We consider the fractional powers of singular (point-like) perturbations of the Laplacian and the singular perturbations of fractional powers of the Laplacian, and we compare two such constructions focusing on their perturbative structure for resolvents and on the local singularity structure of their domains. In application to the linear and non-linear Schrödinger equations for the corresponding operators, we outline a programme of relevant questions that deserve being investigated.

1 aMichelangeli, Alessandro1 aOttolini, Andrea1 aScandone, Raffaele uhttps://doi.org/10.1063/1.503385601226nas a2200193 4500008004100000022001400041245006800055210006500123300001600188490000800204520054000212653002300752653006700775653004400842100002300886700002900909700002300938856007100961 2018 eng d a0022-123600aOn fractional powers of singular perturbations of the Laplacian0 afractional powers of singular perturbations of the Laplacian a1551 - 16020 v2753 aWe qualify a relevant range of fractional powers of the so-called Hamiltonian of point interaction in three dimensions, namely the singular perturbation of the negative Laplacian with a contact interaction supported at the origin. In particular we provide an explicit control of the domain of such a fractional operator and of its decomposition into regular and singular parts. We also qualify the norms of the resulting singular fractional Sobolev spaces and their mutual control with the corresponding classical Sobolev norms.

10aPoint interactions10aRegular and singular component of a point-interaction operator10aSingular perturbations of the Laplacian1 aGeorgiev, Vladimir1 aMichelangeli, Alessandro1 aScandone, Raffaele uhttp://www.sciencedirect.com/science/article/pii/S002212361830104600893nas a2200121 4500008004100000245004200041210004200083300001200125490000700137520055100144100003000695856004600725 2018 eng d00aFramed symplectic sheaves on surfaces0 aFramed symplectic sheaves on surfaces a18500070 v293 aA framed symplectic sheaf on a smooth projective surface $X$ is a torsion-free sheaf $E$ together with a trivialization on a divisor $D \subset X$ and a morphism $\Lambda^2 E \rightarrow \mathcal{O}_X$ satisfying some additional conditions. We construct a moduli space for framed symplectic sheaves on a surface, and present a detailed study for $X =\mathbb{P}_\mathbb{C}^2$. In this case, the moduli space is irreducible and admits an ADHM-type description and a birational proper map onto the space of framed symplectic ideal instantons.

1 aScalise, Jacopo, Vittorio uhttps://doi.org/10.1142/S0129167X1850007601597nas a2200169 4500008004100000245012200041210006900163260002100232300001200253490000700265520094600272100002501218700002201243700001801265700002101283856012301304 2018 eng d00aFree-form deformation, mesh morphing and reduced-order methods: enablers for efficient aerodynamic shape optimisation0 aFreeform deformation mesh morphing and reducedorder methods enab bTaylor & Francis a233-2470 v323 aIn this work, we provide an integrated pipeline for the model-order reduction of turbulent flows around parametrised geometries in aerodynamics. In particular, free-form deformation is applied for geometry parametrisation, whereas two different reduced-order models based on proper orthogonal decomposition (POD) are employed in order to speed-up the full-order simulations: the first method exploits POD with interpolation, while the second one is based on domain decomposition. For the sampling of the parameter space, we adopt a Greedy strategy coupled with Constrained Centroidal Voronoi Tessellations, in order to guarantee a good compromise between space exploration and exploitation. The proposed framework is tested on an industrially relevant application, i.e. the front-bumper morphing of the DrivAer car model, using the finite-volume method for the full-order resolution of the Reynolds-Averaged Navier–Stokes equations.

1 aSalmoiraghi, Filippo1 aScardigli, Angela1 aTelib, Haysam1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/free-form-deformation-mesh-morphing-and-reduced-order-methods-enablers-efficient00581nas a2200133 4500008004100000245004900041210004600090260000900136300001400145490000600159520020400165100002400369856005400393 2018 eng d00aOn fully real eigenconfigurations of tensors0 afully real eigenconfigurations of tensors bSIAM a339–3470 v23 aWe construct generic real symmetric tensors with only real eigenvectors or, equivalently, real homogeneous polynomials with the maximum possible finite number of critical points on the sphere.

1 aKozhasov, Khazhgali uhttps://epubs.siam.org/doi/pdf/10.1137/17M114590201281nas a2200121 4500008004100000245008200041210006900123520085300192100002001045700001701065700002901082856004801111 2017 en d00aFriedrichs systems in a Hilbert space framework: solvability and multiplicity0 aFriedrichs systems in a Hilbert space framework solvability and 3 aThe Friedrichs (1958) theory of positive symmetric systems of first order partial differential equations encompasses many standard equations of mathematical physics, irrespective of their type. This theory was recast in an abstract Hilbert space setting by Ern, Guermond and Caplain (2007), and by Antonić and Burazin (2010). In this work we make a further step, presenting a purely operator-theoretic description of abstract Friedrichs systems, and proving that any pair of abstract Friedrichs operators admits bijective extensions with a signed boundary map. Moreover, we provide suffcient and necessary conditions for existence of infinitely many such pairs of spaces, and by the universal operator extension theory (Grubb, 1968) we get a complete identification of all such pairs, which we illustrate on two concrete one-dimensional examples.1 aAntonić, Nenad1 aErceg, Marko1 aMichelangeli, Alessandro uhttp://preprints.sissa.it/handle/1963/3528001717nas a2200193 4500008004100000245011900041210006900160260001400229520106200243100002401305700002001329700002001349700002101369700002201390700002001412700002201432700001801454856005101472 2016 en d00aA fast virtual surgery platform for many scenarios haemodynamics of patient-specific coronary artery bypass grafts0 afast virtual surgery platform for many scenarios haemodynamics o bSubmitted3 aA fast computational framework is devised to the study of several configurations of patient-specific coronary artery bypass grafts. This is especially useful to perform a sensitivity analysis of the haemodynamics for different flow conditions occurring in native coronary arteries and bypass grafts, the investigation of the progression of the coronary artery disease and the choice of the most appropriate surgical procedure. A complete pipeline, from the acquisition of patientspecific medical images to fast parametrized computational simulations, is proposed. Complex surgical configurations employed in the clinical practice, such as Y-grafts and sequential grafts, are studied. A virtual surgery platform based on model reduction of unsteady Navier Stokes equations for blood dynamics is proposed to carry out sensitivity analyses in a very rapid and reliable way. A specialized geometrical parametrization is employed to compare the effect of stenosis and anastomosis variation on the outcome of the surgery in several relevant cases.1 aBallarin, Francesco1 aFaggiano, Elena1 aManzoni, Andrea1 aRozza, Gianluigi1 aQuarteroni, Alfio1 aIppolito, Sonia1 aScrofani, Roberto1 aAntona, Carlo uhttp://urania.sissa.it/xmlui/handle/1963/3524000965nas a2200169 4500008004100000022001400041245012900055210006900184260000800253300000700261490000700268520041200275100002100687700002200708700001900730856004600749 2016 eng d a1432-083500aFracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: the antiplane case0 aFracture models for elastoplastic materials as limits of gradien cApr a450 v553 aWe study the asymptotic behavior of a variational model for damaged elasto-plastic materials in the case of antiplane shear. The energy functionals we consider depend on a small parameter $\varepsilon$, which forces damage concentration on regions of codimension one. We determine the $\Gamma$-limit as $\varepsilon$ tends to zero and show that it contains an energy term involving the crack opening.

