Mathematics

The course program is divided into three didactic blocks. Each teaching unit includes a part of theory and the related practice sessions. During the course particular attention will be given to the development of the following skills: formalization of abstract reasoning, capacity for analysis and synthesis, problem solving, oral and written communication. Metric spaces (28h). Definition of metric space and examples. Trivial metric, euclidean metric and other metrics in R, Q; discrete metric. The spaces little-elle-2, continuous functions. Sequences in metric spaces: convergent sequences, Cauchy’s sequences, completeness and completion. Sequences defined recursively and the Contraction principle. Topology in metric spaces: open and closed sets, equivalence between compactness and sequential compactness in metric spaces. Sequences and series of functions (28h). Pointwise and uniform convergence, continuity of the uniform limit of continuous functions and change results between limit and integration or derivation. Uniform Cauchy criterion for functions series. Total convergence and Weierstrass criterion. Integrals e derivatives of functions series. Compactness in the space of continuous functions, equilipschitzianity, equicontinuity and equiboundness of sets of continuous functions, Ascoli-Arzelà theorem. Power series in R ad C, radius of convergence, convergence on the boundary. Taylor expansion and Taylor series, real analytic functions. Differential equations (28h). Existence and uniqueness of solution for ordinary differential equations. Equations with separable variables. Maximum existence interval and global existence theorem. Linear differential equations of the first order. Linear differential equations of the second order, structure of the set of solutions of homogeneous and non-homogeneous equations. General integral of linear differential equations of the second order homogeneous with constant coefficients. Similarity method for solving equations non-homogeneous linear second order differentials with constant coefficients.

There are no mandatory preceding exams, however the course requires familiarity with the main topics covered in the basic courses of Calculus I and Linear Algebra, in particular with the following concepts: real numbers, sequences, limits, real functions of real variable, vector space.

The notes written by proff. E.Montefusco and E.Spadaro will be made available at the beginning of the course.

Participation to lectures is recommended, but not mandatory.

The exam aims to assess learning through a written test, which consists of solving problems of the same type as those done in the classroom exercises and/or assigned as a homework load, and an oral test discussing the most relevant topics covered in the course. The written test will last about two hours and may be replaced by two or three intermediate tests lasting two hours. The oral test is non-encouraged for those who do not score at least 18 on the written test. For exemptions, however, it is non-encouraged for those who obtain an average of less than 16. To pass the exam it is necessary to obtain a grade not lower than 18/30. To achieve this, the student must demonstrate to have acquired fairly good skills in the use of concepts introduced during the course, to be able to carry out at least the simplest exercises assigned and to have understood and assimilated the main definitions and the statements of the main results discussed in class. To achieve a score equal to 30/30 cum laude, a student must prove to have acquired an excellent practical skill in problem-solving, an excellent knowledge of all the topics covered during the course and to be able to elaborate them logically and consistently in answering questions.

N. Fusco, P. Marcellini, C. Sbordone, Lezioni di Analisi Matematica 2 , Zanichelli W. Rudin, Principles of Mathematical Analysis, McGraw Hill

Lectures (60%), exercises sessions (40%)

The course program is divided into three didactic blocks. Each teaching unit includes a part of theory and the related practice sessions. During the course particular attention will be given to the development of the following skills: formalization of abstract reasoning, capacity for analysis and synthesis, problem solving, oral and written communication. Metric spaces (28h). Definition of metric space and examples. Trivial metric, euclidean metric and other metrics in R, Q; discrete metric. The spaces little-elle-2, continuous functions. Sequences in metric spaces: convergent sequences, Cauchy’s sequences, completeness and completion. Sequences defined recursively and the Contraction principle. Topology in metric spaces: open and closed sets, equivalence between compactness and sequential compactness in metric spaces. Sequences and series of functions (28h). Pointwise and uniform convergence, continuity of the uniform limit of continuous functions and change results between limit and integration or derivation. Uniform Cauchy criterion for functions series. Total convergence and Weierstrass criterion. Integrals e derivatives of functions series. Compactness in the space of continuous functions, equilipschitzianity, equicontinuity and equiboundness of sets of continuous functions, Ascoli-Arzelà theorem. Power series in R ad C, radius of convergence, convergence on the boundary. Taylor expansion and Taylor series, real analytic functions. Differential equations (28h). Existence and uniqueness of solution for ordinary differential equations. Equations with separable variables. Maximum existence interval and global existence theorem. Linear differential equations of the first order. Linear differential equations of the second order, structure of the set of solutions of homogeneous and non-homogeneous equations. General integral of linear differential equations of the second order homogeneous with constant coefficients. Similarity method for solving equations non-homogeneous linear second order differentials with constant coefficients.

There are no mandatory preceding exams, however the course requires familiarity with the main topics covered in the basic courses of Calculus I and Linear Algebra, in particular with the following concepts: real numbers, sequences, limits, real functions of real variable, vector space.

The notes written by proff. E.Montefusco and E.Spadaro will be made available at the beginning of the course.

Participation to lectures is recommended, but not mandatory.

The exam aims to assess learning through a written test, which consists of solving problems of the same type as those done in the classroom exercises and/or assigned as a homework load, and an oral test discussing the most relevant topics covered in the course. The written test will last about two hours and may be replaced by two or three intermediate tests lasting two hours. The oral test is non-encouraged for those who do not score at least 18 on the written test. For exemptions, however, it is non-encouraged for those who obtain an average of less than 16. To pass the exam it is necessary to obtain a grade not lower than 18/30. To achieve this, the student must demonstrate to have acquired fairly good skills in the use of concepts introduced during the course, to be able to carry out at least the simplest exercises assigned and to have understood and assimilated the main definitions and the statements of the main results discussed in class. To achieve a score equal to 30/30 cum laude, a student must prove to have acquired an excellent practical skill in problem-solving, an excellent knowledge of all the topics covered during the course and to be able to elaborate them logically and consistently in answering questions.

N. Fusco, P. Marcellini, C. Sbordone, Lezioni di Analisi Matematica 2 , Zanichelli W. Rudin, Principles of Mathematical Analysis, McGraw Hill

Lectures (60%), exercises sessions (40%)