We study the average condition number for polynomial eigenvalues of collections of matrices drawn from some random matrix ensembles. In particular, we prove that polynomial eigenvalue problems defined by matrices with random Gaussian entries are very well conditioned on the average.

1 aBeltrán, Carlos1 aKozhasov, Khazhgali uhttps://doi.org/10.1007/s10208-019-09414-201637nas a2200217 4500008004100000022001400041245007700055210006900132300001400201490000800215520097100223653002001194653001501214653001701229653001701246100001901263700002201282700002301304700002101327856007101348 2019 eng d a0022-123600aReducibility of first order linear operators on tori via Moser's theorem0 aReducibility of first order linear operators on tori via Mosers a932 - 9700 v2763 aIn this paper we prove reducibility of a class of first order, quasi-linear, quasi-periodic time dependent PDEs on the torus∂tu+ζ⋅∂xu+a(ωt,x)⋅∂xu=0,x∈Td,ζ∈Rd,ω∈Rν. As a consequence we deduce a stability result on the associated Cauchy problem in Sobolev spaces. By the identification between first order operators and vector fields this problem can be formulated as the problem of finding a change of coordinates which conjugates a weakly perturbed constant vector field on Tν+d to a constant diophantine flow. For this purpose we generalize Moser's straightening theorem: considering smooth perturbations we prove that the corresponding straightening torus diffeomorphism is smooth, under the assumption that the perturbation is small only in some given Sobolev norm and that the initial frequency belongs to some Cantor-like set. In view of applications in KAM theory for PDEs we provide also tame estimates on the change of variables.

10aHyperbolic PDEs10aKAM theory10aNash–Moser10aReducibility1 aFeola, Roberto1 aGiuliani, Filippo1 aMontalto, Riccardo1 aProcesi, Michela uhttp://www.sciencedirect.com/science/article/pii/S002212361830379301650nas a2200157 4500008004100000022001400041245013600055210006900191260000800260520108900268100002501357700002101382700002201403700002001425856004701445 2019 eng d a1618-189100aOn the relaxed area of the graph of discontinuous maps from the plane to the plane taking three values with no symmetry assumptions0 arelaxed area of the graph of discontinuous maps from the plane t cJul3 aIn this paper, we estimate from above the area of the graph of a singular map u taking a disk to three vectors, the vertices of a triangle, and jumping along three $\mathcal{C}^2$-embedded curves that meet transversely at only one point of the disk. We show that the singular part of the relaxed area can be estimated from above by the solution of a Plateau-type problem involving three entangled nonparametric area-minimizing surfaces. The idea is to ``fill the hole'' in the graph of the singular map with a sequence of approximating smooth two-codimensional surfaces of graph-type, by imagining three minimal surfaces, placed vertically over the jump of u, coupled together via a triple point in the target triangle. Such a construction depends on the choice of a target triple point, and on a connection passing through it, which dictate the boundary condition for the three minimal surfaces. We show that the singular part of the relaxed area of u cannot be larger than what we obtain by minimizing over all possible target triple points and all corresponding connections.

1 aBellettini, Giovanni1 aElshorbagy, Alaa1 aPaolini, Maurizio1 aScala, Riccardo uhttps://doi.org/10.1007/s10231-019-00887-000448nas a2200097 4500008004100000245008100041210006900122100002900191700002300220856010700243 2018 eng d00aOn real resonances for the three-dimensional, multi-centre point interaction0 areal resonances for the threedimensional multicentre point inter1 aMichelangeli, Alessandro1 aScandone, Raffaele uhttps://www.math.sissa.it/publication/real-resonances-three-dimensional-multi-centre-point-interaction00849nas a2200157 4500008004100000022001400041245009800055210006900153260000800222300000800230490000700238520036300245100001800608700001900626856004600645 2018 eng d a1432-083500aRecognizing the flat torus among RCD*(0,N) spaces via the study of the first cohomology group0 aRecognizing the flat torus among RCD0N spaces via the study of t cJun a1040 v573 aWe prove that if the dimension of the first cohomology group of a $\mathsf{RCD}^\star (0,N)$ space is $N$, then the space is a flat torus. This generalizes a classical result due to Bochner to the non-smooth setting and also provides a first example where the study of the cohomology groups in such synthetic framework leads to geometric consequences.

1 aGigli, Nicola1 aRigoni, Chiara uhttps://doi.org/10.1007/s00526-018-1377-z00631nas a2200133 4500008004100000245015100041210006900192100002800261700002200289700001700311700002400328700002100352856012400373 2018 eng d00aA Reduced Basis approach for PDEs on parametrized geometries based on the Shifted Boundary Finite Element Method and application to fluid dynamics0 aReduced Basis approach for PDEs on parametrized geometries based1 aKaratzas, Efthymios, N.1 aStabile, Giovanni1 aNouveau, Leo1 aScovazzi, Guglielmo1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/reduced-basis-approach-pdes-parametrized-geometries-based-shifted-boundary-finite00596nas a2200133 4500008004100000245012100041210006900162100002800231700002200259700001600281700002400297700002100321856012000342 2018 eng d00aA Reduced Order Approach for the Embedded Shifted Boundary FEM and a Heat Exchange System on Parametrized Geometries0 aReduced Order Approach for the Embedded Shifted Boundary FEM and1 aKaratzas, Efthymios, N.1 aStabile, Giovanni1 aAtallah, N.1 aScovazzi, Guglielmo1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/reduced-order-approach-embedded-shifted-boundary-fem-and-heat-exchange-system00517nas a2200109 4500008004100000245011200041210006900153100001900222700002200241700002100263856012300284 2018 eng d00aReducibility for a class of weakly dispersive linear operators arising from the Degasperis Procesi equation0 aReducibility for a class of weakly dispersive linear operators a1 aFeola, Roberto1 aGiuliani, Filippo1 aProcesi, Michela uhttps://www.math.sissa.it/publication/reducibility-class-weakly-dispersive-linear-operators-arising-degasperis-procesi01236nas a2200121 4500008004100000245007700041210006900118300001200187490000700199520084400206100001801050856004601068 2018 eng d00aRegularity estimates for scalar conservation laws in one space dimension0 aRegularity estimates for scalar conservation laws in one space d a623-6910 v153 aWe deal with the regularizing effect that, in scalar conservation laws in one space dimension, the nonlinearity of the flux function f has on the entropy solution. More precisely, if the set w : f″(w)≠0 is dense, the regularity of the solution can be expressed in terms of BVΦ spaces, where Φ depends on the nonlinearity of f. If moreover the set w : f″(w) = 0 is finite, under the additional polynomial degeneracy condition at the inflection points, we prove that f′∘ u(t) ∈BV loc(ℝ) for every t > 0 and that this can be improved to SBVloc(ℝ) regularity except an at most countable set of singular times. Finally, we present some examples that show the sharpness of these results and counterexamples to related questions, namely regularity in the kinetic formulation and a property of the fractional BV spaces.

1 aMarconi, Elio uhttps://doi.org/10.1142/S021989161850020000325nas a2200109 4500008004100000245002400041210002400065100001900089700002400108700002100132856006200153 2017 eng d00aRandom spectrahedra0 aRandom spectrahedra1 aBreiding, Paul1 aKozhasov, Khazhgali1 aLerario, Antonio uhttps://www.math.sissa.it/publication/random-spectrahedra00452nas a2200121 4500008004100000245005700041210005700098260002200155490000800177100002300185700002400208856009800232 2017 eng d00aRayleigh–Taylor instability in soft elastic layers0 aRayleigh–Taylor instability in soft elastic layers bThe Royal Society0 v3751 aRiccobelli, Davide1 aCiarletta, Pasquale uhttps://www.math.sissa.it/publication/rayleigh%E2%80%93taylor-instability-soft-elastic-layers01239nas a2200169 4500008004100000022001400041245003900055210003900094260000800133300000700141490000900148520080000157100001900957700002500976700002401001856004401025 2017 eng d a1029-847900aReal topological string amplitudes0 aReal topological string amplitudes cMar a800 v20173 aWe discuss the physical superstring correlation functions in type I theory (or equivalently type II with orientifold) that compute real topological string amplitudes. We consider the correlator corresponding to holomorphic derivative of the real topological amplitude $\mathcal{G_\chi}$, at fixed worldsheet Euler characteristic $\chi$. This corresponds in the low-energy effective action to $\mathcal{N}=2$ Weyl multiplet, appropriately reduced to the orientifold invariant part, and raised to the power $g'= −\chi+ 1$. We show that the physical string correlator gives precisely the holomorphic derivative of topological amplitude. Finally, we apply this method to the standard closed oriented case as well, and prove a similar statement for the topological amplitude $\mathcal{F}_g$.

