Petak, 19.04.2013. u 14 casova, Sala 301F, MI SANU
Akademik Gradimir Milovanovic, Matematicki institut SANU
KVADRATURNI PROCESI GAUSOVOG TIPA NA REALNOJ POLUOSI ZA FUNKCIJE SA EKSPONENCIJALNIM RASTOM U KRAJNJIM TACKAMA INTERVALA

Rezime: Osnovni problem koji se razmatra je tezinska polinomijalna aproksimacija funkcija, definisanih na realnoj poluosi (0,+\infty), koje mogu imati eksponencijalni rast u krajnjim tackama intervala. Preciznije, razmatra se ponasanje Gausovih kvadratura na \mathbb{R}^+ sa neklasicnom tezinskom funkcijom w(x)=exp(-x^{-\alpha}-x^{\beta}), \alpha>0, \beta>1, u vise prostora sa tezinskim uniformnim metrikama, obezbedjujuci konvergenciju formula sa redom najbolje tezinske polinomijalne aproksimacije (pod standardnim pretpostavkama), kao i konvergenciju sa geometrijskom brzinom za funkcije iz C^{\infty}(\mathbb{R}^+). U poredjenju sa nekim dosad poznatim slucajevima eksponencijalnih tezina, ovde nemamo konvergenciju sa optimalnom brzinom u tezinskim L^1-prostorima Soboljeva, sto takodje implicira da niz odgovarajucih Lagranzeovih operatora ne moze biti uniformno ogranicen u tezinskim L^2-prostorima Soboljeva. Prevazilazenje ovog problema moze se postici jednom modifikacijom kvadraturne formule i pritom dokazati konvergencija koja ima isti red kao kod uobicajene Gausove kvadrature za neprekidne funkcije. Stavise, moze se dokazati konvergencija sa redom najbolje tezinske polinomijalne aproksimacije za funkcije iz tezinskog L^1-prostora Soboljeva. Najzad, numericka konstrukcija formula i prevazilazenje problema numericke nestabilnosti se takodje razmatraju.

Petak, 26.04.2013. u 14h sala 301f, MI SANU
Ubertino Battisti, Universita degli Studi di Torino
NON-COMMUTATIVE RESIDUE AND WEYL'S LAW

Abstract: Non-commutative residue or Wodzicki residue was first introduced by M. Wodzicki in 1984 and independently by V. Guillemin in 1985. It was originally defined as the unique trace on the quotient algebra \psi_{cl}(M)/\psi^{-\infty}(M), where \psi_{cl}(M) is s the algebra of classical pseudodifferential operators on the closed manifolds M, and \psi_{cl}^{-\infty}(M) is the set of smoothing operators, that is operators with smoothing kernel. We suppose that the dimension of the manifold is at least two. V. Guillemin introduced this new trace in order to obtain a soft proof of the well known Weyl's law, which describes the asymptotic behaviour of the counting function of a positive densely defined self-adjoint operator with discrete spectrum. Let P:D\subseteq H \rightarrow H be a positive densely defined self-adjoint operator with discrete spectrum \sigma(P)=\{\lambda_j\}_{j\in N}, where each eigenvalue is counted with its multiplicity. The counting function N_P(\lambda) is defined as follows N_P(\lambda)=\sum_{\lambda_j ‹\lambda}1=\sharp\{\lambda_j | \lambda_j ‹ \lambda\}. The counting function, in the case of differential operators on closed manifolds, has been deeply studied in view of its geometric meaning. One of the the main results is the Weyl's law: N_P(\lambda)\sim \lambda^{\frac{m}{n}}C+o(\lambda^{\frac{m}{n}}), \lambda\rightarrow \infty, where n= dim M, m is the order of the operator and C is a constant depending on the principal symbol of P and on the manifold M. V. Guillemin suggested a short proof of Weyl's law using the non commutative residue and a Tauberian Theorem.
We will analyse the analogous problem in three different settings: SG-operators, bisingular operators and globally bisingular operators. The model examples of operators in these classes are respectively SG-operators, (1+|x^2|)(1-\Delta). bisingular operators, P_M\otimes P_N, P_M, P_N being pseudodifferential operators on the closed manifolds M, N, respectively. globally bisingular operators, (|x_1|^2-\Delta_1)\otimes (|x_2|^2-\Delta_2) defined on \mathbb{R}^{n1}\times \mathbb{R}^{n2}. Or, more generally, G_1 \otimes G_2, where G_1(G_2) is a global operator of Shubin type on \mathbb{R}^{n_1}(\mathbb{R}^{n_2}).
Using Tauberian techniques, we will determine in the three cases a Weyl's formula, similar to the one on the closed manifolds.
The talk is based on joint works with S. Coriasco (Universita di Torino), T. Gramchev (Universita di Cagliari), S. Pilipovic (University of Novi Sad) and L. Rodino (Universita di Torino).

REFERENCES

[1] U. Battisti and S. Coriasco. Wodzicki residue for operators on manifolds with cylindrical ends. Ann. Global Anal. Geom., 40(2):223-249, 2011.
[2] U. Battisti. Weyl asymptotics of bisingular operators and Dirichlet divisor problem. Math. Z., 272: 1365-1381, 2012.
[3] U. Battisti, S. Pilipovic, T. Gramchev, and L. Rodino. Globally bisingular elliptic operators. In Operator Theory, Pseudo-Differential Equations, and Mathematical Physics, Operator Theory: Advances and Applications. Birkhauser, Basel, 2013.

Petak, 29.04.2013. u 10h, sala 2, SANU, BGD
(OBRATITE PAZNJU NA DATUM, VREME I MESTO)
Cedric Villani, Institut Henri Poincare, Paris, DOBITNIK FIELDS MEDALJE!
MONGE MEETS RIEMANN

Summary: The author will present how the optimization problem of Monge and Kantorovich can be used to solve problems in Riemannian geometry, in particular, give a synthetic interpretation of Ricci curvature bounds.

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