16 Dicembre, 2009 17:00

Seminario Matematico e Fisico di Milano

On Gauss' linking number and the energy spectrum of magnetic knots

Renzo Ricca, Università di Milano-Bicocca

Dipartimento di Matematica, Università di Milano, Via Saldini

abstract

abstract

By revisiting Gauss work on magnetism we provide [1] a plausible reconstructionof what might have been Gauss own derivation of the linking number
concept and formula. Information on linking number and crossing number are fundamental
in a topological approach to field theory, topology providing a lower bounds on the
energy of the physical system [2]. In the case of knots, for instance, an exact analytical
expression for the minimum energy is found and the energy spectrum determined [3].
These results are fundamental in the classification of knots and links by energy methods and in the search for ground-state energy information in topological field theory [4].



[1] Ricca, R.L., Nipoti, B. On the original derivation and modern interpretations of
Gauss' linking number. To be submitted to J. Knot Theory and Its Ram.


[2] Ricca (2008) Topology bounds energy of knots and links. Proc. R. Soc. A 464, 293-300.


[3] Maggioni, F. and Ricca, R.L. (2009) On the groundstate energy of tight knots. Proc. R. Soc. A 465, 2761-2783.


[4] Ricca, R.L. (Ed.) (2009) Lectures on Topological Fluid Mechanics. Springer-CIME
Lecture Notes in Mathematics 1973. Springer-Verlag. Heidelberg, Germany.

15 Dicembre, 2009 11:00

Seminario Matematico e Fisico di Milano

Mathematical Models for Polymer Degradation and Erosion

Joao Silva Soares, MOX - Dipartimento di Matematica - Politecnico di Milano

Aula Seminari F. Saleri VI Piano MOX- Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

Biodegradable materials are employed in diverse medical applications and offer great promise for future applications. Current applications include non load bearing (drug delivery devices, tissue
engineering scaffolds) and load bearing (sutures, orthopedic implants,etc.) devices. When conditions are simple and the desired time of
performance is fairly well known, a “trial and error” approach to design has been sufficient to produce reasonably reliable devices. With these successes, the enormous potential of biodegradable
materials is being tapped for a wide variety of more complex situations.

Polymer degradation is the chain scission process that breaks polymer chains down to oligomers and finally monomers. Polymers degrade by several different mechanisms, depending on their inherent chemical structure and the environmental conditions to which they are exposed.
Degradation is an irreversible change in the material, i.e. molecular weight reduction, and directly leads to depreciation of the material
properties and eventually results in failure. Extensive degradation leads to erosion, which is the distinct (but related) process of material loss from the polymer bulk. The material lost can be monomers, oligomers, parts of the polymer backbone or even parts of the polymer bulk.

The primary barrier in the development of a rational design process for biodegradable medical devices is the lack of material models that
characterize evolving material properties over time. If such material models existed, medical device designers could employ rational design
tools such as finite element analysis to produce devices that meet design requirements for a desired period, and then gradually fade away
in a controllable manner, not only enhancing desired responses or designed behaviors (e.g. tissue healing or improved drug delivery
kinetics), but also avoiding problems associated with the unnecessary permanent presence of a foreign body.

The present research pushes the development of models enabling the next generation biodegradable material applications and represents the initiation of an iterative process of model formulation, experimental verification, and model refinement that will result in a great enhancement of the design process. Development of modeling, testing, and simulation techniques provides a cost effective resource for therapeutic technology developers, encouraging further industry collaboration, improved medical device designs, expanded medical device markets, and an overall improved health care.

15 Dicembre, 2009 16:15 in punto

Seminario Matematico e Fisico di Milano

Option Pricing with Garch Models: Equivalent Martingale Measures and Stochastic Orders

Fabio Bellini, Università di Milano Bicocca

Aula Seminari F. Saleri VI Piano

Biodegradable materials are employed in diverse medical applications and offer great promise for future applications. Current applications include non load bearing (drug delivery devices, tissue
engineering scaffolds) and load bearing (sutures, orthopedic implants,etc.) devices. When conditions are simple and the desired time of
performance is fairly well known, a “trial and error” approach to design has been sufficient to produce reasonably reliable devices. With these successes, the enormous potential of biodegradable
materials is being tapped for a wide variety of more complex situations.

