To acquire a good basic knowledge of the Lebesgue integration theory in R^n, of the Fourier theory and of the fundamental results in the study of ordinary differential equations.
Curriculum
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Part 2: Fourier Transform in L^1. The L^2 Hilbert Space (on bounded domains and on R^n). Fourier Transform in Schwartz Space. Tempered Distributions. The Fourier Transform in L^2. Plancherel's Theorem.
Part 3: Foundations 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. Maximal and global solutions; globality criteria. Gronwall's lemma and comparison theorems. Linear systems (linear structure, Wronskian); nonhomogeneous systems (variations of constants). Linear systems with constant coefficients (exponential solution). Floquet's theorem.
Paolo Acquistapace, lecture notes of Analisi Matematica Due.
Frank Jones, Lebesgue integration on Euclidean space.
Christopher P. Grant, Lecture notes on the Theory of ordinary differential equations.
Programme
Part 1: Lebesgue Integral in R^n. Definition of L^1 functions. Theorems on limit integration (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.Part 2: Fourier Transform in L^1. The L^2 Hilbert Space (on bounded domains and on R^n). Fourier Transform in Schwartz Space. Tempered Distributions. The Fourier Transform in L^2. Plancherel's Theorem.
Part 3: Foundations 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. Maximal and global solutions; globality criteria. Gronwall's lemma and comparison theorems. Linear systems (linear structure, Wronskian); nonhomogeneous systems (variations of constants). Linear systems with constant coefficients (exponential solution). Floquet's theorem.
Core Documentation
Terence Tao, An Introduction to Measure theory.Paolo Acquistapace, lecture notes of Analisi Matematica Due.
Frank Jones, Lebesgue integration on Euclidean space.
Christopher P. Grant, Lecture notes on the Theory of ordinary differential equations.
Reference Bibliography
Terence Tao, An Introduction to Measure theory. Paolo Acquistapace, lecture notes of Analisi Matematica Due. Frank Jones, Lebesgue integration on Euclidean space. Christopher P. Grant, Lecture notes on the Theory of ordinary differential equations.Attendance
Attendance is not mandatory, but is strongly recommended.Type of evaluation
Written exam with some theoretical and practical exercises + oral exam covering the entire course syllabus, starting with a topic of your choice. The written exam may be replaced by the two midterm exams. teacher profile teaching materials
Part 2: Fourier Transform in L^1. The L^2 Hilbert Space (on bounded domains and on R^n). Fourier Transform in Schwartz Space. Tempered Distributions. The Fourier Transform in L^2. Plancherel's Theorem.
Part 3: Foundations 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. Maximal and global solutions; globality criteria. Gronwall's lemma and comparison theorems. Linear systems (linear structure, Wronskian); nonhomogeneous systems (variations of constants). Linear systems with constant coefficients (exponential solution). Floquet's theorem.
Paolo Acquistapace, lecture notes of Analisi Matematica Due.
Frank Jones, Lebesgue integration on Euclidean space.
Christopher P. Grant, Lecture notes on the Theory of ordinary differential equations.
Mutuazione: 20410609 AM300 - ANALISI MATEMATICA 5 in Matematica L-35 R HAUS EMANUELE
Programme
Part 1: Lebesgue Integral in R^n. Definition of L^1 functions. Theorems on limit integration (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.Part 2: Fourier Transform in L^1. The L^2 Hilbert Space (on bounded domains and on R^n). Fourier Transform in Schwartz Space. Tempered Distributions. The Fourier Transform in L^2. Plancherel's Theorem.
Part 3: Foundations 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. Maximal and global solutions; globality criteria. Gronwall's lemma and comparison theorems. Linear systems (linear structure, Wronskian); nonhomogeneous systems (variations of constants). Linear systems with constant coefficients (exponential solution). Floquet's theorem.
Core Documentation
Terence Tao, An Introduction to Measure theory.Paolo Acquistapace, lecture notes of Analisi Matematica Due.
Frank Jones, Lebesgue integration on Euclidean space.
Christopher P. Grant, Lecture notes on the Theory of ordinary differential equations.
Reference Bibliography
Terence Tao, An Introduction to Measure theory. Paolo Acquistapace, lecture notes of Analisi Matematica Due. Frank Jones, Lebesgue integration on Euclidean space. Christopher P. Grant, Lecture notes on the Theory of ordinary differential equations.Attendance
Attendance is not mandatory, but is strongly recommended.Type of evaluation
Written exam with some theoretical and practical exercises + oral exam covering the entire course syllabus, starting with a topic of your choice. The written exam may be replaced by the two midterm exams.