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 Schr{\"o}dinger equations for the corresponding operators, we outline a programme of relevant questions that deserve being investigated.

}, doi = {10.1063/1.5033856}, url = {https://doi.org/10.1063/1.5033856}, author = {Alessandro Michelangeli and Andrea Ottolini and Raffaele Scandone} } @article {GEORGIEV20181551, title = {On fractional powers of singular perturbations of the Laplacian}, journal = {Journal of Functional Analysis}, volume = {275}, number = {6}, year = {2018}, pages = {1551 - 1602}, abstract = {We 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.

}, keywords = {Point interactions, Regular and singular component of a point-interaction operator, Singular perturbations of the Laplacian}, issn = {0022-1236}, doi = {https://doi.org/10.1016/j.jfa.2018.03.007}, url = {http://www.sciencedirect.com/science/article/pii/S0022123618301046}, author = {Vladimir Georgiev and Alessandro Michelangeli and Raffaele Scandone} } @article {doi:10.1142/S0129167X18500076, title = {Framed symplectic sheaves on surfaces}, journal = {International Journal of Mathematics}, volume = {29}, number = {01}, year = {2018}, pages = {1850007}, abstract = {A 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.

}, doi = {10.1142/S0129167X18500076}, url = {https://doi.org/10.1142/S0129167X18500076}, author = {Jacopo Vittorio Scalise} } @article {SalmoiraghiScardigliTelibRozza2018, title = {Free-form deformation, mesh morphing and reduced-order methods: enablers for efficient aerodynamic shape optimisation}, journal = {International Journal of Computational Fluid Dynamics}, volume = {32}, number = {4-5}, year = {2018}, pages = {233-247}, publisher = {Taylor \& Francis}, abstract = {In 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{\textendash}Stokes equations.

}, doi = {10.1080/10618562.2018.1514115}, author = {Filippo Salmoiraghi and Scardigli, Angela and Telib, Haysam and Gianluigi Rozza} } @article {kozhasov2018fully, title = {On fully real eigenconfigurations of tensors}, journal = {SIAM Journal on Applied Algebra and Geometry}, volume = {2}, number = {2}, year = {2018}, pages = {339{\textendash}347}, publisher = {SIAM}, abstract = {We 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.

}, doi = {10.1137/17M1145902}, url = {https://epubs.siam.org/doi/pdf/10.1137/17M1145902}, author = {Khazhgali Kozhasov} } @article {2017, title = {Friedrichs systems in a Hilbert space framework: solvability and multiplicity}, number = {SISSA;16/2017/MATE}, year = {2017}, abstract = {The 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 Antoni{\'c} 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.}, url = {http://preprints.sissa.it/handle/1963/35280}, author = {Nenad Antoni{\'c} and Marko Erceg and Alessandro Michelangeli} } @article {2016, title = {A fast virtual surgery platform for many scenarios haemodynamics of patient-specific coronary artery bypass grafts}, year = {2016}, institution = {Submitted}, abstract = {A 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.}, url = {http://urania.sissa.it/xmlui/handle/1963/35240}, author = {Francesco Ballarin and Elena Faggiano and Andrea Manzoni and Gianluigi Rozza and Alfio Quarteroni and Sonia Ippolito and Roberto Scrofani and Carlo Antona} } @article {DalMaso2016, title = {Fracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: the antiplane case}, journal = {Calculus of Variations and Partial Differential Equations}, volume = {55}, number = {3}, year = {2016}, month = {Apr}, pages = {45}, abstract = {We 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.

}, issn = {1432-0835}, doi = {10.1007/s00526-016-0981-z}, url = {https://doi.org/10.1007/s00526-016-0981-z}, author = {Gianni Dal Maso and Gianluca Orlando and Rodica Toader} } @mastersthesis {2016, title = {Frames symplectic sheaves on surfaces and their ADHM data}, year = {2016}, school = {SISSA}, abstract = {This 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.}, keywords = {moduli spaces}, author = {Jacopo Vittorio Scalise} } @article {doi:10.1142/S0129167X16500610, title = {A Frobenius theorem for corank-1 continuous distributions in dimensions two and three}, journal = {International Journal of Mathematics}, volume = {27}, number = {08}, year = {2016}, pages = {1650061}, abstract = {We 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.

