Luca ESPOSITO | PRINCIPLES OF ADVANCED MATHEMATICAL ANALYSIS
Luca ESPOSITO PRINCIPLES OF ADVANCED MATHEMATICAL ANALYSIS
cod. 0522200010
PRINCIPLES OF ADVANCED MATHEMATICAL ANALYSIS
| 0522200010 | |
| DEPARTMENT OF MATHEMATICS | |
| EQF7 | |
| MATHEMATICS | |
| 2024/2025 |
| OBBLIGATORIO | |
| YEAR OF COURSE 1 | |
| YEAR OF DIDACTIC SYSTEM 2018 | |
| FULL ACADEMIC YEAR |
| SSD | CFU | HOURS | ACTIVITY | ||
|---|---|---|---|---|---|
| ISTITUZIONI DI ANALISI SUPERIORE (A) | |||||
| MAT/05 | 6 | 48 | LESSONS | ||
| ISTITUZIONI DI ANALISI SUPERIORE (B) | |||||
| MAT/05 | 6 | 48 | LESSONS | ||
| Exam | Date | Session | |
|---|---|---|---|
| ISTITUZIONI DI ANALISI SUPERIORE | 21/01/2025 - 09:00 | SESSIONE DI RECUPERO | |
| ISTITUZIONI DI ANALISI SUPERIORE | 11/02/2025 - 09:00 | SESSIONE DI RECUPERO | |
| ISTITUZIONI DI ANALISI SUPERIORE | 26/02/2025 - 09:00 | SESSIONE DI RECUPERO |
| Objectives | |
|---|---|
| GENERAL OBJECTIVE: THE COURSE AIMS TO PROVIDE STUDENTS WITH BOTH A SOLID THEORETICAL FOUNDATION OF ADVANCED METHODS IN MODERN MATHEMATICAL ANALYSIS AND THE ABILITY TO APPLY THIS KNOWLEDGE IN PRACTICAL CONTEXTS COMMONLY ENCOUNTERED IN THE DEVELOPMENT AND APPLICATIONS OF COMPUTER SCIENCE AND MATHEMATICAL PHYSICS. KNOWLEDGE AND UNDERSTANDING: STUDENTS WILL ACQUIRE A THOROUGH UNDERSTANDING OF THE FUNDAMENTAL CONCEPTS OF MEASURE THEORY AND INTEGRATION THEORY, AS WELL AS THE STRUCTURE OF LEBESGUE SPACES. THEY WILL ALSO LEARN THE BASIC TECHNIQUES OF CALCULUS IN BANACH AND HILBERT SPACES. FURTHERMORE, THEY WILL BE INTRODUCED TO THE THEORY AND METHODS OF COMPLEX VARIABLE FUNCTIONS, FOURIER ANALYSIS, AND ITS APPLICATIONS TO DIFFERENTIAL EQUATIONS. THIS WILL ENABLE THEM TO GRASP THE LANGUAGE AND FUNDAMENTAL TOOLS UNDERLYING ANY MODERN THEORY IN MATHEMATICAL PHYSICS OR COMPUTER SCIENCE THAT UTILIZES THE FUNDAMENTAL TOOLS OF DIFFERENTIAL CALCULUS. ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING: THE STUDENT MUST BE ABLE TO FORMULATE SIMPLE VARIATIONS OF THE THEORETICAL RESULTS LEARNED AND PROVIDE A PROOF OF THEM, AS WELL AS TO APPLY THEM IN APPLICATIVE CONTEXTS INVOLVING: SEQUENCES AND SERIES IN METRIC AND NORMED SPACES, PROJECTIONS AND DISTANCES IN HILBERT SPACES, LAURENT SERIES EXPANSIONS, RESIDUES, FOURIER SERIES AND TRANSFORMS. JUDGMENT AUTONOMY: OBJECTIVE: TO FOSTER THE DEVELOPMENT OF