13 Ottobre, 2005 15:00

Sezione di Probabilità e Statistica Matematica

A NEW AUTOMATIC SAMPLER FOR TRANSDIMENSIONAL MARKOV CHAIN MONTE CARLO

Prof. Giovanni Petris, University of Arkansas, USA

Aula VI piano

abstract

abstract

Modern applied Bayesian statistics relies heavily on simulation methods for the analysis of posterior distributions.
Algorithms to simulate from a Markov chain having a prescribed limiting distribution are nowadays widely available and, for a large class of standard models, easy to implement. The case of
posterior distributions defined on a space of variable dimension, such as those arising in problems of variable selection for regression models, is considerably harder. In fact, although several algorithms to sample from a Markov chain with such a limiting distribution have been proposed in the literature - most notably the so-called Reversible Jump -, they are all difficult to implement on a routine basis and very ill-suited for an automatic use. In the case of reversible jump, the main drawback is the need to devise jump proposals in a clever way, so that the sampler moves reasonably fast between different regions of the support of the posterior. The calculation of the Jacobians entering the acceptance probabilities may also be unappealing for a certain class of practitioners. We propose a new approach to transdimensional Markov chain simulation, based on a simple geometric idea used to map spaces of different dimension onto a space having a fixed dimension. This makes it possible to use standard fixed-dimension samplers also in the transdimensional case. It should be noted that the map mentioned
above can be evaluated in an automatic way based only on the density of the target distribution in regions of different dimension. To make the approach even more automatic, we advocate the use of a multidimensional version of Adaptive Rejection Metropolis Sampling to simulate from the transformed target density. In this way the user has only to specify the set of posterior
densities on the different parts of the variable dimensional support. General purpose functions for implementing the proposed algorithms
have been written for the computer language R.

10 Ottobre, 2005 17:00

Seminario Matematico e Fisico di Milano

Problemi di minimo con dati singolari

Lucio Boccardo, Università di Roma La Sapienza

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

abstract

abstract

Anche senza scomodare il Teorema di De Giorgi, ´e semplice provare l’esistenza del minimo per funzionali integrali del Calcolo delle Variazioni della semplice forma (0.1) J(v) = Z |rv|2 −: 2 Z fv ( aperto limitato di IRN). Essendo l’ambiente adeguato W1,2 0 ( ) e appartenendo v a L 2N N−:2 ( ), serve che f stia in L 2N N+2 ( ). Il minimo u esiste, ´e unico e verica la forma debole dell’equazione di Eulero- Lagrange (0.2) ( −:
u = f , u = 0 @ Poich´e il precedente problema al contorno ammette soluzioni distribuzionali anche se il dato f appartiene anche solo L1( ): (0.3) u 2 W1,q 0 ( ), q < N N −: 1 : Z rur = Z f, 8 2 D( ) potrebbe sorgere la domanda (a me ´e sorta!): il problema (0.3) ´e legato a un qualche problema di minimo, anche se f 2 L1( ) comporta che l’estremo inferiore di J in (0.1) ´e meno infinito? Funzionali convessi pi´u generali del tipo C(v) = Z j(x,rv) −: Z fv Funzionali non convessi del tipo I(v) = Z j(x, |v|,rv) −. Z fv 2 LUCIO BOCCARDO Funzionali vettoriali (Sistemi): lavori in corso. Altri problemi. Dipartimento di Matematica, Universit`a di Roma 1, Piazza A. Moro 2, 00185 Roma tel: 00 39 06 49913202 E-mail address: boccardo@mat.uniroma1.it

4 Ottobre, 2005 11:00

Sezione di Probabilità e Statistica Matematica

Quantum Random Walks on UHF algebras

Dr. Lingaraj Sahu, Università di Greifswald

Aula seminari III piano

abstract

abstract

In the Ph. D. thesis (S-2005), the problem of obtaining Evans-Hudson dilation for strongly continuous quantum dynamical semigroups on
uniformly hyperfinite C*-algebras is investigated and answered partialy. Here we will present the path followed in the last part of
the thesis. Following the work of Lindsay and Parthasarathy (LP-1988) and Attal and Pautrat (AP-2002), the idea is to construct Evans-Hudson flows as a limit of *-homomorphic families parametrize by partitions of positive real line, using toy Fock space language.
These families of *-homomorphisms are called quantum random walks.
For a class of quantum dynamical semigroup, starting from the unbounded generator, we have
constructed quantum random walks and proved that as the width of the partitions goes to zero the quantum random walks converges weakly and
the limit satisfies a quantum stochastic differential equation. As weak limit of
*-homomorphisms the limit is a completely positive flow but strong conergence of the quantumm random walks is not yet proved and homomorohic property of the limit is unknown.
[AP-2002] Attal S. ; Pautrat, Y. : From repeated to continuous quantum interactions, xxx.lanl.gov/math-ph/0311002.

