Dipartimento di Matematica

Prossimi Seminari

• Computational Prediction of Blood Damage
  Marek Behr, Chair for Computational Analysis of Technical Systems Faculty of Mechanical Engineering, RWTH Aachen
  lunedì 1 ottobre 2018 alle ore 14:00, Aula Consiglio VII Piano – Edificio 14, Dipartimento di Matematica POLITECNICO DI MILANO
  ABSTRACT
  ABSTRACT
  

  Modeling and computational analysis play an increasingly important role in bioengineering, particularly in the design of implantable ventricular assist devices (VAD) and other blood-handling devices. Numerical simulation of blood flow and associated physiological phenomena has the potential to shorten the design cycle and give the designers important insights into causes of blood damage and suboptimal performance. A set of modeling techniques is presented which are based on stabilized space-time finite element formulation of the Navier-Stokes equations. Alternate methods that represent the rotating components in an averaged sense using a rotating frame of reference will be discussed. In order to obtain quantitative hemolysis prediction, cumulative tensor-based measures of strain experienced by individual blood cells must be developed; red blood cells under shear can be modeled as deforming droplets, and their deformation tracked throughout the flow volume. The methods are applied to a simplified rotary blood pump, which is currently a subject of an inter-laboratory round-robin study.
  
  Contact: luca.dede@polimi.it
  
• Caputo Evolution Equations with time-nonlocal initial condition
  Lorenzo Toniazzi, University of Warwick
  martedì 9 ottobre 2018 alle ore 15:15, Aula Seminari 3° piano
  ABSTRACT
  ABSTRACT
  

  Consider the Caputo evolution equation (EE) $\partial_t^\beta u =\Delta u$ with initial condition $\phi$ on $\{0\}\times\mathbb R^d$, $\beta\in(0,1)$. As it is well known, the solution reads $u(t,x)=\mathbf E_x[\phi(B_{E_t})]$. Here $B_t$ is a Brownian motion and the independent time-change $E_t$ is an inverse $\beta$-stable subordinator. The fractional kinetic $B_{E_t}$ is a popular model for subdiffusion \cite{Meerschaert2012}, with remarkable universality properties \cite{BC11,Hai18}.\
  We substitute the Caputo fractional derivative $\partial_t^\beta$ with the Marchaud derivative. This results in a natural extension of the Caputo EE featuring a \emph{time-nonlocal initial condition} $\phi$ on $(-\infty,0]\times\mathbb R^d$. We derive the new stochastic representation for the solution, namely $u(t,x)=\mathbf E_x[\phi(-W_t,B_{E_t})]$. This stochastic representation has a pleasing interpretation due to the non-obvious presence of $W_t$, elucidating the notion of time-nonlocal initial conditions. Here $W_t$ denotes the waiting/trapping time of the fractional kinetic $B_{E_t}$. We discuss classical-wellposedness \cite{T18}, and time permitting weak-wellposedness \cite{DYZ17,DTZ18} for the respective extensions of Caputo-type EEs (such as in \cite{chen,HKT17}).
  
  Bibliography:
  
  
  Barlow, \u Cern\’y (2011). Probability theory and related fields, 149.3-4: 639-673.
  
  
  Chen, Kim, Kumagai, Wang (2017). arXiv:1708.05863.
  
  
  Du, Toniazzi, Zhou (2018). Preprint. Submitted in Sept. 2018.
  
  
  Du, Yang, Zhou (2017). Discrete and continuous dynamical systems series B, Vol 22, n. 2.
  
  
  Hairer, Iyer, Koralov, Novikov, Pajor-Gyulai (2018). The Annals of Probability, 46(2), 897-955.
  
  
  Hern\’andez-Hern\’andez, Kolokoltsov, Toniazzi (2017). Chaos, Solitons \& Fractals, 102, 184-196.
  
  
  Meerschaert, Sikorskii (2012). De Gruyter Studies in Mathematics, Book 43.
  
