Code: DT1303   |   ECTS Credits: 3   |   Delivery: Online   |   Taught hours: 18   |   Language: English

Unit Coordinator: Carmela Scalone

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.

Code: DT0708   |   ECTS Credits: 9   |   Delivery: Online   |   Taught hours: 54   |   Language: English

Unit Coordinator: Simone Fagioli, Marco Di Francesco, Emanuela Radici

Content:

Sets, operations with sets, subsets, power set, Cartesian product. Sets of numbers, integers, rational numbers, gentle introduction to real numbers.

A resume of the algebraic properties of rational numbers, ordering in rational numbers. Definition of real numbers through decimal alignment. Absolute value. Intervals. A non-rigorous definition of the separation property or real numbers.

Introduction to functions on arbitrary sets. Image and pre-image. Surjective and injective functions. Composition of functions. The identical function on a set. Invertible functions and their inverse.

Power laws, exponentials, and logarithms in the set of real numbers. 

Cardinality of infinite sets. Countable sets. Cardinality of real numbers.

More on functions of real numbers. Domain of a function. Examples. Operations on the graph of a functions through translations, dilations, and absolute values. Examples of elementary functions. Bounded functions, monotone functions, even and odd functions, periodic functions. Trigonometric functions and their inverse.

Upper bounds and lower bounds. Bounded and unbounded subsets of the real line. Maxima and minima. Supremum and infimum. Examples.

Complex numbers. Cartesian form, real and imaginary part, conjugacy, modulus, operations with complex numbers. Representation on the complex plane. Trigonometric form. De Moivres' formulas. Roots of a complex number. Algebraic equations on complex numbers. Fundamental theorem of algebra.

Polynomials on real numbers. Algebraic and transcendental equations. Trigonometric equations, exponential equations, logarithmic equations. Exercises.

The Cartesian plane. Geometric loci on the plane: straight lines, parabolas, circles, ellipses, hyperboles. Exercises. Solution of nonlinear systems and intersection of geometric loci in the plane.

Solution of algebraic and transcendental inequalities on the real line. Exercises.

Introduction to the study of the graph of a functions of real variables. Domain, zeroes, sign. Exercises

Code: DT0710   |   ECTS Credits: 3   |   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.

Code: DT1014   |   ECTS Credits: 6   |   Delivery: Online   |   Taught hours: 36   |   Language: English

Unit Coordinator: Raffaele D'Ambrosio

Code: DT0712   |   ECTS Credits: 3   |   Delivery: Blended   |   Taught hours: 18   |   Language: English

Unit Coordinator: Juri Di Rocco

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