STUDENTS' JUDGMENT AUTONOMY SO THAT THEY CAN CRITICALLY ANALYZE THE CONCEPTS AND THEORIES OF THE CALCULUS OF VARIATIONS. METHODS: TO STIMULATE STUDENTS TO CRITICALLY EVALUATE THE EVIDENCE AND ARGUMENTS PRESENTED DURING THE COURSE AND TO ENHANCE THEIR DECISION-MAKING CAPACITY IN MATHEMATICAL CONTEXTS. COMMUNICATIVE SKILLS: OBJECTIVE: TO IMPROVE STUDENTS' COMMUNICATIVE SKILLS SO THAT THEY CAN CLEARLY EXPRESS THEIR OWN IDEAS AND ARGUMENTS, BOTH ORALLY AND IN WRITING, USING APPROPRIATE TECHNICAL LANGUAGE. METHODS: TO OFFER STUDENTS THE OPPORTUNITY TO PARTICIPATE ACTIVELY DURING CLASSES, ENCOURAGING THEM TO EXPRESS THEIR IDEAS, ASK QUESTIONS, AND PARTICIPATE IN DISCUSSIONS. FURTHERMORE, TO PROMOTE THE WRITING OF PROOFS THAT REQUIRE CLEAR AND WELL-STRUCTURED COMMUNICATION OF MATHEMATICAL IDEAS. LEARNING ABILITY: OBJECTIVE: TO FOSTER STUDENTS' AUTONOMOUS LEARNING SO THAT THEY CAN DEVELOP EFFECTIVE STUDY STRATEGIES, DEEPEN THEIR UNDERSTANDING OF CONCEPTS, AND APPLY THE ACQUIRED KNOWLEDGE CREATIVELY. METHODS: TO PROVIDE RESOURCES AND TEACHING MATERIALS THAT ALLOW STUDENTS TO DEEPEN THEIR COMPETENCE IN CONCEPTS BEYOND CLASSROOM LECTURES, SUCH AS RECOMMENDED READINGS, ADDITIONAL EXERCISES, AND OPTIONAL ADVANCED TOPICS. FURTHERMORE, TO ENCOURAGE ACTIVE PARTICIPATION OF STUDENTS THROUGH CRITICAL ANALYSIS OF MATHEMATICAL PROBLEMS AND INDEPENDENT RESEARCH OF SOLUTIONS. |
| Prerequisites | |
|---|---|
| KNOWLEDGE OF THE THEORY OF FUNCTIONS OF SEVERAL VARIABLES. MEASURE AND RIEMANN INTEGRALS IN R^N. BASIC TOPOLOGY. |
| Contents | |
|---|---|
| PART I (48 HOURS LESSON) - 1.TOPOLOGY OF METRIC AND STANDARD SPACES (4 HOURS LESSON). BANACH SPACES AND CONTINUOUS FUNCTION SPACES [GI] (2 HOURS LESSON). ASCOLI-ARZELÀ THEOREM (2 HOURS LESSON). (TOTAL 8 HOURS LESSON) 2.LEBESGUE MEASUREMENT AND INTEGRATION THEORY. POSITIVE BOREL MEASUREMENTS (12 HOURS LESSON). MONOTONE CLASSES AND MEASUREMENT EXTENSION THEOREM (4HOURS LESSON) [CA]. SPACES LP [RU] (6 HOURS LESSON). CONVOLUTION AND REGULARIZATION (6 HOURS LESSON). RIESZ-FRÉCHET-KOLMOGOROV THEOREM [BR] (2 HOURS LESSON). (TOTAL 30 HOURS LESSON) 3.HILBERT SPACES [RU](10 HOURS LESSON) PART II (40 HOURS LESSON + 8 HOURS EXERCISE) - 1. THE COMPLEX PLAN. DERIVABILITY IN THE COMPLEX SENSE (4 HOURS LESSON). INTEGRATION IN THE COMPLEX FIELD (4 HOURS LESSON). CAUCHY INTEGRAL THEOREM (2 