Overview: “Pre-Master’s Foundation Programme in Applied Mathematics”
Depending on the student's undergraduate study programmes and on the education system in their country of origin, students enrolling in these three programmes may feature very diverse set of skills in disciplines that characterise these Master's programmes. 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 programmes mentioned above.
Modules :: Pre-Master's Foundation Programme in Applied Mathematics (PMFP-AM)
ECTS Credits: 3 | Dates: 2 - 27 August 2021 | Delivery: Online | Taught hours: 18 | Language: English
Unit Coordinator: Bruno Rubino
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.
ECTS Credits: 8 | Dates: 2 August - 24 September 2021 | Delivery: Online | Taught hours: 48 | Language: English
Unit Coordinator: Marco Di Francesco, Corrado Lattanzio, Rosella Sampalmieri
Propositional logic. Propositional calculus.
Sets, set operations, relations, functions. 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 trigonetric form, basic properties, powers, complex roots, fundamental theorem of algebra.
Sequences of real numbers: monotone sequences, 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.
ECTS Credits: 3 | Dates: 30 August - 17 September 2021 | Delivery: Online | Taught hours: 18 | Language: English
Unit Coordinator: Riccardo Aragona
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.
ECTS Credits: 3 | Dates: 27 September - 17 December 2021 | Delivery: Blended | Taught hours: 18 | Language: English
Unit Coordinator: Luca Forlizzi
Algorithms, programs and programming languages.
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
ECTS Credits: 3 | Dates: 17 January - 31 March 2022 | Delivery: Blended | Taught hours: 18 | Language: English
Unit Coordinator: Antonio Cicone
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.