1 aNarain, K., S.1 aPiazzalunga, Nicolò1 aTanzini, Alessandro uhttps://doi.org/10.1007/JHEP03(2017)08002504nas a2200157 4500008004100000245005700041210005700098260001200155300000800167490000600175520201600181100001502197700002202212700002102234856009102255 2017 eng d00aReduced Basis Methods for Uncertainty Quantification0 aReduced Basis Methods for Uncertainty Quantification c08/2017 a8690 v53 aIn this work we review a reduced basis method for the solution of uncertainty quantification problems. Based on the basic setting of an elliptic partial differential equation with random input, we introduce the key ingredients of the reduced basis method, including proper orthogonal decomposition and greedy algorithms for the construction of the reduced basis functions, a priori and a posteriori error estimates for the reduced basis approximations, as well as its computational advantages and weaknesses in comparison with a stochastic collocation method [I. Babuška, F. Nobile, and R. Tempone, *SIAM Rev.*, 52 (2010), pp. 317--355]. We demonstrate its computational efficiency and accuracy for a benchmark problem with parameters ranging from a few to a few hundred dimensions. Generalizations to more complex models and applications to uncertainty quantification problems in risk prediction, evaluation of statistical moments, Bayesian inversion, and optimal control under uncertainty are also presented to illustrate how to use the reduced basis method in practice. Further challenges, advancements, and research opportunities are outlined.

Read More: http://epubs.siam.org/doi/abs/10.1137/151004550

POD–Galerkin reduced-order models (ROMs) for fluid-structure interaction problems (incompressible fluid and thin structure) are proposed in this paper. Both the high-fidelity and reduced-order methods are based on a Chorin-Temam operator-splitting approach. Two different reduced-order methods are proposed, which differ on velocity continuity condition, imposed weakly or strongly, respectively. The resulting ROMs are tested and compared on a representative haemodynamics test case characterized by wave propagation, in order to assess the capabilities of the proposed strategies.

1 aBallarin, Francesco1 aRozza, Gianluigi1 aMaday, Yvon1 aBenner, Peter1 aOhlberger, Mario1 aPatera, Anthony1 aRozza, Gianluigi1 aUrban, Karsten uhttps://www.math.sissa.it/node/1294801221nas a2200097 4500008004100000245007700041210006900118520087000187100001801057856004801075 2017 en d00aRegularity estimates for scalar conservation laws in one space dimension0 aRegularity estimates for scalar conservation laws in one space d3 aIn this paper we deal with the regularizing effect that, in a scalar conservation laws in one space dimension, the nonlinearity of the flux function ƒ has on the entropy solution. More precisely, if the set ⟨w : ƒ " (w) ≠ 0⟩ is dense, the regularity of the solution can be expressed in terms of BV Ф spaces, where Ф depends on the nonlinearity of ƒ. If moreover the set ⟨w : ƒ " (w) = 0⟩ is finite, under the additional polynomial degeneracy condition at the inflection points, we prove that ƒ' 0 u(t) ∈ BVloc (R) for every t > 0 and that this can be improved to SBVloc (R) regularity except an at most countable set of singular times. Finally we present some examples that shows the sharpness of these results and counterexamples to related questions, namely regularity in the kinetic formulation and a property of the fractional BV spaces.1 aMarconi, Elio uhttp://preprints.sissa.it/handle/1963/3529100917nas a2200157 4500008004100000020002200041245008300063210006900146260004400215300001400259520035500273100002400628700002900652700002900681856004900710 2017 eng d a978-3-319-58904-600aRemarks on the Derivation of Gross-Pitaevskii Equation with Magnetic Laplacian0 aRemarks on the Derivation of GrossPitaevskii Equation with Magne aChambSpringer International Publishing a257–2663 aThe effective dynamics for a Bose-Einstein condensate in the regime of high dilution and subject to an external magnetic field is governed by a magnetic Gross-Pitaevskii equation. We elucidate the steps needed to adapt to the magnetic case the proof of the derivation of the Gross-Pitaevskii equation within the ``projection counting'' scheme.

1 aOlgiati, Alessandro1 aMichelangeli, Alessandro1 aDell'Antonio, Gianfausto uhttps://doi.org/10.1007/978-3-319-58904-6_1501691nas a2200169 4500008004100000245008700041210006900128260001800197300000600215490000600221520116500227100002101392700001901413700001701432700002101449856005101470 2016 en d00aA Reduced Basis Approach for Modeling the Movement of Nuclear Reactor Control Rods0 aReduced Basis Approach for Modeling the Movement of Nuclear Reac bASMEc02/2016 a80 v23 aThis work presents a reduced order model (ROM) aimed at simulating nuclear reactor control rods movement and featuring fast-running prediction of reactivity and neutron flux distribution as well. In particular, the reduced basis (RB) method (built upon a high-fidelity finite element (FE) approximation) has been employed. The neutronics has been modeled according to a parametrized stationary version of the multigroup neutron diffusion equation, which can be formulated as a generalized eigenvalue problem. Within the RB framework, the centroidal Voronoi tessellation is employed as a sampling technique due to the possibility of a hierarchical parameter space exploration, without relying on a “classical” a posteriori error estimation, and saving an important amount of computational time in the offline phase. Here, the proposed ROM is capable of correctly predicting, with respect to the high-fidelity FE approximation, both the reactivity and neutron flux shape. In this way, a computational speedup of at least three orders of magnitude is achieved. If a higher precision is required, the number of employed basis functions (BFs) must be increased.1 aSartori, Alberto1 aCammi, Antonio1 aLuzzi, Lelio1 aRozza, Gianluigi uhttp://urania.sissa.it/xmlui/handle/1963/3519201826nas a2200145 4500008004100000245012700041210006900168260001600237520129800253100002101551700001901572700001701591700002101608856005101629 2016 en d00aReduced basis approaches in time-dependent noncoercive settings for modelling the movement of nuclear reactor control rods0 aReduced basis approaches in timedependent noncoercive settings f bSISSAc20163 aIn this work, two approaches, based on the certified Reduced Basis method, have been developed for simulating the movement of nuclear reactor control rods, in time-dependent non-coercive settings featuring a 3D geometrical framework. In particular, in a first approach, a piece-wise affine transformation based on subdomains division has been implemented for modelling the movement of one control rod. In the second approach, a “staircase” strategy has been adopted for simulating the movement of all the three rods featured by the nuclear reactor chosen as case study. The neutron kinetics has been modelled according to the so-called multi-group neutron diffusion, which, in the present case, is a set of ten coupled parametrized parabolic equations (two energy groups for the neutron flux, and eight for the precursors). Both the reduced order models, developed according to the two approaches, provided a very good accuracy compared with high-fidelity results, assumed as “truth” solutions. At the same time, the computational speed-up in the Online phase, with respect to the fine “truth” finite element discretization, achievable by both the proposed approaches is at least of three orders of magnitude, allowing a real-time simulation of the rod movement and control.

1 aSartori, Alberto1 aCammi, Antonio1 aLuzzi, Lelio1 aRozza, Gianluigi uhttp://urania.sissa.it/xmlui/handle/1963/3496301905nas a2200157 4500008004100000245012000041210006900161260002200230300000800252490000700260520128900267100002101556700002201577700002101599856012701620 2016 en d00aReduced basis method and domain decomposition for elliptic problems in networks and complex parametrized geometries0 aReduced basis method and domain decomposition for elliptic probl bElsevierc01/2016 a4300 v713 aThe aim of this work is to solve parametrized partial differential equations in computational domains represented by networks of repetitive geometries by combining reduced basis and domain decomposition techniques. The main idea behind this approach is to compute once, locally and for few reference shapes, some representative finite element solutions for different values of the parameters and with a set of different suitable boundary conditions on the boundaries: these functions will represent the basis of a reduced space where the global solution is sought for. The continuity of the latter is assured by a classical domain decomposition approach. Test results on Poisson problem show the flexibility of the proposed method in which accuracy and computational time may be tuned by varying the number of reduced basis functions employed, or the set of boundary conditions used for defining locally the basis functions. The proposed approach simplifies the pre-computation of the reduced basis space by splitting the global problem into smaller local subproblems. Thanks to this feature, it allows dealing with arbitrarily complex network and features more flexibility than a classical global reduced basis approximation where the topology of the geometry is fixed.1 aIapichino, Laura1 aQuarteroni, Alfio1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/reduced-basis-method-and-domain-decomposition-elliptic-problems-networks-and-complex01201nas a2200145 4500008004100000245005000041210005000091260003400141300001400175490000700189520075600196100002200952700003100974856005001005 2016 eng d00aRefined node polynomials via long edge graphs0 aRefined node polynomials via long edge graphs bInternational Press of Boston a193–2340 v103 aThe generating functions of the Severi degrees for sufficiently ample line bundles on algebraic surfaces are multiplicative in the topological invariants of the surface and the line bundle. Recently new proofs of this fact were given for toric surfaces by Block, Colley, Kennedy and Liu, Osserman, using tropical geometry and in particular the combinatorial tool of long-edged graphs. In the first part of this paper these results are for $\mathbb{P}^2$ and rational ruled surfaces generalised to refined Severi degrees. In the second part of the paper we give a number of mostly conjectural generalisations of this result to singular surfaces, and curves with prescribed multiple points. The formulas involve modular forms and theta functions.