  
  Toniazzi (2018). To appear in: Journal of Mathematical Analysis and Applications. arXiv:1805.02464.
• Statistical modeling and monitoring of product and process quality in Additive Manufacturing: opportunities and challenges
  Bianca Maria Colosimo, Dipartimento di Meccanica, Politecnico di Milano
  giovedì 11 ottobre 2018 alle ore 14:00,  Aula Consiglio VII Piano – Edificio 14, Dipartimento di Matematica POLITECNICO DI MILANO
  ABSTRACT
  ABSTRACT
  

  Additive Manufacturing (AM), commonly known as three-dimensional printing, is widely recognized as a disruptive technology, and has the potential to fundamentally change the nature of future manufacturing. Building products layer-by-layer, AM represents a paradigm shift in manufacturing with many industrial applications. It enables production of huge varieties of customized products with considerable geometric complexity, and the same time, with extended capabilities and functional performances.
  Despite tremendous enthusiasm, AM faces major research challenges for widespread adoption of this innovative technology. Specifically, addressing the unique challenges associated with quality engineering of AM processes is crucial to the eventual success of AM. This talk presents an overview of quality-related issues for AM processes and products, focusing on opportunities and challenges in quality inspection, monitoring, control.
  
  Contact: piercesare.secchi@polimi.it
  
• Elastic waves in soft tissues: inverse analysis, experiments, simulations, validation
  Michel Destrade, Chair of Applied Mathematics at NUI Galway
  giovedì 18 ottobre 2018 alle ore 14:00, Aula Consiglio VII Piano – Edificio 14, Dipartimento di Matematica POLITECNICO DI MILANO
  ABSTRACT
  ABSTRACT
  

  Biological soft tissues and soft gels are difficult to study and model mathematically. Bioengineers often see them as engineering materials and try to evaluate their mechanical properties with standard testing protocols, such as tensile testing, simple shear, torsion, etc. These processes are destructive for tissues, as a sample is taken out of the body and placed in a device. The resulting measured parameters and models are expected to be very different from their in vivo counterparts. To test soft tissues properly, non¬-destructively, and non¬-invasively, we can rely on elastic waves. We can study the influence of pre¬stress on their speed and obtain the nonlinear elastic parameter by inverse analysis. This idea forms the basis of the theory of acousto-¬elasticity, which can be dated back to early works of Brillouin, and has been used successfully in the past for “hard” elastic solids such as rocks and metals. With this talk, we will explore the extension of acousto-¬elasticity to “soft” elastic solids, which can be subjected to large deformations in service. We will look at theoretical, numerical, experimental, and even clinical results, generated in particular on gels, brain, breast, and skin.
  
  Contact: pasquale.ciarletta@polimi.it
  
• An overview of some mathematical and computational problems in Network Science
  Michele Benzi, Scuola Normale Superiore, Pisa
  giovedì 22 novembre 2018 alle ore 14:00,  Aula Consiglio VII Piano – Edificio 14, Dipartimento di Matematica POLITECNICO DI MILANO
  ABSTRACT
  ABSTRACT
  

  Network Science is a rapidly growing interdisciplinary area at the intersection of mathematics, computer science, and a multitude of disciplines from the natural and life sciences to the social sciences and even the humanities. Network analysis methods are now widely used in proteomics, in the study of social networks (both human and animal), in finance, in ecology, in bibliometric studies, in archeology, and in a host of other fields. In this talk I will introduce the audience to some of the mathematical and computational problems and methods of complex networks, with an emphasis on the basic notions of centrality and communicability. More specifically, I will describe some of the problems in large-scale numerical linear algebra arising in this area, and how they differ from the corresponding problems encountered in more traditional applications of numerical analysis. The talk will be accessible to students, requiring only a modest background in linear algebra, numerical analysis and graph theory.
  
  Contacts: paola.antonietti@polimi.it, paolo.zunino@polimi.it
  
• BIRATIONAL EQUIVALENCE OF ALGEBRAIC VARIETIES
  Shigefumi Mori, Kyoto University Institute of Advanced Study
  lunedì 26 novembre 2018 alle ore 16:30, Aula Chisini, Diparimento di Matematica, Via C. Saldini 50

Seminari Passati

• Coupled problems in Environmental and medical applications
  Hiroshi Suito, Mathematical Science Group, Advanced Institute for Materials Research, Tohoku University, Japan
  martedì 18 settembre 2018 alle ore 14:00, Aula Consiglio VII Piano – Edificio 14, Dipartimento di Matematica POLITECNICO DI MILANO
  ABSTRACT
  

  Fluid–structure interaction (FSI) problems arise under various circumstances including environmental and medical applications. This talk firstly presents numerical simulations for motions of aquatic plants interacting with surrounding fluids using immersed boundary method (IBM) with finite-difference approximation on fixed meshes. We also present simulations for blood flows in aorta solved on finite-element meshes moving with the vessel wall. Aortic aneurysms and aortic coarctations are considered here. Geometrical characteristics such as curvature and torsion help us to understand the essential difference for morphologies among patients. The latter problem is followed by a machine learning approach by which wall shear stress and oscillatory shear index distributions are estimated using geometrical characteristics of the vessels.
  