1 aDal Maso, Gianni1 aOrlando, Gianluca1 aToader, Rodica uhttps://doi.org/10.1007/s00526-016-0981-z01114nas a2200121 4500008004100000245006200041210006200103260001000165520067200175653001800847100003000865856009700895 2016 en d00aFrames symplectic sheaves on surfaces and their ADHM data0 aFrames symplectic sheaves on surfaces and their ADHM data bSISSA3 aThis dissertation is centered on the moduli space of what we call framed symplectic sheaves on a surface, compactifying the corresponding moduli space of framed principal SP−bundles. It contains the construction of the moduli space, which is carried out for every smooth projective surface X with a big and nef framing divisor, and a study of its deformation theory. We also develop an in-depth analysis of the examples X = P2 and X = Blp (P2 ), showing that the corresponding moduli spaces enjoy an ADHM-type description. In the former case, we prove irreducibility of the space and exhibit a relation with the space of framed ideal instantons on S4 in type C.10amoduli spaces1 aScalise, Jacopo, Vittorio uhttps://www.math.sissa.it/publication/frames-symplectic-sheaves-surfaces-and-their-adhm-data00789nas a2200145 4500008004100000245009000041210006900131300001200200490000700212520031400219100002200533700001800555700002400573856004600597 2016 eng d00aA Frobenius theorem for corank-1 continuous distributions in dimensions two and three0 aFrobenius theorem for corank1 continuous distributions in dimens a16500610 v273 aWe formulate a notion of (uniform) asymptotic involutivity and show that it implies (unique) integrability of corank-1 continuous distributions in dimensions three or less. This generalizes and extends a classical Frobenius theorem, which says that an involutive C1 distribution is uniquely integrable.

1 aLuzzatto, Stefano1 aTüreli, Sina1 aWar, Khadim, Mbacke uhttps://doi.org/10.1142/S0129167X1650061001820nas a2200169 4500008004100000245015600041210006900197520118400266100002401450700002001474700002001494700002001514700002201534700002101556700002201577856005101599 2015 en d00aFast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a POD-Galerkin method and a vascular shape parametrization0 aFast simulations of patientspecific haemodynamics of coronary ar3 aIn this work a reduced-order computational framework for the study of haemodynamics in three-dimensional patient-specific configurations of coronary artery bypass grafts dealing with a wide range of scenarios is proposed. We combine several efficient algorithms to face at the same time both the geometrical complexity involved in the description of the vascular network and the huge computational cost entailed by time dependent patient-specific flow simulations. Medical imaging procedures allow to reconstruct patient-specific configurations from clinical data. A centerlines-based parametrization is proposed to efficiently handle geometrical variations. POD–Galerkin reduced-order models are employed to cut down large computational costs. This computational framework allows to characterize blood flows for different physical and geometrical variations relevant in the clinical practice, such as stenosis factors and anastomosis variations, in a rapid and reliable way. Several numerical results are discussed, highlighting the computational performance of the proposed framework, as well as its capability to perform sensitivity analysis studies, so far out of reach.1 aBallarin, Francesco1 aFaggiano, Elena1 aIppolito, Sonia1 aManzoni, Andrea1 aQuarteroni, Alfio1 aRozza, Gianluigi1 aScrofani, Roberto uhttp://urania.sissa.it/xmlui/handle/1963/3462301899nas a2200133 4500008004300000245010100043210006900144520142800213100002101641700001701662700001701679700001801696856005101714 2015 en_Ud 00aFEM SUPG stabilisation of mixed isoparametric BEMs: application to linearised free surface flows0 aFEM SUPG stabilisation of mixed isoparametric BEMs application t3 aIn finite element formulations, transport dominated problems are often stabilised through the Streamline-Upwind-Petrov–Galerkin (SUPG) method. Its application is straightforward when the problem at hand is solved using Galerkin methods. Applications of boundary integral formulations often resort to collocation techniques which are computationally more tractable. In this framework, the Galerkin method and the stabilisation may still be used to successfully apply boundary conditions and resolve instabilities that are frequently observed in transport dominated problems. We apply this technique to an adaptive collocation boundary element method for the solution of stationary potential flows, where we solve a mixed Poisson problem in boundary integral form, with the addition of linearised free surface boundary conditions. We use a mixed boundary element formulation to allow for different finite dimensional spaces describing the flow potential and its normal derivative, and we validate our method simulating the flow around both a submerged body and a surface piercing body. The coupling of mixed surface finite elements and strongly consistent stabilisation techniques with boundary elements opens up the possibility to use non conformal unstructured grids with local refinement, without introducing the inconsistencies of other stabilisation techniques based on up-winding and finite difference schemes.

