Quantitative Methods

A.Y. 2021/2022
Overall hours
Learning objectives
The purpose is that students learn the main mathematical and computational tools needed for formal methods in advanced courses for Environmental Science, and other life sciences. The course serves mostly to refresh students' knowledge in certain topics, and to ensure that all students taking the advanced courses have a common mathematical level.
Expected learning outcomes
Students should develop an understanding of the dynamical systems with application in the environmental science and the knowledge of optimization methods.
Course syllabus and organization

Single session

Lesson period
First semester
Live lectures (on line) in the form of interactive lessons recorded and made available to students. Asynchronous lectures for deepen some topics. On line Tutoring for teaching support.
Course syllabus
Review basic calculus one real variable.
Linear Algebra and applications. Real vector spaces. Linear combination, dependence and linear independence. Basis and dimension in R^n. Algebra of vectors, inner product and Norm. Matrix algebra (inverse, rank, derivatives, eigenvalues, diagonalization and factorization). Graph theory and applications.
Calculus. Real functions on Rn (continuity, differentiability, implicit function theorem, basic fixed point theorem, gradient).

Optimization. First and Second order conditions for unconstrained problems. Constrained optimization: equality constraints and Lagrange Multipliers. Inequality constraints. Linear programming.
Discrete and continuous dynamical systems with applications.
[Computational methods. Basic numerical methods for discrete and continuous dynamical systems. MATLAB or R Laboratory].
Prerequisites for admission
Prerequisites for this course include a good knowledge of the mathematical tools presented in a basic Calculus course and a Basic Linear Algebra course.
Teaching methods
Teaching method consists in starting the course with the revision of some basic notions of Calculus introducing students to some mathematical models in life sciences. This activity is developed by considering more complex methods and phenomena. The main issues of optimization and computational methods are then presented. Part of the activity could be carried out using a computer based laboratory.
Teaching Resources
REFERENCE TEXTS (not mandatory)
K. Sydsaeter, P. Hammond, A. Strom, A. Carvajal, Essential Mathematics for Economic Analysis, Pearson, 2016
E. Salinelli, F. Tomarelli, Discrete-Dynamical Models, Springer, 2014, ISBN: 978-3-319-02290-1

Lecture notes be uploaded on the course web site (http://ariel.unimi.it)
Assessment methods and Criteria
There are two components to the final grade: problem sets and test,
and a project. The contribution of each component to the course grade is as follows:
Problem sets 40%
Tests 30%
Final project 30%
MAT/08 - NUMERICAL ANALYSIS - University credits: 6
Practicals: 32 hours
Lessons: 32 hours
Appointment by email
Office or online (by videocall)