26 Settembre, 2003 11:30

Sezione di Geometria, Algebra e loro applicazioni

Emumerative problems inspired by Mayer's theory of cluster integrals in thermodynamics.

Pierre Leroux, Université du Québec a Montréal

Dipartimento di Matematica - Politecnico di Milano

23 Settembre, 2003 17:00

Seminario Matematico e Fisico di Milano

Ferromagnetic and Anti-Ferromagnetic Systems

Jalal Shatah, Courant Institute of Mathematical Sciences, NY (U.S.A.)

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

abstract

abstract

Dispersive geometric evolution equations have received much attention in the past few decades. Some of these equations describe the dynamics of micromagnetics, vortex motion in liquid crystals, or vibrations of certain crystalline lattices. In this talk we will present some recent results on the equations describing ferromagnetic and anti-ferromagnetic materials concerning existence of solutions and the relation between anti-ferromagnetic equations (Schrodinger maps) and wave maps.

22 Settembre, 2003 15:00

Sezione di Analisi

Equazioni di Volterra stocastiche

Stefano Bonaccorsi, Universita di Trento

Dipartimento di Matematica

Dispersive geometric evolution equations have received much attention in the past few decades. Some of these equations describe the dynamics of micromagnetics, vortex motion in liquid crystals, or vibrations of certain crystalline lattices. In this talk we will present some recent results on the equations describing ferromagnetic and anti-ferromagnetic materials concerning existence of solutions and the relation between anti-ferromagnetic equations (Schrodinger maps) and wave maps.

22 Settembre, 2003 17:00

Seminario Matematico e Fisico di Milano

Differential Geometry and the Control of Nonlinear Systems

Kurt Schlacher, Johannes Kepler Universität, Linz (Austria)

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

abstract

abstract

Control theory makes significant progress, since nonlinear systems and differential geometry have been brought together about 30 years ago. The main advantage is that a description of the control system is avalaible, which does not require the choice of special coordinates. Now, many important properties of dynamic systems can be characterized in a geometric way. Within this framework nonlinear systems are considered as geometric objects, defined on smooth manifolds. E.g., a nonlinear control system can be identified with a submanifold of a certain geometric structure. Non observable or non accesible systems generate a foliation, and so on. This talk gives an introduction to the geometric description of nonlinear control systems, described by a set of ordinary differential equations. It starts with a short overview, where time invariant and time variant systems with or without control input are characterized by its geometric properties. At the same time a short introduction to the required mathematical tools will be presented. Having this preliminaries at our disposal we develop the concept of accessibility and observability by geometric ideas. It will turn our that this approach is not confined to explicit systems, on the contrary it can be generalized to more complex ones. Fortunately these methods are not only useful for the analysis of systems, where equivalence means that solutions of one system can be transformed to solutions of the other and vice versa. Using the idea that a dynamic system is also a certain sybmanifold, we show that equivalent systems describe the same submanifold in different coordinates only. From a systems designer's point of view, it is important to construct equivalent systems such that the control loop design can be reduced to an already solved problem Finally, this talk finishes with an industrial application of the present methods. The hydraulic gap control of stands in steel rolling mills is a challenging task because of the intrinsic nonlinear behaviour of these systems. We present new design ideas and controllers and prove the performance by simulations and measurements taken in mills located in the US and Europe.

18 Settembre, 2003 15:00

Seminario Matematico e Fisico di Milano

Finite element approximation to infinite Prandtl number Boussinesq equations with temperature dependent viscosity and its application to Earth s mantle convection problem

Masahisa TABATA, Department of Mathematical Sciences- Kyushu Univer

aula seminari MOX-6° piano dip. mat.

abstract

abstract

A stabilized finite element scheme for infinite Prandtl number Boussinesq
equations with temperature dependent viscosity is analyzed.
The domain is a spherical shell and the P1-element is employed for every
unknown function.
The finite element solution is proved to converge to the exact one in the
first order of the time increment and the mesh size.
The scheme is applied to Earth s mantle convection problems with
viscosities strongly dependent on the temperature and some numerical
results are shown.