1 aGiuliani, Nicola1 aMola, Andrea1 aHeltai, Luca1 aFormaggia, L. uhttp://urania.sissa.it/xmlui/handle/1963/3446601351nas a2200133 4500008004100000245008400041210007000125260003900195520087300234100001901107700001901126700002101145856005101166 2014 en d00aFinite dimensional Kadomtsev-Petviashvili τ-functions. I. Finite Grassmannians0 aFinite dimensional KadomtsevPetviashvili τfunctions I Finite Gra bAmerican Institute of Physics Inc.3 aWe study τ-functions of the Kadomtsev-Petviashvili hierarchy in terms of abelian group actions on finite dimensional Grassmannians, viewed as subquotients of the Hilbert space Grassmannians of Sato, Segal, and Wilson. A determinantal formula of Gekhtman and Kasman involving exponentials of finite dimensional matrices is shown to follow naturally from such reductions. All reduced flows of exponential type generated by matrices with arbitrary nondegenerate Jordan forms are derived, both in the Grassmannian setting and within the fermionic operator formalism. A slightly more general determinantal formula involving resolvents of the matrices generating the flow, valid on the big cell of the Grassmannian, is also derived. An explicit expression is deduced for the Plücker coordinates appearing as coefficients in the Schur function expansion of the τ-function.1 aBalogh, Ferenc1 aFonseca, Tiago1 aHarnad, John, P. uhttp://urania.sissa.it/xmlui/handle/1963/3495201202nas a2200145 4500008004100000245010600041210006900147260001000216520062900226653002300855100001700878700001700895700002200912856012200934 2014 en d00aA fully nonlinear potential model for ship hydrodynamics directly interfaced with CAD data structures0 afully nonlinear potential model for ship hydrodynamics directly bSISSA3 aWe present a model for ship hydrodynamics simulations currently under development at SISSA. The model employs potential flow theory and fully nonlinear free surface boundary conditions. The spatial discretization of the equations is performed by means of a collocation BEM. This gives rise to a Differential Algbraic Equations (DAE) system, solved using an implicit BDF scheme to time advance the solution. The model has been implemented into a C++ software able to automatically generate the computational grids from the CAD geometry of the hull. Numerical results on Kriso KCS and KVLCC2 hulls are presented and discussed.10aship hydrodynamics1 aMola, Andrea1 aHeltai, Luca1 aDeSimone, Antonio uhttps://www.math.sissa.it/publication/fully-nonlinear-potential-model-ship-hydrodynamics-directly-interfaced-cad-data01604nas a2200133 4500008004100000245010100041210006900142260001900211490000800230520099000238653009201228100002101320856012901341 2014 eng d00aFundamentals of Reduced Basis Method for problems governed by parametrized PDEs and applications0 aFundamentals of Reduced Basis Method for problems governed by pa aWienbSpringer0 v5543 aIn this chapter we consider Reduced Basis (RB) approximations of parametrized Partial Differential Equations (PDEs). The the idea behind RB is to decouple the generation and projection stages (Offline/Online computational procedures) of the approximation process in order to solve parametrized PDEs in a fast, inexpensive and reliable way. The RB method, especially applied to 3D problems, allows great computational savings with respect to the classical Galerkin Finite Element (FE) Method. The standard FE method is typically ill suited to (i) iterative contexts like optimization, sensitivity analysis and many-queries in general, and (ii) real time evaluation. We consider for simplicity coercive PDEs. We discuss all the steps to set up a RB approximation, either from an analytical and a numerical point of view. Then we present an application of the RB method to a steady thermal conductivity problem in heat transfer with emphasis on geometrical and physical parameters.

10areduced basis method, linear elasticity, heat transfer, error bounds, parametrized PDEs1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/fundamentals-reduced-basis-method-problems-governed-parametrized-pdes-and-applications01391nas a2200109 4500008004100000245006200041210006200103260001000165520104500175100002501220856003601245 2013 en d00aFields of bounded deformation for mesoscopic dislocations0 aFields of bounded deformation for mesoscopic dislocations bSISSA3 aIn this paper we discuss the consequences of the distributional approach to dislocations in terms of the mathematical properties\\r\\nof the auxiliary model fields such as displacement and displacement gradient which are obtained directly from \\r\\nthe main model field here considered as the linear strain. We show that these fields cannot be introduced rigourously without \\r\\nthe introduction of gauge fields, or equivalently, without cuts in the Riemann foliation associated to the dislocated crystal.\\r\\nIn a second step we show that the space of bounded deformations follows from the distributional approach in a natural way and \\r\\ndiscuss the reasons why it is adequate to model dislocations. The case of dislocation clusters is also addressed, as it represents an important issue in industrial crystal growth while from a mathematical point of view, peculiar phenomena might appear at the set of accumulation points. \\r\\nThe elastic-plastic decomposition of the strain within this approach is also given a precise meaning.1 aVan Goethem, Nicolas uhttp://hdl.handle.net/1963/637800902nas a2200121 4500008004100000245005300041210005200094260004800146520050700194100002100701700002200722856003600744 2013 en d00aFracture models as Gamma-limits of damage models0 aFracture models as Gammalimits of damage models bAmerican Institute of Mathematical Sciences3 aWe analyze the asymptotic behavior of a variational model for damaged elastic materials. This model depends on two small parameters, which govern the width of the damaged regions and the minimum elasticity constant attained in the damaged regions. When these parameters tend to zero, we find that the corresponding functionals Gamma-converge to a functional related to fracture mechanics. The corresponding problem is brittle or cohesive, depending on the asymptotic ratio of the two parameters.

