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Course 2021-2022 a.y.

30551 - ADVANCED ANALYSIS AND OPTIMIZATION - MODULE 1

BAI
Department of Decision Sciences

Course taught in English

Go to class group/s: 27

BAI (5 credits - I sem. - OB  |  MAT/05)
Course Director:
ANTONIO DE ROSA

Classes: 27 (I sem.)
Instructors:
Class 27: ANTONIO DE ROSA


Suggested background knowledge

For a fruitful and effective learning experience, it is recommended a preliminary knowledge of basic calculus (limits, derivatives, integrals), of vector spaces, of linear maps and of matrix calculus.


Mission & Content Summary
MISSION

The purpose of this course is to give rigorous mathematical tools and methods in the study of ordinary differential equations and, more generally, dynamical systems. This is necessary in a challenging degree in Mathematical and Computing Sciences. The theories and applications encountered in this course will create a strong foundation for studying nonlinear systems of ODEs with application to simple real-world and physical models.

CONTENT SUMMARY
  • Linear algebra: eigenvalues, eigenvectors, diagonalization, quadratic forms, Jordan decomposition.
  • Spaces of continuous functions: uniform convergence, completeness, compactness (Arzelà-Ascoli theorem). Series of functions, power series.
  • Models and examples of ODEs. Trajectories, gradient flows and autonomous systems.
  • Elementary techniques for solving simple differential equations.
  • Linear systems of ODE's: general results and structure of the space of solutions, exponential matrix.
  • Nonlinear systems of ODEs - local theory: contraction theorem, existence, Gronwall lemma, uniqueness.
  • Nonlinear systems of ODEs - global theory: comparison and stability for Cauchy problems, maximal solutions, level sets, nullclines and trapping regions. Continuity and differentiability properties of solutions.

Intended Learning Outcomes (ILO)
KNOWLEDGE AND UNDERSTANDING
At the end of the course student will be able to...
  • Know fundamental notions in linear algebra: eigenvalues, eigenvectors, diagonalization, quadratic forms, Jordan decomposition.
  • Express basic notions and results about spaces of continuous functions.
  • Understanding the relevance of the fundamental theorems for ODEs and dynamical systems: existence, uniqueness and stability. 
     
APPLYING KNOWLEDGE AND UNDERSTANDING
At the end of the course student will be able to...
  • Solve linear systems of ODEs.
  • Provide qualitative description of the solutions of nonlinear ODEs and dynamical systems.
  • Make use of the presented methodological tools in applied sciences such as computer science and physics.
     

Teaching methods
  • Face-to-face lectures
  • Online lectures
  • Exercises (exercises, database, software etc.)
DETAILS

Online lectures have the same conceptual role as face-to-face lectures. The actual blend of face- to-face lectures and online lectures will mainly depend on external constraints.

 

Exercise sessions (again: both face-to face and online) are dedicated to the application of the main theoretical results obtained during lectures to problems and exercises of various nature.

 


Assessment methods
  Continuous assessment Partial exams General exam
  • Written individual exam (traditional/online)
  •   x x
    ATTENDING AND NOT ATTENDING STUDENTS

    Students will be evaluated on the basis of written exams, which can be taken in one of the two following ways. 


    The exam can be split in two partial exams. Each partial may contain multiple-choice questions and open-answer questions; each partial weighs for one-half of the final mark. Multiple-choice questions mainly aim at evaluating the knowledge of the fundamental mathematical notions and the ability to apply these notions to the solution of simple problems and exercises, while open-answer questions mainly aim at evaluating: 

    • The ability to articulate the knowledge of mathematical notions in a conceptually and formally correct way, adequately using definitions, theorems and proofs.
    • The ability to actively search for deductive ideas that are fit to prove possible links between the properties of mathematical objects. 
    • The ability to apply mathematical notions to the solution of more complex problems and exercises.

     

    The exam can also be taken as a single general exam, which contains both multiple-choice questions and open-answer questions. The general exam covers the whole syllabus of the course and it can be taken in one of the four general sessions scheduled in the academic year. This option is mainly meant for students who have withdrawn from the two-partials procedure or could not follow it. Each type of question contributes in a specific way to the assessment of the students' acquired knowledge. Multiple-choice questions mainly aim at evaluating the knowledge of the fundamental mathematical notions and the ability to apply these notions to the solution of simple problems and exercises, while open-answer questions mainly aim at evaluating: 

    • The ability to articulate the knowledge of mathematical notions in a conceptually and formally correct way, adequately using definitions, theorems and proofs.
    • The ability to actively search for deductive ideas that are fit to prove possible links between the properties of mathematical objects. 
    • The ability to apply mathematical notions to the solution of more complex problems and exercises. 

     

    We will take care to obtain final grades whose distribution follows the grade distribution that is recommended by Università Bocconi. 
     


    Teaching materials
    ATTENDING AND NOT ATTENDING STUDENTS
    • Schaeffer, David G.; Cain, John W.: Ordinary differential equations: basics and beyond. Texts in Applied Mathematics, 65. Springer, New York, 2018 (brossura dell'edizione 2016).xxx+542 pp. ISBN: 978-1-4939-8184-7
    • Giaquinta, Mariano; Modica, Giuseppe. Mathematical analysis. Linear and metric structures and continuity. Birkhäuser Boston, Inc., Boston, MA, 2007.(brossura) xx+465 pp. ISBN: 978--08176-4375-1
    Last change 28/06/2021 11:10