In this paper we prove that, within the framework of $\textsf{RCD}^\star(K,N)$ spaces with $N<\infty$, the entropic cost (i.e. the minimal value of the Schrödinger problem) admits:A threefold dynamical variational representation, in the spirit of the Benamou–Brenier formula for the Wasserstein distance; A Hamilton–Jacobi–Bellman dual representation, in line with Bobkov–Gentil–Ledoux and Otto–Villani results on the duality between Hamilton–Jacobi and continuity equation for optimal transport;A Kantorovich-type duality formula, where the Hopf–Lax semigroup is replaced by a suitable `entropic' counterpart.We thus provide a complete and unifying picture of the equivalent variational representations of the Schrödinger problem as well as a perfect parallelism with the analogous formulas for the Wasserstein distance. Riemannian manifolds with Ricci curvature bounded from below are a relevant class of $\textsf{RCD}^*(K,N)$ spaces and our results are new even in this setting.

1 aGigli, Nicola1 aTamanini, Luca uhttps://doi.org/10.1007/s00440-019-00909-102204nas a2200229 4500008004100000022001400041245009200055210006900147300001400216490000800230520153700238653000801775653001801783653000801801653001501809653000801824653002501832653001701857653000801874100002101882856007101903 2019 eng d a0010-465500aBlackNUFFT: Modular customizable black box hybrid parallelization of type 3 NUFFT in 3D0 aBlackNUFFT Modular customizable black box hybrid parallelization a324 - 3350 v2353 aMany applications benefit from an efficient Discrete Fourier Transform (DFT) between arbitrarily spaced points. The Non Uniform Fast Fourier Transform reduces the computational cost of such operation from O(N2) to O(NlogN) exploiting gridding algorithms and a standard Fast Fourier Transform on an equi-spaced grid. The parallelization of the NUFFT of type 3 (between arbitrary points in space and frequency) still poses some challenges: we present a novel and flexible hybrid parallelization in a MPI-multithreaded environment exploiting existing HPC libraries on modern architectures. To ensure the reliability of the developed library, we exploit continuous integration strategies using Travis CI. We present performance analyses to prove the effectiveness of our implementation, possible extensions to the existing library, and an application of NUFFT type 3 to MRI image processing. Program summary Program Title: BlackNUFFT Program Files doi: http://dx.doi.org/10.17632/vxfj6x2p8x.1 Licensing provisions: LGPL Programming language: C++ External routines/libraries: deal.II , FFTW, PFFT Nature of problem: Provide a modular and extensible implementation of a parallel Non Uniform Fast Fourier Transform of type 3. Solution method: Use of hybrid shared distributed memory paradigm to achieve high level of efficiency. We exploit existing HPC library following best practices in scientific computing (as continuous integration via TravisCI) to reach higher complexities and guarantee the accuracy of the solution proposed.

10aC++10aExtensibility10aFFT10aModularity10aMPI10aMRI image processing10aNUFFT type 310aTBB1 aGiuliani, Nicola uhttp://www.sciencedirect.com/science/article/pii/S001046551830353900784nas a2200157 4500008004100000022001400041245006600055210006600121520024700187653002800434653002100462653003000483100001800513700002400531856007100555 2019 eng d a0723-086900aDifferential structure associated to axiomatic Sobolev spaces0 aDifferential structure associated to axiomatic Sobolev spaces3 aThe aim of this note is to explain in which sense an axiomatic Sobolev space over a general metric measure space (à la Gol’dshtein–Troyanov) induces – under suitable locality assumptions – a first-order differential structure.

10aAxiomatic Sobolev space10aCotangent module10aLocality of differentials1 aGigli, Nicola1 aPasqualetto, Enrico uhttp://www.sciencedirect.com/science/article/pii/S072308691830097502567nas a2200169 4500008004100000245009100041210006900132520193100201100001702132700001902149700002102168700002502189700001902214700002102233700002102254856012202275 2019 eng d00aEfficient Reduction in Shape Parameter Space Dimension for Ship Propeller Blade Design0 aEfficient Reduction in Shape Parameter Space Dimension for Ship 3 aIn this work, we present the results of a ship propeller design optimization campaign carried out in the framework of the research project PRELICA, funded by the Friuli Venezia Giulia regional government. The main idea of this work is to operate on a multidisciplinary level to identify propeller shapes that lead to reduced tip vortex-induced pressure and increased efficiency without altering the thrust. First, a specific tool for the bottom-up construction of parameterized propeller blade geometries has been developed. The algorithm proposed operates with a user defined number of arbitrary shaped or NACA airfoil sections, and employs arbitrary degree NURBS to represent the chord, pitch, skew and rake distribution as a function of the blade radial coordinate. The control points of such curves have been modified to generate, in a fully automated way, a family of blade geometries depending on as many as 20 shape parameters. Such geometries have then been used to carry out potential flow simulations with the Boundary Element Method based software PROCAL. Given the high number of parameters considered, such a preliminary stage allowed for a fast evaluation of the performance of several hundreds of shapes. In addition, the data obtained from the potential flow simulation allowed for the application of a parameter space reduction methodology based on active subspaces (AS) property, which suggested that the main propeller performance indices are, at a first but rather accurate approximation, only depending on a single parameter which is a linear combination of all the original geometric ones. AS analysis has also been used to carry out a constrained optimization exploiting response surface method in the reduced parameter space, and a sensitivity analysis based on such surrogate model. The few selected shapes were finally used to set up high fidelity RANS simulations and select an optimal shape.

1 aMola, Andrea1 aTezzele, Marco1 aGadalla, Mahmoud1 aValdenazzi, Federica1 aGrassi, Davide1 aPadovan, Roberta1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/efficient-reduction-shape-parameter-space-dimension-ship-propeller-blade-design01140nas a2200205 4500008004100000022001400041245005800055210005500113520049400168653002800662653002300690653002100713653002500734653002500759100001700784700002400801700001900825700001900844856007100863 2019 eng d a0304-414900aAn entropic interpolation proof of the HWI inequality0 aentropic interpolation proof of the HWI inequality3 aThe HWI inequality is an “interpolation”inequality between the Entropy H, the Fisher information I and the Wasserstein distance W. We present a pathwise proof of the HWI inequality which is obtained through a zero noise limit of the Schrödinger problem. Our approach consists in making rigorous the Otto–Villani heuristics in Otto and Villani (2000) taking advantage of the entropic interpolations, which are regular both in space and time, rather than the displacement ones.

