the master program.
LEARNING OUTCOMES.
At the end of the course, the student should:
1. know the advanced mathematical notions used in the field of (applied) mathematical analysis, as measure theory, Sobolev Spaces, distributions, and their applications to the theory of linear and non-linear partial differential equations;
2. understand and be able to explain thesis and proofs in advanced mathematical analysis;
3. have strengthened the logic and computational skills;
4. be able to read and understand other mathematical texts on related topics.

Content:

Distributions. Locally integrable functions. The space of test function D(Ω). 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. Fourier transform and tempered distributions. Convolution between distributions and regularization of distributions. Denseness of D(Ω) in D′(Ω).

Sobolev spaces. Definition of weak derivatives and his motivation. Sobolev spaces Wk,p(Ω) and their properties. Interior and global approximation by smooth functions. Extensions. Traces. Embeddings theorems: Gagliardo–Nirenberg–Sobolev inequality and 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. Embedding theorem for p = n. Characterization of the dual space H-1.

Second order parabolic equations. Definition of parabolic operator. Weak solutions for linear parabolic equations. Existence of weak solutions: Galerkin approximation, construction of approximating solutions, energy estimates, existence and uniqueness of solutions.

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.

Measures. System of sets, Positive Measures, Outer Measures, Construction of Measures, Signed Measures, Borel and Radon Measures
Integration. Measurable Functions, Simple Functions, Convergence Almost Everywhere, Integral of Measurable Functions, Convergences of Integrals, Fubini-Tonelli Theorems.
Differentiation. The Radon-Nikodym Theorem, Differentiation on Euclidean space, Differentiation of the real line.
Radon measures and continuous functions. Spaces of continuous functions, Riesz Theorem.

Pre-requisites:

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.

Reading list:

- L. Ambrosio, G. Da Prato, A. Mennucci. Introduction to measure theory and integration. Edizioni della Normale.

- L. Ambrosio, N. Fusco, D. Pallara. Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs.

- V.I. Bogachev. Measure theory, Volume I, Springer.

- H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, Springer.

- P. Cannarsa, T. D’Aprile. Introduction to Measure Theory and Functional Analysis. Springer.

- C.M. Dafermos. Hyperbolic Conservation Laws in Continuum Physics, Springer.

- L.C. Evans. Partial Differential Equations. Graduate Studies in Mathematics, Vol. 19, AMS.

- L. Evans, R. Gariepy. Measure Theory and Fine Properties of Functions, CRC Press. Revised Edition.

- G.B. Folland. Real analysis: Modern techniques and their applications. New York Wiley

- G. Gilardi. Analisi 3. McGraw–Hill.

- L. Grafakos, Classical Fourier Analysis. Springer.

- V.S. Vladimirov. Equations of Mathematical Physics. Marcel Dekker, Inc

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English   |   

Unit Coordinator: Simone Fagioli

Aims:

• To learn the basics in the mathematical modelling of population dynamics.
• To provide a mathematical description of ODE models in population dynamics and the interpretation of the qualitative behaviour of the solutions
• To get the basic notions in mathematical models in epidemiology and reaction kinetics.
• To learn the mathematical modelling of population models in heterogeneous environment, described by partial differential equations.
• To deal with advanced models in biology such as chemotaxis models and structured dynamics equations.
• To get a sound background in reaction diffusion phenomena, Turing instability, and pattern formation.

Content:

Continuous Population Models for Single Species. Continuous Growth Models. Delay models. Linear Analysis of Delay Population Models: Periodic Solutions.
Continuous models for Interacting Populations. Predator-Prey Models: Lotka-Volterra Systems. Realistic Predator–Prey Models. Competition Models: Principle of Competitive Exclusion. Mutualism or Symbiosis 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
Matematical models for tumor growth.
Structured population dynamics. An example in ecology: competition for resources. Continuous traits. Evolutionary stable strategy in a continuous model.

Pre-requisites:

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).

Reading list:

Murray - Mathematical Biology I&II

Lecturers lecture notes

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English   |   

Unit Coordinator: Marco Di Francesco, Antonio Esposito

Aims:

At the end of the course, the student will be familiar with multi-agent systems models of discrete type (both deterministic and stochastic), with their meso-scopic formulation, with their continuum formulation, both of first and second order, and with their formulation on a graph. The students will acquire the mathematical techniques to solve those models suitably and will be able to use those models in various interdisciplinary applications and to adapt them to specific problems in contexts of interest in health-care systems such as diagnostics and imaging, neural networks, genetics, epidemiology, dynamic data management, and biological aggregation phenomena in physiology.

Content:

• Discrete particle systems for interacting agents. Models with external field. Models with nonlocal aggregation/repulsion forces. Models with alignment, self-propulsion and friction. Swarms models (Vicsek) . Opinion models (Sznajd, Krause). Examples of asymptotic behaviour. The stochastic case.
• Control for discrete models. Mean-field games. Application to optimisation problems. Many species models and models with species transitions. Applications in genetics, imaging, and data science.
• Complementary topics of abstract measure theory. Measure topologies. Transport of measures.
• Mesoscopic models of Vlasov type. Derivation as mean field limits from discrete particle models. Formal derivation of continuum second order models. Derivation of first order models in friction dominated regimes.
• Derivation of linear and nonlinear diffusion models from particle systems. Existence of solutions to the nonlinear diffusion equation. Asymptotic self-similar behavior. 
• Aggregation-diffusion equations. Existence of solutions with linear diffusion. Existence in the  diffusion-less case and formation of clusters in finite time. Stationary states with quadratic diffusion. The case of many species. Application to epidemiology.
• Introduction to graph modelling of nonlocal type. Applications to neural networks.
• Probabilistic label-switching models and applications.