1 aDal Maso, Gianni1 aIurlano, Flaviana uhttp://hdl.handle.net/1963/422502234nas a2200109 4500008004100000245004000041210004000081520191700121100001602038700002002054856005002074 2013 en d00aFramed sheaves on projective stacks0 aFramed sheaves on projective stacks3 aGiven a normal projective irreducible stack $\mathscr X$ over an algebraically closed field of characteristic zero we consider {\em framed sheaves} on $\mathscr X$, i.e., pairs $(\mathcal E,\phi_{\mathcal E})$, where $\mathcal E$ is a coherent sheaf on $\mathscr X$ and $\phi_{\mathcal E}$ is a morphism from $\mathcal E$ to a fixed coherent sheaf $\mathcal F$. After introducing a suitable notion of (semi)stability, we construct a projective scheme, which is a moduli space for semistable framed sheaves with fixed Hilbert polynomial, and an open subset of it, which is a fine moduli space for stable framed sheaves. If $\mathscr X$ is a projective irreducible orbifold of dimension two and $\mathcal F$ a locally free sheaf on a smooth divisor $\mathscr D\subset \mathscr X$ satisfying certain conditions, we consider {\em $(\mathscr{D}, \mathcal{F})$-framed sheaves}, i.e., framed sheaves $(\mathcal E,\phi_{\mathcal E})$ with $\mathcal E$ a torsion-free sheaf which is locally free in a neighborhood of $\mathscr D$, and ${\phi_{\mathcal{E}}}_{\vert \mathscr{D}}$ an isomorphism. These pairs are $\mu$-stable for a suitable choice of a parameter entering the (semi)stability condition, and of the polarization of $\mathscr X$. This implies the existence of a fine moduli space parameterizing isomorphism classes of $(\mathscr{D}, \mathcal{F})$-framed sheaves on $\mathscr{X}$ with fixed Hilbert polynomial, which is a quasi-projective scheme. In an appendix we develop the example of stacky Hirzebruch surfaces. This is the first paper of a project aimed to provide an algebro-geometric approach to the study of gauge theories on a wide class of 4-dimensional Riemannian manifolds by means of framed sheaves on ``stacky" compactifications of them. In particular, in a subsequent paper \cite{art:bruzzopedrinisalaszabo2013} these results are used to study gauge theories on ALE spaces of type $A_k$.1 aBruzzo, Ugo1 aSala, Francesco uhttp://urania.sissa.it/xmlui/handle/1963/743801318nas a2200121 4500008004100000245007900041210006900120520083000189100002001019700002201039700002101061856011401082 2013 eng d00aFree Form Deformation Techniques Applied to 3D Shape Optimization Problems0 aFree Form Deformation Techniques Applied to 3D Shape Optimizatio3 aThe purpose of this work is to analyse and study an efficient parametrization technique for a 3D shape optimization problem. After a brief review of the techniques and approaches already available in literature, we recall the Free Form Deformation parametrization, a technique which proved to be efficient and at the same time versatile, allowing to manage complex shapes even with few parameters. We tested and studied the FFD technique by establishing a path, from the geometry definition, to the method implementation, and finally to the simulation and to the optimization of the shape. In particular, we have studied a bulb and a rudder of a race sailing boat as model applications, where we have tested a complete procedure from Computer-Aided-Design to build the geometrical model to discretization and mesh generation.1 aKoshakji, Anwar1 aQuarteroni, Alfio1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/free-form-deformation-techniques-applied-3d-shape-optimization-problems01179nas a2200133 4500008004100000245006000041210005500101260001000156520076500166653004100931100001600972700002100988856003601009 2012 en d00aA formula for Popp\'s volume in sub-Riemannian geometry0 aformula for Popps volume in subRiemannian geometry bSISSA3 aFor an equiregular sub-Riemannian manifold M, Popp\'s volume is a smooth\r\nvolume which is canonically associated with the sub-Riemannian structure, and\r\nit is a natural generalization of the Riemannian one. In this paper we prove a\r\ngeneral formula for Popp\'s volume, written in terms of a frame adapted to the\r\nsub-Riemannian distribution. As a first application of this result, we prove an\r\nexplicit formula for the canonical sub-Laplacian, namely the one associated\r\nwith Popp\'s volume. Finally, we discuss sub-Riemannian isometries, and we prove\r\nthat they preserve Popp\'s volume. We also show that, under some hypotheses on\r\nthe action of the isometry group of M, Popp\'s volume is essentially the unique\r\nvolume with such a property.10asubriemannian, volume, Popp, control1 aRizzi, Luca1 aBarilari, Davide uhttp://hdl.handle.net/1963/650100484nas a2200133 4500008004100000022001400041245009400055210006900149300001400218490000800232100001900240700002000259856007100279 2012 eng d a0010-361600aFredholm determinants and pole-free solutions to the noncommutative Painlevé II equation0 aFredholm determinants and polefree solutions to the noncommutati a793–8330 v3091 aBertola, Marco1 aCafasso, Mattia uhttp://0-dx.doi.org.mercury.concordia.ca/10.1007/s00220-011-1383-x00909nas a2200109 4500008004100000245007800041210006900119260001300188520054100201100002100742856003600763 2012 en d00aFrobenius manifold for the dispersionless Kadomtsev-Petviashvili equation0 aFrobenius manifold for the dispersionless KadomtsevPetviashvili bSpringer3 aWe consider a Frobenius structure associated with the dispersionless\\r\\nKadomtsev-Petviashvili equation. This is done, essentially, by applying a\\r\\ncontinuous analogue of the finite dimensional theory in the space of Schwartz\\r\\nfunctions on the line. The potential of the Frobenius manifold is found to be a\\r\\nlogarithmic potential with quadratic external field. Following the construction\\r\\nof the principal hierarchy, we construct a set of infinitely many commuting\\r\\nflows, which extends the classical dKP hierarchy.1 aRaimondo, Andrea uhttp://hdl.handle.net/1963/604001763nas a2200169 4500008004100000245010700041210006900148260001000217520118200227653002601409653002901435653003501464100001701499700001701516700002401533856003601557 2012 en d00aA Fully Coupled Immersed Finite Element Method for Fluid Structure Interaction via the Deal.II Library0 aFully Coupled Immersed Finite Element Method for Fluid Structure bSISSA3 aWe present the implementation of a solution scheme for fluid-structure\\r\\ninteraction problems via the finite element software library deal.II. The\\r\\nsolution scheme is an immersed finite element method in which two independent discretizations are used for the fluid and immersed deformable body. In this type of formulation the support of the equations of motion of the fluid is extended to cover the union of the solid and fluid domains. The equations of motion over the extended solution domain govern the flow of a fluid under the action of a body force field. This body force field informs the fluid of the presence of the immersed solid. The velocity field of the immersed solid is the restriction over the immersed domain of the velocity field in the extended equations of motion. The focus of this paper is to show how the determination of the motion of the immersed domain is carried out in practice. We show that our implementation is general, that is, it is not dependent on a specific choice of the finite element spaces over the immersed solid and the extended fluid domains. We present some preliminary results concerning the accuracy of the proposed method.10aFinite Element Method10aImmersed Boundary Method10aImmersed Finite Element Method1 aHeltai, Luca1 aRoy, Saswati1 aCostanzo, Francesco uhttp://hdl.handle.net/1963/625500878nas a2200109 4500008004100000245008900041210006900130260002200199520048900221100002200710856003600732 2011 en d00aFracture and plastic models as Gamma-limits of damage models under different regimes0 aFracture and plastic models as Gammalimits of damage models unde bWalter de Gruyter3 aWe consider a variational model for damaged elastic materials. This model depends on three small parameters, which are related to the cost of the damage, to the width of the damaged regions, and to the minimum elasticity constant attained in the damaged regions. As these parameters tend to zero, our models Gamma-converge to a model for brittle fracture, for fracture with a cohesive zone, or for perfect plasticity, depending on the asymptotic ratios of the three parameters.

