Antonio VITOLO | PRINCIPLES OF ADVANCED MATHEMATICAL ANALYSIS

cod. 0522200010

PRINCIPLES OF ADVANCED MATHEMATICAL ANALYSIS

	0522200010
	DIPARTIMENTO DI MATEMATICA

	MATHEMATICS
	2015/2016

PERCORSO COMUNE

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2010
	ANNUALE

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
1	ISTITUZIONI DI ANALISI SUPERIORE MODULO A
	MAT/05	6	48	LESSONS	COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
2	ISTITUZIONI DI ANALISI SUPERIORE MODULO B
	MAT/05	6	48	LESSONS	COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS

	Objectives
	1. KNOWLEDGE AND UNDERSTANDING: THE COURSE IS FOCUSED ON ADVANCED CONCEPTS OF MATHEMATICAL ANALYSIS, TO BE LEARNED AND UNDERSTOOD BY THE STUDENTS, AS LEBESGUE MEASURE AND INTEGRATION, BANACH AND HILBERT SPACES, FUNCTIONS OF COMPLEX VARIABLE, AND MATHEMATICAL METHODS AND MODEL OF HIGHER LEVEL AS FOURIER ANALYSIS AND PDES. 2. APPLYING KNOWLEDGE AND UNDERSTANDING: THE STUDENT WILL APPRECIATE THE RELEVANCE OF THE LEARNED THEORETICAL RESULTS, TO FORMULATE SIMPLE VARIANTS AND GIVE A PROOF, AND IN ADDITION TO UNDERSTAND THE MEANING OF VARIABLES AND OF COMPUTATIONS IN THE APPLICATIONS. IN PARTICULAR, HE WILL BE ABLE TO VERIFY WHETHER ASSUMPTIONS OF THEOREMS ARE FULFILLED IN CASE STUDIES, TO STUDY THE CONVERGENCE OF SEQUENCES AND SERIES IN METRIC AND NORMED SPACES, TO CALCULATE PROJECTIONS AND DISTANCES IN HILBERT SPACES, TO DEVELOP A COMPLEX FUNCTION IN LAURENT SERIES, TO COMPUTE INTEGRALS BY THE RESIDUES METHOD, TO DEVELOP A FUNCTION IN FOURIER SERIES AND TO COMPUTE FOURIER TRANSFORMS. 3. MAKING JUDGEMENTS: THE STUDENT WILL BE ABLE TO DISCUSS THE ASSUMPTIONS AND THE CONSEQUENCES OF A THEOREM, TO CHOOSE THE MOST APPROPRIATE ARGUMENT OF PROOF AND COMPUTATION TECHNIQUES, TO ESTABLISH THE PLAUSIBILITY OF A RESULT AND TO ELABORATE AUTONOMOUS METHODS OF CHECKING. 4. COMMUNICATION SKILLS: THE STUDENT WILL BE ABLE TO COMMUNICATE AND EXPRESS WITH RIGOROUS AND EFFECTIVE EVIDENCE THE RESULTS OF THE LEARNED TOPICS AND TO PRESENT THEM WITH THE SUITABLE MOTIVATIONS IN THE FRAMEWORK OF RESPECTIVE THEORIES. 5. LEARNING SKILLS: THE STUDENT WILL GET A MATHEMATICAL BACKGROUND WHICH ALLOWS HIM TO LEARN MORE COMPLEX MATHEMATICAL CONCEPTS AND MORE GENERALLY SCIENTIFIC SUBJECTS WHICH USE ADVANCED MATHEMATICAL TOOLS.