}, doi = {10.1142/S0129167X16500610}, url = {https://doi.org/10.1142/S0129167X16500610}, author = {Stefano Luzzatto and Sina T{\"u}reli and Khadim Mbacke War} } @article {2015, title = {Fast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a POD-Galerkin method and a vascular shape parametrization}, year = {2015}, abstract = {In 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{\textendash}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.}, url = {http://urania.sissa.it/xmlui/handle/1963/34623}, author = {Francesco Ballarin and Elena Faggiano and Sonia Ippolito and Andrea Manzoni and Alfio Quarteroni and Gianluigi Rozza and Roberto Scrofani} } @article {2015, title = {FEM SUPG stabilisation of mixed isoparametric BEMs: application to linearised free surface flows}, journal = {Engineering Analysis with Boundary Elements 59 (2015), pp. 8-22}, year = {2015}, abstract = {In finite element formulations, transport dominated problems are often stabilised through the Streamline-Upwind-Petrov{\textendash}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.

}, doi = {10.1016/j.enganabound.2015.04.006}, url = {http://urania.sissa.it/xmlui/handle/1963/34466}, author = {Nicola Giuliani and Andrea Mola and Luca Heltai and L. Formaggia} } @article {2014, title = {Finite dimensional Kadomtsev-Petviashvili τ-functions. I. Finite Grassmannians}, number = {Journal of Mathematical Physics;volume 55; issue 8; article number 1.4890818;}, year = {2014}, publisher = {American Institute of Physics Inc.}, abstract = {We 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 Pl{\"u}cker coordinates appearing as coefficients in the Schur function expansion of the τ-function.}, doi = {10.1063/1.4890818}, url = {http://urania.sissa.it/xmlui/handle/1963/34952}, author = {Ferenc Balogh and Tiago Fonseca and John P. Harnad} } @proceedings {2014, title = {A fully nonlinear potential model for ship hydrodynamics directly interfaced with CAD data structures}, year = {2014}, publisher = {SISSA}, abstract = {We 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.}, keywords = {ship hydrodynamics}, author = {Andrea Mola and Luca Heltai and Antonio DeSimone} } @inbook {2014, title = {Fundamentals of Reduced Basis Method for problems governed by parametrized PDEs and applications}, booktitle = {Separated representations and PGD-based model reduction : fundamentals and applications}, series = { CISM International Centre for Mechanical Sciences}, volume = {554}, year = {2014}, publisher = {Springer}, organization = {Springer}, chapter = {4}, address = {Wien}, abstract = {In 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.

}, keywords = {reduced basis method, linear elasticity, heat transfer, error bounds, parametrized PDEs}, doi = {10.1007/978-3-7091-1794-1_4}, author = {Gianluigi Rozza} } @article {2013, title = {Fields of bounded deformation for mesoscopic dislocations}, year = {2013}, publisher = {SISSA}, abstract = {In 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.}, url = {http://hdl.handle.net/1963/6378}, author = {Nicolas Van Goethem} } @article {2013, title = {Fracture models as Gamma-limits of damage models}, journal = {Communications on Pure and Applied Analysis 12 (2013) 1657-1686}, number = {SISSA;25/2011/M}, year = {2013}, publisher = {American Institute of Mathematical Sciences}, abstract = {We 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.