1 aIurlano, Flaviana uhttp://hdl.handle.net/1963/506901148nas a2200133 4500008004100000245009700041210006900138260001300207520069800220100002200918700002200940700001600962856003600978 2010 en d00aFeedback schemes for radiation damping suppression in NMR: a control-theoretical perspective0 aFeedback schemes for radiation damping suppression in NMR a cont bElsevier3 aIn NMR spectroscopy, the collective measurement is weakly invasive and its back-action is called radiation damping. The aim of this paper is to provide a control-theoretical analysis of the problem of suppressing this radiation damping. It is shown that the two feedback schemes commonly used in the NMR practice correspond one to a high gain oputput feedback for the simple case of maintaining the spin 1/2 in its inverted state, and the second to a 2-degree of freedom control design with a prefeedback that exactly cancels the radiation damping field. A general high gain feedback stabilization design not requiring the knowledge of the radiation damping time constant is also investigated.1 aAltafini, Claudio1 aCappellaro, Paola1 aCory, David uhttp://hdl.handle.net/1963/438400439nas a2200133 4500008004100000022001400041245006200055210006100117300001400178490000700192100001900199700001600218856007100234 2010 eng d a0176-427600aFirst colonization of a hard-edge in random matrix theory0 aFirst colonization of a hardedge in random matrix theory a231–2570 v311 aBertola, Marco1 aLee, S., Y. uhttp://0-dx.doi.org.mercury.concordia.ca/10.1007/s00365-009-9052-400938nas a2200109 4500008004300000245007800043210006900121520056500190100001700755700002000772856003600792 2009 en_Ud 00aFamilies of Monads and Instantons from a Noncommutative ADHM Construction0 aFamilies of Monads and Instantons from a Noncommutative ADHM Con3 aWe give a \\\\theta-deformed version of the ADHM construction of SU(2) instantons with arbitrary topological charge on the sphere S^4. Classically the instanton gauge fields are constructed from suitable monad data; we show that in the deformed case the set of monads is itself a noncommutative space. We use these monads to construct noncommutative `families\\\' of SU(2) instantons on the deformed sphere S^4_\\\\theta. We also compute the topological charge of each of the families. Finally we discuss what it means for such families to be gauge equivalent.1 aBrain, Simon1 aLandi, Giovanni uhttp://hdl.handle.net/1963/347800454nas a2200133 4500008004100000022001400041245006900055210006900124300001400193490000700207100001900214700001600233856007100249 2009 eng d a0176-427600aFirst colonization of a spectral outpost in random matrix theory0 aFirst colonization of a spectral outpost in random matrix theory a225–2630 v301 aBertola, Marco1 aLee, S., Y. uhttp://0-dx.doi.org.mercury.concordia.ca/10.1007/s00365-008-9026-y08321nas a2200145 4500008004100000245008100041210006900122260001300191300001600204490000700220520785400227100003008081700001908111856004508130 2009 eng d00aFoliations of small tubes in Riemannian manifolds by capillary minimal discs0 aFoliations of small tubes in Riemannian manifolds by capillary m bElsevier a4422–44400 v703 aLetting be an embedded curve in a Riemannian manifold , we prove the existence of minimal disc-type surfaces centered at inside the surface of revolution of around , having small radius, and intersecting it with constant angles. In particular we obtain that small tubular neighborhoods can be foliated by minimal discs.