1 aGöttsche, Lothar1 aKikwai, Benjamin, Kipkirui uhttp://dx.doi.org/10.4310/CNTP.2016.v10.n2.a201365nas a2200145 4500008004100000245009200041210006900133300000900202490000700211520090200218100002301120700002101143700001601164856003901180 2016 eng d00aRenormalization for Autonomous Nearly Incompressible BV Vector Fields in Two Dimensions0 aRenormalization for Autonomous Nearly Incompressible BV Vector F a1-330 v483 aGiven a bounded autonomous vector field $b \colon \mathbb{R}^d \to \mathbb{R}^d$, we study the uniqueness of bounded solutions to the initial value problem for the related transport equation \begin{equation*} \partial_t u + b \cdot \nabla u= 0. \end{equation*} We are interested in the case where $b$ is of class BV and it is nearly incompressible. Assuming that the ambient space has dimension $d=2$, we prove uniqueness of weak solutions to the transport equation. The starting point of the present work is the result which has been obtained in [7] (where the steady case is treated). Our proof is based on splitting the equation onto a suitable partition of the plane: this technique was introduced in [3], using the results on the structure of level sets of Lipschitz maps obtained in [1]. Furthermore, in order to construct the partition, we use Ambrosio's superposition principle [4].

1 aBianchini, Stefano1 aBonicatto, Paolo1 aGusev, N.A. uhttps://doi.org/10.1137/15M100738000485nas a2200145 4500008004100000022001400041245008800055210006900143300001700212490000800229100001900237700001600256700002200272856004500294 2016 eng d a1364-502100aRogue waves in multiphase solutions of the focusing nonlinear Schrödinger equation0 aRogue waves in multiphase solutions of the focusing nonlinear Sc a20160340, 120 v4721 aBertola, Marco1 aEl, Gennady1 aTovbis, Alexander uhttp://dx.doi.org/10.1098/rspa.2016.034002042nas a2200217 4500008004100000022001400041245010200055210006900157490003500226520122900261653002501490653002101515653002501536653002701561653002501588653001601613100002201629700002101651700002201672856013001694 2015 eng d a1019-716800aReduced basis approximation and a-posteriori error estimation for the coupled Stokes-Darcy system0 aReduced basis approximation and aposteriori error estimation for0 vspecial issue for MoRePaS 20123 aThe coupling of a free flow with a flow through porous media has many potential applications in several fields related with computational science and engineering, such as blood flows, environmental problems or food technologies. We present a reduced basis method for such coupled problems. The reduced basis method is a model order reduction method applied in the context of parametrized systems. Our approach is based on a heterogeneous domain decomposition formulation, namely the Stokes-Darcy problem. Thanks to an offline/online-decomposition, computational times can be drastically reduced. At the same time the induced error can be bounded by fast evaluable a-posteriori error bounds. In the offline-phase the proposed algorithms make use of the decomposed problem structure. Rigorous a-posteriori error bounds are developed, indicating the accuracy of certain lifting operators used in the offline-phase as well as the accuracy of the reduced coupled system. Also, a strategy separately bounding pressure and velocity errors is extended. Numerical experiments dealing with groundwater flow scenarios demonstrate the efficiency of the approach as well as the limitations regarding a-posteriori error estimation.

10aDomain decomposition10aError estimation10aNon-coercive problem10aPorous medium equation10aReduced basis method10aStokes flow1 aMartini, Immanuel1 aRozza, Gianluigi1 aHaasdonk, Bernard uhttps://www.math.sissa.it/publication/reduced-basis-approximation-and-posteriori-error-estimation-coupled-stokes-darcy-system01086nas a2200133 4500008004100000245009800041210006900139300001400208490000800222520055000230100001900780700002100799856013200820 2015 eng d00aReduced basis approximation of parametrized advection-diffusion PDEs with high Péclet number0 aReduced basis approximation of parametrized advectiondiffusion P a419–4260 v1033 aIn this work we show some results about the reduced basis approximation of advection dominated parametrized problems, i.e. advection-diffusion problems with high Péclet number. These problems are of great importance in several engineering applications and it is well known that their numerical approximation can be affected by instability phenomena. In this work we compare two possible stabilization strategies in the framework of the reduced basis method, by showing numerical results obtained for a steady advection-diffusion problem.

1 aPacciarini, P.1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/reduced-basis-approximation-parametrized-advection-diffusion-pdes-high-p%C3%A9clet-number01235nas a2200145 4500008004100000245010300041210006900144300001400213490000700227520066400234100002000898700002000918700002100938856013000959 2015 eng d00aReduced basis approximation of parametrized optimal flow control problems for the Stokes equations0 aReduced basis approximation of parametrized optimal flow control a319–3360 v693 aThis paper extends the reduced basis method for the solution of parametrized optimal control problems presented in Negri et al. (2013) to the case of noncoercive (elliptic) equations, such as the Stokes equations. We discuss both the theoretical properties-with particular emphasis on the stability of the resulting double nested saddle-point problems and on aggregated error estimates-and the computational aspects of the method. Then, we apply it to solve a benchmark vorticity minimization problem for a parametrized bluff body immersed in a two or a three-dimensional flow through boundary control, demonstrating the effectivity of the methodology.

1 aNegri, Federico1 aManzoni, Andrea1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/reduced-basis-approximation-parametrized-optimal-flow-control-problems-stokes-equations02449nas a2200121 4500008004100000245012900041210006900170520189900239100002002138700002502158700001702183856012702200 2015 en d00aReduced Basis Isogeometric Methods (RB-IGA) for the real-time simulation of potential flows about parametrized NACA airfoils0 aReduced Basis Isogeometric Methods RBIGA for the realtime simula3 aWe present a Reduced Basis (RB) method based on Isogeometric Analysis (IGA) for the rapid and reliable evaluation of PDE systems characterized by complex geometrical features. At the current state of the art, this is the first case of coupling between RB and IGA methods. The construction of the RB method relies on an Isogeometric Boundary Element Method (IGA-BEM) as the high-fidelity technique, allowing a direct interface with Computer Aided Design (CAD) tools. A suitable Empirical Interpolation Method (EIM) ensures an efficient offline/online decomposition between the construction and the evaluation of the RB method. We consider the real-time simulation of potential flows past airfoils, parametrized with respect to the angle of attack and the NACA number identifying their shape, and we provide a validation of our methodology with respect to experimental data and reference numerical codes, showing in both cases a very good agreement.We present a Reduced Basis (RB) method based on Isogeometric Analysis (IGA) for the rapid and reliable evaluation of PDE systems characterized by complex geometrical features. At the current state of the art, this is the first case of coupling between RB and IGA methods. The construction of the RB method relies on an Isogeometric Boundary Element Method (IGA-BEM) as the high-fidelity technique, allowing a direct interface with Computer Aided Design (CAD) tools. A suitable Empirical Interpolation Method (EIM) ensures an efficient offline/online decomposition between the construction and the evaluation of the RB method. We consider the real-time simulation of potential flows past airfoils, parametrized with respect to the angle of attack and the NACA number identifying their shape, and we provide a validation of our methodology with respect to experimental data and reference numerical codes, showing in both cases a very good agreement.1 aManzoni, Andrea1 aSalmoiraghi, Filippo1 aHeltai, Luca uhttps://www.math.sissa.it/publication/reduced-basis-isogeometric-methods-rb-iga-real-time-simulation-potential-flows-about01509nas a2200121 4500008004100000245012400041210006900165260001000234520098200244653002001226100001801246856012301264 2015 en d00aThe relaxed area of maps from the plane to the plane with a line discontinuity, and the role of semicartesian surfaces.0 arelaxed area of maps from the plane to the plane with a line dis bSISSA3 aIn this thesis we study the relaxation of the area functional w.r.t. the L^1 topology of a map from a bounded planar domain with values in the plane and jumping on a segment. We estimate from above the singular contribution of this functional due to the presence of the jump in terms of the infimum of the area among a suitable family of surfaces that we call semicartesian surfaces. In our analysis, we also introduce a different notion of area, namely the relaxation of the area w.r.t. a convergence stronger than the L^1 convergence, whose singular contribution is completely characterized in terms of suitable semicartesian area minimizing problems. We propose also some examples of maps for which the two notions of relaxation are different: these examples underline the highly non-local behaviour of the L^1-relaxation, and justify the introduction of the other functional. Some result about the existence of a semicartesian area-minimizing surface is also provided.10aArea functional1 aTealdi, Lucia uhttps://www.math.sissa.it/publication/relaxed-area-maps-plane-plane-line-discontinuity-and-role-semicartesian-surfaces00953nas a2200121 4500008004100000245009700041210006900138520050800207100001800715700002500733700002200758856005100780 2015 en d00aResults on the minimization of the Dirichlet functional among semicartesian parametrizations0 aResults on the minimization of the Dirichlet functional among se3 aWe start to investigate the existence of conformal minimizers for the Dirichlet functional in the setting of the so-called semicartesian parametrizations, adapting to this context some techniques used in solving the classical Plateau's problem. The final goal is to find area minimizing semicartesian parametrizations spanning a Jordan curve obtained as union of two graphs; this problem appeared in the study of the relaxed area functional for maps from the plane to the plane jumping on a line.

