Mathematical Modelling and Programming

The course is offered during the first term of the PhD and Master’s Programmes in Science and Management of Climate Change. It is one of the core courses designed to provide students with the mathematical and computational tools needed to understand the behaviour of socioeconomic and environmental systems and to apply dynamic models to describe and predict their functioning and evolution. The course provides a foundation for environmental and socioeconomic modelling, which will be further developed through applications in subsequent courses.

1. Knowledge and understanding:
* Understand the main mathematical tools required for the formulation and analysis of models of environmental and socioeconomic systems;
* Understand the formulation, main properties, and elementary solution methods of ordinary differential equations and difference equations;
* Understand the fundamental concepts of dynamical systems theory;
* Understand and interpret complex phenomena from a systems perspective.
2. Applying knowledge and understanding:
* Select and use appropriate mathematical tools to describe and analyse environmental and socioeconomic systems;
* Formulate and analyse simple mathematical models applied to environmental, socioeconomic, and climate-related problems;
* Apply cross-cutting mathematical tools to the analysis of climate change problems.

3. Making judgements:
* Identify the strengths and limitations of mathematical models used in environmental and socioeconomic modelling.

The student is expected to have a thorough command of the following topics: set theory, elementary algebra, first- and second-degree equations and inequalities (including inequalities involving absolute values), trigonometry, analytic geometry, functions and their properties (domain, codomain, composition and inverse functions, exponential and logarithmic functions).
A basic knowledge of vectors, matrices, and systems of linear equations is also assumed.
In addition, students are required to have at least a basic knowledge of the fundamental concepts of calculus: limits, derivatives of functions (including applications to monotonicity, convexity, and optimization), integrals, and function analysis.

It is ESSENTIAL that students begin the course already possessing these skills.

1. Introduction: course presentation and assessment of prior knowledge.
2. Elements of mathematical analysis and linear algebra: derivatives, integrals, Taylor series, vectors, matrices, linear systems, eigenvalues and eigenvectors, multivariable functions, gradient, Jacobian matrix, and Hessian matrix.
3. Introduction to optimization and linear programming.
4. Ordinary Differential Equations: formulation, main types, Cauchy problem, existence and uniqueness, elementary solution methods, and second-order differential equations.
5. Difference equations and continuous- and discrete-time dynamical systems.
6. Analysis of dynamical systems: equilibria, stability, linear systems, phase space, phase portraits, and qualitative methods. Introduction to Euler and Runge-Kutta numerical methods and to dynamic optimization.
7. Nonlinear dynamics and complex systems: linearization, introduction to chaos and catastrophe theory, fractal geometry, emergent properties, and networks of systems.

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

The instructor is responsible for ensuring the authenticity and originality of all examinations and coursework. In cases of suspected academic misconduct, an additional on-site assessment may be required during the exams, which may differ from the standard format.

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.

The suggested program is a rough estimate; the course content will depend in part on the average level of preparation of the class.

This subject deals with topics related to the macro-area "Natural capital and environmental quality" and contributes to the achievement of one or more goals of U. N. Agenda for Sustainable Development