2 Maggio, 2005 15:00

Seminario Matematico e Fisico di Milano

Alcuni aspetti simplettici della teoria dei nodi

Mauro Spera, Università di Padova

Dipartimento di Matematica e Applicazioni - Università degli Studi di Milano Bicocca - Via Cozzi, 53 - Aula 3014

abstract

abstract

"n questo seminario verranno esposti alcuni risultati (ottenuti in collaborazione con Alberto Besana (Dipartimento di Matematica ""F.Enriques"", Università di Milano)) riguardanti un'interpretazione meccanica del writhing number (o writhe, o numero di avvitamento, o di Tait) di un nodo, rispetto ad una sua proiezione regolare piana, ricorrendo ad un analogo infinito-dimensionale della teoria di Maslov- Hörmander. Il cambiamento di writhe corrisponde al passaggio attraverso il ciclo di Maslov (caustica), rappresentato da nodi con un solo punto doppio in un'opportuna sottovarietà lagrangiana della varietà simplettica dei nodi (debolmente singolari) introdotta da J.L. Brylinski, ottenuta tramite una semplice costruzione geometrica (nodi che si avvitano su un cono avente come direttrice una prefissata proiezione piana). La funzione generatrice è rappresentata da un'azione di Chern-Simons abeliana (elicità, in meccanica dei fluidi) con l'aggiunta di un termine di sorgente, dipendente dal nodo. Le connessioni (abeliane) forniscono le variabili ausiliarie. L'elicità gioca il ruolo della segnatura dell'operatore Hessiano rispetto alle suddette variabili. Verranno inoltre discusse, tempo permettendo, questioni di quantizzazione geometrica e si mostrerà come la condizione di quantizzazione di Feynman-Onsager della teoria quantistica dei vortici emerga quale condizione di Bohr-Sommerfeld imposta ad una diversa sottovarietà lagrangiana (nodi sulla sfera unitaria)."

20 Aprile, 2005 12:30

Seminari di Cultura Matematica

Acusmetria. Il suono visibile

F. Rampichini, M. Maiocchi, E. Lariani, varie

aula T2.2

"n questo seminario verranno esposti alcuni risultati (ottenuti in collaborazione con Alberto Besana (Dipartimento di Matematica ""F.Enriques"", Università di Milano)) riguardanti un'interpretazione meccanica del writhing number (o writhe, o numero di avvitamento, o di Tait) di un nodo, rispetto ad una sua proiezione regolare piana, ricorrendo ad un analogo infinito-dimensionale della teoria di Maslov- Hörmander. Il cambiamento di writhe corrisponde al passaggio attraverso il ciclo di Maslov (caustica), rappresentato da nodi con un solo punto doppio in un'opportuna sottovarietà lagrangiana della varietà simplettica dei nodi (debolmente singolari) introdotta da J.L. Brylinski, ottenuta tramite una semplice costruzione geometrica (nodi che si avvitano su un cono avente come direttrice una prefissata proiezione piana). La funzione generatrice è rappresentata da un'azione di Chern-Simons abeliana (elicità, in meccanica dei fluidi) con l'aggiunta di un termine di sorgente, dipendente dal nodo. Le connessioni (abeliane) forniscono le variabili ausiliarie. L'elicità gioca il ruolo della segnatura dell'operatore Hessiano rispetto alle suddette variabili. Verranno inoltre discusse, tempo permettendo, questioni di quantizzazione geometrica e si mostrerà come la condizione di quantizzazione di Feynman-Onsager della teoria quantistica dei vortici emerga quale condizione di Bohr-Sommerfeld imposta ad una diversa sottovarietà lagrangiana (nodi sulla sfera unitaria)."

18 Aprile, 2005 17:00

Seminario Matematico e Fisico di Milano

Long-time behaviour of nonlinearly damped semilinear wave equation

Igor Chueshov, Kharkov National University (Ucraina)

Dipartimento di Matematica - Università degli Studi - Via Saldini 50 - Milano - Sala di Rappresentanza

abstract

abstract

We present a new approach for proving of existence and finite-dimensionality of global attractors for infinite-dimensional dissipative systems generated by abstract nonlinear second order in time evolution equations. This approach is based on far reaching generalizations of the Ceron-Lopes theorem on asymptotic compactness and Ladyzhenskaya's theorem on the dimension of invariant sets. An application of our results to nonlinear damped wave equations allow us to obtain new results pertaining to structure and properties of global attractors for nonlinear waves.