1 aTealdi, Lucia1 aBellettini, Giovanni1 aPaolini, Maurizio uhttp://urania.sissa.it/xmlui/handle/1963/3448801420nas a2200133 4500008004100000245009400041210006900135260001000204520094900214100002401163700002401187700002501211856005001236 2015 en d00aRigidity of three-dimensional lattices and dimension reduction in heterogeneous nanowires0 aRigidity of threedimensional lattices and dimension reduction in bSISSA3 aIn the context of nanowire heterostructures we perform a discrete to continuum limit of the corresponding free energy by means of Γ-convergence techniques. Nearest neighbours are identified by employing the notions of Voronoi diagrams and Delaunay triangulations. The scaling of the nanowire is done in such a way that we perform not only a continuum limit but a dimension reduction simultaneously. The main part of the proof is a discrete geometric rigidity result that we announced in an earlier work and show here in detail for a variety of three-dimensional lattices. We perform the passage from discrete to continuum twice: once for a system that compensates a lattice mismatch between two parts of the heterogeneous nanowire without defects and once for a system that creates dislocations. It turns out that we can verify the experimentally observed fact that the nanowires show dislocations when the radius of the specimen is large1 aLazzaroni, Giuliano1 aPalombaro, Mariapia1 aSchlomerkemper, Anja uhttp://urania.sissa.it/xmlui/handle/1963/749401353nas a2200145 4500008004100000245007400041210006900115260001000184520088100194100002401075700002001099700001901119700001901138856005001157 2014 en d00aRate-independent damage in thermo-viscoelastic materials with inertia0 aRateindependent damage in thermoviscoelastic materials with iner bSISSA3 aWe present a model for rate-independent, unidirectional, partial damage in visco-elastic materials with inertia and thermal effects. The damage process is modeled by means of an internal variable, governed by a rate-independent flow rule. The heat equation and the momentum balance for the displacements are coupled in a highly nonlinear way. Our assumptions on the corresponding energy functional also comprise the case of the Ambrosio-Tortorelli phase-field model (without passage to the brittle limit). We discuss a suitable weak formulation and prove an existence theorem obtained with the aid of a (partially) decoupled time-discrete scheme and variational convergence methods. We also carry out the asymptotic analysis for vanishing viscosity and inertia and obtain a fully rate-independent limit model for displacements and damage, which is Independent of temperature.1 aLazzaroni, Giuliano1 aRossi, Riccarda1 aThomas, Marita1 aToader, Rodica uhttp://urania.sissa.it/xmlui/handle/1963/744401330nas a2200121 4500008004100000245006100041210006000102260001900162520091900181653003501100100002301135856005001158 2014 en d00aRational curves and instantons on the Fano threefold Y_50 aRational curves and instantons on the Fano threefold Y5 barXiv preprint3 aThis thesis is an investigation of the moduli spaces of instanton bundles on the Fano threefold Y_5 (a linear section of Gr(2,5)). It contains new proofs of classical facts about lines, conics and cubics on Y_5, and about linear sections of Y_5. The main original results are a Grauert-Mülich theorem for the splitting type of instantons on conics, a bound to the splitting type of instantons on lines and an SL_2-equivariant description of the moduli space in charge 2 and 3. Using these results we prove the existence of a unique SL_2-equivariant instanton of minimal charge and we show that for all instantons of charge 2 the divisor of jumping lines is smooth. In charge 3, we provide examples of instantons with reducible divisor of jumping lines. Finally, we construct a natural compactification for the moduli space of instantons of charge 3, together with a small resolution of singularities for it.10aModuli space of vector bundles1 aSanna, Giangiacomo uhttp://urania.sissa.it/xmlui/handle/1963/748200566nas a2200133 4500008004100000245010000041210006900141300001000210100002100220700002200241700002100263700002100284856012700305 2014 eng d00aReduced basis method for the Stokes equations in decomposable domains using greedy optimization0 aReduced basis method for the Stokes equations in decomposable do a1–71 aIapichino, Laura1 aQuarteroni, Alfio1 aRozza, Gianluigi1 aVolkwein, Stefan uhttps://www.math.sissa.it/publication/reduced-basis-method-stokes-equations-decomposable-domains-using-greedy-optimization01878nam a2200181 4500008004100000022002200041245006700063210006700130250000600197260002100203300000800224490000600232520123600238653007801474100002201552700002101574856010101595 2014 eng d a978-3-319-02089-100aReduced Order Methods for Modeling and Computational Reduction0 aReduced Order Methods for Modeling and Computational Reduction a1 aMilanobSpringer a3340 v93 aThis monograph addresses the state of the art of reduced order methods for modeling and computational reduction of complex parametrized systems, governed by ordinary and/or partial differential equations, with a special emphasis on real time computing techniques and applications in computational mechanics, bioengineering and computer graphics.

Several topics are covered, including: design, optimization, and control theory in real-time with applications in engineering; data assimilation, geometry registration, and parameter estimation with special attention to real-time computing in biomedical engineering and computational physics; real-time visualization of physics-based simulations in computer science; the treatment of high-dimensional problems in state space, physical space, or parameter space; the interactions between different model reduction and dimensionality reduction approaches; the development of general error estimation frameworks which take into account both model and discretization effects.

This book is primarily addressed to computational scientists interested in computational reduction techniques for large scale differential problems.

10areduced order methods, MOR, ROM, POD, RB, greedy, CFD, Numerical Analysis1 aQuarteroni, Alfio1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/reduced-order-methods-modeling-and-computational-reduction01688nas a2200193 4500008004100000020002000041245009500061210006900156250004400225260008500269300002800354520096400382100002101346700001901367700001901386700001701405700002101422856005101443 2014 en d a978-079184595-000aA reduced order model for multi-group time-dependent parametrized reactor spatial kinetics0 areduced order model for multigroup timedependent parametrized re aAmerican Society Mechanical Engineering aPrague, Czech RepublicbAmerican Society of Mechanical Engineers (ASME)c07/2014 aV005T17A048-V005T17A0483 a

In this work, a Reduced Order Model (ROM) for multigroup time-dependent parametrized reactor spatial kinetics is presented. The Reduced Basis method (built upon a high-fidelity "truth" finite element approximation) has been applied to model the neutronics behavior of a parametrized system composed by a control rod surrounded by fissile material. The neutron kinetics has been described by means of a parametrized multi-group diffusion equation where the height of the control rod (i.e., how much the rod is inserted) plays the role of the varying parameter. In order to model a continuous movement of the rod, a piecewise affine transformation based on subdomain division has been implemented. The proposed ROM is capable to efficiently reproduce the neutron flux distribution allowing to take into account the spatial effects induced by the movement of the control rod with a computational speed-up of 30000 times, with respect to the "truth" model.