Polymer degradation is the chain scission process that breaks polymer chains down to oligomers and finally monomers. Polymers degrade by several different mechanisms, depending on their inherent chemical structure and the environmental conditions to which they are exposed.
Degradation is an irreversible change in the material, i.e. molecular weight reduction, and directly leads to depreciation of the material
properties and eventually results in failure. Extensive degradation leads to erosion, which is the distinct (but related) process of material loss from the polymer bulk. The material lost can be monomers, oligomers, parts of the polymer backbone or even parts of the polymer bulk.

The primary barrier in the development of a rational design process for biodegradable medical devices is the lack of material models that
characterize evolving material properties over time. If such material models existed, medical device designers could employ rational design
tools such as finite element analysis to produce devices that meet design requirements for a desired period, and then gradually fade away
in a controllable manner, not only enhancing desired responses or designed behaviors (e.g. tissue healing or improved drug delivery
kinetics), but also avoiding problems associated with the unnecessary permanent presence of a foreign body.

The present research pushes the development of models enabling the next generation biodegradable material applications and represents the initiation of an iterative process of model formulation, experimental verification, and model refinement that will result in a great enhancement of the design process. Development of modeling, testing, and simulation techniques provides a cost effective resource for therapeutic technology developers, encouraging further industry collaboration, improved medical device designs, expanded medical device markets, and an overall improved health care.

15 Dicembre, 2009 16:15 in punto

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

Hardy-Rellich type inequalities with boundary terms and applications.

Elvise Berchio, Politecnico di Milano

aula seminari III piano

abstract

abstract

We present a family of Hardy-Rellich type inequalities having boundary terms and where the optimal constants are not necessarily the classical Hardy-Rellich ones. When the domain is the unit ball the exact values of the constants are performed. We exploit this fact to study the regularity of the “extremal solution” to some semilinear elliptic problems under Steklov boundary conditions.

11 Dicembre, 2009 14:00

Sezione di Geometria, Algebra e loro applicazioni

Quick algorithms for state-minimization of finite automata

Boris Melnikov, State University of Togliattigrad

Aula seminari III piano

We present a family of Hardy-Rellich type inequalities having boundary terms and where the optimal constants are not necessarily the classical Hardy-Rellich ones. When the domain is the unit ball the exact values of the constants are performed. We exploit this fact to study the regularity of the “extremal solution” to some semilinear elliptic problems under Steklov boundary conditions.

9 Dicembre, 2009 14:30 in punto

Sezione di Probabilità Quantistica

Covariant positive operator valued measures and instruments: an overview.

Alessandro Toigo, Politecnico di Milano

Aula seminari III piano

abstract

abstract

Covariance with respect to some symmetry group representation is a natural strong request imposed upon the whole set of measurements that can be performed on a quantum system, and, as such, it greatly simplifies their mathematical description. In this seminar, we present some general structure theorems for covariant positive operator valued measures and instruments, and apply them to two classes of measurements arising in quantum physics: the position and momentum measurements for a quantum particle, and the phase observables for the photon.

2 Dicembre, 2009 15:15

Seminari FDS

Problemi di misura del cielo: la vela quadrabile fiorentina

Luisa Rossi, Politecnico di Milano

Aula Consiglio - VII piano

Covariance with respect to some symmetry group representation is a natural strong request imposed upon the whole set of measurements that can be performed on a quantum system, and, as such, it greatly simplifies their mathematical description. In this seminar, we present some general structure theorems for covariant positive operator valued measures and instruments, and apply them to two classes of measurements arising in quantum physics: the position and momentum measurements for a quantum particle, and the phase observables for the photon.