10aEntropic interpolations10aFisher information10aRelative entropy10aSchrödinger problem10aWasserstein distance1 aGentil, Ivan1 aLéonard, Christian1 aRipani, Luigia1 aTamanini, Luca uhttp://www.sciencedirect.com/science/article/pii/S030441491830345402145nas a2200157 4500008004100000245008600041210006900127260003000196300001500226490000800241520163000249100002001879700002001899700002101919856004701940 2019 eng d00aIsomonodromy deformations at an irregular singularity with coalescing eigenvalues0 aIsomonodromy deformations at an irregular singularity with coale bDuke University Pressc04 a967–11080 v1683 aWe consider an n×n linear system of ODEs with an irregular singularity of Poincar\'e rank 1 at z=∞, holomorphically depending on parameter t within a polydisc in Cn centred at t=0. The eigenvalues of the leading matrix at z=∞ coalesce along a locus Δ contained in the polydisc, passing through t=0. Namely, z=∞ is a resonant irregular singularity for t∈Δ. We analyse the case when the leading matrix remains diagonalisable at Δ. We discuss the existence of fundamental matrix solutions, their asymptotics, Stokes phenomenon and monodromy data as t varies in the polydisc, and their limits for t tending to points of Δ. When the deformation is isomonodromic away from Δ, it is well known that a fundamental matrix solution has singularities at Δ. When the system also has a Fuchsian singularity at z=0, we show under minimal vanishing conditions on the residue matrix at z=0 that isomonodromic deformations can be extended to the whole polydisc, including Δ, in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisc. These data can be computed just by considering the system at fixed t=0. Conversely, if the t-dependent system is isomonodromic in a small domain contained in the polydisc not intersecting Δ, if the entries of the Stokes matrices with indices corresponding to coalescing eigenvalues vanish, then we show that Δ is not a branching locus for the fundamental matrix solutions. The importance of these results for the analytic theory of Frobenius Manifolds is explained. An application to Painlev\'e equations is discussed.

1 aCotti, Giordano1 aDubrovin, Boris1 aGuzzetti, Davide uhttps://doi.org/10.1215/00127094-2018-005900644nas a2200145 4500008004100000245006000041210005800101260003400159300001400193490000700207520015800214100001800372700001900390856008900409 2019 eng d00aA Note About the Strong Maximum Principle on RCD Spaces0 aNote About the Strong Maximum Principle on RCD Spaces bCanadian Mathematical Society a259–2660 v623 aWe give a direct proof of the strong maximum principle on finite dimensional RCD spaces based on the Laplacian comparison of the squared distance.

1 aGigli, Nicola1 aRigoni, Chiara uhttps://www.math.sissa.it/publication/note-about-strong-maximum-principle-rcd-spaces00394nas a2200109 4500008004100000245004900041210004800090100002000138700001800158700002400176856008400200 2019 eng d00aQuasi-continuous vector fields on RCD spaces0 aQuasicontinuous vector fields on RCD spaces1 aDebin, Clément1 aGigli, Nicola1 aPasqualetto, Enrico uhttps://www.math.sissa.it/publication/quasi-continuous-vector-fields-rcd-spaces01637nas a2200217 4500008004100000022001400041245007700055210006900132300001400201490000800215520097100223653002001194653001501214653001701229653001701246100001901263700002201282700002301304700002101327856007101348 2019 eng d a0022-123600aReducibility of first order linear operators on tori via Moser's theorem0 aReducibility of first order linear operators on tori via Mosers a932 - 9700 v2763 aIn this paper we prove reducibility of a class of first order, quasi-linear, quasi-periodic time dependent PDEs on the torus∂tu+ζ⋅∂xu+a(ωt,x)⋅∂xu=0,x∈Td,ζ∈Rd,ω∈Rν. As a consequence we deduce a stability result on the associated Cauchy problem in Sobolev spaces. By the identification between first order operators and vector fields this problem can be formulated as the problem of finding a change of coordinates which conjugates a weakly perturbed constant vector field on Tν+d to a constant diophantine flow. For this purpose we generalize Moser's straightening theorem: considering smooth perturbations we prove that the corresponding straightening torus diffeomorphism is smooth, under the assumption that the perturbation is small only in some given Sobolev norm and that the initial frequency belongs to some Cantor-like set. In view of applications in KAM theory for PDEs we provide also tame estimates on the change of variables.

10aHyperbolic PDEs10aKAM theory10aNash–Moser10aReducibility1 aFeola, Roberto1 aGiuliani, Filippo1 aMontalto, Riccardo1 aProcesi, Michela uhttp://www.sciencedirect.com/science/article/pii/S002212361830379300556nas a2200121 4500008004100000245010600041210006900147260002000216100002200236700002400258700002300282856012900305 2018 eng d00aA Comparison Between Active Strain and Active Stress in Transversely Isotropic Hyperelastic Materials0 aComparison Between Active Strain and Active Stress in Transverse bSpringer Nature1 aGiantesio, Giulia1 aMusesti, Alessandro1 aRiccobelli, Davide uhttps://www.math.sissa.it/publication/comparison-between-active-strain-and-active-stress-transversely-isotropic-hyperelastic00540nas a2200145 4500008004100000245007700041210006900118300001400187490000600201100002100207700002100228700002200249700001700271856010600288 2018 eng d00adeal2lkit: A toolkit library for high performance programming in deal.II0 adeal2lkit A toolkit library for high performance programming in a318–3270 v71 aSartori, Alberto1 aGiuliani, Nicola1 aBardelloni, Mauro1 aHeltai, Luca uhttps://www.math.sissa.it/publication/deal2lkit-toolkit-library-high-performance-programming-dealii-000702nas a2200253 4500008004100000245003700041210003000078100002200108700001800130700001700148700001800165700002100183700001900204700002200223700001800245700001700263700002300280700002400303700002000327700002400347700001700371700001700388856004300405 2018 eng d00aThe deal.II Library, Version 9.00 adealII Library Version 901 aAlzetta, Giovanni1 aArndt, Daniel1 aBangerth, W.1 aBoddu, Vishal1 aBrands, Benjamin1 aDavydov, Denis1 aGassmöller, Rene1 aHeister, Timo1 aHeltai, Luca1 aKormann, Katharina1 aKronbichler, Martin1 aMaier, Matthias1 aPelteret, Jean-Paul1 aTurcksin, B.1 aWells, David uhttps://doi.org/10.1515/jnma-2018-005400395nas a2200109 4500008004100000245004700041210004700088100001800135700002400153700002600177856008200203 2018 eng d00aDifferential of metric valued Sobolev maps0 aDifferential of metric valued Sobolev maps1 aGigli, Nicola1 aPasqualetto, Enrico1 aSoultanis, Elefterios uhttps://www.math.sissa.it/publication/differential-metric-valued-sobolev-maps00547nas a2200145 4500008004100000245013400041210006900175260004400244300001100288490000700299100001900306700002000325700001700345856003900362 2018 eng d00aA distributed lagrange formulation of the finite element immersed boundary method for fluids interacting with compressible solids0 adistributed lagrange formulation of the finite element immersed aChambSpringer International Publishing a1–210 v161 aBoffi, Daniele1 aGastaldi, Lucia1 aHeltai, Luca uhttps://arxiv.org/abs/1712.02545v101226nas 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/S002212361830104600800nas a2200121 4500008004100000245006300041210005900104520039700163100002000560700002900580700002100609856004800630 2018 en d00aOn Geometric Quantum Confinement in Grushin-Like Manifolds0 aGeometric Quantum Confinement in GrushinLike Manifolds3 aWe study the problem of so-called geometric quantum confinement in a class of two-dimensional incomplete Riemannian manifold with metric of Grushin type. We employ a constant-fibre direct integral scheme, in combination with Weyl's analysis in each fibre, thus fully characterising the regimes of presence and absence of essential self-adjointness of the associated Laplace-Beltrami operator.1 aGallone, Matteo1 aMichelangeli, Alessandro1 aPozzoli, Eugenio uhttp://preprints.sissa.it/handle/1963/3532201340nas a2200109 4500008004100000245005100041210005100092520099000143100002001133700002901153856004801182 2018 en d00aHydrogenoid Spectra with Central Perturbations0 aHydrogenoid Spectra with Central Perturbations3 aThrough the Kreĭn-Višik-Birman extension scheme, unlike the previous classical analysis based on von Neumann's theory, we reproduce the construction and classification of all self-adjoint realisations of two intimately related models: the three-dimensional hydrogenoid-like Hamiltonians with singular perturbation supported at the centre (the nucleus), and the Schördinger operators on the halfline with Coulomb potentials centred at the origin. These two problems are technically equivalent, albeit sometimes treated by their own in the the literature. Based on such scheme, we then recover the formula to determine the eigenvalues of each self-adjoint extension, which are corrections to the non-relativistic hydrogenoid energy levels.We discuss in which respect the Kreĭn-Višik-Birman scheme is somehow more natural in yielding the typical boundary condition of self-adjointness at the centre of the perturbation and in identifying the eigenvalues of each extension.1 aGallone, Matteo1 aMichelangeli, Alessandro uhttp://preprints.sissa.it/handle/1963/3532101258nas a2200133 4500008004100000245006300041210006300104260001000167520083800177100002001015700002001035700002101055856004801076 2018 en d00aLocal moduli of semisimple Frobenius coalescent structures0 aLocal moduli of semisimple Frobenius coalescent structures bSISSA3 aThere is a conjectural relation, formulated by the second author, between the enumerative geometry of a wide class of smooth projective varieties and their derived category of coherent sheaves. In particular, there is an increasing interest for an explicit description of certain local invariants, called monodromy data, of semisimple quantum cohomologies in terms of characteristic classes of exceptional collections in the derived categories. Being intentioned to address this problem, which, to our opinion, is still not well understood, we have realized that some issues in the theory of Frobenius manifolds need to be preliminarily clarified, and that an extension of the theory itself is necessary, in view of the fact that quantum cohomologies of certain classes of homogeneous spaces may show a coalescence phenomenon.