}, doi = {10.3934/cpaa.2013.12.1657}, url = {http://hdl.handle.net/1963/4225}, author = {Gianni Dal Maso and Flaviana Iurlano} } @article {2013, title = {Framed sheaves on projective stacks}, year = {2013}, abstract = {Given 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 {\textquoteleft}{\textquoteleft}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 $A_k$.}, url = {http://urania.sissa.it/xmlui/handle/1963/7438}, author = {Ugo Bruzzo and Francesco Sala} } @article {KoshakjiQuarteroniRozza2013, title = {Free Form Deformation Techniques Applied to 3D Shape Optimization Problems}, journal = {Communications in Applied and Industrial Mathematics}, year = {2013}, abstract = {The 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.}, doi = {10.1685/journal.caim.452}, author = {Anwar Koshakji and Alfio Quarteroni and Gianluigi Rozza} } @article {2012, title = {A formula for Popp\'s volume in sub-Riemannian geometry}, journal = {Analysis and Geometry in Metric Spaces, vol. 1 (2012), pages : 42-57}, number = {arXiv:1211.2325;}, year = {2012}, note = {16 pages, minor revisions}, publisher = {SISSA}, abstract = {For 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.}, keywords = {subriemannian, volume, Popp, control}, doi = {10.2478/agms-2012-0004}, url = {http://hdl.handle.net/1963/6501}, author = {Luca Rizzi and Davide Barilari} } @article {BertolaCafasso3, title = {Fredholm determinants and pole-free solutions to the noncommutative Painlev{\'e} II equation}, journal = {Comm. Math. Phys.}, volume = {309}, number = {3}, year = {2012}, pages = {793{\textendash}833}, issn = {0010-3616}, doi = {10.1007/s00220-011-1383-x}, url = {http://0-dx.doi.org.mercury.concordia.ca/10.1007/s00220-011-1383-x}, author = {Marco Bertola and Mattia Cafasso} } @article {2012, title = {Frobenius manifold for the dispersionless Kadomtsev-Petviashvili equation}, journal = {Communications in Mathematical Physics 311 (2012) 557-594}, number = {arXiv:1008.2128;}, year = {2012}, publisher = {Springer}, abstract = {We 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.}, doi = {10.1007/s00220-012-1470-7}, url = {http://hdl.handle.net/1963/6040}, author = {Andrea Raimondo} } @article {2012, title = {A Fully Coupled Immersed Finite Element Method for Fluid Structure Interaction via the Deal.II Library}, number = {arXiv:1209.2811;}, year = {2012}, note = {28 pages, 9 figures}, institution = {SISSA}, abstract = {We 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.}, keywords = {Finite Element Method, Immersed Boundary Method, Immersed Finite Element Method}, url = {http://hdl.handle.net/1963/6255}, author = {Luca Heltai and Saswati Roy and Francesco Costanzo} } @article {10226, title = {Fracture and plastic models as Gamma-limits of damage models under different regimes}, journal = {Advances in Calculus of Variations., to appear.}, number = {SISSA;59/2011/M}, year = {2011}, publisher = {Walter de Gruyter}, abstract = {We 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.

}, doi = {10.1515/acv-2011-0011}, url = {http://hdl.handle.net/1963/5069}, author = {Flaviana Iurlano} } @article {2010, title = {Feedback schemes for radiation damping suppression in NMR: a control-theoretical perspective}, journal = {Systems and Control Letters, 59 (12):782-786, 2010}, year = {2010}, publisher = {Elsevier}, abstract = {In 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.}, doi = {10.1016/j.sysconle.2010.09.004}, url = {http://hdl.handle.net/1963/4384}, author = {Claudio Altafini and Paola Cappellaro and David Cory} } @article {Bertola:wj, title = {First colonization of a hard-edge in random matrix theory}, journal = {Constr. Approx.}, volume = {31}, number = {2}, year = {2010}, pages = {231{\textendash}257}, issn = {0176-4276}, doi = {10.1007/s00365-009-9052-4}, url = {http://0-dx.doi.org.mercury.concordia.ca/10.1007/s00365-009-9052-4}, author = {Marco Bertola and Lee, S. Y.} } @article {2009, title = {Families of Monads and Instantons from a Noncommutative ADHM Construction}, number = {SISSA;59/2008/FM}, year = {2009}, abstract = {We 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 {\textquoteleft}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.}, url = {http://hdl.handle.net/1963/3478}, author = {Simon Brain and Giovanni Landi} } @article {Bertola:Lee, title = {First colonization of a spectral outpost in random matrix theory}, journal = {Constr. Approx.}, volume = {30}, number = {2}, year = {2009}, pages = {225{\textendash}263}, issn = {0176-4276}, doi = {10.1007/s00365-008-9026-y}, url = {http://0-dx.doi.org.mercury.concordia.ca/10.1007/s00365-008-9026-y}, author = {Marco Bertola and Lee, S. Y.} } @article {fall2009foliations, title = {Foliations of small tubes in Riemannian manifolds by capillary minimal discs}, journal = {Nonlinear Analysis: Theory, Methods \& Applications}, volume = {70}, number = {12}, year = {2009}, pages = {4422{\textendash}4440}, publisher = {Elsevier}, abstract = {Letting\ \ 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.