1 aSartori, Alberto1 aBaroli, Davide1 aCammi, Antonio1 aLuzzi, Lelio1 aRozza, Gianluigi uhttp://urania.sissa.it/xmlui/handle/1963/3512300458nas a2200133 4500008004100000245007200041210006900113260001000182653003000192100002200222700002300244700002100267856003600288 2014 en d00aReduction on characteristics for continuous of a scalar balance law0 aReduction on characteristics for continuous of a scalar balance bSISSA10aMethod of characteristics1 aAlberti, Giovanni1 aBianchini, Stefano1 aCaravenna, Laura uhttp://hdl.handle.net/1963/656200545nas a2200109 4500008004100000245004500041210004300086260001300129520022100142100002100363856005100384 2014 en d00aA Review of the Sixth Painlevé Equation0 aReview of the Sixth Painlevé Equation bSpringer3 aFor the Painlevé VI transcendents, we provide a unitary description of the critical behaviours, the connection formulae, their complete tabulation, and the asymptotic distribution of poles close to a critical point.1 aGuzzetti, Davide uhttp://urania.sissa.it/xmlui/handle/1963/3465801083nas a2200121 4500008004100000245012700041210006900168260002900237520052100266100002200787700002200809856013000831 2014 en d00aA robotic crawler exploiting directional frictional interactions: experiments, numerics, and derivation of a reduced model0 arobotic crawler exploiting directional frictional interactions e bRoyal Society Publishing3 aWe present experimental and numerical results for a model crawler which is able to extract net positional changes from reciprocal shape changes, i.e. ‘breathing-like’ deformations, thanks to directional, frictional interactions with a textured solid substrate, mediated by flexible inclined feet. We also present a simple reduced model that captures the essential features of the kinematics and energetics of the gait, and compare its predictions with the results from experiments and from numerical simulations.1 aNoselli, Giovanni1 aDeSimone, Antonio uhttps://www.math.sissa.it/publication/robotic-crawler-exploiting-directional-frictional-interactions-experiments-numerics-and02183nas a2200145 4500008004100000245015300041210006900194260001300263520163200276653003401908100002101942700001801963700002001981856003602001 2013 en d00aReduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants0 aReduced basis approximation and a posteriori error estimation fo bSpringer3 aIn this paper we review and we extend the reduced basis approximation and a posteriori error estimation for steady Stokes flows in a ffinely parametrized geometries, focusing on the role played by the Brezzi\\\'s and Babu ska\\\'s stability constants. The crucial ingredients of the methodology are a Galerkin projection onto a low-dimensional space of basis functions properly selected, an a ne parametric dependence enabling to perform competitive Off ine-Online splitting in the computational\\r\\nprocedure and a rigorous a posteriori error estimation on eld variables.\\r\\nThe combination of these three factors yields substantial computational savings which are at the basis of an e fficient model order reduction, ideally suited for real-time simulation and many-query contexts (e.g. optimization, control or parameter identi cation). In particular, in this work we focus on i) the stability of the reduced basis approximation based on the Brezzi\\\'s saddle point theory and the introduction of a supremizer operator on the pressure terms, ii) a rigorous a posteriori error estimation procedure for velocity and pressure elds based on the Babu ska\\\'s inf-sup constant (including residuals calculations), iii) the computation of a lower bound of the stability constant, and iv) di erent options for the reduced basis spaces construction. We present some illustrative results for both\\r\\ninterior and external steady Stokes flows in parametrized geometries representing two parametrized classical Poiseuille and Couette \\r\\nflows, a channel contraction and a simple flow control problem around a curved obstacle.10aparametrized Stokes equations1 aRozza, Gianluigi1 aHuynh, Phuong1 aManzoni, Andrea uhttp://hdl.handle.net/1963/633900531nas a2200121 4500008004100000245011700041210006900158300001100227490000700238100001800245700002100263856012500284 2013 eng d00aReduced Basis Approximation for the Structural-Acoustic Design based on Energy Finite Element Analysis (RB-EFEA)0 aReduced Basis Approximation for the StructuralAcoustic Design ba a98-1150 v481 aDevaud, Denis1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/reduced-basis-approximation-structural-acoustic-design-based-energy-finite-element01701nas a2200157 4500008004100000245007600041210006900117300001800186490000700204520113900211100002001350700002101370700002001391700002201411856011001433 2013 eng d00aReduced basis method for parametrized elliptic optimal control problems0 aReduced basis method for parametrized elliptic optimal control p aA2316–A23400 v353 aWe propose a suitable model reduction paradigm-the certified reduced basis method (RB)-for the rapid and reliable solution of parametrized optimal control problems governed by partial differential equations. In particular, we develop the methodology for parametrized quadratic optimization problems with elliptic equations as a constraint and infinite-dimensional control variable. First, we recast the optimal control problem in the framework of saddle-point problems in order to take advantage of the already developed RB theory for Stokes-type problems. Then, the usual ingredients of the RB methodology are called into play: a Galerkin projection onto a low-dimensional space of basis functions properly selected by an adaptive procedure; an affine parametric dependence enabling one to perform competitive offline-online splitting in the computational procedure; and an efficient and rigorous a posteriori error estimate on the state, control, and adjoint variables as well as on the cost functional. Finally, we address some numerical tests that confirm our theoretical results and show the efficiency of the proposed technique.1 aNegri, Federico1 aRozza, Gianluigi1 aManzoni, Andrea1 aQuarteroni, Alfio uhttps://www.math.sissa.it/publication/reduced-basis-method-parametrized-elliptic-optimal-control-problems00548nas a2200133 4500008004100000245009200041210006900133260001000202100001800212700002000230700002200250700002100272856012100293 2013 en d00aA Reduced Computational and Geometrical Framework for Inverse Problems in Haemodynamics0 aReduced Computational and Geometrical Framework for Inverse Prob bSISSA1 aLassila, Toni1 aManzoni, Andrea1 aQuarteroni, Alfio1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/reduced-computational-and-geometrical-framework-inverse-problems-haemodynamics00568nas a2200133 4500008004100000245010500041210006900146260001000215100001800225700002000243700002200263700002100285856012800306 2013 en d00aA reduced-order strategy for solving inverse Bayesian identification problems in physiological flows0 areducedorder strategy for solving inverse Bayesian identificatio bSISSA1 aLassila, Toni1 aManzoni, Andrea1 aQuarteroni, Alfio1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/reduced-order-strategy-solving-inverse-bayesian-identification-problems-physiological00490nas a2200121 4500008004100000245007900041210006900120260001000189100001800199700002000217700002100237856011000258 2013 en d00aReduction Strategies for Shape Dependent Inverse Problems in Haemodynamics0 aReduction Strategies for Shape Dependent Inverse Problems in Hae bSISSA1 aLassila, Toni1 aManzoni, Andrea1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/reduction-strategies-shape-dependent-inverse-problems-haemodynamics00793nas a2200145 4500008004100000245004800041210004800089260003500137300001200172490000600184520038200190100002200572700002000594856003300614 2013 en d00aRemarks on the Moser–Trudinger inequality0 aRemarks on the Moser–Trudinger inequality bAdvances in Nonlinear Analysis a389-4250 v23 aWe extend the Moser-Trudinger inequality to any Euclidean domain satisfying Poincaré's inequality. We find out that the same equivalence does not hold in general for conformal metrics on the unit ball, showing counterexamples. We also study the existence of extremals for the Moser-Trudinger inequalities for unbounded domains, proving it for the infinite planar strip.

1 aMancini, Gabriele1 aBattaglia, Luca uhttp://edoc.unibas.ch/43974/01651nas a2200133 4500008004100000245008400041210006900125520120500194653002101399100002101420700002001441700002001461856003601481 2012 en d00aReduction strategies for PDE-constrained oprimization problems in Haemodynamics0 aReduction strategies for PDEconstrained oprimization problems in3 aSolving optimal control problems for many different scenarios obtained by varying a set of parameters in the state system is a computationally extensive task. In this paper we present a new reduced framework for the formulation, the analysis and the numerical solution of parametrized PDE-constrained optimization problems. This framework is based on a suitable saddle-point formulation of the optimal control problem and exploits the reduced basis method for the rapid and reliable solution of parametrized PDEs, leading to a relevant computational reduction with respect to traditional discretization techniques such as the finite element method. This allows a very efficient evaluation of state solutions and cost functionals, leading to an effective solution of repeated optimal control problems, even on domains of variable shape, for which a further (geometrical) reduction is pursued, relying on flexible shape parametrization techniques. This setting is applied to the solution of two problems arising from haemodynamics, dealing with both data reconstruction and data assimilation over domains of variable shape,\\r\\nwhich can be recast in a common PDE-constrained optimization formulation.10ainverse problems1 aRozza, Gianluigi1 aManzoni, Andrea1 aNegri, Federico uhttp://hdl.handle.net/1963/633800498nas a2200121 4500008004100000245013300041210006900174260003300243300001400276490000700290100002300297856005600320 2012 eng d00aResonance at the first eigenvalue for first-order systems in the plane: vanishing Hamiltonians and the Landesman-Lazer condition0 aResonance at the first eigenvalue for firstorder systems in the bKhayyam Publishing, Inc.c05 a505–5260 v251 aGarrione, Maurizio uhttps://projecteuclid.org:443/euclid.die/135601267602076nas a2200145 4500008004100000245004700041210004700088520166300135653001801798100001901816700001701835700002001852700002201872856003601894 2012 en d00aReverse engineering the euglenoid movement0 aReverse engineering the euglenoid movement3 aEuglenids exhibit an unconventional motility strategy amongst unicellular eukaryotes, consisting of large-amplitude highly concerted deformations of the entire body (euglenoid movement or metaboly). A plastic cell envelope called pellicle mediates these deformations. Unlike ciliary or flagellar motility, the biophysics of this mode is not well understood, including its efficiency and molecular machinery. We quantitatively examine video recordings of four euglenids executing such motions with statistical learning methods. This analysis reveals strokes of high uniformity in shape and pace. We then interpret the observations in the light of a theory for the pellicle kinematics, providing a precise understanding of the link between local actuation by pellicle shear and shape control. We systematically understand common observations, such as the helical conformations of the pellicle, and identify previously unnoticed features of metaboly. While two of our euglenids execute their stroke at constant body volume, the other two exhibit deviations of about 20% from their average volume, challenging current models of low Reynolds number locomotion. We find that the active pellicle shear deformations causing shape changes can reach 340%, and estimate the velocity of the molecular motors. Moreover, we find that metaboly accomplishes locomotion at hydrodynamic efficiencies comparable to those of ciliates and flagellates. Our results suggest new quantitative experiments, provide insight into the evolutionary history of euglenids, and suggest that the pellicle may serve as a model for engineered active surfaces with applications in microfluidics.10amicroswimmers1 aArroyo, Marino1 aHeltai, Luca1 aMillán, Daniel1 aDeSimone, Antonio uhttp://hdl.handle.net/1963/644400581nas a2200121 4500008004100000245004500041210004300086260001000129520024000139653002300379100002100402856003600423 2012 en d00aA Review on The Sixth Painlevé Equation0 aReview on The Sixth Painlevé Equation bSISSA3 aFor the Painlev\\\'e 6 transcendents, we provide a unitary description of the\r\ncritical behaviours, the connection formulae, their complete tabulation, and\r\nthe asymptotic distribution of the poles close to a critical point.