1. KNOWLEDGE AND UNDERSTANDING: THE COURSE IS FOCUSED ON ADVANCED CONCEPTS OF MATHEMATICAL ANALYSIS, TO BE LEARNED AND UNDERSTOOD BY THE STUDENTS, AS LEBESGUE MEASURE AND INTEGRATION, BANACH AND HILBERT SPACES, FUNCTIONS OF COMPLEX VARIABLE, AND MATHEMATICAL METHODS AND MODEL OF HIGHER LEVEL AS FOURIER ANALYSIS AND PDES.
2. APPLYING KNOWLEDGE AND UNDERSTANDING: THE STUDENT WILL APPRECIATE THE RELEVANCE OF THE LEARNED THEORETICAL RESULTS, TO FORMULATE SIMPLE VARIANTS AND GIVE A PROOF, AND IN ADDITION TO UNDERSTAND THE MEANING OF VARIABLES AND OF COMPUTATIONS IN THE APPLICATIONS. IN PARTICULAR, HE WILL BE ABLE TO VERIFY WHETHER ASSUMPTIONS OF THEOREMS ARE FULFILLED IN CASE STUDIES, TO STUDY THE CONVERGENCE OF SEQUENCES AND SERIES IN METRIC AND NORMED SPACES, TO CALCULATE PROJECTIONS AND DISTANCES IN HILBERT SPACES, TO DEVELOP A COMPLEX FUNCTION IN LAURENT SERIES, TO COMPUTE INTEGRALS BY THE RESIDUES METHOD, TO DEVELOP A FUNCTION IN FOURIER SERIES AND TO COMPUTE FOURIER TRANSFORMS.
3. MAKING JUDGEMENTS: THE STUDENT WILL BE ABLE TO DISCUSS THE ASSUMPTIONS AND THE CONSEQUENCES OF A THEOREM, TO CHOOSE THE MOST APPROPRIATE ARGUMENT OF PROOF AND COMPUTATION TECHNIQUES, TO ESTABLISH THE PLAUSIBILITY OF A RESULT AND TO ELABORATE AUTONOMOUS METHODS OF CHECKING.
4. COMMUNICATION SKILLS: THE STUDENT WILL BE ABLE TO COMMUNICATE AND EXPRESS WITH RIGOROUS AND EFFECTIVE EVIDENCE THE RESULTS OF THE LEARNED TOPICS AND TO PRESENT THEM WITH THE SUITABLE MOTIVATIONS IN THE FRAMEWORK OF RESPECTIVE THEORIES.
5. LEARNING SKILLS: THE STUDENT WILL GET A MATHEMATICAL BACKGROUND WHICH ALLOWS HIM TO LEARN MORE COMPLEX MATHEMATICAL CONCEPTS AND MORE GENERALLY SCIENTIFIC SUBJECTS WHICH USE ADVANCED MATHEMATICAL TOOLS.

	Prerequisites
	KNOWLEDGE OF THE THEORY OF FUNCTIONS OF ONE AND SEVERAL REAL VARIABLES. MEASURE AND INTEGRATION IN RN. BASIC TOPOLOGY.

	Contents
	PART A (6 CREDITS) - 1.TOPOLOGIA. SPAZI METRICI E NORMATI. SPAZI DI BANACH. SPAZI DI FUNZIONI CONTINUE [GI]. TEOREMA DI ASCOLI-ARZELÀ [LN]. 2.TEORIA DELLA MISURA E DELL’INTEGRAZIONE DI LEBESGUE. MISURE DI BOREL POSITIVE E TEOREMA DI RAPPRESENTAZIONE DI RIESZ. SPAZI LP [RU]. CONVOLUZIONE E REGOLARIZZAZIONE. TEOREMA DI RIESZ-FRÉCHET-KOLMOGOROV [BR]. 3.SPAZI DI HILBERT [RU]. SERIE DI FOURIER. [GI]. APPLICATIONS TO BOUNDARY VALUE PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS (PDE) [LN]. PART B (6 CREDITS) - 1.THE COMPLEX PLANE. COMPLEX DIFFERENTIATION. INTEGRATION ON THE COMPLEX PLANE. CAUCHY INTEGRAL THEOREM [CO/GR]. 2.CAUCHY INTEGRAL FORMULA AND APPLICATIONS. HOLOMORFIC FUNCTIONS. IDENTITY PRINCIPLE. LAURENT SERIES. CLASSIFICATION OF ISOLATED SINGULARITIES. RESIDUES THEORY [CO/GR]. INDEX OF A CURVE [CO]. ARGUMENT PRINCIPLE. EULER FUNCTIONS [CO/GR]. 3.FOURIER TRANSFORM. L1 THEORY AND INVERSION FORMULA. L2 THEORY AND PLANCHEREL THEOREM [RU]. APPLICATIONS TO INITIAL VALUE PROBLEMS FOR PDE’S [LN].

Contents

PART A (6 CREDITS) -
1.TOPOLOGIA. SPAZI METRICI E NORMATI. SPAZI DI BANACH. SPAZI DI FUNZIONI CONTINUE [GI]. TEOREMA DI ASCOLI-ARZELÀ [LN].
2.TEORIA DELLA MISURA E DELL’INTEGRAZIONE DI LEBESGUE. MISURE DI BOREL POSITIVE E TEOREMA DI RAPPRESENTAZIONE DI RIESZ. SPAZI LP [RU]. CONVOLUZIONE E REGOLARIZZAZIONE. TEOREMA DI RIESZ-FRÉCHET-KOLMOGOROV [BR].
3.SPAZI DI HILBERT [RU]. SERIE DI FOURIER. [GI]. APPLICATIONS TO BOUNDARY VALUE PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS (PDE) [LN].

PART B (6 CREDITS) -
1.THE COMPLEX PLANE. COMPLEX DIFFERENTIATION. INTEGRATION ON THE COMPLEX PLANE. CAUCHY INTEGRAL THEOREM [CO/GR].
2.CAUCHY INTEGRAL FORMULA AND APPLICATIONS. HOLOMORFIC FUNCTIONS. IDENTITY PRINCIPLE. LAURENT SERIES. CLASSIFICATION OF ISOLATED SINGULARITIES. RESIDUES THEORY [CO/GR]. INDEX OF A CURVE [CO]. ARGUMENT PRINCIPLE. EULER FUNCTIONS [CO/GR].
3.FOURIER TRANSFORM. L1 THEORY AND INVERSION FORMULA. L2 THEORY AND PLANCHEREL THEOREM [RU]. APPLICATIONS TO INITIAL VALUE PROBLEMS FOR PDE’S [LN].