3 Ottobre, 2005 14:30

Sezione di Probabilità e Statistica Matematica

Feynman’s quantum walks

dott. Dario Tamascelli, Università degli Studi di Milano

aula MOX VI piano

abstract

abstract

Nel calcolatore di Feynman il meccanismo di temporizzazione, o clock, è scalato a livello quantistico ed è incluso nel sistema di calcolo: esso consiste di una particella, o cursore, che si muove di moto inerziale su un reticolo regolare o un grafo. Il clock è quindi accoppiato ad aggiuntivi gradi di libertà che realizzano il registro di ingresso/uscita.
In questo seminario il modello di calcolatore di Feynman viene presentato come non solo un’anticipazione, ma addirittura una generalizzazione del paradigma di calcolo basato sui quantum walks a tempo continuo che molta attenzione ha attirato in tempi recenti.
Particolare attenzione viene dedicata ai problemi di sincronizzazione del segnale di temporizzazione, alla possibilità di usare salti
condizionati del cursore per iterare subroutine quantistiche ed alle strategie di preparazione e lettura del dispositivo di calcolo suggerite
da considerazioni di carattere dinamico ed entropico, che motivano l’introduzione della nozione di costo in probabilità di un algoritmo.

21 Settembre, 2005 

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

The shape of extremal functions in Poincare-Sobolev-type inequalities

Tobias Weth, Univsità di Giessen, Germania

Nel calcolatore di Feynman il meccanismo di temporizzazione, o clock, è scalato a livello quantistico ed è incluso nel sistema di calcolo: esso consiste di una particella, o cursore, che si muove di moto inerziale su un reticolo regolare o un grafo. Il clock è quindi accoppiato ad aggiuntivi gradi di libertà che realizzano il registro di ingresso/uscita.
In questo seminario il modello di calcolatore di Feynman viene presentato come non solo un’anticipazione, ma addirittura una generalizzazione del paradigma di calcolo basato sui quantum walks a tempo continuo che molta attenzione ha attirato in tempi recenti.
Particolare attenzione viene dedicata ai problemi di sincronizzazione del segnale di temporizzazione, alla possibilità di usare salti
condizionati del cursore per iterare subroutine quantistiche ed alle strategie di preparazione e lettura del dispositivo di calcolo suggerite
da considerazioni di carattere dinamico ed entropico, che motivano l’introduzione della nozione di costo in probabilità di un algoritmo.

19 Settembre, 2005 14:30

Sezione di Probabilità e Statistica Matematica

Quasi-free Quantum Stochastic Calculus with Infinite Dimensional Noise

Nicholas Weatherall, Nottingham University

aula seminari III piano

abstract

abstract

I describe a Quasi-free Quantum Stochastic Calculus with Infinite Dimensional Noise on a "Double" Fock Space. This generalises the calculus for finite dimensional noise developed by J.M. Lindsay. We use the approach using the classical Skorohod integral. In particular, we develop a theory of iterated integration as a method of solving QSDE for quantum flows. This utilises an algebraic characterisation of iterative integrability which allows a characterisation of *-homomorphic solutions.

22 Luglio, 2005 14:00

Sezione di Probabilità e Statistica Matematica

Changes in the Hemodynamic Stresses Occurring During the Enlargement ofAbdominal Aortic Aneurysms