10aPainlevé equation1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/652500516nas a2200145 4500008004100000022001400041245008700055210007100142300001600213490000800229653002300237100001900260700002000279856007100299 2012 eng d a0167-278900aRiemann–Hilbert approach to multi-time processes: The Airy and the Pearcey cases0 aRiemann–Hilbert approach to multitime processes The Airy and the a2237 - 22450 v24110aIntegrable kernels1 aBertola, Marco1 aCafasso, Mattia uhttp://www.sciencedirect.com/science/article/pii/S016727891200011500821nas a2200121 4500008004100000245007700041210006900118520040700187100002500594700002300619700002100642856003600663 2012 en d00aOn robust Lie-algebraic stability conditions for switched linear systems0 arobust Liealgebraic stability conditions for switched linear sys3 aThis paper presents new sufficient conditions for exponential stability of switched linear systems under arbitrary switching, which involve the commutators (Lie brackets) among the given matrices generating the switched system. The main novelty feature of these stability criteria is that, unlike their earlier counterparts, they are robust with respect to small perturbations of the system parameters.1 aAgrachev, Andrei, A.1 aBaryshnikov, Yurij1 aLiberzon, Daniel uhttp://hdl.handle.net/1963/645501079nas a2200121 4500008004100000245007600041210006900117300001400186490000700200520062100207100002300828856010600851 2011 eng d00aResonance and Landesman-Lazer conditions for first order systems in R^20 aResonance and LandesmanLazer conditions for first order systems a153–1600 v663 aThe first part of the paper surveys the concept of resonance for $T$-periodic nonlinear problems. In the second part, some new results about existence conditions for nonlinear planar systems are presented. In particular, the Landesman-Lazer conditions are generalized to systems in $\mathbbR^2$ where the nonlinearity interacts with two resonant Hamiltonians. Such results apply to second order equations, generalizing previous theorems by Fabry [4] (for the undamped case), and Frederickson-Lazer [9] (for the case with friction). The results have been obtained with A. Fonda, and have been published in [8].

1 aGarrione, Maurizio uhttps://www.math.sissa.it/publication/resonance-and-landesman-lazer-conditions-first-order-systems-r201464nas a2200193 4500008004100000022001400041245012500055210006900180300001600249490000700265520078200272653003201054653003301086653001401119653002001133100002301153700002301176856007101199 2011 eng d a0362-546X00aResonance and rotation numbers for planar Hamiltonian systems: Multiplicity results via the Poincaré–Birkhoff theorem0 aResonance and rotation numbers for planar Hamiltonian systems Mu a4166 - 41850 v743 aIn the general setting of a planar first order system (0.1)u′=G(t,u),u∈R2, with G:[0,T]×R2→R2, we study the relationships between some classical nonresonance conditions (including the Landesman–Lazer one) — at infinity and, in the unforced case, i.e. G(t,0)≡0, at zero — and the rotation numbers of “large” and “small” solutions of (0.1), respectively. Such estimates are then used to establish, via the Poincaré–Birkhoff fixed point theorem, new multiplicity results for T-periodic solutions of unforced planar Hamiltonian systems Ju′=∇uH(t,u) and unforced undamped scalar second order equations x″+g(t,x)=0. In particular, by means of the Landesman–Lazer condition, we obtain sharp conclusions when the system is resonant at infinity.

