Programme

Part 1: Introduction to Lebesgue's theory in R^n
Definition of L^1 functions.
Theorems on the integration of limits (monotone convergence, dominated convergence, Fatou's lemma).
Completeness of L^1 (Riesz-Fischer Theorem).
Iterated integrals and Fubini's theorem.
Measurable functions and Lebesgue measure.
Convolution and regularization.
Theorem of the change of variables in R^n.
Divergence theorem in R^n.

Part 2: Fourier in L^2
The Hilbert space L^2 (on bounded domains and on R^n).
Fourier series and transforms in L^2.

Part 3: Fundamentals of the theory of ordinary differential equations
Examples and classes of ordinary differential equations.
Local existence and uniqueness theorem (Picard-Lindelof).
Lipschitz dependence on initial data.
Maximum and global solutions; globality criteria.
Linear systems (linear structure, Wronskian); non-homogeneous systems (variations of constants; Liouville's theorem.
Linear systems with constant coefficients (exponential solution).
Jordan canonical form and qualitative analysis of solutions).
Flows. Variational equation. Parameter dependence C^k.
Introduction to qualitative analysis.
Phase space.
Use of Fourier theory in differential equations (outline).

Core Documentation

During the lessons, typed notes will be provided.

Type of delivery of the course

Lectures and exercises in class. All the material of the program will be explained in class. The lessons / exercises will include a continuous dialogue with the students: the feedback from the students during the course is a fundamental tool for the success of the course itself. In the event of an extension of the health emergency from COVID-19, all the provisions (of the State and of the Roma Tre University) governing the methods of carrying out educational activities will be implemented. In particular, distance learning may be necessary.

Attendance

Attendance is optional and the understanding of the text adopted is sufficient for the full use of the course. Of course, attendance is desirable and STRONGLY recommended, as the interaction between teacher and students is a fundamental and unrepeatable teaching tool.

Type of evaluation

The evaluation is based on a written test and an oral exam. The written part will mainly focus on part 2 and 3.

Programme

Part 1: Introduction to Lebesgue's theory in R^n
Definition of L^1 functions.
Theorems on the integration of limits (monotone convergence, dominated convergence, Fatou's lemma).
Completeness of L^1 (Riesz-Fischer Theorem).
Iterated integrals and Fubini's theorem.
Measurable functions and Lebesgue measure.
Convolution and regularization.
Theorem of the change of variables in R^n.
Divergence theorem in R^n.

Part 2: Fourier in L^2
The Hilbert space L^2 (on bounded domains and on R^n).
Fourier series and transforms in L^2.

Part 3: Fundamentals of the theory of ordinary differential equations
Examples and classes of ordinary differential equations.
Local existence and uniqueness theorem (Picard-Lindelof).
Lipschitz dependence on initial data.
Maximum and global solutions; globality criteria.
Linear systems (linear structure, Wronskian); non-homogeneous systems (variations of constants; Liouville's theorem.
Linear systems with constant coefficients (exponential solution).
Jordan canonical form and qualitative analysis of solutions).
Flows. Variational equation. Parameter dependence C^k.
Introduction to qualitative analysis.
Phase space.
Use of Fourier theory in differential equations (outline).

Core Documentation

During the lessons, typed notes will be provided.

Type of delivery of the course

Lectures and exercises in class. All the material of the program will be explained in class. The lessons / exercises will include a continuous dialogue with the students: the feedback from the students during the course is a fundamental tool for the success of the course itself. In the event of an extension of the health emergency from COVID-19, all the provisions (of the State and of the Roma Tre University) governing the methods of carrying out educational activities will be implemented. In particular, distance learning may be necessary.

Attendance

Attendance is optional and the understanding of the text adopted is sufficient for the full use of the course. Of course, attendance is desirable and STRONGLY recommended, as the interaction between teacher and students is a fundamental and unrepeatable teaching tool.

Type of evaluation

The evaluation is based on a written test and an oral exam. The written part will mainly focus on part 2 and 3.

Programme

Part 1: Introduction to Lebesgue's theory in R^n
Definition of L^1 functions.
Theorems on the integration of limits (monotone convergence, dominated convergence, Fatou's lemma).
Completeness of L^1 (Riesz-Fischer Theorem).
Iterated integrals and Fubini's theorem.
Measurable functions and Lebesgue measure.
Convolution and regularization.
Theorem of the change of variables in R^n.
Divergence theorem in R^n.

Part 2: Fourier in L^2
The Hilbert space L^2 (on bounded domains and on R^n).
Fourier series and transforms in L^2.

Part 3: Fundamentals of the theory of ordinary differential equations
Examples and classes of ordinary differential equations.
Local existence and uniqueness theorem (Picard-Lindelof).
Lipschitz dependence on initial data.
Maximum and global solutions; globality criteria.
Linear systems (linear structure, Wronskian); non-homogeneous systems (variations of constants; Liouville's theorem.
Linear systems with constant coefficients (exponential solution).
Jordan canonical form and qualitative analysis of solutions).
Flows. Variational equation. Parameter dependence C^k.
Introduction to qualitative analysis.
Phase space.
Use of Fourier theory in differential equations (outline).

Core Documentation

During the lessons, typed notes will be provided.

Type of delivery of the course

Lectures and exercises in class. All the material of the program will be explained in class. The lessons / exercises will include a continuous dialogue with the students: the feedback from the students during the course is a fundamental tool for the success of the course itself. In the event of an extension of the health emergency from COVID-19, all the provisions (of the State and of the Roma Tre University) governing the methods of carrying out educational activities will be implemented. In particular, distance learning may be necessary.

Attendance

Attendance is optional and the understanding of the text adopted is sufficient for the full use of the course. Of course, attendance is desirable and STRONGLY recommended, as the interaction between teacher and students is a fundamental and unrepeatable teaching tool.

Type of evaluation

The evaluation is based on a written test and an oral exam. The written part will mainly focus on part 2 and 3.