Pre MSc Foundation Programme in Applied Mathematics

The Pre-Master's Foundation Programme (PMFP) in Applied Mathematics aims at homogenizing the competencies portfolios of prospective students of the two Master's Programmes in Mathematical Modelling and Mathematical Engineering at the University of L'Aquila, which include the Erasmus Mundus program "InterMaths - Interdisciplinary Mathematics", the joint Master's program "MathMods", and the "InterMaths" Double Degree program.

Depending on the student's undergraduate study programs and on the education system in their country of origin, students enrolling in these three programs may feature a very diverse set of skills in disciplines that characterize these Master's programs. The PMFP in Applied Mathematics is designed to address this issue by covering specific competencies in both theoretical mathematics (Real Analysis and Linear Algebra) and computer programming. As for theoretical mathematics, the main goal of the PMFP is bridging the gap between "calculus" and "real analysis", a typical issue arising quite often for prospective MSc students with a very "applied" background.

The PMFM will include very basic topics of real analysis enabling the students to deal with infinitesimal calculus with a rigorous "real analysis" perspective (including the use of rigorous mathematical proofs). On the other hand, students with a strong "theoretical" background sometimes lack basic programming and computational skills. Hence, the PMFP provides a basic introduction to computer programming and in particular to the computing environment "MATLAB", which is widely used in the numerical analysis courses of the MSc programs mentioned above.

Modules

Part 1

• A Crash Course on Linear Algebra

Linear spaces, linear dependence, bases of a linear space, dimension of a linear space, linear subspaces.

Matrices, basic operations with matrices, change of coordinates, determinants, rank. A brief account on linear systems and Gauss elimination.

Diagonalisation of squared matrices, eigenvalues, eigenvectors. Inner products, bilinear forms, and quadratic forms.

• Differential Equations: Foundations

General introduction to differential equations, Cauchy problems.

Existence and uniqueness of solutions. Peano’s and Cauchy’s theorems. Examples, Peano’s brush.

Introduction to linear differential equations. Examples.

A brief outline of qualitative analysis of Cauchy problems. Comparison of solutions, maximal solutions, global existence of solutions, blow-up of solutions. Examples.

• Real Analysis: Foundations

Propositional logic. Propositional calculus.

Sets, set operations, relations, functions. The cardinality of sets, countable sets, uncountable sets. Elementary number sets. Integers and rationals. Induction principle.

More on functions: injective and surjective functions, invertible functions, image, and pre-image.

The set of real numbers. Separation axiom, Dedekind cuts. Infimum and supremum. Archimedean property. Complex numbers: cartesian and trigonometric form, basic properties, powers, complex roots, fundamental theorem of algebra.

Sequences of real numbers: monotone sequences, the convergence of a sequence, subsequences, limsup and liminf of a sequence, Bolzano-Weierstrass theorem.

Introduction to functions of real numbers. Elementary functions: exponential and logarithmic function, trigonometric functions, irrational functions. Monotone functions.

The topology of real numbers: intervals, half-lines, open sets, closed sets. The topology of the Euclidean space Rn: balls, open and closed sets. Compact sets in the Euclidean space.

Part 2

• Introduction to MATLAB

The MATLAB environment, Basic computer programming, Variables and constants, operators and simple calculations, Formulas, and functions. MATLAB toolboxes.

Matrix and linear algebra review, Vectors, and matrices in MATLAB, Matrix operations, and functions in MATLAB.

Algorithms and structures, MATLAB scripts and functions (m-files), Simple sequential algorithms, Control structures (if...then, loops).

Reading and writing data, file handling, Personalized functions, MATLAB graphic functions. Interactive hands-on-sessions.

• Introduction to Programming

Algorithms, programs, and programming languages.

The learning environment for the Python programming language and Turtle Graphics. Commands and sequences of commands. Writing and executing a program.

Definite iteration. Procedures: defining and calling Python functions. Procedures with parameters.

Variables and objects. Basic data types in Python. Expressions.

Selection, recursion, and indefinite iteration.

Basic data structures in Python: tuples, strings, lists, dictionaries.