  Contact: alfio.quarteroni@polimi.it
  
  ABSTRACT
• An inverse boundary value problem arising from cardiac electrophisiology
  Luca Ratti, Politecnico di Milano
  martedì 18 settembre 2018 alle ore 15:15, Politecnico di Milano, Dipartimento di Matematica, Aula Seminari 3° Piano
  ABSTRACT
  

  The cardiac electrical activity can be comprehensively described throughout the monodomain model, consisting of a semilinear parabolic equation coupled with a nonlinear ordinary differential equation.
  
  In my talk, I will introduce the inverse problem of identifying conductivity inhomogeneities in the monodomain system, taking advantage of data acquired on the boundary of the domain. Due to the complexity of the task, I will first tackle the stationary counterpart of the problem, regarding which it is possible to formulate well-posedness results both for the forward and for the inverse problem, and to rigorously introduce reconstruction procedures. Similar results are then generalized to the full complexity of the original model.
  
  
  Throughout the presentation, I will focus on the problem of localizing small size inhomogeneities, as well as arbitrarily large ones, by means of the constraint optimization of a suitable misfit functional. The first task is achieved by relying on an asymptotic expansion of the boundary voltage with respect to the size of the inclusion, and employing tools from the topological optimization theory. The second issue is analyzed by means of the regularization theory of inverse problems and introducing a convenient relaxation of the optimization problem. The theoretical results are supported by numerical experiments, which are exhaustively reported.
  
  
  This is a joint work with Elena Beretta, Cristina Cerutti, Cecilia Cavaterra, Andrea Manzoni and Marco Verani.
  ABSTRACT
• Rotation number of the linear Schrödinger equation with discontinuous almost periodic potentials
  Zhe Zhou, Chinese Academy of Sciences, Beijing
  giovedì 13 settembre 2018 alle ore 15:00, Aula Seminari 3° piano
  ABSTRACT
  

  In this talk, based on the celebrated paper [R. Johnson and J. Moser, Comm. Math. Phys., 1982], we will recover the rotation numbers of the Schrödinger equation. The essential elements in the proof are positive homogeneity and almost periodicity. From this point of view, the concept of rotation numbers may be introduced in the case of discontinuous potentials. Moreover, we will show the optimal estimate of rotation numbers in such case.
  ABSTRACT
• KöNIG’S PROBLEM FOR ABELIAN PERMUTATION GROUPS
  Andrzej Kisielewicz, Uniwersytet Wroc?awski, Wydzia? Matematyki i Informatyki
  martedì 11 settembre 2018 alle ore 14:15, Aula seminari III piano
  ABSTRACT
  

  König’s problem for permutation groups concerns the following question: Given a permutation group P = (P, X) acting on a finite set X, is there a graph G=(G, X) with the set of vertices X, such its automorphisms are precisely permutations in P? König’s problem is to find a necessary and sufficient conditions for a permutation group P to be the automorphism groups of some graph.
  
  There exist permutation groups that are not the automorphism groups of any graph (for example, alternating groups or groups generated by a single cyclic permutation). So far, this version of König’s problem (known also as the concrete version) has been solved only for regular permutation groups, cyclic permutation groups (generated by a single permutation), and partially, for abelian permutation groups.
  
  In this talk we demonstrate however that the result by Zelikovskij [3] concerning König’s problem for abelian permutation groups, reported in a recent survey [2], is false. We argue that a more natural setting for this problem is that concerning the automorphism groups of edge-colored graphs. Our main result, based on techniques applied in [1], provides a characterization of those abelian permutation groups that are the automorphism groups of edge-colored graphs and shows, in addition, that each such group can be represented by an edge-colored graph using no more than 4 colors.
  
  References
  
  
  [1] M. Grech, A. Kisielewicz, Symmetry groups of boolean functions, European J. Combin. 40 (2014) 1-10.
  
  [2] J. Morris, Automorphism Groups of Circulant Graphs – a Survey, in: Bondy A., Fonlupt J., Fouquet JL., Fournier JC., Ramrez Alfonsn J.L. (eds) Graph Theory in Paris. Trends in Mathematics. Birkhuser Basel 2006, pp. 311-325.
  