1 aCotti, Giordano1 aDubrovin, Boris1 aGuzzetti, Davide uhttp://preprints.sissa.it/handle/1963/3530400583nas a2200145 4500008004100000245008400041210006900125653003700194100002100231700001300252700002100265700002100286700001900307856011100326 2018 eng d00aA Localized Reduced-Order Modeling Approach for PDEs with Bifurcating Solutions0 aLocalized ReducedOrder Modeling Approach for PDEs with Bifurcati10aMathematics - Numerical Analysis1 aHess, Martin, W.1 aAlla, A.1 aQuaini, Annalisa1 aRozza, Gianluigi1 aGunzburger, M. uhttps://www.math.sissa.it/publication/localized-reduced-order-modeling-approach-pdes-bifurcating-solutions01777nas a2200157 4500008004100000245013300041210006900174260003000243520120300273100001901476700001701495700002101512700001701533700002101550856004801571 2018 eng d00aModel Order Reduction by means of Active Subspaces and Dynamic Mode Decomposition for Parametric Hull Shape Design Hydrodynamics0 aModel Order Reduction by means of Active Subspaces and Dynamic M aTrieste, ItalybIOS Press3 aWe present the results of the application of a parameter space reduction methodology based on active subspaces (AS) to the hull hydrodynamic design problem. Several parametric deformations of an initial hull shape are considered to assess the influence of the shape parameters on the hull wave resistance. Such problem is relevant at the preliminary stages of the ship design, when several flow simulations are carried out by the engineers to establish a certain sensibility with respect to the parameters, which might result in a high number of time consuming hydrodynamic simulations. The main idea of this work is to employ the AS to identify possible lower dimensional structures in the parameter space. The complete pipeline involves the use of free form deformation to parametrize and deform the hull shape, the full order solver based on unsteady potential flow theory with fully nonlinear free surface treatment directly interfaced with CAD, the use of dynamic mode decomposition to reconstruct the final steady state given only few snapshots of the simulation, and the reduction of the parameter space by AS, and shared subspace. Response surface method is used to minimize the total drag.1 aTezzele, Marco1 aDemo, Nicola1 aGadalla, Mahmoud1 aMola, Andrea1 aRozza, Gianluigi uhttp://ebooks.iospress.nl/publication/4927000361nas a2200097 4500008004100000245005400041210004700095100001800142700002400160856007900184 2018 eng d00aOn the notion of parallel transport on RCD spaces0 anotion of parallel transport on RCD spaces1 aGigli, Nicola1 aPasqualetto, Enrico uhttps://www.math.sissa.it/publication/notion-parallel-transport-rcd-spaces01177nas a2200145 4500008004100000245008600041210006900127300001300196490000800209520068400217100001800901700002100919700002100940856007000961 2018 eng d00aNumerical study of the Kadomtsev-Petviashvili equation and dispersive shock waves0 aNumerical study of the KadomtsevPetviashvili equation and disper a201704580 v4743 aA detailed numerical study of the long time behaviour of dispersive shock waves in solutions to the Kadomtsev–Petviashvili (KP) I equation is presented. It is shown that modulated lump solutions emerge from the dispersive shock waves. For the description of dispersive shock waves, Whitham modulation equations for KP are obtained. It is shown that the modulation equations near the soliton line are hyperbolic for the KPII equation while they are elliptic for the KPI equation leading to a focusing effect and the formation of lumps. Such a behaviour is similar to the appearance of breathers for the focusing nonlinear Schrödinger equation in the semiclassical limit.

1 aGrava, Tamara1 aKlein, Christian1 aPitton, Giuseppe uhttps://royalsocietypublishing.org/doi/abs/10.1098/rspa.2017.045801288nas a2200145 4500008004100000245008600041210007000127260004400197490000700241520070800248100001900956700003200975700001801007856011701025 2018 eng d00aPainlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane0 aPainlevé IV Critical Asymptotics for Orthogonal Polynomials in t bNational Academy of Sciences of Ukraine0 v143 aWe study the asymptotic behaviour of orthogonal polynomials in the complex plane that are associated to a certain normal matrix model. The model depends on a parameter and the asymptotic distribution of the eigenvalues undergoes a transition for a special value of the parameter, where it develops a corner-type singularity. In the double scaling limit near the transition we determine the asymptotic behaviour of the orthogonal polynomials in terms of a solution of the Painlev´e IV equation. We determine the Fredholm determinant associated to such solution and we compute it numerically on the real line, showing also that the corresponding Painlev´e transcendent is pole-free on a semiaxis.

