ANALYTICAL AND STOCHASTIC MATHEMATICAL METHODS FOR ENGINEERING

Second cycle degree in MATHEMATICAL ENGINEERING (Ord. 2017)

Campus: PADOVA

Language: English

Teaching period: First Semester

Lecturer: GIORGIA CALLEGARO

Number of ECTS credits allocated: 12

Syllabus
 Prerequisites: None Examination methods: Final examination based on: written examination. Course unit contents: 1. Lebesgue Measure and Integral — Lebesgue measure and integral on R^d, limit theorems, dependence by parameters, reduction formula and change of variables. Introduction to abstract measure and integral. 2. Normed spaces — Concept of norm; uniform, L^1 and L^p norms; sequences, limits, completeness and Banach spaces. Concept of scalar product and Hilbert space; the space L^2; separable spaces and concept of orthonormal base. 3. Fundamentals of Complex Analysis — Power series in C and elementary analytic functions; concept of holomorphic function, Cauchy-Riemann equations, analyticity of holomorphic functions; isolated singularities; residue theorem and applications. 4. Introduction to Fourier Analysis — Fourier series: Euler formulas, convergence and sum (point-wise, uniform, L^2). Fourier transform: L^1 definition, convolution and approximate units, inversion formula. Schwarz functions and L^2 Fourier transform. Applications. 5. Introduction to probability — probability spaces, axioms of probability, conditional probabilities, independence of events. 6. Random variables (discrete and continuous) — definition, expectation and moments. Examples of random variables and applications, with a focus on Gaussian random variables. 7. Random vectors. 8. Characteristic function. 9. Convergence of random variables: weak, in probability, in L^p, almost sure. 10. The law of large numbers and the central limit theorem with applications. 11. Conditional expectation. 12. Martingales in discrete time.