Anne-Virginie Salsac, Labor. d Hydrodyn. (LadHyX)-Ecole Polytechnique

Aula seminari MOX - 6° piano dip. di matematica

abstract

abstract

This research seeks to quantify the role that mechanical stimuli play during
the growth of abdominal aortic aneurysms (AAA). The formation of disturbed
patterns of wall shear stresses (WSS) and gradients of wall shear stresses
(GWSS) has been shown to affect the wall structural integrity, through
their effect on the morphology and functions of the endothelial cells (EC)
and circulating blood cells.
Particle Image Velocimetry measurements of the pulsatile blood flow have
been performed in aneurysm models, while changing systematically their
geometric parameters. The parametric study shows that the flow separates
from the wall even at early stages of the disease (dilatation < 50%).
A large vortex ring forms in symmetric AAA, leading to the formation of two
distinct region: a region of flow detachment, with low oscillatory WSS, and
a downstream region of flow reattachment, with large negative WSS and
sustained GWSS.
The loss of symmetry in the geometry engenders a helical flow. The
dominant vortex, whose strength increases with the asymmetry parameter,
is shed from
the most bulged wall (anterior). It results in the formation of a large
recirculating region, where ECs are subjected to quasi-steady reversed
WSS of low magnitude, while the posterior wall is exposed to positive WSS.
Lagrangian tracking of blood cells inside the aneurysm models shows a
dramatic increase in the cell residence time as the aneurysm grows. While
recirculating, cells experience higher shear stress, which may lead to
their activation. The flow vortical structure convects the cells towards
the wall, increasing the probability for cell deposition and ipso facto for
an intraluminal thrombus to form.

14 Luglio, 2005 

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

A phase- field model for phase transitions in nonequilibrium systems

Peter Galenko, Institute of Space Simulation, Koeln, Germany

This research seeks to quantify the role that mechanical stimuli play during
the growth of abdominal aortic aneurysms (AAA). The formation of disturbed
patterns of wall shear stresses (WSS) and gradients of wall shear stresses
(GWSS) has been shown to affect the wall structural integrity, through
their effect on the morphology and functions of the endothelial cells (EC)
and circulating blood cells.
Particle Image Velocimetry measurements of the pulsatile blood flow have
been performed in aneurysm models, while changing systematically their
geometric parameters. The parametric study shows that the flow separates
from the wall even at early stages of the disease (dilatation < 50%).
A large vortex ring forms in symmetric AAA, leading to the formation of two
distinct region: a region of flow detachment, with low oscillatory WSS, and
a downstream region of flow reattachment, with large negative WSS and
sustained GWSS.
The loss of symmetry in the geometry engenders a helical flow. The
dominant vortex, whose strength increases with the asymmetry parameter,
is shed from
the most bulged wall (anterior). It results in the formation of a large
recirculating region, where ECs are subjected to quasi-steady reversed
WSS of low magnitude, while the posterior wall is exposed to positive WSS.
Lagrangian tracking of blood cells inside the aneurysm models shows a
dramatic increase in the cell residence time as the aneurysm grows. While
recirculating, cells experience higher shear stress, which may lead to
their activation. The flow vortical structure convects the cells towards
the wall, increasing the probability for cell deposition and ipso facto for
an intraluminal thrombus to form.

14 Luglio, 2005 

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

Remarks on boundary control manifolds

Walter Littman, School of Mathematics, University of Minnesota, Minneapolis

This research seeks to quantify the role that mechanical stimuli play during
the growth of abdominal aortic aneurysms (AAA). The formation of disturbed
patterns of wall shear stresses (WSS) and gradients of wall shear stresses
(GWSS) has been shown to affect the wall structural integrity, through
their effect on the morphology and functions of the endothelial cells (EC)
and circulating blood cells.
Particle Image Velocimetry measurements of the pulsatile blood flow have
been performed in aneurysm models, while changing systematically their
geometric parameters. The parametric study shows that the flow separates
from the wall even at early stages of the disease (dilatation < 50%).
A large vortex ring forms in symmetric AAA, leading to the formation of two
distinct region: a region of flow detachment, with low oscillatory WSS, and
a downstream region of flow reattachment, with large negative WSS and
sustained GWSS.
The loss of symmetry in the geometry engenders a helical flow. The
dominant vortex, whose strength increases with the asymmetry parameter,
is shed from
the most bulged wall (anterior). It results in the formation of a large
recirculating region, where ECs are subjected to quasi-steady reversed
WSS of low magnitude, while the posterior wall is exposed to positive WSS.
Lagrangian tracking of blood cells inside the aneurysm models shows a
dramatic increase in the cell residence time as the aneurysm grows. While
recirculating, cells experience higher shear stress, which may lead to
their activation. The flow vortical structure convects the cells towards
the wall, increasing the probability for cell deposition and ipso facto for
an intraluminal thrombus to form.

7 Luglio, 2005 17:00

Seminario Matematico e Fisico di Milano

Rational knots, rational tangles and DNA

Louis H. Kauffman, University of Illinois, Chicago (Stati Uniti)

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

abstract

abstract

This talk will give elementary proofs (due to Jay Goldman, Sofia Lambropoulou and LK) of the Conway classification of rational tangles, and we will sketch the classification of rational knots and links by using methods developed with Sofia Lambropoulou. This is very beautiful elementary topoloogy that applies to the study of DNA recombination. The talk will discuss how this knot theory is used to study DNA.