	Teaching Methods
	FRONTAL LECTURES. CLASS DISCUSSIONS ABOUT CONCEPTS AND APPLICATIONS. HOMEWORKS.

	Verification of learning
	THE EXAM CONSISTS OF TWO PARTS: A WRITTEN TEST WITH THEORETICAL AND NUMERICAL EXERCISES; AN ORAL EXAM WITH CONCEPTUAL AND TECHNICAL QUESTIONS CONCERNING THE CONTENT OF THE COURSE.

	Texts
	[GI] E. GIUSTI, ANALISI MATEMATICA 2, BOLLATI BORINGHIERI ED. 1984 [CAP. 1; 2] [RU] W. RUDIN, REAL AND COMPLEX ANALYSIS, INTERNATIONAL SERIES IN PURE AND APPLIED MATHEMATICS, MCGRAW – HILL LONDON [CAP. 1; 2; 3; 4; 9] [BR] H. BREZIS, FUNCTIONAL ANALYSIS, SOBOLEV SPACES AND PARTIAL DIFFERENTIAL EQUATIONS, UNIVERSITEXT, SPRINGER BERLIN [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] OR, ALTERNATIVELY, [GR] D. GRECO, COMPLEMENTI DI ANALISI, LIGUORI ED. 1980 [PARTE I] [LN] LECTURE NOTES NOTE. EACH SUBJECT OF THE COURSE HAS BEEN ASSOCIATED TO ONE ONLY REFERRING TEXT, EVEN IF IT IS DEALT WITH IN MORE THAN ONE LISTED TEXTS, WHICH IT IS WORTH AS WELL TO REFER TO FOR BETTER KNOWLEDGE.

Texts

[GI] E. GIUSTI, ANALISI MATEMATICA 2, BOLLATI BORINGHIERI ED. 1984 [CAP. 1; 2]
[RU] W. RUDIN, REAL AND COMPLEX ANALYSIS, INTERNATIONAL SERIES IN PURE AND APPLIED MATHEMATICS, MCGRAW – HILL LONDON [CAP. 1; 2; 3; 4; 9]
[BR] H. BREZIS, FUNCTIONAL ANALYSIS, SOBOLEV SPACES AND PARTIAL DIFFERENTIAL EQUATIONS, UNIVERSITEXT, SPRINGER BERLIN [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] OR, ALTERNATIVELY,
[GR] D. GRECO, COMPLEMENTI DI ANALISI, LIGUORI ED. 1980 [PARTE I]
[LN] LECTURE NOTES
NOTE. EACH SUBJECT OF THE COURSE HAS BEEN ASSOCIATED TO ONE ONLY REFERRING TEXT, EVEN IF IT IS DEALT WITH IN MORE THAN ONE LISTED TEXTS, WHICH IT IS WORTH AS WELL TO REFER TO FOR BETTER KNOWLEDGE.

	More Information
	THE LECTURER WILL DELIVER MATERIAL AND INFORMATION VIA WEB AND THROUGH SHARED ELECTRONIC FOLDERS. THE STUDENTS WILL HAVE THE OPPORTUNITY TO ATTEND LECTURES AND SEMINARS ON TOPIC RELATED WITH SUBJECTS OF THE COURSE IN THE FRAMEWORK OF OTHER TEACHING PROGRAMMES OF THE DEPARTMENT OF MATHEMATICS SUCH AS THE PHD PROGRAMME AS WELL MINI-COURSES OF VISITING PROFESSORS, ON THE OCCASION. AS A PART OF THE COURSE, POSSIBLY YOUNG TUTORS OF THE DEPARTMENT OF MATEMATHICS WILL DELIVER SUPPORT SESSIONS.

More Information

THE LECTURER WILL DELIVER MATERIAL AND INFORMATION VIA WEB AND THROUGH SHARED ELECTRONIC FOLDERS.
THE STUDENTS WILL HAVE THE OPPORTUNITY TO ATTEND LECTURES AND SEMINARS ON TOPIC RELATED WITH SUBJECTS OF THE COURSE IN THE FRAMEWORK OF OTHER TEACHING PROGRAMMES OF THE DEPARTMENT OF MATHEMATICS SUCH AS THE PHD PROGRAMME AS WELL MINI-COURSES OF VISITING PROFESSORS, ON THE OCCASION.
AS A PART OF THE COURSE, POSSIBLY YOUNG TUTORS OF THE DEPARTMENT OF MATEMATHICS WILL DELIVER SUPPORT SESSIONS.

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Antonio VITOLO | PRINCIPLES OF ADVANCED MATHEMATICAL ANALYSIS