18 Settembre, 2003 

Sezione di Geometria, Algebra e loro applicazioni

Faà di Bruno formula and determinantal identities

Chu Wenchang, Univ. degli Studi di Lecce

CNR, via Bassini 15, Milano

A stabilized finite element scheme for infinite Prandtl number Boussinesq
equations with temperature dependent viscosity is analyzed.
The domain is a spherical shell and the P1-element is employed for every
unknown function.
The finite element solution is proved to converge to the exact one in the
first order of the time increment and the mesh size.
The scheme is applied to Earth s mantle convection problems with
viscosities strongly dependent on the temperature and some numerical
results are shown.

16 Settembre, 2003 15:00

Sezione di Geometria, Algebra e loro applicazioni

Simulazione della circolazione feto-placentare: una possibile applicazione di modelli multiscala?

Caterina Guiot, Dip Neuroscienze-Sez Fisiologia-Lab Fisica medica

Aula Seminari MOX - 6° piano dip. di matematica

abstract

abstract

An increased interest towards the feto-placental venous circulation has
been recently devoted by clinicians, mainly regarding the umbilical
cord (morphological structure and feasibility of quantitative
measurements of the venous blood flow with Doppler US), the Ductus
Venosus and the inferior vena cava.
To investigate the haemodynamic changes leading to pathologies we
have developed a lumped mathematical model (implemented using SIMULINK ™)
simulating the feto-placental circulation from the inlet of
the umbilical artery (where the pressure waveform is assumed to be
known) across the placenta, the intraamniotic and intrahepatic umbilical
vein, the ductus venosus and the inferior vena cava until the fetal
atrium, where again a pressure waveform is assumed.
The main original assumptions of the lumped model are that the
umbilical vein behaves as a collapsible tube subject to an external
pressure exerted by the arteries which are coiled together inside the
cord and that the ductus venosus can dilate and, being in parallel with
the portal-hepatic system, can actively modify the outlet resistance
for the umbilical vein. All these features, however, could be described
with greater detail provided more sophisticated mathematical and numerical
treatments are properly included in the basic model.
The potential for model improvements due to a multiscale approach will
be discussed and evaluated.

15 Settembre, 2003 17:00

Seminario Matematico e Fisico di Milano

Blow-up for Nonlinear Hyperbolic Problems

Stanislav I. Pohozaev, Steklov Institute of Mathematics, Mosca (Russia)

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

abstract

abstract

"By using the concept of nonlinear capacity we suggest the general approach to the blow-up phenomena in nonlinear problems. This approach does not use the maximum principle arguments and properties of the fundamental solutions of related differential operators. That fact gives the possibility to consider a wide classes of nonlinear problems and obtain the sharp results for blow-up problems including the estimates for the ""blow-up time"". Finally we construct the ""Mendeleev table"" for nolinear problems. I am going to demonstrate this approach for nonlinear (multidimensional) hyperbolic problems. The results are obtained jointly with E.Mitidieri, L.Veron and A.Tesei."

28 Agosto, 2003 17:00

Seminario Matematico e Fisico di Milano

On a class of nonvariational elliptic problems

Djairo De Figueiredo, UNICAMP, Campinas (Brasile)

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

abstract

abstract

A priori bounds for positive solutions of a class of nonvariational elliptic systems is obtained by the blow-up method. Some Liouville type results are discussed.

1 Luglio, 2003 17:00

Seminario Matematico e Fisico di Milano

Dynamics of the Discretized FitzHugh-Nagumo Equation

Yakov Pesin, Penn State University, PA (U.S.A.)

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

abstract

abstract

I will consider the FitzHugh-Nagumo PDE. It is well-known in neuroscience and is used to describe the propagation of voltage impulse through a nurve axion. Its discrete version provides a competing model that I discuss in the talk. I present some results on the dynamics of the evolution operator on the space of traveling wave solutions and in particular, show that this dynamics changes from Morse-Smale type to a chaotic attractor to a horseshoe as a leading parameter (corresponding to the Reynolds number) of the system varies.