1 aFall, Mouhamed, Moustapha1 aMercuri, Carlo uhttps://doi.org/10.1016/j.na.2008.10.02400501nas a2200157 4500008004100000245009500041210007100136260001000207300001400217490000700231100001800238700001700256700001700273700001600290856003700306 2008 eng d00aFluid–structure interaction problems in free surface flows: Application to boat dynamics0 aFluid–structure interaction problems in free surface flows Appli bWiley a965–9780 v561 aFormaggia, L.1 aMiglio, Edie1 aMola, Andrea1 aParolini, N uhttps://doi.org/10.1002/fld.158300637nas a2200109 4500008004300000245004900043210004900092520031300141100001300454700002400467856003600491 2008 en_Ud 00aForced Vibrations of a Nonhomogeneous String0 aForced Vibrations of a Nonhomogeneous String3 aWe prove existence of vibrations of a nonhomogeneous string under a nonlinear time periodic forcing term in the case in which the forcing frequency avoids resonances with the vibration modes of the string (nonresonant case). The proof relies on a Lyapunov-Schmidt reduction and a Nash-Moser iteration scheme.1 aBaldi, P1 aBerti, Massimiliano uhttp://hdl.handle.net/1963/264300701nas a2200121 4500008004300000245009900043210006900142520027900211100002000490700001500510700001800525856003600543 2008 en_Ud 00aFrobenius Manifolds and Central Invariants for the Drinfeld - Sokolov Bihamiltonian Structures0 aFrobenius Manifolds and Central Invariants for the Drinfeld Soko3 aThe Drinfeld - Sokolov construction associates a hierarchy of bihamiltonian integrable systems with every untwisted affine Lie algebra. We compute the complete set of invariants of the related bihamiltonian structures with respect to the group of Miura type transformations.1 aDubrovin, Boris1 aSi-Qi, Liu1 aYoujin, Zhang uhttp://hdl.handle.net/1963/252301559nas a2200157 4500008004300000245008100043210006900124520105300193100001801246700001801264700002501282700002001307700001901327700001901346856003601365 2008 en_Ud 00aFulde-Ferrell-Larkin-Ovchinnikov pairing in one-dimensional optical lattices0 aFuldeFerrellLarkinOvchinnikov pairing in onedimensional optical 3 aSpin-polarized attractive Fermi gases in one-dimensional (1D) optical lattices are expected to be remarkably good candidates for the observation of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. We model these systems with an attractive Hubbard model with population imbalance. By means of the density-matrix renormalization-group method we compute the pairing correlations as well as the static spin and charge structure factors in the whole range from weak to strong coupling. We demonstrate that pairing correlations exhibit quasi-long range order and oscillations at the wave number expected from FFLO theory. However, we also show by numerically computing the mixed spin-charge static structure factor that charge and spin degrees of freedom appear to be coupled already for small imbalance. We discuss the consequences of this coupling for the observation of the FFLO phase, as well as for the stabilization of the quasi-long range order into long-range order by coupling many identical 1D systems, as in quasi-1D optical lattices.

1 aRizzi, Matteo1 aPolini, Marco1 aCazalilla, Miguel A.1 aBakhtiari, M.R.1 aTosi, Mario P.1 aFazio, Rosario uhttp://hdl.handle.net/1963/269402024nas a2200109 4500008004300000245009300043210006900136520163000205100002501835700001801860856003601878 2007 en_Ud 00aOn feedback classification of control-affine systems with one and two-dimensional inputs0 afeedback classification of controlaffine systems with one and tw3 aThe paper is devoted to the local classification of generic control-affine systems on an n-dimensional manifold with scalar input for any n>3 or with two inputs for n=4 and n=5, up to state-feedback transformations, preserving the affine structure. First using the Poincare series of moduli numbers we introduce the intrinsic numbers of functional moduli of each prescribed number of variables on which a classification problem depends. In order to classify affine systems with scalar input we associate with such a system the canonical frame by normalizing some structural functions in a commutative relation of the vector fields, which define our control system. Then, using this canonical frame, we introduce the canonical coordinates and find a complete system of state-feedback invariants of the system. It also gives automatically the micro-local (i.e. local in state-input space) classification of the generic non-affine n-dimensional control system with scalar input for n>2. Further we show how the problem of feedback-equivalence of affine systems with two-dimensional input in state space of dimensions 4 and 5 can be reduced to the same problem for affine systems with scalar input. In order to make this reduction we distinguish the subsystem of our control system, consisting of the directions of all extremals in dimension 4 and all abnormal extremals in dimension 5 of the time optimal problem, defined by the original control system. In each classification problem under consideration we find the intrinsic numbers of functional moduli of each prescribed number of variables according to its Poincare series.1 aAgrachev, Andrei, A.1 aZelenko, Igor uhttp://hdl.handle.net/1963/218600760nas a2200097 4500008004300000245003700043210003700080520048700117100002200604856003600626 2007 en_Ud 00aFeedback control of spin systems0 aFeedback control of spin systems3 aThe feedback stabilization problem for ensembles of coupled spin 1/2 systems is discussed from a control theoretic perspective. The noninvasive nature of the bulk measurement allows for a fully unitary and deterministic closed loop. The Lyapunov-based feedback design presented does not require spins that are selectively addressable. With this method, it is possible to obtain control inputs also for difficult tasks, like suppressing undesired couplings in identical spin systems.1 aAltafini, Claudio uhttp://hdl.handle.net/1963/180801194nas a2200097 4500008004300000245010500043210006900148520082100217100002201038856003601060 2007 en_Ud 00aFeedback stabilization of quantum ensembles: a global convergence analysis on complex flag manifolds0 aFeedback stabilization of quantum ensembles a global convergence3 aIn an N-level quantum mechanical system, the problem of unitary feedback stabilization of mixed density operators to periodic orbits admits a natural Lyapunov-based time-varying feedback design. A global description of the domain of attraction of the closed-loop system can be provided based on a \\\"root-space\\\"-like structure of the space of density operators. This convex set foliates as a complex flag manifold where each leaf is identified with the coadjoint orbit of the eigenvalues of the density operator. The converging