1 aBertola, Marco1 aElias Rebelo, José Gustavo1 aGrava, Tamara uhttps://www.math.sissa.it/publication/painlev%C3%A9-iv-critical-asymptotics-orthogonal-polynomials-complex-plane00509nas a2200133 4500008004100000245012700041210006900168300001400237490000600251100002100257700001700278700002200295856005800317 2018 eng d00aPredicting and Optimizing Microswimmer Performance from the Hydrodynamics of Its Components: The Relevance of Interactions0 aPredicting and Optimizing Microswimmer Performance from the Hydr a410–4240 v51 aGiuliani, Nicola1 aHeltai, Luca1 aDeSimone, Antonio uhttps://www.ncbi.nlm.nih.gov/pmc/articles/PMC6094362/00849nas a2200157 4500008004100000022001400041245009800055210006900153260000800222300000800230490000700238520036300245100001800608700001900626856004600645 2018 eng d a1432-083500aRecognizing the flat torus among RCD*(0,N) spaces via the study of the first cohomology group0 aRecognizing the flat torus among RCD0N spaces via the study of t cJun a1040 v573 aWe prove that if the dimension of the first cohomology group of a $\mathsf{RCD}^\star (0,N)$ space is $N$, then the space is a flat torus. This generalizes a classical result due to Bochner to the non-smooth setting and also provides a first example where the study of the cohomology groups in such synthetic framework leads to geometric consequences.

1 aGigli, Nicola1 aRigoni, Chiara uhttps://doi.org/10.1007/s00526-018-1377-z00517nas a2200109 4500008004100000245011200041210006900153100001900222700002200241700002100263856012300284 2018 eng d00aReducibility for a class of weakly dispersive linear operators arising from the Degasperis Procesi equation0 aReducibility for a class of weakly dispersive linear operators a1 aFeola, Roberto1 aGiuliani, Filippo1 aProcesi, Michela uhttps://www.math.sissa.it/publication/reducibility-class-weakly-dispersive-linear-operators-arising-degasperis-procesi00433nas a2200121 4500008004100000245006100041210005800102300001400160490000700174100001800181700001900199856009300218 2018 eng d00aSecond order differentiation formula on RCD(K, N) spaces0 aSecond order differentiation formula on RCDK N spaces a377–3860 v291 aGigli, Nicola1 aTamanini, Luca uhttps://www.math.sissa.it/publication/second-order-differentiation-formula-rcdk-n-spaces00386nas a2200097 4500008004100000245006100041210005700102100001800159700001900177856009200196 2018 eng d00aSecond order differentiation formula on RCD*(K,N) spaces0 aSecond order differentiation formula on RCDKN spaces1 aGigli, Nicola1 aTamanini, Luca uhttps://www.math.sissa.it/publication/second-order-differentiation-formula-rcdkn-spaces01912nas a2200157 4500008004100000245009800041210006900139260003000208520136800238100001701606700001901623700002101642700002201663700002101685856004801706 2018 eng d00aShape Optimization by means of Proper Orthogonal Decomposition and Dynamic Mode Decomposition0 aShape Optimization by means of Proper Orthogonal Decomposition a aTrieste, ItalybIOS Press3 aShape optimization is a challenging task in many engineering fields, since the numerical solutions of parametric system may be computationally expensive. This work presents a novel optimization procedure based on reduced order modeling, applied to a naval hull design problem. The advantage introduced by this method is that the solution for a specific parameter can be expressed as the combination of few numerical solutions computed at properly chosen parametric points. The reduced model is built using the proper orthogonal decomposition with interpolation (PODI) method. We use the free form deformation (FFD) for an automated perturbation of the shape, and the finite volume method to simulate the multiphase incompressible flow around the deformed hulls. Further computational reduction is done by the dynamic mode decomposition (DMD) technique: from few high dimensional snapshots, the system evolution is reconstructed and the final state of the simulation is faithfully approximated. Finally the global optimization algorithm iterates over the reduced space: the approximated drag and lift coefficients are projected to the hull surface, hence the resistance is evaluated for the new hulls until the convergence to the optimal shape is achieved. We will present the results obtained applying the described procedure to a typical Fincantieri cruise ship.1 aDemo, Nicola1 aTezzele, Marco1 aGustin, Gianluca1 aLavini, Gianpiero1 aRozza, Gianluigi uhttp://ebooks.iospress.nl/publication/4922901093nas a2200145 4500008004100000245009200041210006900133300001300202490000800215520058600223100002100809700002300830700002400853856007000877 2018 eng d00aSymplectic invariants for parabolic orbits and cusp singularities of integrable systems0 aSymplectic invariants for parabolic orbits and cusp singularitie a201704240 v3763 aWe discuss normal forms and symplectic invariants of parabolic orbits and cuspidal tori in integrable Hamiltonian systems with two degrees of freedom. Such singularities appear in many integrable systems in geometry and mathematical physics and can be considered as the simplest example of degenerate singularities. We also suggest some new techniques which apparently can be used for studying symplectic invariants of degenerate singularities of more general type. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

1 aBolsinov, Alexey1 aGuglielmi, Lorenzo1 aKudryavtseva, Elena uhttps://royalsocietypublishing.org/doi/abs/10.1098/rsta.2017.042400574nas a2200133 4500008004100000245012000041210007000161300001200231490000800243100002100251700001700272700001700289856013400306 2018 eng d00aπ-BEM : A flexible parallel implementation for adaptive , geometry aware , and high order boundary element methods0 aπBEM A flexible parallel implementation for adaptive geometry aw a39–580 v1211 aGiuliani, Nicola1 aMola, Andrea1 aHeltai, Luca uhttps://www.math.sissa.it/publication/%CF%80-bem-flexible-parallel-implementation-adaptive-geometry-aware-and-high-order-boundary01295nas a2200133 4500008004100000245006800041210006800109300001200177490000700189520087800196100002001074700002101094856004601115 2017 eng d00aAnalytic geometry of semisimple coalescent Frobenius structures0 aAnalytic geometry of semisimple coalescent Frobenius structures a17400040 v063 aWe present some results of a joint paper with Dubrovin (see references), as exposed at the Workshop “Asymptotic and Computational Aspects of Complex Differential Equations” at the CRM in Pisa, in February 2017. The analytical description of semisimple Frobenius manifolds is extended at semisimple coalescence points, namely points with some coalescing canonical coordinates although the corresponding Frobenius algebra is semisimple. After summarizing and revisiting the theory of the monodromy local invariants of semisimple Frobenius manifolds, as introduced by Dubrovin, it is shown how the definition of monodromy data can be extended also at semisimple coalescence points. Furthermore, a local Isomonodromy theorem at semisimple coalescence points is presented. Some examples of computation are taken from the quantum cohomologies of complex Grassmannians.