1 Dicembre, 2009 16:15 in punto

Seminari FDS

Estimation of the instantaneous volatility and detection of volatility jumps

Monique Pontier, Toulouse Universite Paul Sabatier

Aula Seminari VI Piano

Covariance with respect to some symmetry group representation is a natural strong request imposed upon the whole set of measurements that can be performed on a quantum system, and, as such, it greatly simplifies their mathematical description. In this seminar, we present some general structure theorems for covariant positive operator valued measures and instruments, and apply them to two classes of measurements arising in quantum physics: the position and momentum measurements for a quantum particle, and the phase observables for the photon.

1 Dicembre, 2009 16:15 in punto

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

On the asymptotic behaviour of some solutions of the fast diffusion equation.

Gabriele Grillo, Politecnico di Torino

Aula seminari terzo piano

abstract

abstract

The so-called Barenblatt profiles are known to be explicit solution to the fast diffusion equation. Such solutions play, in the study of such equation, a role similar to the one played by the Gaussian solutions when dealing with the heat equation. In this talk we shall in fact show that, in suitable senses, certain classes of solutions to the fast diffusion equation converge to the Barenblatt profiles, giving explicit rates of convergence. Such rates are related to the best constant, explicitly determined, in a suitable Hardy-Poincaré inequality. A particular case, in which such constant vanishes, shows a polynomial decay instead of an exponential one. Such behaviour is proved using a geometric interpretation of the linearized evolution, the Li-Yau theory on the heat kernel on manifolds with nonnegative Ricci curvature, some weighted Nash inequalities and, finally, an appropriate use of the parabolic Harnack inequality.

This is a report of joint works with M. Bonforte, J.L. Vazquez and, in part, A. Blanchet e J. Dolbeault.

30 Novembre, 2009 14:15

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

The spectral element method: large scale applications to earthquake and vibration engineering

Roberto Paolucci, Politecnico di Milano - Dept. of Structural Engineering

Aula Seminari F. Saleri VI Piano MOX- Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

The spectral element method is introduced as an effective tool to perform large-scale numerical simulations of elastic wave propagation and dynamic soil-structure interaction. The theoretical formulation was first proposed in the late 90s (Faccioli et al., 1997), where the Galerkin method applied to the dynamic linear equilibrium problem was tailored in terms of spectral approximation of the spatial domain. Specifically, the computational domain is split into hexahedral elements where the unknown displacement field is approximated by a Lagrange polynomial of degree N, and each volume integral is evaluated numerically by the Legendre-Gauss-Lobatto (LGL) quadrature formula. This way, on one hand the exponential accuracy of the spectral method is preserved and, on the other hand, the computational effort in evaluating the integrals is minimized, since the resulting mass matrix is diagonal.
More recently, the research carried out at Department of Structural Engineering, Politecnico di Milano, and at CRS4, Sardinia, lead to major improvements of the numerical code in terms of pre- and post-processing tools, and of the implementation in parallel architectures to achieve high-performance computing. The resulting numerical code GeoELSE (GeoELastodynamics by Spectral Elements, see web site geoelse.stru.polimi.it) presently enables one to carry out numerical simulations of elastic wave propagation in 3D media under various types of excitation, including an earthquake source, moving loads, vibrating machines etc. Among its specific features, GeoELSE presently includes paraxial boundary conditions, linear visco-elastic and non-linear elastic materials, domain reduction method.
Several examples of application are provided, including: (i) the seismic response of the Grenoble valley in the near-field of a moderate earthquake (Stupazzini et al., 2009); (ii) the seismic response of a railway viaduct; (iii) the analysis of ground vibrations induced by high-speed trains (Paolucci and Spinelli, 2006).


References
- Faccioli E., F. Maggio, R. Paolucci, A. Quarteroni (1997). “2D and 3D elastic wave propagation by a pseudo-spectral domain decomposition method”, Journal of Seismology, Vol. 1, 237-251.
- Paolucci R., D. Spinelli (2006). “Ground motion induced by train passage”, ASCE Journal of Engineering Mechanics, Vol. 132, 201-210.
- Stupazzini M., R. Paolucci, H. Igel (2009). “Near-fault earthquake ground motion simulation in the Grenoble Valley by a high-performance spectral element code”. Bulletin of the Seismological Society of America, Vol. 99(1), 286-301.