  [3] A. Z. Zelikovskij, Konigs problem for Abelian permutation groups, Izv. Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk 5 (1989), 34-39.
  ABSTRACT
• A new paradigm for geometric modeling: Pythagorean Hodograph (PH) B-Spline curves
  Gudrun Albrecht, National University of Colombia
  martedì 31 luglio 2018 alle ore 14:00, Aula Seminari ‘Saleri’ VI Piano MOX-Dipartimento di Matematica, Politecnico di Milano – Edificio 14
  ABSTRACT
  

  We introduce the new class of planar Pythagorean-Hodograph (PH) B–Spline curves. They can be seen as a generalization of the well-known class of planar Pythagorean-Hodograph (PH) Bézier curves, presented by R. Farouki and T. Sakkalis in 1990, including the latter ones as special cases. Pythagorean-Hodograph B–Spline curves are nonuniform parametric B–Spline curves whose arc-length is a B–Spline function as well. An important consequence of this special property is that the offsets of Pythagorean-Hodograph B–Spline curves are non-uniform rational B–Spline (NURBS) curves. Thus, although Pythagorean-Hodograph B–Spline curves have fewer degrees of freedom than general B–Spline curves of the same degree, they offer unique advantages for computer-aided design and manufacturing, robotics, motion control, path planning, computer graphics, animation, and related fields. After providing a general definition for this new class of planar parametric curves, we !
  present useful formulae for their construction and discuss their remarkable attractive properties. Then we provide a method to determine within the set of all PH B–Splines the one that is closest to a given reference spline having the same degree and knot partition.
  
  Contact franca.calio@polimi.it
  
  
  ABSTRACT
• Optimal control of treatment time in a diffuse interface model of tumor growth and related issues
  Elisabetta Rocca, Università di Pavia
  giovedì 28 giugno 2018 alle ore 11:15, Aula Seminari ‘Saleri’ VI Piano MOX-Dipartimento di Matematica, Politecnico di Milano – Edificio 14
  ABSTRACT
  

  We consider an optimal control problem for a diffuse interface model of tumor growth. The state equations couples a Cahn-Hilliard equation and a reaction-diffusion equation, which models the growth of a tumor in the presence of a nutrient and surrounded by host tissue. The introduction of cytotoxic drugs into the system serves to eliminate the tumor cells and in this setting the concentration of the cytotoxic drugs will act as the control variable. Furthermore, we allow the objective functional to depend on a free time variable, which represents the unknown treatment time to be optimized. As a result, we obtain first order necessary optimality conditions for both the cytotoxic concentration and the treatment time
  
  Contact: pasquale.ciarletta@polimi.it
  
  ABSTRACT
• VAPORIZING AND FREEZING THE RIEMANN ZETA FUNCTION
  Terence Tao, University of California, Los Angeles
  venerdì 22 giugno 2018 alle ore 14:30, Edificio U4, P.zza della Scienza, 4, Aula Luisella Sironi
  ABSTRACT
  

  In 1950, de Bruijn studied the effect of evolving the Riemann zeta function (or more precisely, a closely related function known as the Riemann xi function) by the (backwards) heat equation. His analysis, together with later work by Newman, showed that there existed a finite constant Lambda, at most 1/2 in value, such that the Riemann hypothesis for this evolved function was true at times greater than or equal to Lambda, and false below that threshold. Thus the Riemann hypothesis for the zeta function is equivalent to Lambda being non-positive. Recently, in joint work with Brad Rodgers, I was able to establish the complementary estimate that Lambda is non-negative, confirming a conjecture of Newman; thus, the Riemann hypothesis for zeta, if true, is only “barely so”. The proof relies on an analysis of the dynamics of zeroes of entire functions under heat flow; it turns out that as one evolves forward in time, the zeroes “freeze” into approximate arithmetic progressions, while if one evolves backwards, the zeroes “vaporize” to leave the critical line. In followup work in an online collaborative “Polymath” project, the upper bound on Lambda has also been improved. We describe these results and their proofs in this talk.
  
  ABSTRACT
• Approximating the true time weighted return
  Marco Guzzetti, Politecnico di Milano
  mercoledì 20 giugno 2018 alle ore 12:15 precise, Aula seminari del Terzo piano

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