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

Content:

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

Content:

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

Content:

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

Content:

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

Content:

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.