Dipartimento di Scienze di Base e Applicate per l'Ingegneria

Discrete Harmonic Analysis Fabio Scarabotti; 20 hours

Tuesday room  1E  10:00-12:00 Friday room 1E  10:00-12:00

Starting data : Friday November 13

Program

The Discrete Fourier Transform (DFT), the discrete Fourier transform on the hypercube, the Fast Fourier Transform (FFT). Applications to random walks: the Ehrenfest diffusion model, convergence to the stationary distribution and the Diaconis cutoff phenomenon. Graph Laplacians and the isoperimetric constant. The inequalities of Alon-Milman and Dodziuk.
Asymptotic behaviour of the spectral gap: the Alon-Boppana-Serre theorem.
Expanders and Ramanujan graphs. The Margulis construction.

References

Ceccherini-Silberstein, Tullio; Scarabotti, Fabio; Tolli, Filippo Harmonic analysis on finite groups. Representation theory, Gelfand pairs and Markov chains. Cambridge Studies in Advanced Mathematics, 108. Cambridge University Press, Cambridge, 2008.

Ceccherini-Silberstein, Tullio; Scarabotti, Fabio; Tolli, Filippo Harmonic analysis on finite abelian groups and finite fields. Book in preparation.

Davidoff, Giuliana; Sarnak, Peter; Valette, Alain Elementary number theory, group theory, and Ramanujan graphs. London Mathematical Society Student Texts, 55. Cambridge University Press, Cambridge, 2003.

Hoory, Shlomo; Linial, Nathan; Wigderson, Avi Expander graphs and their applications. Bull. Amer. Math. Soc. (N.S.) 43 (2006), no. 4, 439-561.

Stein, Elias M.; Shakarchi, Rami Fourier analysis. An introduction. Princeton Lectures in Analysis, 1. Princeton University Press, Princeton, NJ, 2003.

Terras, Audrey Fourier analysis on finite groups and applications. London Mathematical Society Student Texts, 43. Cambridge University Press, Cambridge, 1999.

 

"Introduction to Sobolev Spaces and Differential Equations"  Prof. Daniele Andreucci

2016-03-07 Mon 11:00-13:00

2016-03-09 Wed 11:00-13:00

2016-03-14 Mon 11:00-13:00

2016-03-22 Tue 10:15-12:15

2016-03-31 Thu 14:00-16:00

2016-04-04 Mon 11:00-13:00

Program

Lebesgue measure. Lp spaces. Convergence in Lp spaces.  Weak derivatives. Sobolev spaces. Approximation. Traces.  Sobolev embeddings. Hilbert spaces. Representation of linear operators. Lax Milgram theorem.

 

"Integro-differential equations"   Prof. Daniela Sforza

06/4   11-13 room 1B1

11/4  11-13 room 1B1

13/4  11-13 room 1B1

18/4  11-13 room 1B1

20/4  11-13 room 1B1

Program

How to define viscoelastic partial integro-differential equations, by introducing in the wave equation memory terms. Structure and meaning  of a memory term: an integral convolution between the unknown function and a kernel characterizing the viscoelastic material. The assumption of fading memory in viscoelasticity theory. Admissible classes of integral kernels. Representations  by means of Fourier series of the solutions of integrodifferential equations when integral kernels are decreasing exponential functions. Existence and regularity of solutions of viscoelastic equations in the case of more general integral kernels. Dissipation of the energy.

 

"Elliptic equation: general existence results"   Prof. Daniela Giachetti

June- July 2016

Program

Linear and semilinear elliptic equations. Lax-Milgram’s and Stampacchia’s Theorems.Nonlinear elliptic equations. The Leray-Lions existence Theorem. Some notions on spectral analysis for linear operator and applications to some semilinear equations.Maximum principle and strong maximum principle. Singular elliptic equations. Problems with source terms in L^1 or measures

 

Constrained Optimization Prof. Paola Loreti

Starting days

2016-04-18 Mon 14:00    via Eudossiana room 22

2016-04-21 Thu  15:45    via Eudossiana room 22

Syllabus

Convex sets and convex functions.   Convexity and optimization. Constrained optimization. Constrained qualification. Karush-Kuhn-Tucker conditions. Examples of optimal control problems. The value function. The dynamic programming principle and the Hamilton-Jacobi-Bellman equation