Pre-requisites:

Ordinary differential equations, real and functional analysis

Reading list:

Lecture notes will be provided.

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English   |   

Unit Coordinator: Raffaele D'Ambrosio

Aims:

Learning objectives.

The aim of the course is to provide knowledge and skills in the field of numerical modelling in the epidemiological field, creating an effective bridge between the understanding of the continuous model, its qualitative properties and their counterparts in the discrete.

Learning outcomes.

At the end of the course, the student should

- have a deep knowledge and understanding of the most relevant computational techniques for the treatment of models in the epidemiological field, together with aspects related to their implementation in accurate and efficient mathematical software;
- demonstrate the ability to evaluate the most appropriate discretization in relation to the problem to be solved and ability in theoretical analysis and mathematical software design;
- demonstrate ability to read and understand other texts on related topics.

Content:

- Compartmental models in Epidemiology.
- Stochastic models.
- Epidemiological models based on pds.
- An introduction to network models for the spread of epidemics. 

For each theme, the presentation will be twofold: presenting the continuous model, its qualitative properties and, jointly, presenting the most appropriate numerical approach to the approximation of solutions and the discrete conservation of the properties of the model. In addition, advanced techniques of numerical linear algebra and optimization for large linear systems arising from semi-discretization of PDEs in epidemiology will be presented.

Pre-requisites:

Basic knowledge of Numerical Analysis, differential equations, linear algebra.

Reading list:

- F. Brauer et al., Mathematical Epidemiology, Springer-Verlag (2008).
- J. D. Lambert, Numerical methods for ordinary differential systems: the initial value problem, John Wiley (1991).
- A. Quarteroni, Numerical Models for Differential Problems, Springer (2017).
- E. Isaacson, H.Keller, Analysis of numerical methods, Dover Publications (1994).
- D.J. Higham and P. E. Kloeden, An Introduction to the Numerical Simulation of Stochastic Differential Equations, SIAM, 2021.
--Walter Gander, Martin J. Gander, Felix Kwok. Scientific Computing
An introduction using Maple and MATLAB. Springer.

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English   |   

Unit Coordinator: Alessandro Borri

Aims:

Systems biology is an emerging research area, which aims at providing mathematical models helping to understand the dynamic interactions occurring within and among cells. This course provides the basic mathematical tools to model and analyze gene transcription as well as biochemical reaction networks: the most important network motifs are investigated exploiting both deterministic and stochastic approaches.

Content:

• Review of basic concepts of biology and probability; deterministic vs. stochastic approach.
• Transcription factors and transcription networks; gene expression: transcription and translation; separation of time scales and the Quasi-Steady-State Approximation (QSSA); activators and repressors; Hill and logic input functions; dynamics of simple regulation and switching production in ordinary differential equation (ODE) form; response time; mass action kinetics and conservation equation; the Michaelis-Menten kinetics; cooperativity; dependent and independent bindings; multi-dimensional input functions.
• Network Motifs: real and randomized networks; the Erdos-Renyi (ER) model; graph properties; negative autoregulation (NAR) and stability analysis; three-node network motifs: feedback loops (FBLs) and feedforward loops (FFLs); coherent and incoherent FFLs; sign-sensitive delay, output pulse generation, speed up of response time, biphasic response curves in I1-FFL; review of the other types of FFL; examples: Arabinose system, Galactose system, flagella motor genes in E. Coli.
• Introduction to Numerical Simulation in Systems Biology; MATLAB: introduction and basic notions; integration of non-stiff differential equations via ode45; randomized computation, Monte Carlo simulation, construction of randomized networks (E-R models) and digraphs.
• General Transcription and Interaction Networks: Single-Input Module (SIM), Multi-Output Feedforward Loop, LIFO and FIFO temporal programs, bi-fans and Densely Overlapping Regulons (DORs), interlocked FFLs; positive autoregulation (PAR) and response slow-down; two-node positive feedback: double-positive (lock-on) and double-negative (toggle-switch) circuits; three-node motives; protein-protein interaction (PPI) networks and hybrid motifs; network motifs in real-world networks: food webs, social networks, electrical circuits, neuronal networks. Examples: sporulation of bacterium Bacillus subtilis, phosphorylation, response to pain stimuli in humans.
• Biological Oscillations: the role of negative feedback and delay; time-separation, feedback strength and cooperativity; noise-induced and delay oscillations; bistability and hysteresis; robustness in biological circuits; models at a higher level of organization: tissues and organs; the glucose-insulin feedback loop.
• The stochastic approach in Systems Biology: motivation, Chemical Reaction Networks, Continuous-Time Markov Chains; the Chemical Master Equation (CME) and its properties, characterization of the CME stationary distribution, the macroscopic equation, one-step processes; stochastic simulation in Systems Biology, inverse sampling method, Gillespie Stochastic Simulation Algorithm (SSA); examples.
• Mesoscopic models and noise: extrinsic vs intrinsic noise, bursty reactions, the Chemical Langevin Equation, the Wiener Process, the Euler-Maruyama integration of Stochastic Differential Equations (SDEs).
• Biological and biomedical examples: metabolism, tumors, epidemics, pharmacokinetics and pharmacodynamics (PKPD), environmental pollution, ecology.

Pre-requisites:

Basic notions of mathematical analysis, dynamical systems and probability.

Reading list:

- Uri Alon, An introduction to systems biology: design principles of
biological circuits, CRC press, 2019.
- E. Klipp, W. Liebermeister, C. Wierling, and A. Kowald, Systems biology: a textbook, John Wiley & Sons, 2016.
- J. D. Murray, Mathematical biology, 3rd edition. New York: Springer, 2001.