15 Aprile, 2005 14:45

Sezione di Probabilità e Statistica Matematica

On the Strong Approximation of Jump-Di®usion Processes

Nicola Bruti Liberati, Technical University of Sydney

Aula VI Piano

abstract

abstract

In Financial modelling, Filtering and other areas
the underlying dynamics are often specified via stochastic differential equations (SDEs) of jump-diffusion type. The class
of jump-diffusion SDEs that admits explicit solutions is rather
limited. Consequently, there is a need for the systematic use of
discrete time approximations in corresponding simulations. This
paper presents a survey and new results on strong numerical
schemes for SDEs of jump-diffusion type. These are relevant for
scenario analysis, filtering and hedge simulation in finance. It
provides a convergence theorem for the construction of strong
approximations of any given order of convergence for SDEs driven
by Wiener processes and Poisson random measures. The paper
covers also derivative free, drift-implicit and jump adapted strong
approximations. For the commutative case particular schemes are
obtained. Finally, a numerical study on the accuracy of several
strong schemes is presented.

14 Aprile, 2005 

Sezione di Fisica Matematica

Spectral Methods and the Resolution of the Gibbs Phenomenon

Professor Bernie Shizgal, University of British Columbia (Canada)

Aula Seminari MOX VI Piano

abstract

abstract

Spectral methods involve the expansion of the solution of some partial differential equation or integral equation in a set of basis functions. The basis functions chosen are the Fourier functions, Chebyshev, Hermite and nonclassical polynomials. Pseudospectral (or collocation) methods determine the solution at the set of quadrature points for classical and nonclassical polynomials. Recent work on the use of nonclassical basis functions will be briefly dicussed.
It is well known that the expansion of an analytic nonperiodic function on a finite interval in a Fourier series leads to spurious oscillations at the interval boundaries. This result is known as the Gibbs phenomenon. The present talk describes a new method for the resolution of the Gibbs phenomenon (Shizgal and Jung, J. Comput. Appl. Math. 161, 41 (2003); Jung and Shizgal, J. Comput. Appl. Math. 172, 131-151(2004) ) which follows on the reconstruction method of Gottlieb and coworkers (SIAM Rev. 39, 644 (1997)) based on Gegenbauer polynomials. We refer to their approach as the direct method and to the new methodology as the inverse method. The direct method requires that certain conditions are met concerning lambda in the weight function (1-x^2)^{\lambda-1/2}, the number of Fourier coefficients, N and the number of Gegenbauer polynomials, m. We show that the new inverse method can give exact results for polynomials independent of lambda and with m=N. The paper presents several numerical examples applied to a single domain or to subdomains of the main domain so as to illustrate the superiority of the inverse method in comparison with the direct method. Image resolution in two dimensions will also be discussed and illustrated with several examples including a resolution of the Shepp-Logan phantom image.

14 Aprile, 2005 

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

Global attractors of non-autonomous hyperbolic equations with singularly oscillating external forces

Vladimir Chepyzhov, Russian Academy of Sciences, Mosca

Spectral methods involve the expansion of the solution of some partial differential equation or integral equation in a set of basis functions. The basis functions chosen are the Fourier functions, Chebyshev, Hermite and nonclassical polynomials. Pseudospectral (or collocation) methods determine the solution at the set of quadrature points for classical and nonclassical polynomials. Recent work on the use of nonclassical basis functions will be briefly dicussed.
It is well known that the expansion of an analytic nonperiodic function on a finite interval in a Fourier series leads to spurious oscillations at the interval boundaries. This result is known as the Gibbs phenomenon. The present talk describes a new method for the resolution of the Gibbs phenomenon (Shizgal and Jung, J. Comput. Appl. Math. 161, 41 (2003); Jung and Shizgal, J. Comput. Appl. Math. 172, 131-151(2004) ) which follows on the reconstruction method of Gottlieb and coworkers (SIAM Rev. 39, 644 (1997)) based on Gegenbauer polynomials. We refer to their approach as the direct method and to the new methodology as the inverse method. The direct method requires that certain conditions are met concerning lambda in the weight function (1-x^2)^{\lambda-1/2}, the number of Fourier coefficients, N and the number of Gegenbauer polynomials, m. We show that the new inverse method can give exact results for polynomials independent of lambda and with m=N. The paper presents several numerical examples applied to a single domain or to subdomains of the main domain so as to illustrate the superiority of the inverse method in comparison with the direct method. Image resolution in two dimensions will also be discussed and illustrated with several examples including a resolution of the Shepp-Logan phantom image.