conditions are time-independent but depend from the topology of the flag manifold: it is shown that the closed loop must have a number of equilibria at least equal to the Euler characteristic of the manifold, thus imposing obstructions of topological nature to global stabilizability.1 aAltafini, Claudio uhttp://hdl.handle.net/1963/172900873nas a2200133 4500008004300000245009700043210006900140520040800209100002500617700001900642700002100661700002100682856003600703 2007 en_Ud 00aOn finite-dimensional projections of distributions for solutions of randomly forced PDE\\\'s0 afinitedimensional projections of distributions for solutions of 3 aThe paper is devoted to studying the image of probability measures on a Hilbert space under finite-dimensional analytic maps. We establish sufficient conditions under which the image of a measure has a density with respect to the Lebesgue measure and continuously depends on the map. The results obtained are applied to the 2D Navier-Stokes equations perturbed by various random forces of low dimension.1 aAgrachev, Andrei, A.1 aKuksin, Sergei1 aSarychev, Andrey1 aShirikyan, Armen uhttp://hdl.handle.net/1963/201200868nas a2200109 4500008004300000245007300043210006900116520049600185100002400681700001700705856003600722 2006 en_Ud 00aForced vibrations of wave equations with non-monotone nonlinearities0 aForced vibrations of wave equations with nonmonotone nonlinearit3 aWe prove existence and regularity of periodic in time solutions of completely resonant nonlinear forced wave equations with Dirichlet boundary conditions for a large class of non-monotone forcing terms. Our approach is based on a variational Lyapunov-Schmidt reduction. It turns out that the infinite dimensional bifurcation equation exhibits an intrinsic lack of compactness. We solve it via a minimization argument and a-priori estimate methods inspired to regularity theory of Rabinowitz.1 aBerti, Massimiliano1 aBiasco, Luca uhttp://hdl.handle.net/1963/216001285nas a2200097 4500008004300000245007100043210006700114520095200181100001801133856003601151 2006 en_Ud 00aFundamental form and Cartan tensor of (2,5)-distributions coincide0 aFundamental form and Cartan tensor of 25distributions coincide3 aIn our previous paper for generic rank 2 vector distributions on n-dimensional manifold (n greater or equal to 5) we constructed a special differential invariant, the fundamental form. In the case n=5 this differential invariant has the same algebraic nature, as the covariant binary biquadratic form, constructed by E.Cartan in 1910, using his ``reduction- prolongation\\\'\\\' procedure (we call this form Cartan\\\'s tensor). In the present paper we prove that our fundamental form coincides (up to constant factor -35) with Cartan\\\'s tensor. This result explains geometric reason for existence of Cartan\\\'s tensor (originally this tensor was obtained by very sophisticated algebraic manipulations) and gives the true analogs of this tensor in Riemannian geometry. In addition, as a part of the proof, we obtain a new useful formula for Cartan\\\'s tensor in terms of structural functions of any frame naturally adapted to the distribution.1 aZelenko, Igor uhttp://hdl.handle.net/1963/218700716nas a2200121 4500008004100000245009400041210006900135260001300204520029900217100002000516700002200536856003600558 2005 en d00aA fourth order uniformization theorem on some four manifolds with large total Q-curvature0 afourth order uniformization theorem on some four manifolds with bElsevier3 aGiven a four-dimensional manifold (M,g), we study the existence of a conformal metric for which the Q-curvature, associated to a conformally invariant fourth-order operator (the Paneitz operator), is constant. Using a topological argument, we obtain a new result in cases which were still open.1 aDjadli, Zindine1 aMalchiodi, Andrea uhttp://hdl.handle.net/1963/486801246nas a2200121 4500008004100000245005100041210005100092260001800143520089000161100001701051700002001068856003601088 2004 en d00aFredholm modules for quantum euclidean spheres0 aFredholm modules for quantum euclidean spheres bSISSA Library3 aThe quantum Euclidean spheres, $S_q^{N-1}$, are (noncommutative) homogeneous spaces of quantum orthogonal groups, $\\\\SO_q(N)$. The *-algebra $A(S^{N-1}_q)$ of polynomial functions on each of these is given by generators and relations which can be expressed in terms of a self-adjoint, unipotent matrix. We explicitly construct complete sets of generators for the K-theory (by nontrivial self-adjoint idempotents and unitaries) and the K-homology (by nontrivial Fredholm modules) of the spheres $S_q^{N-1}$. We also construct the corresponding Chern characters in cyclic homology and cohomology and compute the pairing of K-theory with K-homology. On odd spheres (i. e., for N even) we exhibit unbounded Fredholm modules by means of a natural unbounded operator D which, while failing to have compact resolvent, has bounded commutators with all elements in the algebra $A(S^{N-1}_q)$.1 aHawkins, Eli1 aLandi, Giovanni uhttp://hdl.handle.net/1963/163600803nas a2200109 4500008004100000245008400041210006900125260001800194520042700212100001800639856003600657 2003 en d00aA finite element approximation of the Griffith\\\'s model in fracture mechanics0 afinite element approximation of the Griffiths model in fracture bSISSA Library3 aThe Griffith model for the mechanics of fractures in brittle materials is consider in the weak formulation of SBD spaces. We suggest an approximation, in the sense of Gamma-convergence, by a sequence of discrete functionals defined on finite elements spaces over structured and adaptive triangulations. The quasi-static evolution for boundary value problems is also taken into account and some numerical results are shown.1 aNegri, Matteo uhttp://hdl.handle.net/1963/154800422nas a2200121 4500008004100000022001400041245005900055210005600114300001400170490000800184100001900192856008900211 2003 eng d a0550-321300aFree energy of the two-matrix model/dToda tau-function0 aFree energy of the twomatrix modeldToda taufunction a435–4610 v6691 aBertola, Marco uhttps://www.math.sissa.it/publication/free-energy-two-matrix-modeldtoda-tau-function00795nas a2200121 4500008004300000245006000043210006000103260004800163520038200211100002400593700002000617856003600637 2002 en_Ud 00aFast Arnold diffusion in systems with three time scales0 aFast Arnold diffusion in systems with three time scales bAmerican Institute of Mathematical Sciences3 aWe consider the problem of Arnold Diffusion for nearly integrable partially isochronous Hamiltonian systems with three time scales. By means of a careful shadowing analysis, based on a variational technique, we prove that, along special directions, Arnold diffusion takes place