Dipartimento di Matematica

Abstract

It is known that positive solutions to $\Delta_p u + u^{p^*-1}=0$ in $\mathbb{R}^n$, with $n \geq 3$ and $1
We provide a new approach to this problem which allows us to give a complete classification of the solutions in an anisotropic setting as well as to a suitable generalization of the problem in convex cones.

This is a joint work with A. Figalli and A. Roncoroni.

Abstract

Advances in numerical techniques and the ever increasing computational power have rendered the execution of forward models of total heart function feasible. Using such models based on clinical images and parameterized to reflect a given patient's physiology, are a highly promising approach to comprehensively and quantitatively characterize cardiovascular function in a given patient. Such models are anticipated to play a pivotal role in future precision medicine as a method to stratify diseases, optimize therapeutic procedures, predict outcomes and thus better inform clinical decision making.
However, to translate modeling into a clinically applicable modality a number of key challenges have to be addressed. In particular, expensive computational models must be made efficient enough to be compatible with clinical time frames. This can be addressed either with hierarchical models of varying complexity which are cheaper to evaluate, by using computational efficient techniques such as spatio-temporal adaptivity, or by exploiting the power of new HPC hardware through massive parallelization or the use of accelerators. Further, the etiology of most cardiac pathologies comprises Multiphysics aspects, requiring the coupling of various physics, which may be characterized by very different space and time scales, rendering their coupling a challenging endeavor. Finally and most importantly, to be of clinical utility generic models must be specialized based on clinical data, which requires complex parameterization and data assimilation procedures to match model behavior with clinical observations.
In this presentation, I will give an overview of our multi-physics forward modelling framework and our recent work on m personalising models using clinical data.

contact: alfio.quarteroni@polimi.it

This seminar is organized within the ERC-2016-ADG Research project iHEART - An Integrated Heart Model for the simulation of the cardiac function, that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 740132)

Abstract

Questo seminario vuole condividere i primi risultati di un lavoro di ricerca sui significati e i possibili usi didattici dei meme matematici. I meme sono oggetti digitali – tipicamente di natura umoristica – creati dagli utenti e condivisi in modo virale nel web: essi vengono dal pianeta social, ma recenti ricerche mostrano che possano dare un contributo alla didattica della matematica, facendo leva su ciò che gli studenti conoscono. Nel seminario sarà prima proposto un costrutto teorico per identificare gli elementi che compongono il sistema di significati veicolati da un meme e verranno condivisi esempi ed esperienze didattiche.

Abstract

: The first part of the talk will provide a self-contained introduction to the formal deduction of the quantum hydrodynamics (QHD) in the form of an Euler Dispersive Irrotational compressible fluid system.
We will then introduce in a possible simple way the a macroscopic approach to various related to superfluids due to the Russian school by Landau, Khalaktikov and other important models in superconductivity and semiconductor devices.
Then we show important mathematical difficulties related to the presence of vacuum and explain their physical counterpart, even in relation with quantum vortices and Bose Einstein condensation. We also will relate our analysis to the Gross-Pitaevskii model.
We will show how to develop a mathematical theory in 2-D and 3-D, consistent with the physics for the problem of large data weak solutions, in the energy norm. All the theory, including irrotationality, is formulated by using observable quantities (density and linear momentum), avoiding the need of defining the velocity fields. The methods are based on a ad hoc" polar factorization, dispersive analysis and local smoothing. The initial data are restricted to be momenta of a wave funtiona.
It has been recently developed a large data 1-D theory purely hydrodynamical of weal solutions with strong stability.
In conclusion we will just mention several other related problems connected to quantum vortices, dispersive shocks and the presence of magnetic fields.

Contatto: paola.antonietti@polimi.it

Abstract

Scopo dell’intervento è presentare il percorso didattico innovativo, in continuità fra vari ordini scolastici, sul tema dell’invertibilità. Partendo dalle formule inverse delle funzioni elementari, l’argomento sottende la soluzione delle equazioni elementari ed è la chiave per affrontare i modelli differenziali.
Il tema sarà trattato in interazione dinamica fra Realtà e Matematica.

Nota: questo seminario sarà una lezione-laboratorio e avrà durata un’ora e mezzo.