1 aCotti, Giordano1 aGuzzetti, Davide uhttps://doi.org/10.1142/S201032631740004400875nas a2200193 4500008004100000022001400041245006900055210006600124300001600190490000800206520024700214653002900461653002400490653002300514653003300537100002200570700001800592856007100610 2017 eng d a0022-039600aAn avoiding cones condition for the Poincaré–Birkhoff Theorem0 aavoiding cones condition for the Poincaré–Birkhoff Theorem a1064 - 10840 v2623 aWe provide a geometric assumption which unifies and generalizes the conditions proposed in [11], [12], so to obtain a higher dimensional version of the Poincaré–Birkhoff fixed point Theorem for Poincaré maps of Hamiltonian systems.

10aAvoiding cones condition10aHamiltonian systems10aPeriodic solutions10aPoincaré–Birkhoff theorem1 aFonda, Alessandro1 aGidoni, Paolo uhttp://www.sciencedirect.com/science/article/pii/S002203961630327801132nas a2200109 4500008004100000245006100041210006000102520076300162100002000925700002900945856004800974 2017 en d00aDiscrete spectra for critical Dirac-Coulomb Hamiltonians0 aDiscrete spectra for critical DiracCoulomb Hamiltonians3 aThe one-particle Dirac Hamiltonian with Coulomb interaction is known to be realised, in a regime of large (critical) couplings, by an infinite multiplicity of distinct self-adjoint operators, including a distinguished physically most natural one. For the latter, Sommerfeld’s celebrated fine structure formula provides the well-known expression for the eigenvalues in the gap of the continuum spectrum. Exploiting our recent general classification of all other self-adjoint realisations, we generalise Sommerfeld’s formula so as to determine the discrete spectrum of all other self-adjoint versions of the Dirac-Coulomb Hamiltonian. Such discrete spectra display naturally a fibred structure, whose bundle covers the whole gap of the continuum spectrum.1 aGallone, Matteo1 aMichelangeli, Alessandro uhttp://preprints.sissa.it/handle/1963/3530001320nas a2200133 4500008004100000245008300041210006900124300001400193490000700207520089100214100001801105700002201123856004101145 2017 eng d00aOn the genesis of directional friction through bristle-like mediating elements0 agenesis of directional friction through bristlelike mediating el a1023-10460 v233 aWe propose an explanation of the genesis of directional dry friction, as emergent property of the oscillations produced in a bristle-like mediating element by the interaction with microscale fluctuations on the surface. Mathematically, we extend a convergence result by Mielke, for Prandtl–Tomlinson-like systems, considering also non-homothetic scalings of a wiggly potential. This allows us to apply the result to some simple mechanical models, that exemplify the interaction of a bristle with a surface having small fluctuations. We find that the resulting friction is the product of two factors: a geometric one, depending on the bristle angle and on the fluctuation profile, and a energetic one, proportional to the normal force exchanged between the bristle-like element and the surface. Finally, we apply our result to discuss the with the nap/against the nap asymmetry.

1 aGidoni, Paolo1 aDeSimone, Antonio uhttps://doi.org/10.1051/cocv/201703000713nas a2200157 4500008004100000245004400041210004000085520026500125653001200390653001000402653004000412100002000452700002400472700001800496856004100514 2017 eng d00aThe injectivity radius of Lie manifolds0 ainjectivity radius of Lie manifolds3 aWe prove in a direct, geometric way that for any compatible Riemannian metric on a Lie manifold the injectivity radius is positive

10a(58J40)10a53C2110aMathematics - Differential Geometry1 aAntonini, Paolo1 aDe Philippis, Guido1 aGigli, Nicola uhttps://arxiv.org/pdf/1707.07595.pdf00742nas a2200121 4500008004100000245006300041210006000104520033800164100002000502700002900522700002100551856004800572 2017 en d00aKrein-Visik-Birman self-adjoint extension theory revisited0 aKreinVisikBirman selfadjoint extension theory revisited3 aThe core results of the so-called KreIn-Visik-Birman theory of self-adjoint extensions of semi-bounded symmetric operators are reproduced, both in their original and in a more modern formulation, within a comprehensive discussion that includes missing details, elucidative steps, and intermediate results of independent interest.1 aGallone, Matteo1 aMichelangeli, Alessandro1 aOttolini, Andrea uhttp://preprints.sissa.it/handle/1963/3528601350nas a2200193 4500008004100000022001400041245007200055210006900127300001600196490000800212520074100220653001800961653000800979653002400987653002301011653002901034100002201063856007101085 2017 eng d a0022-039600aQuasi-periodic solutions for quasi-linear generalized KdV equations0 aQuasiperiodic solutions for quasilinear generalized KdV equation a5052 - 51320 v2623 aWe prove the existence of Cantor families of small amplitude, linearly stable, quasi-periodic solutions of quasi-linear autonomous Hamiltonian generalized KdV equations. We consider the most general quasi-linear quadratic nonlinearity. The proof is based on an iterative Nash–Moser algorithm. To initialize this scheme, we need to perform a bifurcation analysis taking into account the strongly perturbative effects of the nonlinearity near the origin. In particular, we implement a weak version of the Birkhoff normal form method. The inversion of the linearized operators at each step of the iteration is achieved by pseudo-differential techniques, linear Birkhoff normal form algorithms and a linear KAM reducibility scheme.