}, doi = {10.1016/j.na.2008.10.024}, url = {https://doi.org/10.1016/j.na.2008.10.024}, author = {Fall, Mouhamed Moustapha and Mercuri, Carlo} } @article {Formaggia2008, title = {Fluid{\textendash}structure interaction problems in free surface flows: Application to boat dynamics}, journal = {International Journal for Numerical Methods in Fluids}, volume = {56}, number = {8}, year = {2008}, pages = {965{\textendash}978}, publisher = {Wiley}, doi = {10.1002/fld.1583}, url = {https://doi.org/10.1002/fld.1583}, author = {L. Formaggia and Edie Miglio and Andrea Mola and N Parolini} } @article {2008, title = {Forced Vibrations of a Nonhomogeneous String}, journal = {SIAM J. Math. Anal. 40 (2008) 382-412}, number = {SISSA;36/2006/M}, year = {2008}, abstract = {We 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.}, doi = {10.1137/060665038}, url = {http://hdl.handle.net/1963/2643}, author = {P Baldi and Massimiliano Berti} } @article {2008, title = {Frobenius Manifolds and Central Invariants for the Drinfeld - Sokolov Bihamiltonian Structures}, journal = {Adv. Math. 219 (2008) 780-837}, number = {arXiv.org;0710.3115}, year = {2008}, abstract = {The 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.}, doi = {10.1016/j.aim.2008.06.009}, url = {http://hdl.handle.net/1963/2523}, author = {Boris Dubrovin and Liu Si-Qi and Zhang Youjin} } @article {2008, title = {Fulde-Ferrell-Larkin-Ovchinnikov pairing in one-dimensional optical lattices}, journal = {Phys. Rev. B 77 (2008) 245105}, number = {arXiv.org;0712.3364v1}, year = {2008}, abstract = {Spin-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.