10aMultiple periodic solutions10aPoincaré–Birkhoff theorem10aResonance10aRotation number1 aBoscaggin, Alberto1 aGarrione, Maurizio uhttp://www.sciencedirect.com/science/article/pii/S0362546X1100181700741nas a2200133 4500008004300000245009200043210006900135260001900204520028800223100001800511700002100529700002100550856003600571 2010 en_Ud 00aThe reductions of the dispersionless 2D Toda hierarchy and their Hamiltonian structures0 areductions of the dispersionless 2D Toda hierarchy and their Ham bIOP Publishing3 aWe study finite-dimensional reductions of the dispersionless 2D Toda hierarchy showing that the consistency conditions for such reductions are given by a system of radial Loewner equations. We then construct their Hamiltonian structures, following an approach proposed by Ferapontov.1 aCarlet, Guido1 aLorenzoni, Paolo1 aRaimondo, Andrea uhttp://hdl.handle.net/1963/384601121nas a2200121 4500008004300000245007400043210006900117260003700186520069600223100002200919700002200941856003600963 2010 en_Ud 00aRiemann-Roch theorems and elliptic genus for virtually smooth schemes0 aRiemannRoch theorems and elliptic genus for virtually smooth sch bMathematical Sciences Publishers3 aFor a proper scheme X with a fixed 1-perfect obstruction theory, we define virtual versions of holomorphic Euler characteristic, chi y-genus, and elliptic genus; they are deformation invariant, and extend the usual definition in the smooth case. We prove virtual versions of the Grothendieck-Riemann-Roch and Hirzebruch-Riemann-Roch theorems. We show that the virtual chi y-genus is a polynomial, and use this to define a virtual topological Euler characteristic. We prove that the virtual elliptic genus satisfies a Jacobi modularity property; we state and prove a localization theorem in the toric equivariant case. We show how some of our results apply to moduli spaces of stable sheaves.1 aFantechi, Barbara1 aGöttsche, Lothar uhttp://hdl.handle.net/1963/388801810nas a2200121 4500008004300000245007000043210006600113520141500179100001901594700002201613700001701635856003601652 2010 en_Ud 00aThe role of membrane viscosity in the dynamics of fluid membranes0 arole of membrane viscosity in the dynamics of fluid membranes3 aFluid membranes made out of lipid bilayers are the fundamental separation structure in eukaryotic cells. Many physiological processes rely on dramatic shape and topological changes (e.g. fusion, fission) of fluid membrane systems. Fluidity is key to the versatility and constant reorganization of lipid bilayers. Here, we study the role of the membrane intrinsic viscosity, arising from the friction of the lipid molecules as they rearrange to accommodate shape changes, in the dynamics of morphological changes of fluid vesicles. In particular, we analyze the competition between the membrane viscosity and the viscosity of the bulk fluid surrounding the vesicle as the dominant dissipative mechanism. We consider the relaxation dynamics of fluid vesicles put in an out-of-equilibrium state, but conclusions can be drawn regarding the kinetics or power consumption in regulated shape changes in the cell. On the basis of numerical calculations, we find that the dynamics arising from the membrane viscosity are qualitatively different from the dynamics arising from the bulk viscosity. When these two dissipation mechanisms are put in competition, we find that for small vesicles the membrane dissipation dominates, with a relaxation time that scales as the size of the vesicle to the power 2. For large vesicles, the bulk dissipation dominates, and the exponent in the relaxation time vs. size relation is 3.1 aArroyo, Marino1 aDeSimone, Antonio1 aHeltai, Luca uhttp://hdl.handle.net/1963/393000456nas a2200133 4500008004100000022001400041245006800055210006800123300001400191490000800205100001900213700001900232856007100251 2009 eng d a0021-904500aRegularity of a vector potential problem and its spectral curve0 aRegularity of a vector potential problem and its spectral curve a353–3700 v1611 aBalogh, Ferenc1 aBertola, Marco uhttp://0-dx.doi.org.mercury.concordia.ca/10.1016/j.jat.2008.10.01001411nas a2200121 4500008004300000245004300043210004300086260003000129520105300159100001901212700002201231856003601253 2009 en_Ud 00aRelaxation dynamics of fluid membranes0 aRelaxation dynamics of fluid membranes bAmerican Physical Society3 aWe study the effect of membrane viscosity in the dynamics of liquid membranes-possibly with free or internal boundaries-driven by conservative forces (curvature elasticity and line tension) and dragged by the bulk dissipation of the ambient fluid and the friction occurring when the amphiphilic molecules move relative to each other. To this end, we formulate a continuum model which includes a form of the governing equations for a two-dimensional viscous fluid moving on a curved, time-evolving surface. The effect of membrane viscosity has received very limited attention in previous continuum studies of the dynamics of fluid membranes, although recent coarse-grained discrete simulations suggest its importance. By applying our model to the study of vesiculation and membrane fusion in a simplified geometry, we conclude that membrane viscosity plays a dominant role in the relaxation dynamics of fluid membranes of sizes comparable to those found in eukaryotic cells, and is not negligible in many large synthetic systems of current interest.1 aArroyo, Marino1 aDeSimone, Antonio uhttp://hdl.handle.net/1963/361800444nas a2200121 4500008004300000245009700043210006900140100001700209700001800226700002200244700002000266856003600286 2008 en_Ud 00aRelaxation of some transversally isotropic energies and applications to smectic A elastomers0 aRelaxation of some transversally isotropic energies and applicat1 aAdams, James1 aConti, Sergio1 aDeSimone, Antonio1 aDolzmann, Georg uhttp://hdl.handle.net/1963/191200973nas a2200109 4500008004300000245006600043210006600109520061200175100002200787700001800809856003600827 2007 en_Ud 00aReciprocal transformations and flat metrics on Hurwitz spaces0 aReciprocal transformations and flat metrics on Hurwitz spaces3 aWe consider hydrodynamic systems which possess a local Hamiltonian structure of Dubrovin-Novikov type. To such a system there are also associated an infinite number of nonlocal Hamiltonian structures. We give necessary and sufficient conditions so that, after a nonlinear transformation of the independent variables, the reciprocal system still possesses a local Hamiltonian structure of Dubrovin-Novikov type. We show that, under our hypotheses, bi-hamiltonicity is preserved by the reciprocal transformation. Finally we apply such results to reciprocal systems of genus g Whitham-KdV modulation equations.1 aAbenda, Simonetta1 aGrava, Tamara uhttp://hdl.handle.net/1963/221001299nas a2200097 4500008004300000245006000043210005900103520097400162100002901136856003601165 2007 en_Ud 00aReduced density matrices and Bose-Einstein condensation0 aReduced density matrices and BoseEinstein condensation3 aEmergence and applications of the ubiquitous tool of reduced density matrices in the rigorous analysis of Bose Einstein condensation is reviewed, and new related results are added. The need and the nature of scaling limits of infinitely many particles is discussed, which imposes that a physically meaningful and mathematically well-posed definition of asymptotic condensation is placed at the level of marginals.\\nThe topic of correlations in the condensed state is addressed in order to show their influence at this level of marginals, both in the true condensed state and in the suitable trial functions one introduces to approximate the many-body structure and energy. Complete condensation is shown to be equivalently defined at any fixed k-body level, both for pure and mixed states. Further, it is proven to be equivalent to some other characterizations in terms of asymptotic factorization of the many-body state, which are currently present in the literature.1 aMichelangeli, Alessandro uhttp://hdl.handle.net/1963/198600931nas a2200121 4500008004100000245007500041210006800116260001000184520053900194100002000733700002000753856003600773 2007 en d00aOn the reductions and classical solutions of the Schlesinger equations0 areductions and classical solutions of the Schlesinger equations bSISSA3 aThe Schlesinger equations S(n,m) describe monodromy preserving\\r\\ndeformations of order m Fuchsian systems with n+1 poles. They\\r\\ncan be considered as a family of commuting time-dependent Hamiltonian\\r\\nsystems on the direct product of n copies of m×m matrix algebras\\r\\nequipped with the standard linear Poisson bracket. In this paper we address\\r\\nthe problem of reduction of particular solutions of “more complicated”\\r\\nSchlesinger equations S(n,m) to “simpler” S(n′,m′) having n′ < n\\r\\nor m′ < m.1 aDubrovin, Boris1 aMazzocco, Marta uhttp://hdl.handle.net/1963/647200600nas a2200097 4500008004300000245005300043210004500096520030500141100002000446856003600466 2007 en_Ud 00aOn the regularity of weak solutions to H-systems0 aregularity of weak solutions to Hsystems3 aAbstract. In this paper we prove that every weak solution to the H-surface equation is locally bounded, provided the prescibed mean curvatore H is asymptotic to a constant at infinity (with a suitable decay rate). No smoothness ssumptions are required on H. We consider also the Dirichlet problem....1 aMusina, Roberta uhttp://hdl.handle.net/1963/175300981nas a2200097 4500008004300000245008200043210006900125520062400194100002900818856003600847 2007 en_Ud 00aRole of scaling limits in the rigorous analysis of Bose-Einstein condensation0 aRole of scaling limits in the rigorous analysis of BoseEinstein 3 aIn the context of the rigorous analysis of Bose-Einstein condensation, recent achievements have been obtained in the form of asymptotic results when some appropriate scaling is performed in the Hamiltonian, and the limit of infinite number of particles is taken. In particular, two modified thermodynamic limits of infinite dilution turned out to provide an insight in this analysis, the so-\\ncalled Gross-Pitaevskii limit and the related Tomas-Fermi limit. Here such scalings are discussed with respect to their physical and mathematical motivations, and to the currently known results obtained within this framework.1 aMichelangeli, Alessandro uhttp://hdl.handle.net/1963/198400669nas a2200109 4500008004300000245010800043210007000151520026200221100002400483700001600507856003600523 2006 en_Ud 00aRadial solutions concentrating on spheres of nonlinear Schrödinger equations with vanishing potentials0 aRadial solutions concentrating on spheres of nonlinear Schröding3 aWe prove the existence of radial solutions of 1.2) concentrating at a sphere for potentials which might be zero and might decay to zero at\\r\\ninfinity. The proofs use a perturbation technique in a variational setting, through a Lyapunov-Schmidt reduction.1 aAmbrosetti, Antonio1 aRuiz, David uhttp://hdl.handle.net/1963/175500420nas a2200133 4500008004300000020002200043245005300065210005300118100002200171700002100193700002000214700001600234856003600250 2006 en_Ud a978-0-12-480874-400aRecent analytical developments in micromagnetics0 aRecent analytical developments in micromagnetics1 aDeSimone, Antonio1 aKohn, Robert, V.1 aMüller, Stefan1 aOtto, Felix uhttp://hdl.handle.net/1963/223001165nas a2200109 4500008004300000245005900043210005900102520081400161100002200975700002200997856003601019 2006 en_Ud 00aReflection symmetries for multiqubit density operators0 aReflection symmetries for multiqubit density operators3 aFor multiqubit density operators in a suitable tensorial basis, we show that a number of nonunitary operations used in the detection and synthesis of entanglement are classifiable as reflection symmetries, i.e., orientation changing rotations. While one-qubit reflections correspond to antiunitary symmetries, as is known for example from the partial transposition criterion, reflections on the joint density of two or more qubits are not accounted for by the Wigner Theorem and are well-posed only for sufficiently mixed states. One example of such nonlocal reflections is the unconditional NOT operation on a multiparty density, i.e., an operation yelding another density and such that the sum of the two is the identity operator. This nonphysical operation is admissible only for sufficiently mixed states.1 aAltafini, Claudio1 aHavel, Timothy F. uhttp://hdl.handle.net/1963/212101167nas a2200109 4500008004100000245010100041210006900142260001300211520077600224100002101000856003601021 2005 en d00aRegularity properties of optimal trajectories of single-input control systems in dimension three0 aRegularity properties of optimal trajectories of singleinput con bSpringer3 aLet q=f(q)+ug(q) be a smooth control system on a three-dimensional manifold. Given a point q 0 of the manifold at which the iterated Lie brackets of f and g satisfy some prescribed independence condition, we analyze the structure of a control function u(t) corresponding to a time-optimal trajectory lying in a neighborhood of q 0. The control turns out to be the concatenation of some bang-bang and some singular arcs. More general optimality criteria than time-optimality are considered. The paper is a step toward to the analysis of generic single-input systems affine in the control in dimension 3. The main techniques used are second-order optimality conditions and, in particular, the index of the second variation of the switching times for bang-bang trajectories.1 aSigalotti, Mario uhttp://hdl.handle.net/1963/479401711nas