10aKAM for PDE's10aKdV10aNash–Moser theory10aQuasi-linear PDE's10aQuasi-periodic solutions1 aGiuliani, Filippo uhttp://www.sciencedirect.com/science/article/pii/S002203961730048700410nas a2200097 4500008004100000245006900041210006500110100001800175700001900193856010000212 2017 eng d00aSecond order differentiation formula on compact RCD*(K,N) spaces0 aSecond order differentiation formula on compact RCDKN spaces1 aGigli, Nicola1 aTamanini, Luca uhttps://www.math.sissa.it/publication/second-order-differentiation-formula-compact-rcdkn-spaces00941nas a2200109 4500008004100000245006900041210006800110260001000178520057200188100002000760856005100780 2017 en d00aSelf-Adjoint Extensions of Dirac Operator with Coulomb Potential0 aSelfAdjoint Extensions of Dirac Operator with Coulomb Potential bSISSA3 aIn this note we give a concise review of the present state-of-art for the problem of self-adjoint realisations for the Dirac operator with a Coulomb-like singular scalar potential V(x) = Ø(x)I4. We try to follow the historical and conceptual path that leads to the present understanding of the problem and to highlight the techniques employed and the main ideas. In the final part we outline a few major open questions that concern the topical problem of the multiplicity of self-adjoint realisations of the model, and which are worth addressing in the future.1 aGallone, Matteo uhttp://urania.sissa.it/xmlui/handle/1963/3527301120nas a2200109 4500008004100000245008000041210006900121520072300190100002000913700002900933856004800962 2017 en d00aSelf-adjoint realisations of the Dirac-Coulomb Hamiltonian for heavy nuclei0 aSelfadjoint realisations of the DiracCoulomb Hamiltonian for hea3 aWe derive a classification of the self-adjoint extensions of the three-dimensional Dirac-Coulomb operator in the critical regime of the Coulomb coupling. Our approach is solely based upon the KreĬn-Višik- Birman extension scheme, or also on Grubb's universal classification theory, as opposite to previous works within the standard von Neu- mann framework. This let the boundary condition of self-adjointness emerge, neatly and intrinsically, as a multiplicative constraint between regular and singular part of the functions in the domain of the exten- sion, the multiplicative constant giving also immediate information on the invertibility property and on the resolvent and spectral gap of the extension.1 aGallone, Matteo1 aMichelangeli, Alessandro uhttp://preprints.sissa.it/handle/1963/3528701538nas a2200157 4500008004100000022001400041245009300055210006900148260000800217300001400225490000700239520104800246100001801294700002201312856004601334 2017 eng d a1572-964800aStasis domains and slip surfaces in the locomotion of a bio-inspired two-segment crawler0 aStasis domains and slip surfaces in the locomotion of a bioinspi cFeb a587–6010 v523 aWe formulate and solve the locomotion problem for a bio-inspired crawler consisting of two active elastic segments (i.e., capable of changing their rest lengths), resting on three supports providing directional frictional interactions. The problem consists in finding the motion produced by a given, slow actuation history. By focusing on the tensions in the elastic segments, we show that the evolution laws for the system are entirely analogous to the flow rules of elasto-plasticity. In particular, sliding of the supports and hence motion cannot occur when the tensions are in the interior of certain convex regions (stasis domains), while support sliding (and hence motion) can only take place when the tensions are on the boundary of such regions (slip surfaces). We solve the locomotion problem explicitly in a few interesting examples. In particular, we show that, for a suitable range of the friction parameters, specific choices of the actuation strategy can lead to net displacements also in the direction of higher friction.

1 aGidoni, Paolo1 aDeSimone, Antonio uhttps://doi.org/10.1007/s11012-016-0408-000795nas a2200241 4500008004100000245011200041210006900153260003500222300001100257490000800268100001800276700001800294700001600312700002200328700001900350700002300369700002200392700002200414700001800436700001800454700002100472856006000493 2017 eng d00aUniversality of the Peregrine Soliton in the Focusing Dynamics of the Cubic Nonlinear Schrödinger Equation0 aUniversality of the Peregrine Soliton in the Focusing Dynamics o bAmerican Physical SocietycJul a0339010 v1191 aTikan, Alexey1 aBillet, Cyril1 aEl, Gennady1 aTovbis, Alexander1 aBertola, Marco1 aSylvestre, Thibaut1 aGustave, Francois1 aRandoux, Stephane1 aGenty, Goëry1 aSuret, Pierre1 aDudley, John, M. uhttps://link.aps.org/doi/10.1103/PhysRevLett.119.03390100407nas a2200097 4500008004100000245006600041210006600107100001800173700002400191856009400215 2016 eng d00aBehaviour of the reference measure on RCD spaces under charts0 aBehaviour of the reference measure on RCD spaces under charts1 aGigli, Nicola1 aPasqualetto, Enrico uhttps://www.math.sissa.it/publication/behaviour-reference-measure-rcd-spaces-under-charts00471nas a2200097 4500008004100000245009600041210006900137100001800206700002400224856012500248 2016 eng d00aEquivalence of two different notions of tangent bundle on rectifiable metric measure spaces0 aEquivalence of two different notions of tangent bundle on rectif1 aGigli, Nicola1 aPasqualetto, Enrico uhttps://www.math.sissa.it/publication/equivalence-two-different-notions-tangent-bundle-rectifiable-metric-measure-spaces00945nas a2200157 4500008004100000022001400041245007900055210007200134260000800206300001600214490000800230520046300238100002200701700001800723856004600741 2016 eng d a1618-189100aGeneralizing the Poincaré–Miranda theorem: the avoiding cones condition0 aGeneralizing the Poincaré–Miranda theorem the avoiding cones con cAug a1347–13710 v1953 aAfter proposing a variant of the Poincaré–Bohl theorem, we extend the Poincaré–Miranda theorem in several directions, by introducing an avoiding cones condition. We are thus able to deal with functions defined on various types of convex domains, and situations where the topological degree may be different from \$\$\backslashpm \$\$±1. An illustrative application is provided for the study of functionals having degenerate multi-saddle points.

1 aFonda, Alessandro1 aGidoni, Paolo uhttps://doi.org/10.1007/s10231-015-0519-600883nas a2200157 4500008004100000245005000041210005000091260001500141300001400156490000600170520040100176100002200577700002300599700001800622856008500640 2016 eng d00aPeriodic perturbations of Hamiltonian systems0 aPeriodic perturbations of Hamiltonian systems bDe Gruyter a367–3820 v53 aWe prove existence and multiplicity results for periodic solutions of Hamiltonian systems, by the use of a higher dimensional version of the Poincaré–Birkhoff fixed point theorem. The first part of the paper deals with periodic perturbations of a completely integrable system, while in the second part we focus on some suitable global conditions, so to deal with weakly coupled systems.

1 aFonda, Alessandro1 aGarrione, Maurizio1 aGidoni, Paolo uhttps://www.math.sissa.it/publication/periodic-perturbations-hamiltonian-systems00912nas a2200229 4500008004100000020002200041245004000063210004000103260004400143300001100187520024800198100002100446700002400467700002000491700001800511700002000529700002200549700001900571700002000590700002400610856004800634 2016 eng d a978-3-319-29116-100aPimsner Algebras and Circle Bundles0 aPimsner Algebras and Circle Bundles aChambSpringer International Publishing a1–253 aWe report on the connections between noncommutative principal circle bundles, Pimsner algebras and strongly graded algebras. We illustrate several results with examples of quantum weighted projective and lens spaces and θ-deformations.

1 aArici, Francesca1 aD'Andrea, Francesco1 aLandi, Giovanni1 aAlpay, Daniel1 aCipriani, Fabio1 aColombo, Fabrizio1 aGuido, Daniele1 aSabadini, Irene1 aSauvageot, Jean-Luc uhttps://doi.org/10.1007/978-3-319-29116-1_101201nas a2200145 4500008004100000245005000041210005000091260003400141300001400175490000700189520075600196100002200952700003100974856005001005 2016 eng d00aRefined node polynomials via long edge graphs0 aRefined node polynomials via long edge graphs bInternational Press of Boston a193–2340 v103 aThe generating functions of the Severi degrees for sufficiently ample line bundles on algebraic surfaces are multiplicative in the topological invariants of the surface and the line bundle. Recently new proofs of this fact were given for toric surfaces by Block, Colley, Kennedy and Liu, Osserman, using tropical geometry and in particular the combinatorial tool of long-edged graphs. In the first part of this paper these results are for $\mathbb{P}^2$ and rational ruled surfaces generalised to refined Severi degrees. In the second part of the paper we give a number of mostly conjectural generalisations of this result to singular surfaces, and curves with prescribed multiple points. The formulas involve modular forms and theta functions.