}, doi = {10.1103/PhysRevB.77.245105}, url = {http://hdl.handle.net/1963/2694}, author = {Matteo Rizzi and Marco Polini and Miguel A. Cazalilla and M.R. Bakhtiari and Mario P. Tosi and Rosario Fazio} } @article {2007, title = {On feedback classification of control-affine systems with one and two-dimensional inputs}, journal = {SIAM J. Control Optim. 46 (2007) 1431-1460}, number = {arXiv.org;math/050231v2}, year = {2007}, abstract = {The 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.}, doi = {10.1137/050623711}, url = {http://hdl.handle.net/1963/2186}, author = {Andrei A. Agrachev and Igor Zelenko} } @article {2007, title = {Feedback control of spin systems}, number = {SISSA;01/2006/M}, year = {2007}, note = {Proc. of the 44th IEEE Conf. on Decision and Control,. Seville, Spain, December 2005 (to appear).}, abstract = {The 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.}, doi = {10.1007/s11128-006-0038-x}, url = {http://hdl.handle.net/1963/1808}, author = {Claudio Altafini} } @article {2007, title = {Feedback stabilization of quantum ensembles: a global convergence analysis on complex flag manifolds}, journal = {45th IEEE Conference on Decision and Control (2007) 2471-2476}, number = {SISSA;65/2005/M}, year = {2007}, abstract = {In 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.}, doi = {10.1109/CDC.2006.377774}, url = {http://hdl.handle.net/1963/1729}, author = {Claudio Altafini} } @article {2007, title = {On finite-dimensional projections of distributions for solutions of randomly forced PDE\\\'s}, journal = {Ann. Inst. Henri Poincare-Prob. Stat. 43 (2007) 399-415}, number = {arXiv.org;math/0603295}, year = {2007}, abstract = {The 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.}, doi = {10.1016/j.anihpb.2006.06.001}, url = {http://hdl.handle.net/1963/2012}, author = {Andrei A. Agrachev and Sergei Kuksin and Andrey Sarychev and Armen Shirikyan} } @article {2006, title = {Forced vibrations of wave equations with non-monotone nonlinearities}, journal = {Ann. Inst. H. Poincar{\'e} Anal. Non Lin{\'e}aire 23 (2006) 439-474}, number = {arXiv.org;math/0410619v1}, year = {2006}, abstract = {We 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.}, doi = {10.1016/j.anihpc.2005.05.004}, url = {http://hdl.handle.net/1963/2160}, author = {Massimiliano Berti and Luca Biasco} } @article {2006, title = {Fundamental form and Cartan tensor of (2,5)-distributions coincide}, journal = {J. Dyn. Control Syst. 12 (2006) 247-276}, number = {arXiv.org;math/0402195v2}, year = {2006}, abstract = {In 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 {\textquoteleft}{\textquoteleft}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.}, doi = {10.1007/s10450-006-0383-1}, url = {http://hdl.handle.net/1963/2187}, author = {Igor Zelenko} } @article {2005, title = {A fourth order uniformization theorem on some four manifolds with large total Q-curvature}, journal = {C. R. Acad. Sci. Paris, Ser. I 340 (2005) 341-346.}, year = {2005}, publisher = {Elsevier}, abstract = {Given 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.}, doi = {10.1016/j.crma.2005.01.013}, url = {http://hdl.handle.net/1963/4868}, author = {Zindine Djadli and Andrea Malchiodi} } @article {2004, title = {Fredholm modules for quantum euclidean spheres}, journal = {J. Geom. Phys. 49 (2004) 272-293}, number = {SISSA;66/2002/FM}, year = {2004}, publisher = {SISSA Library}, abstract = {The 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 $A(S^{N-1}_q)$.}, doi = {10.1016/S0393-0440(03)00092-5}, url = {http://hdl.handle.net/1963/1636}, author = {Eli Hawkins and Giovanni Landi} } @article {2003, title = {A finite element approximation of the Griffith\\\'s model in fracture mechanics}, journal = {Numer. Math., 2003, 95, 653}, number = {SISSA;91/00/M}, year = {2003}, publisher = {SISSA Library}, abstract = {The 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.}, doi = {10.1007/s00211-003-0456-y}, url = {http://hdl.handle.net/1963/1548}, author = {Matteo Negri} } @article {Bertola:FreeEnergyNPB, title = {Free energy of the two-matrix model/dToda tau-function}, journal = {Nuclear Phys. B}, volume = {669}, number = {3}, year = {2003}, pages = {435{\textendash}461}, issn = {0550-3213}, author = {Marco Bertola} } @article {2002, title = {Fast Arnold diffusion in systems with three time scales}, journal = {Discrete Contin. Dyn. Syst. 8 (2002) 795-811}, number = {SISSA;21/2001/M}, year = {2002}, publisher = {American Institute of Mathematical Sciences}, abstract = {We 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.}, url = {http://hdl.handle.net/1963/3058}, author = {Massimiliano Berti and Philippe Bolle} } @article {2002, title = {Flow Stability of Patchy Vector Fields and Robust Feedback Stabilization}, journal = {SIAM J. Control Optim. 