Abstract

What if physicians and surgeons could virtually analyze their patients’ health and plan therapies and surgeries using the same advanced modeling and simulation technology that the automotive, aerospace, energy and hi-tech industries rely on to test their product before they are built? As early as 1906, researchers first began suggesting the solution of continuum mechanics problems by modeling the body with a lattice of elastic bars and employing frame analysis methods. In 1941, Courant recognized piecewise polynomial interpolation over triangular subregions as a Rayleigh-Ritz solution of variational problems. Since there were no computers at the time, neither approach was practical and Courant’s work was largely forgotten until engineers had independently developed it. By 1953, structural engineers were solving matrix stiffness equations with digital computers. The widespread use of finite element methods in engineering began with the classic papers by Turner et al. and Argyris and Kelsey. The name “finite element” was coined in 1960, and the method began to be recognized as mathematically rigorous by 1963. The creation of subject-specific biventricular finite element models has been a long-term endeavor within the biomedical engineering community. Using high resolution (0.3 × 0.3 × 0.8 mm) ex-vivo data, we constructed precise fully subject-specific biventricular finite-element models of healthy and failing swine hearts. Each model includes fully subject-specific geometries, myofiber architecture and, in the case of the failing heart, fibrotic tissue distribution. Each model was calibrated using subject-specific experimental data and compared with independent in-vivo strain data obtained from echocardiography. Our methods produced highly detailed representations of swine hearts that function mechanically in a remarkably similar manner to the in-vivo subject-specific strains on a global and regional comparison. The degree of subject-specificity included in the models represents a milestone for modeling efforts that captures realism of the whole heart. This study establishes a foundation for future computational studies that can apply these validated methods to advance cardiac mechanics research.

Contatto: alfio.quarteroni@polimi.it

This seminar is organized within the ERC-2016-ADG Research project iHEART - An Integrated Heart Model for
the simulation of the cardiac function, that has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 740132)

Abstract

Dopo aver introdotto il concetto di geodetica su una superficie e aver brevemente discusso le geometrie non euclidee, si passerà a una descrizione della relatività (ristretta e generale). Lo scopo è quello di mostrare che in relatività generale avviene una fusione tra geometria e fisica: lo spazio non si può più concepire come un palcoscenico immutabile nel quale vengono poi introdotti i corpi, come nelle teorie classiche. In conclusione, se il tempo lo permetterà, si accennerà all'osservazione delle onde gravitazionali

Abstract

In questo seminario intendo esporre un argomento molto importante della geometria dello spazio, anche se, talvolta, questo viene trascurato, o presentato superficialmente, nei programmi della scuola superiore: i poliedri. Dopo avere introdotto la loro definizione e le loro principali proprietà, sarà mostrata la loro presenza nella natura, nella vita quotidiana e nell’arte, partendo dall’antica Grecia fino ad arrivare ai nostri giorni. Anzitutto saranno trattati i poliedri regolari; successivamente sarà introdotta l’importante famiglia, soprattutto nelle applicazioni, dei poliedri archimedei. Infine, saranno presentati gli interessanti poliedri di Goldberg

Abstract

L’obiettivo del seminario è riflettere su come, a partire dai dati restituiti dal Servizio Nazionale di Valutazione, si possano intraprendere azioni di consolidamento e miglioramento all’interno dalla propria classe, e non concentrarsi su cosa fare per preparare le prove INVALSI. Questo viene realizzato esplicitando il legame delle domande con le Indicazioni Nazionali e con le prassi didattiche, e utilizzando gli opportuni costrutti della didattica. Particolare rilievo verrà dato al fatto che gli obiettivi di apprendimento alla base delle pratiche d'aula sono fissati dalle Indicazioni Nazionali e non dalle prove che invece forniscono informazioni su come questi obiettivi sono raggiunti

Abstract

The ability to measure the heart, its shape, its structure and its function across multiple spatial and temporal scales continues to grow. Interpreting this data remains challenging. Computational biophysical models of the heart allow us to quantitatively link and interpret these large disparate data sets within the context of known cardiac physiology and invariable physical constraints. Within these models, we can infer unobservable states, propose and test new hypothesis and predict how systems will respond to challenges increasing our ability to interrogate and understand biological systems. We are increasingly applying this approach to modelling human hearts to investigate clinical applications. In this presentation, I will give an overview on our modelling work simulating anthracycline-induced heart failure, how we are using models of individual patients to study cardiac resynchronisation therapy and how we are using simulations to characterise the anatomy and pathophysiology of atrial fibrillation patients. Finally, I will present some of our preliminary results on simulating the four-chamber heart to begin simulating the interactions between atrial and ventricular function.

Contact: alfio.quarteroni@polimi.it

This seminar is organized within the ERC-2016-ADG Research project iHEART - An Integrated Heart Model for the simulation of the cardiac function, that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 740132)

Abstract

Some materials may naturally form discontinuities such as cracks as a result of scale effects and long range interactions. Peridynamic models such behavior introducing a new nonlocal framework for the basic equations of continuum mechanics. In this lecture we consider a nonlinear peridynamic model and discuss its well-posedness in suitable fractional Sobolev spaces.
Those results were obtained in collaboration with S. Dipierro (Perth), F. Maddalena (Bari) and E. Valdinoci (Perth).