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

1 aBianchini, Stefano1 aBonicatto, Paolo1 aGusev, N.A. uhttps://doi.org/10.1137/15M100738000745nas a2200121 4500008004100000245005600041210005600097260001000153520032000163653003100483100001800514856009100532 2016 en d00aTwo explorations in Dynamical Systems and Mechanics0 aTwo explorations in Dynamical Systems and Mechanics bSISSA3 aThis thesis contains the work done by Paolo Gidoni during the doctorate programme in Matematical Analysis at SISSA, under the supervision of A. Fonda and A. DeSimone. The thesis is composed of two parts: "Avoiding cones conditions and higher dimensional twist" and "Directional friction in bio-inspired locomotion".10aPoincaré-Birkhoff Theorem1 aGidoni, Paolo uhttps://www.math.sissa.it/publication/two-explorations-dynamical-systems-and-mechanics01640nas a2200145 4500008004100000245007700041210006900118260001000187520116500197100002101362700002101383700002201404700001701426856005101443 2015 en d00aDeal2lkit: a Toolkit Library for High Performance Programming in deal.II0 aDeal2lkit a Toolkit Library for High Performance Programming in bSISSA3 aWe present version 1.0.0 of the deal2lkit (deal.II ToolKit) library. deal2lkit is a collection of modules and classes for the general purpose finite element library deal.II. Its principal aim is to provide a high level interface, controlled via parameter files, for those steps that are common in all finite element programs: mesh generation, selection of the finite element type, application of boundary conditions and many others. Each module can be used as a building block independently on the others, and can be integrated in existing finite element codes based on deal.II, drastically reducing the size of programs, rendering their use automatically parametrised, and reducing the overall time-to-market of finite element programming. Moreover, deal2lkit features interfaces with the SUNDIALS (SUite of Nonlinear and DIfferential/ALgebraic equation Solvers) and ASSIMP (Open Asset Import Library) libraries. Some examples are provided which show the aim and scopes of deal2lkit. The deal2lkit library is released under the GNU Lesser General Public License (LGPL) and can be retrieved from the deal2lkit repository https://github.com/mathLab/deal2lkit.1 aSartori, Alberto1 aGiuliani, Nicola1 aBardelloni, Mauro1 aHeltai, Luca uhttp://urania.sissa.it/xmlui/handle/1963/3500600480nas a2200133 4500008004100000022001400041245011900055210006900174300001500243490000700258100001900265700002200284856004000306 2015 eng d a0022-248800aA degeneration of two-phase solutions of the focusing nonlinear Schrödinger equation via Riemann-Hilbert problems0 adegeneration of twophase solutions of the focusing nonlinear Sch a061507, 170 v561 aBertola, Marco1 aGiavedoni, Pietro uhttp://dx.doi.org/10.1063/1.492236201899nas 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/1963/3446601188nas a2200157 4500008004100000245005800041210005700099260003700156300001600193490000700209520065100216100002500867700002400892700002100916856009300937 2015 eng d00aGeodesics and horizontal-path spaces in Carnot groups0 aGeodesics and horizontalpath spaces in Carnot groups bMathematical Sciences Publishers a1569–16300 v193 aWe study properties of the space of horizontal paths joining the origin with a vertical point on a generic two-step Carnot group. The energy is a Morse-Bott functional on paths and its critical points (sub-Riemannian geodesics) appear in families (compact critical manifolds) with controlled topology. We study the asymptotic of the number of critical manifolds as the energy grows. The topology of the horizontal-path space is also investigated, and we find asymptotic results for the total Betti number of the sublevels of the energy as it goes to infinity. We interpret these results as local invariants of the sub-Riemannian structure.

1 aAgrachev, Andrei, A.1 aGentile, Alessandro1 aLerario, Antonio uhttps://www.math.sissa.it/publication/geodesics-and-horizontal-path-spaces-carnot-groups01765nas a2200217 4500008004100000022001400041245005300055210005300108300001400161490000700175520110000182653002201282653002501304653002801329653003001357653002701387100002201414700001801436700002201454856007101476 2015 eng d a0022-509600aLiquid crystal elastomer strips as soft crawlers0 aLiquid crystal elastomer strips as soft crawlers a254 - 2720 v843 aIn this paper, we speculate on a possible application of Liquid Crystal Elastomers to the field of soft robotics. In particular, we study a concept for limbless locomotion that is amenable to miniaturisation. For this purpose, we formulate and solve the evolution equations for a strip of nematic elastomer, subject to directional frictional interactions with a flat solid substrate, and cyclically actuated by a spatially uniform, time-periodic stimulus (e.g., temperature change). The presence of frictional forces that are sensitive to the direction of sliding transforms reciprocal, ‘breathing-like’ deformations into directed forward motion. We derive formulas quantifying this motion in the case of distributed friction, by solving a differential inclusion for the displacement field. The simpler case of concentrated frictional interactions at the two ends of the strip is also solved, in order to provide a benchmark to compare the continuously distributed case with a finite-dimensional benchmark. We also provide explicit formulas for the axial force along the crawler body.

10aCrawling motility10aDirectional surfaces10aFrictional interactions10aLiquid crystal elastomers10aSoft biomimetic robots1 aDeSimone, Antonio1 aGidoni, Paolo1 aNoselli, Giovanni uhttp://www.sciencedirect.com/science/article/pii/S002250961530043000827nas a2200193 4500008004100000022001400041245005300055210005100108300001200159490000800171520026300179653002100442653001500463653002000478653002400498100002200522700001800544856007100562 2015 eng d a0362-546X00aA permanence theorem for local dynamical systems0 apermanence theorem for local dynamical systems a73 - 810 v1213 aWe provide a necessary and sufficient condition for permanence related to a local dynamical system on a suitable topological space. We then present an illustrative application to a Lotka–Volterra predator–prey model with intraspecific competition.