41 (2002) 1455-1476}, number = {SISSA;71/2001/M}, year = {2002}, publisher = {SIAM}, abstract = {The 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.}, doi = {10.1137/S0363012901391676}, url = {http://hdl.handle.net/1963/3073}, author = {Fabio Ancona and Alberto Bressan} } @article {2002, title = {Following a path of varying curvature as an output regulation problem}, journal = {IEEE Trans. Automatic Control 47 (2002) 1551-1556}, year = {2002}, publisher = {IEEE}, abstract = {Given 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.}, doi = {10.1109/TAC.2002.802750}, url = {http://hdl.handle.net/1963/3143}, author = {Claudio Altafini} } @article {2002, title = {A functional analysis approach to Arnold diffusion}, journal = {Ann. Inst. H. Poincare Anal. Non Lineaire 19 (2002) 395-450}, year = {2002}, publisher = {Elsevier}, abstract = {We 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.}, doi = {10.1016/S0294-1449(01)00084-1}, url = {http://hdl.handle.net/1963/3151}, author = {Massimiliano Berti and Philippe Bolle} } @article {2001, title = {Finite Difference Approximation of Free Discontinuity Problems}, journal = {Proc. Royal Soc. Edinb. Ser. A 131 (2001), no. 3, 567-595}, number = {SISSA;14/99/M}, year = {2001}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1228}, author = {Massimo Gobbino and Maria Giovanna Mora} } @article {2001, title = {A Fourier transform for sheaves on real tori. I. The equivalence Sky(T)~ Loc (T)}, journal = {J. Geom. Phys. 39 (2001), no. 2, 174--182}, number = {SISSA;68/00/FM}, year = {2001}, publisher = {SISSA Library}, doi = {10.1016/S0393-0440(01)00009-2}, url = {http://hdl.handle.net/1963/1526}, author = {Ugo Bruzzo and Giovanni Marelli and Fabio Pioli} } @mastersthesis {2001, title = {Free-discontinuity problems: calibration and approximation of solutions}, year = {2001}, school = {SISSA}, keywords = {Calibration of solutions}, url = {http://hdl.handle.net/1963/5398}, author = {Massimiliano Morini} } @article {Bertola:Frobenius1, title = {Frobenius manifold structure on orbit space of Jacobi groups. I}, journal = {Differential Geom. Appl.}, volume = {13}, number = {1}, year = {2000}, pages = {19{\textendash}41}, issn = {0926-2245}, author = {Marco Bertola} } @article {Bertola:Frobenius2, title = {Frobenius manifold structure on orbit space of Jacobi groups. II}, journal = {Differential Geom. Appl.}, volume = {13}, number = {3}, year = {2000}, pages = {213{\textendash}233}, issn = {0926-2245}, author = {Marco Bertola} } @article {2000, title = {Functionals depending on curvatures with constraints}, journal = {Rend. Sem. Mat. Univ. Padova 104 (2000), 173--199}, number = {SISSA;85/99/M}, year = {2000}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1299}, author = {Maria Giovanna Mora and Massimiliano Morini} } @article {1999, title = {Frobenius manifolds and Virasoro constraints}, journal = {Selecta Math. (N.S.) 5 (1999) 423-466}, number = {SISSA;84/98/FM}, year = {1999}, publisher = {Springer}, abstract = {For 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.}, doi = {10.1007/s000290050053}, url = {http://hdl.handle.net/1963/2883}, author = {Boris Dubrovin and Zhang Youjin} } @inbook {1997, title = {Flat pencils of metrics and Frobenius manifolds}, booktitle = {Integrable systems and algebraic geometry : proceedings of the Taniguchi symposium 1997, Kobe, June 30 - July 4, 1997 and Research Institute for Mathematical Sciences, Kyoto University, July 7 - 11, 1997 / eds. M.-H. Saito, Y. Shimizu and K. Ueno. - Sing}, number = {arXiv.org;math/9803106v1}, year = {1997}, publisher = {World Scientific}, organization = {World Scientific}, abstract = {This 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.}, isbn = {981-02-3266-7}, url = {http://hdl.handle.net/1963/3237}, author = {Boris Dubrovin} } @inbook {1997, title = {Functionals of the Peierls - Fr{\"o}hlich Type and the Variational Principle for the Whitham Equations}, booktitle = {Solitons, geometry, and topology : on the crossroad / V. M. Buchstaber, S. P. Novikov editors.- Providence : American Mathematical Society, 1997. - ( American mathematical society translations. Series 2. - vol. 179). - pages : 35-44}, number = {American mathematical society translations. Series 2.;v.179}, year = {1997}, publisher = {American Mathematical Society}, organization = {American Mathematical Society}, isbn = {0821806661}, url = {http://hdl.handle.net/1963/6485}, author = {Boris Dubrovin} } @article {1985, title = {Flat connections for Lax hierarchies on coadjoint orbits}, journal = {Phys. Lett. A 108 (1985), no. 7, 311-314}, number = {SISSA;73/85/MP}, year = {1985}, publisher = {SISSA Library}, doi = {10.1016/0375-9601(85)90102-1}, url = {http://hdl.handle.net/1963/460}, author = {Giovanni Landi and Sergio De Filippo} }