HOURS LESSON)[CO/GR]. (TOTAL 10 HOURS LESSON) 2. CAUCHY'S INTEGRAL FORMULA AND APPLICATIONS (4 HOURS LESSON). ANALYTICAL FUNCTIONS (2 HOURS LESSON). PRINCIPLES OF IDENTITY (2 HOURS LESSON). LAURENT SERIES (4 HOURS LESSON). CLASSIFICATION OF ISOLATED SINGULARITIES (2 HOURS LESSON). RESIDUE THEORY [CO/GR (2 HOURS LESSON + 4 HOURS EXERCISE)]. [CO/GR]. (TOTAL 16 HOURS LESSON + 4 HOURS EXERCISE) 3.FOURIER SERIES. [GI]. APPLICATION TO BOUNDARY VALUE PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS (PDE). (6 HOURS LESSON + 4 HOURS EXERCISE) 4. FOURIER TRANSFORM. L1 THEORY AND INVERSION FORMULA. L2 THEORY AND PLANCHEREL THEOREM [RU]. APPLICATION TO INITIAL VALUE PROBLEMS FOR PDE. (8 HOURS LESSON) |
| Teaching Methods | |
|---|---|
| THE COURSE INCLUDES 96 HOURS LESSON DEVIDED IN 88 HOURS OF THEORETICAL LESSONS AND 8 HOURS OF EXERCISE. THE WAY IN WHICH THE ACQUIRED KNOWLEDGE CAN BE USED FOR THE SOLUTION OF PROBLEMS CONNECTED TO THE TOPICS COVERED WILL ALSO BE ILLUSTRATED. PARTICIPATION IN THE FRONTAL TEACHING IS STRONGLY RECOMMENDED. |
| Verification of learning | |
|---|---|
| THE EXAM CONSISTS OF AN ORAL TEST AIMED AT ASSESSING THE WHOLE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED IN THE LESSONS WITH CONCEPTUAL AND TECHNICAL QUESTIONS ON THE TOPICS CONSIDERED IN THE LESSONS. DURING THE ORAL EXAMINATION THE CANDIDATE WILL ALSO BE ASKED TO CARRY OUT AN EXERCISE OF THE SAME TYPE AS THOSE CARRIED OUT IN LESSON. HONOR MAY BE GIVEN TO STUDENTS WHO DEMONSTRATE THEY ARE ABLE TO INDEPENDENTLY APPLY THE KNOWLEDGE AND SKILLS ACQUIRED EVEN IN CONTEXTS DIFFERENT FROM THOSE PROPOSED IN LESSON. |
| Texts | |
|---|---|
| [CA] P.CANNARSA, T.D'APRILE, INTRODUZIONE ALLA TEORIA DELLA MISURA E ALL'ANALISI FUNZIONALE, SPRINGER 2008 [CAP. 1] [GI] E. GIUSTI, ANALISI MATEMATICA 2, BOLLATI BORINGHIERI ED. 1984 [CAP. 1; 2] [RU] W. RUDIN, ANALISI REALE E COMPLESSA, BORINGHIERI [CAP. 1; 2; 3; 4; 9] [BR] H. BREZIS, ANALISI FUNZIONALE (TEORIA E APPLICAZIONI), LIGUORI [CAP. 4: $4,5] [CO] J.B. CONWAY, FUNCTIONS OF ONE COMPLEX VARIABLE, GTM, SPRINGER-VERLAG 2ND ED. [CAP. 1; 3: $1,2; 4; 5; 7: $5,7,8] O IN ALTERNATIVA [GR] D. GRECO, COMPLEMENTI DI ANALISI, LIGUORI ED. 1980 [PARTE I] |
| More Information | |
|---|---|
| WEB: HTTPS://DOCENTI.UNISA.IT/003512/RISORSE |
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