Mathematical Modelling and Programming

The course is part of the first trimester of the PhD and Master's Programs in Science and Management of Climate Change. It is one of the core courses, providing students with the mathematical and computational tools needed to understand socioeconomic and environmental systems and apply quantitative models to describe and predict their functioning and evolution. The course is a cornerstone for understanding environmental and socioeconomic modeling, which will be explored in greater depth through applications in subsequent courses.

1. Knowledge and Understanding:
* Understand the fundamentals of single- and multivariable calculus, linear algebra, and optimization.
* Acquire the foundations for understanding and solving optimization problems.
* Be able to define and describe complex phenomena related to environmental and socioeconomic systems, interpreting them through calculus.
* Acquire the theoretical and methodological tools necessary to develop a systemic view of environmental and socioeconomic phenomena.
2. Ability to Apply Knowledge and Understanding:
* Be able to select and use appropriate mathematical tools to describe and analyze complex phenomena.
* Solve applied problems using mathematical models.
* Be able to address specific applied problems related to climate change with transversal mathematical tools.
3. Judgment:
* Be able to identify the potential and limitations of mathematical models used in environmental and socioeconomic modeling.

Students are expected to have a thorough understanding of the following topics: set theory, elementary algebra, equations and inequalities (both first- and second-degree, including absolute-value inequalities), analytic geometry, functions and their properties (domain, codomain, exponential and logarithmic functions). Knowledge of the main concepts of vectors, matrices, and systems of linear equations is assumed.

Students are also expected to have at least a basic understanding of the fundamentals of calculus (derivatives of functions, and applications to monotonicity, optimization, convexity, ...), which will however be revisited and explored further during the course.

It is ESSENTIAL that students arrive at the course already possessing these skills

1. Introduzione:
- Presentazione del corso, verifica conoscenze pregresse.
2. Elementi di calcolo:
- funzioni e loro proprietà
- derivate, ed applicazioni (ottimizzazione, convessità);
- Definizione di funzione multivariata, esempi. Derivate parziali. Gradiente, matrice Jacobiana ed Hessiana.
- Ottimizzazione libera e vincolata, ad una e più variabli. Cenno ai moltiplicatori di Lagrange;
3. Algebra lineare:
- vettori, prodotto scalare e vettoriale, matrici, somma e prodotto tra matrici, matrice trasposta, determinante;
- Risoluzione sistemi di equazioni lineari;
- Autovalori, autovettori.
4. Modelli di Sistemi Dinamici (opzizonale):
- Definizione di sistema dinamico, sistema complesso, sistemi a tempo continuo e discreto, stabilità, equilibri, in particolare a tempo discreto;
- Equazioni differenziali Ordinarie (EDO) 1: tipi di EDO, problema di Cauchy, esistenza e unicità delle soluzioni;
- Sistemi dinamici lineari;
- Cenni a dinamiche non lineari, teoria del caos ed attrattori, struttura frattale.

Attendance at the lectures is mandatory.
Relevant teaching materials will be provided during the lectures and made available on Moodle.
Some reference books will be recommended throughout the course.

The assessment will include a final written exam, possibly followed by an oral examination. During the course, some intermediate assessment tests may be administered.

written and oral

For this course the votes will be expressed in thirtieths

The lectures will take place in the classroom, with explanations given on the board and supported by multimedia materials to enhance intuition and understanding of the concepts.