14 Aprile, 2005 

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

Some asymptotic problems in optimal transportation

Guy Bouchittè, Dip. di Matematica Università de Toulon

Spectral methods involve the expansion of the solution of some partial differential equation or integral equation in a set of basis functions. The basis functions chosen are the Fourier functions, Chebyshev, Hermite and nonclassical polynomials. Pseudospectral (or collocation) methods determine the solution at the set of quadrature points for classical and nonclassical polynomials. Recent work on the use of nonclassical basis functions will be briefly dicussed.
It is well known that the expansion of an analytic nonperiodic function on a finite interval in a Fourier series leads to spurious oscillations at the interval boundaries. This result is known as the Gibbs phenomenon. The present talk describes a new method for the resolution of the Gibbs phenomenon (Shizgal and Jung, J. Comput. Appl. Math. 161, 41 (2003); Jung and Shizgal, J. Comput. Appl. Math. 172, 131-151(2004) ) which follows on the reconstruction method of Gottlieb and coworkers (SIAM Rev. 39, 644 (1997)) based on Gegenbauer polynomials. We refer to their approach as the direct method and to the new methodology as the inverse method. The direct method requires that certain conditions are met concerning lambda in the weight function (1-x^2)^{\lambda-1/2}, the number of Fourier coefficients, N and the number of Gegenbauer polynomials, m. We show that the new inverse method can give exact results for polynomials independent of lambda and with m=N. The paper presents several numerical examples applied to a single domain or to subdomains of the main domain so as to illustrate the superiority of the inverse method in comparison with the direct method. Image resolution in two dimensions will also be discussed and illustrated with several examples including a resolution of the Shepp-Logan phantom image.

13 Aprile, 2005 17:00

Seminario Matematico e Fisico di Milano

A geometric problem and the Hopf Lemma

Louis Nirenberg, Courant Institute, New York (Stati Uniti)

Dipartimento di Matematica - Università degli Studi - Via Saldini 50 - Milano - Sala di Rappresentanza

Spectral methods involve the expansion of the solution of some partial differential equation or integral equation in a set of basis functions. The basis functions chosen are the Fourier functions, Chebyshev, Hermite and nonclassical polynomials. Pseudospectral (or collocation) methods determine the solution at the set of quadrature points for classical and nonclassical polynomials. Recent work on the use of nonclassical basis functions will be briefly dicussed.
It is well known that the expansion of an analytic nonperiodic function on a finite interval in a Fourier series leads to spurious oscillations at the interval boundaries. This result is known as the Gibbs phenomenon. The present talk describes a new method for the resolution of the Gibbs phenomenon (Shizgal and Jung, J. Comput. Appl. Math. 161, 41 (2003); Jung and Shizgal, J. Comput. Appl. Math. 172, 131-151(2004) ) which follows on the reconstruction method of Gottlieb and coworkers (SIAM Rev. 39, 644 (1997)) based on Gegenbauer polynomials. We refer to their approach as the direct method and to the new methodology as the inverse method. The direct method requires that certain conditions are met concerning lambda in the weight function (1-x^2)^{\lambda-1/2}, the number of Fourier coefficients, N and the number of Gegenbauer polynomials, m. We show that the new inverse method can give exact results for polynomials independent of lambda and with m=N. The paper presents several numerical examples applied to a single domain or to subdomains of the main domain so as to illustrate the superiority of the inverse method in comparison with the direct method. Image resolution in two dimensions will also be discussed and illustrated with several examples including a resolution of the Shepp-Logan phantom image.

11 Aprile, 2005 17:00

Seminario Matematico e Fisico di Milano

Homogenization of highly contrasted photonic crystals: plasma frequency and artificial magnetism

Guy Bouchitté, Université de Toulon et du Var (Francia)

Dipartimento di Matematica - Politecnico di Milano - Via Bonardi 9 - Milano - Aula Seminari III piano

abstract

abstract

"We consider a domain of R3 filled periodically (period η

6 Aprile, 2005 12:30

Seminari di Cultura Matematica

Personaggi e idee dell'Analisi Matematica: Da Cavalieri a De Giorgi (I parte)

S. Salsa, Dip. di Matematica

aula T2.2

"We consider a domain of R3 filled periodically (period η