with fast (polynomial) speed, even though the \\\"splitting determinant\\\" is exponentially small.1 aBerti, Massimiliano1 aBolle, Philippe uhttp://hdl.handle.net/1963/305800809nas a2200121 4500008004300000245007700043210006900120260000900189520041400198100001800612700002100630856003600651 2002 en_Ud 00aFlow Stability of Patchy Vector Fields and Robust Feedback Stabilization0 aFlow Stability of Patchy Vector Fields and Robust Feedback Stabi bSIAM3 aThe paper is concerned with patchy vector fields, a class of discontinuous, piecewise smooth vector fields that were introduced in AB to study feedback stabilization problems. We prove the stability of the corresponding solution set w.r.t. a wide class of impulsive perturbations. These results yield the robusteness of patchy feedback controls in the presence of measurement errors and external disturbances.1 aAncona, Fabio1 aBressan, Alberto uhttp://hdl.handle.net/1963/307300768nas a2200109 4500008004300000245007400043210006900117260000900186520040500195100002200600856003600622 2002 en_Ud 00aFollowing a path of varying curvature as an output regulation problem0 aFollowing a path of varying curvature as an output regulation pr bIEEE3 aGiven a path of nonconstant curvature, local asymptotic stability can be proven for the general n trailer whenever the curvature can be considered as the output of an exogenous dynamical system. The controllers that provide convergence to zero of the tracking error chosen for the path-following problem are composed of a prefeedback that input-output linearizes the system, plus a linear controller.1 aAltafini, Claudio uhttp://hdl.handle.net/1963/314300697nas a2200121 4500008004300000245005500043210005300098260001300151520033100164100002400495700002000519856003600539 2002 en_Ud 00aA functional analysis approach to Arnold diffusion0 afunctional analysis approach to Arnold diffusion bElsevier3 aWe discuss in the context of nearly integrable Hamiltonian systems a functional analysis approach to the \\\"splitting of separatrices\\\" and to the \\\"shadowing problem\\\". As an application we apply our method to the problem of Arnold Diffusion for nearly integrable partially isochronous systems improving known results.1 aBerti, Massimiliano1 aBolle, Philippe uhttp://hdl.handle.net/1963/315100385nas a2200109 4500008004100000245006700041210006700108260001800175100002100193700002500214856003600239 2001 en d00aFinite Difference Approximation of Free Discontinuity Problems0 aFinite Difference Approximation of Free Discontinuity Problems bSISSA Library1 aGobbino, Massimo1 aMora, Maria Giovanna uhttp://hdl.handle.net/1963/122800426nas a2200121 4500008004100000245008500041210006900126260001800195100001600213700002200229700001700251856003600268 2001 en d00aA Fourier transform for sheaves on real tori. I. The equivalence Sky(T)~ Loc (T)0 aFourier transform for sheaves on real tori I The equivalence Sky bSISSA Library1 aBruzzo, Ugo1 aMarelli, Giovanni1 aPioli, Fabio uhttp://hdl.handle.net/1963/152600396nas a2200109 4500008004100000245007600041210006900117260001000186653002900196100002500225856003600250 2001 en d00aFree-discontinuity problems: calibration and approximation of solutions0 aFreediscontinuity problems calibration and approximation of solu bSISSA10aCalibration of solutions1 aMorini, Massimiliano uhttp://hdl.handle.net/1963/539800449nas a2200121 4500008004100000022001400041245006800055210006700123300001200190490000700202100001900209856009900228 2000 eng d a0926-224500aFrobenius manifold structure on orbit space of Jacobi groups. I0 aFrobenius manifold structure on orbit space of Jacobi groups I a19–410 v131 aBertola, Marco uhttps://www.math.sissa.it/publication/frobenius-manifold-structure-orbit-space-jacobi-groups-i00454nas a2200121 4500008004100000022001400041245006900055210006800124300001400192490000700206100001900213856010000232 2000 eng d a0926-224500aFrobenius manifold structure on orbit space of Jacobi groups. II0 aFrobenius manifold structure on orbit space of Jacobi groups II a213–2330 v131 aBertola, Marco uhttps://www.math.sissa.it/publication/frobenius-manifold-structure-orbit-space-jacobi-groups-ii00369nas a2200109 4500008004100000245005700041210005700098260001800155100002500173700002500198856003600223 2000 en d00aFunctionals depending on curvatures with constraints0 aFunctionals depending on curvatures with constraints bSISSA Library1 aMora, Maria Giovanna1 aMorini, Massimiliano uhttp://hdl.handle.net/1963/129900734nas a2200121 4500008004300000245004900043210004900092260001300141520038400154100002000538700001800558856003600576 1999 en_Ud 00aFrobenius manifolds and Virasoro constraints0 aFrobenius manifolds and Virasoro constraints bSpringer3 aFor an arbitrary Frobenius manifold a system of Virasoro constraints is constructed. In the semisimple case these constraints are proved to hold true in the genus one approximation. Particularly, the genus $\\\\leq 1$ Virasoro conjecture of T.Eguchi, K.Hori, M.Jinzenji, and C.-S.Xiong and of S.Katz is proved for smooth projective varieties having semisimple quantum cohomology.1 aDubrovin, Boris1 aYoujin, Zhang uhttp://hdl.handle.net/1963/288300967nas a2200121 4500008004300000020001800043245005200061210005200113260002100165520060300186100002000789856003600809 1997 en_Ud a981-02-3266-700aFlat pencils of metrics and Frobenius manifolds0 aFlat pencils of metrics and Frobenius manifolds bWorld Scientific3 aThis paper is based on the author\\\'s talk at 1997 Taniguchi Symposium \\\"Integrable Systems and Algebraic Geometry\\\". We consider an approach to the theory of Frobenius manifolds based on the geometry of flat pencils of contravariant metrics. It is shown that, under certain homogeneity assumptions, these two objects are identical. The flat pencils of contravariant metrics on a manifold $M$ appear naturally in the classification of bihamiltonian structures of hydrodynamics type on the loop space $L(M)$. This elucidates the relations between Frobenius manifolds and integrable hierarchies.1 aDubrovin, Boris uhttp://hdl.handle.net/1963/323700430nas a2200109 4500008004100000020001500041245010400056210007000160260003400230100002000264856003600284 1997 en d a082180666100aFunctionals of the Peierls - Fröhlich Type and the Variational Principle for the Whitham Equations0 aFunctionals of the Peierls Fröhlich Type and the Variational Pri bAmerican Mathematical Society1 aDubrovin, Boris uhttp://hdl.handle.net/1963/648500369nas a2200109 4500008004100000245006100041210006100102260001800163100002000181700002300201856003500224 1985 en d00aFlat connections for Lax hierarchies on coadjoint orbits0 aFlat connections for Lax hierarchies on coadjoint orbits bSISSA Library1 aLandi, Giovanni1 aDe Filippo, Sergio uhttp://hdl.handle.net/1963/460