a2200109 4500008004300000245014800043210006900191260001700260520126600277100002201543856003601565 2004 en_Ud 00aReduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite Riemannian metric0 aReduction by group symmetry of second order variational problems bEDP Sciences3 aFor a Riemannian structure on a semidirect product of Lie groups, the variational problems can be reduced using the group symmetry. Choosing the Levi-Civita connection of a positive definite metric tensor, instead of any of the canonical connections for the Lie group, simplifies the reduction of the variations but complicates the expression for the Lie algebra valued covariant derivatives. The origin of the discrepancy is in the semidirect product structure, which implies that the Riemannian exponential map and the Lie group exponential map do not coincide. The consequence is that the reduced equations look more complicated than the original ones. The main scope of this paper is to treat the reduction of second order variational problems (corresponding to geometric splines) on such semidirect products of Lie groups. Due to the semidirect structure, a number of extra terms appears in the reduction, terms that are calculated explicitely. The result is used to compute the necessary conditions of an optimal control problem for a simple mechanical control system having invariant Lagrangian equal to the kinetic energy corresponding to the metric tensor. As an example, the case of a rigid body on the Special Euclidean group is considered in detail.1 aAltafini, Claudio uhttp://hdl.handle.net/1963/352101168nas a2200097 4500008004300000245006600043210006600109520083700175100002201012856003601034 2004 en_Ud 00aRepresenting multiqubit unitary evolutions via Stokes tensors0 aRepresenting multiqubit unitary evolutions via Stokes tensors3 aFor the Stokes tensor parametrization of a multiqubit density operator, we provide an explicit formulation of the corresponding unitary dynamics at the infinitesimal level. The main advantage of this formalism (clearly reminiscent of the ideas of ``coherences\\\'\\\' and ``coupling Hamiltonians\\\'\\\' of spin systems) is that the pattern of correlation between qubits and the pattern of infinitesimal correlation are highlighted simultaneously and can be used constructively for qubit manipulation. For example, it allows to compute explicitly a Rodrigues\\\' formula for the one-parameter orbits of nonlocal Hamiltonians. The result is easily generalizable to orbits of Cartan subalgebras and allows to express the Cartan decomposition of unitary propagators as a linear action directly in terms of the infinitesimal generators.1 aAltafini, Claudio uhttp://hdl.handle.net/1963/230701180nas a2200121 4500008004300000245007900043210006900122260001700191520077400208100001700982700002300999856003601022 2004 en_Ud 00aResonance of minimizers for n-level quantum systems with an arbitrary cost0 aResonance of minimizers for nlevel quantum systems with an arbit bEDP Sciences3 aWe consider an optimal control problem describing a laser-induced population transfer on a $n$-level quantum system.\\nFor a convex cost depending only on the moduli of controls (i.e. the lasers intensities), we prove that there always exists a minimizer in resonance. This permits to justify some strategies used in experimental physics. It is also quite important because it permits to reduce remarkably the complexity of the problem (and extend some of our previous results for $n=2$ and $n=3$): instead of looking for minimizers on the sphere $S^{2n-1}\\\\subset\\\\C^n$ one is reduced to look just for minimizers on the sphere $S^{n-1}\\\\subset \\\\R^n$. Moreover, for the reduced problem, we investigate on the question of existence of strict abnormal minimizer.1 aBoscain, Ugo1 aCharlot, Grégoire uhttp://hdl.handle.net/1963/291000388nas a2200097 4500008004300000245008700043210006900130260003500199100002000234856003600254 2004 en_Ud 00aThe role of the spectrum of the Laplace operator on \\\\S2 in the H-bubble problem0 arole of the spectrum of the Laplace operator on S2 in the Hbubbl bHebrew University Magnes Press1 aMusina, Roberta uhttp://hdl.handle.net/1963/289401012nas a2200121 4500008004300000245005300043210005300096260001300149520064100162100002200803700002900825856003600854 2004 en_Ud 00aRotating Singular Perturbations of the Laplacian0 aRotating Singular Perturbations of the Laplacian bSpringer3 aWe study a system of a quantum particle interacting with a singular time-dependent uniformly rotating potential in 2 and 3 dimensions: in particular we consider an interaction with support on a point (rotating point interaction) and on a set of codimension 1 (rotating blade). We prove the existence of the Hamiltonians of such systems as suitable self-adjoint operators and we give an explicit expression for their unitary semigroups. Moreover we analyze the asymptotic limit of large angular velocity and we prove strong convergence of the time-dependent propagator to some one-parameter unitary group as (\\\\omega \\\\to \\\\infty).1 aCorreggi, Michele1 aDell'Antonio, Gianfausto uhttp://hdl.handle.net/1963/294501129nas a2200133 4500008004100000245005300041210004600094260001800140520073800158100002000896700002000916700002300936856003600959 2002 en d00aOn the reachability of quantized control systems0 areachability of quantized control systems bSISSA Library3 aIn this paper, we study control systems whose input sets are quantized, i.e., finite or regularly distributed on a mesh. We specifically focus on problems relating to the structure of the reachable set of such systems, which may turn out to be either dense or discrete. We report results on the reachable set of linear quantized systems, and on a particular but interesting class of nonlinear systems, i.e., nonholonomic chained-form systems. For such systems, we provide a complete characterization of the reachable set, and, in case the set is discrete, a computable method to completely and succinctly describe its structure. Implications and open problems in the analysis and synthesis of quantized control systems are addressed.1 aBicchi, Antonio1 aMarigo, Alessia1 aPiccoli, Benedetto uhttp://hdl.handle.net/1963/150101047nas a2200109 4500008004300000245006200043210005800105260001300163520070300176100002200879856003600901 2002 en_Ud 00aThe reachable set of a linear endogenous switching system0 areachable set of a linear endogenous switching system bElsevier3 aIn this work, switching systems are named endogenous when their switching pattern is controllable. Linear endogenous switching systems can be considered as a particular class of bilinear control systems. The key idea is that both types of systems are equivalent to polysystems, i.e. to systems whose flow is piecewise smooth. The reachable set of a linear endogenous switching system can be studied consequently. The main result is that, in general, it has the structure of a semigroup, even when the Lie algebra rank condition is satisfied since the logic inputs cannot reverse the direction of the flow. The adaptation of existing controllability criteria for bilinear systems is straightforward.1 aAltafini, Claudio uhttp://hdl.handle.net/1963/314200851nas a2200145 4500008004300000245005500043210005500098260001000153520041100163100002200574700001600596700003100612700002600643856003600669 2002 en_Ud 00aRelatively stable bundles over elliptic fibrations0 aRelatively stable bundles over elliptic fibrations bWiley3 aWe consider a relative Fourier-Mukai transform defined on elliptic fibrations over an arbitrary normal base scheme. This is used to construct relative Atiyah sheaves and generalize Atiyah\\\'s and Tu\\\'s results about semistable sheaves over elliptic curves to the case of elliptic fibrations. Moreover we show that this transform preserves relative (semi)stability of sheaves of positive relative degree.1 aBartocci, Claudio1 aBruzzo, Ugo1 aHernandez Ruiperez, Daniel1 aMunoz Porras, Jose M. uhttp://hdl.handle.net/1963/313201019nas a2200133 4500008004300000245006700043210006700110260000900177520060000186100002000786700002300806700002000829856003600849 2000 en_Ud 00aReachability Analysis for a Class of Quantized Control Systems0 aReachability Analysis for a Class of Quantized Control Systems bIEEE3 aWe study control systems whose input sets are quantized. We specifically focus on problems relating to the structure of the reachable set of such systems, which may turn out to be either dense or discrete. We report on some results on the reachable set of linear quantized systems, and study in detail an interesting class of nonlinear systems, forming the discrete counterpart of driftless nonholonomic continuous systems. For such systems, we provide a complete characterization of the reachable set, and, in the case the set is discrete, a computable method to describe its lattice structure.1 aMarigo, Alessia1 aPiccoli, Benedetto1 aBicchi, Antonio uhttp://hdl.handle.net/1963/351800781nas a2200133 4500008004300000245011000043210006900153260001300222520032300235100002100558700001800579700001400597856003600611 2000 en_Ud 00aReduction of bi-Hamiltonian systems and separation of variables: an example from the Boussinesq hierarchy0 aReduction of biHamiltonian systems and separation of variables a bSpringer3 aWe discuss the Boussinesq system with $t_5$ stationary, within a general framework for the analysis of stationary flows of n-Gel\\\'fand-Dickey hierarchies. We show how a careful use of its bihamiltonian structure can be used to provide a set of separation coordinates for the corresponding Hamilton--Jacobi equations.1 aFalqui, Gregorio1 aMagri, Franco1 aTondo, G. uhttp://hdl.handle.net/1963/321901450nas a2200121 4500008004300000245006400043210006400107260000900171520106500180100002301245700002401268856003601292 2000 en_Ud 00aRegular Synthesis and Sufficiency Conditions for Optimality0 aRegular Synthesis and Sufficiency Conditions for Optimality bSIAM3 aWe propose a definition of \\\"regular synthesis\\\" that is more general than those suggested by other authors such as Boltyanskii and Brunovsky, and an even more general notion of \\\"regular presynthesis.\\\" We give a complete proof of the corresponding sufficiency theorem, a slightly weaker version of which had been stated in an earlier article, with only a rough outline of the proof. We illustrate the strength of our result by showing that the optimal synthesis for the famous Fuller problem satisfies our hypotheses. We also compare our concept of synthesis with the simpler notion of a \\\"family of solutions of the closed-loop equation arising from an optimal feedback law,\\\" and show by means of examples why the latter is inadequate, and why the difficulty cannot be resolved byusing other concepts of solution--such as Filippov solutions, or the limits of sample-and-hold solutions recently proposed as feedback solutions by Clarke, Ledyaev, Subbotin and Sontag -for equations with a non-Lipschitz and possibly discontinuous right-hand side.1 aPiccoli, Benedetto1 aSussmann, Hector J. uhttp://hdl.handle.net/1963/321300914nas a2200133 4500008004300000245007500043210006900118260002100187520047300208100002200681700002200703700001900725856003600744 2000 en_Ud 00aA Remark on One-Dimensional Many-Body Problems with Point Interactions0 aRemark on OneDimensional ManyBody Problems with Point Interactio bWorld Scientific3 aThe integrability of one dimensional quantum mechanical many-body problems with general contact interactions is extensively studied. It is shown that besides the pure (repulsive or attractive) $\\\\delta$-function interaction there is another singular point interactions which gives rise to a new one-parameter family of integrable quantum mechanical many-body systems. The bound states and scattering matrices are calculated for both bosonic and fermionic statistics.1 aAlbeverio, Sergio1 aDabrowski, Ludwik1 aFei, Shao-Ming uhttp://hdl.handle.net/1963/321400981nas a2200121 4500008004100000245011300041210007000154260001000224520054300234100002000777700002600797856003600823 1999 en d00aRecurrent procedure for the determination of the free energy ε^2 expansion in the topological string theory0 aRecurrent procedure for the determination of the free energy ε2 bSISSA3 aWe present here the iteration procedure for the determination of free energy ǫ2-expansion using the theory of KdV - type equations. In our approach we use the conservation laws for KdV - type equations depending explicitly on times t1, t2, . . . to find the ǫ2-expansion of u(x, t1, t2, . . .) after the infinite number of shifts of u(x, 0, 0, . . .) ≡ x along t1, t2, . . . in recurrent form. The formulas for the free energy expansion are just obtained then as a result of quite simple integration procedure applied to un(x).

1 aDubrovin, Boris1 aMaltsev, Andrei, Ya A uhttp://hdl.handle.net/1963/648900470nas a2200133 4500008004100000245007500041210006900116260003700185100002100222700002000243700001800263700001900281856003600300 1999 en d00aRenormalized solutions of elliptic equations with general measure data0 aRenormalized solutions of elliptic equations with general measur bScuola Normale Superiore di Pisa1 aDal Maso, Gianni1 aMurat, Francois1 aOrsina, Luigi1 aPrignet, Alain uhttp://hdl.handle.net/1963/123600377nas a2200097 4500008004100000245009200041210006900133260001800202100002400220856003500244 1981 en d00aRecent advances in the study of the existence of periodic orbits of Hamiltonian systems0 aRecent advances in the study of the existence of periodic orbits bSISSA Library1 aAmbrosetti, Antonio uhttp://hdl.handle.net/1963/159