30551  ADVANCED ANALYSIS AND OPTIMIZATION  MODULE 1
Department of Decision Sciences
ANTONIO DE ROSA
Suggested background knowledge
Mission & Content Summary
MISSION
CONTENT SUMMARY
 Linear algebra: eigenvalues, eigenvectors, diagonalization, quadratic forms, Jordan decomposition.
 Spaces of continuous functions: uniform convergence, completeness, compactness (ArzelàAscoli 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
 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
 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
 Facetoface lectures
 Online lectures
 Exercises (exercises, database, software etc.)
DETAILS
Online lectures have the same conceptual role as facetoface lectures. The actual blend of face toface lectures and online lectures will mainly depend on external constraints.
Exercise sessions (again: both faceto 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  


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 multiplechoice questions and openanswer questions; each partial weighs for onehalf of the final mark. Multiplechoice 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 openanswer 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 multiplechoice questions and openanswer 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 twopartials procedure or could not follow it. Each type of question contributes in a specific way to the assessment of the students' acquired knowledge. Multiplechoice 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 openanswer 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: 9781493981847
 Giaquinta, Mariano; Modica, Giuseppe. Mathematical analysis. Linear and metric structures and continuity. Birkhäuser Boston, Inc., Boston, MA, 2007.(brossura) xx+465 pp. ISBN: 9780817643751