10aLotka–Volterra10apermanence10aPredator–prey10aUniform persistence1 aFonda, Alessandro1 aGidoni, Paolo uhttp://www.sciencedirect.com/science/article/pii/S0362546X1400333200836nas a2200121 4500008004100000245009000041210007300131260001300204520040500217100001600622700002500638856005100663 2014 en d00aApproximate Hermitian–Yang–Mills structures on semistable principal Higgs bundles0 aApproximate Hermitian–Yang–Mills structures on semistable princi bSpringer3 aWe generalize the Hitchin-Kobayashi correspondence between semistability and the existence of approximate Hermitian-Yang-Mills structures to the case of principal Higgs bundles. We prove that a principal Higgs bundle on a compact Kaehler manifold, with structure group a connected linear algebraic reductive group, is semistable if and only if it admits an approximate Hermitian-Yang-Mills structure.1 aBruzzo, Ugo1 aGrana-Otero, Beatriz uhttp://urania.sissa.it/xmlui/handle/1963/3464500469nas a2200145 4500008004100000022001400041245007300055210006900128300001400197490000800211100001900219700001700238700002000255856004800275 2014 eng d a0010-361600aCauchy-Laguerre two-matrix model and the Meijer-G random point field0 aCauchyLaguerre twomatrix model and the MeijerG random point fiel a111–1440 v3261 aBertola, Marco1 aGekhtman, M.1 aSzmigielski, J. uhttp://dx.doi.org/10.1007/s00220-013-1833-801580nas a2200157 4500008004100000245009300041210006900134260002800203520103900231100001901270700001901289700002101308700002001329700002201349856005101371 2014 en d00aConformal invariants from nodal sets. I. negative eigenvalues and curvature prescription0 aConformal invariants from nodal sets I negative eigenvalues and bOxford University Press3 aIn this paper, we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators; more specifically, the Graham, Jenne, Mason, and Sparling (GJMS) operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a version in conformal geometry of Courant's Nodal Domain Theorem. We also show that on any manifold of dimension n≥3, there exist many metrics for which our invariants are nontrivial. We prove that the Yamabe operator can have an arbitrarily large number of negative eigenvalues on any manifold of dimension n≥3. We obtain similar results for some higher order GJMS operators on some Einstein and Heisenberg manifolds. We describe the invariants arising from the Yamabe and Paneitz operators associated to left-invariant metrics on Heisenberg manifolds. Finally, in Appendix, the second named author and Andrea Malchiodi study the Q-curvature prescription problems for noncritical Q-curvatures.1 aGover, Rod, R.1 aCanzani, Yaiza1 aJakobson, Dmitry1 aPonge, Raphaël1 aMalchiodi, Andrea uhttp://urania.sissa.it/xmlui/handle/1963/3512801733nas a2200217 4500008004100000022001400041245003700055210003700092300001200129490000700141520111900148653002901267653001901296653002201315653002501337653002001362100001801382700002201400700002201422856007101444 2014 eng d a0020-746200aCrawling on directional surfaces0 aCrawling on directional surfaces a65 - 730 v613 aIn this paper we study crawling locomotion based on directional frictional interactions, namely, frictional forces that are sensitive to the sign of the sliding velocity. Surface interactions of this type are common in biology, where they arise from the presence of inclined hairs or scales at the crawler/substrate interface, leading to low resistance when sliding ‘along the grain’, and high resistance when sliding ‘against the grain’. This asymmetry can be exploited for locomotion, in a way analogous to what is done in cross-country skiing (classic style, diagonal stride). We focus on a model system, namely, a continuous one-dimensional crawler and provide a detailed study of the motion resulting from several strategies of shape change. In particular, we provide explicit formulae for the displacements attainable with reciprocal extensions and contractions (breathing), or through the propagation of extension or contraction waves. We believe that our results will prove particularly helpful for the study of biological crawling motility and for the design of bio-mimetic crawling robots.

10aBio-mimetic micro-robots10aCell migration10aCrawling motility10aDirectional surfaces10aSelf-propulsion1 aGidoni, Paolo1 aNoselli, Giovanni1 aDeSimone, Antonio uhttp://www.sciencedirect.com/science/article/pii/S002074621400021301955nas a2200145 4500008004100000245009100041210006900132260006400201520139800265100002701663700002201690700002101712700002501733856005101758 2014 en d00aAn effective model for nematic liquid crystal composites with ferromagnetic inclusions0 aeffective model for nematic liquid crystal composites with ferro bSociety for Industrial and Applied Mathematics Publications3 aMolecules of a nematic liquid crystal respond to an applied magnetic field by reorienting themselves in the direction of the field. Since the dielectric anisotropy of a nematic is small, it takes relatively large fields to elicit a significant liquid crystal response. The interaction may be enhanced in colloidal suspensions of ferromagnetic particles in a liquid crystalline matrix- ferronematics-as proposed by Brochard and de Gennes in 1970. The ability of these particles to align with the field and simultaneously cause reorientation of the nematic molecules greatly increases the magnetic response of the mixture. Essentially the particles provide an easy axis of magnetization that interacts with the liquid crystal via surface anchoring. We derive an expression for the effective energy of ferronematic in the dilute limit, that is, when the number of particles tends to infinity while their total volume fraction tends to zero. The total energy of the mixture is assumed to be the sum of the bulk elastic liquid crystal contribution, the anchoring energy of the liquid crystal on the surfaces of the particles, and the magnetic energy of interaction between the particles and the applied magnetic field. The homogenized limiting ferronematic energy is obtained rigorously using a variational approach. It generalizes formal expressions previously reported in the physical literature.1 aCalderer, Maria, Carme1 aDeSimone, Antonio1 aGolovaty, Dmitry1 aPanchenko, Alexander uhttp://urania.sissa.it/xmlui/handle/1963/3494000545nas a2200109 4500008004100000245004500041210004300086260001300129520022100142100002100363856005100384 2014 en d00aA Review of the Sixth Painlevé Equation0 aReview of the Sixth Painlevé Equation bSpringer3 aFor the Painlevé VI transcendents, we provide a unitary description of the critical behaviours, the connection formulae, their complete tabulation, and the asymptotic distribution of poles close to a critical point.1 aGuzzetti, Davide uhttp://urania.sissa.it/xmlui/handle/1963/3465800993nas a2200121 4500008004100000245006500041210006500106260003000171520058100201100001600782700002200798856005100820 2014 en d00aSpontaneous division and motility in active nematic droplets0 aSpontaneous division and motility in active nematic droplets bAmerican Physical Society3 aWe investigate the mechanics of an active droplet endowed with internal nematic order and surrounded by an isotropic Newtonian fluid. Using numerical simulations we demonstrate that, due to the interplay between the active stresses and the defective geometry of the nematic director, this system exhibits two of the fundamental functions of living cells: spontaneous division and motility, by means of self-generated hydrodynamic flows. These behaviors can be selectively activated by controlling a single physical parameter, namely, an active variant of the capillary number.1 aGiomi, Luca1 aDeSimone, Antonio uhttp://urania.sissa.it/xmlui/handle/1963/3490200470nas a2200109 4500008004100000245008400041210006900125260001000194100002300204700001600227856011700243 2014 en d00aSteady nearly incompressible vector elds in 2D: chain rule and renormalization0 aSteady nearly incompressible vector elds in 2D chain rule and re bSISSA1 aBianchini, Stefano1 aGusev, N.A. uhttps://www.math.sissa.it/publication/steady-nearly-incompressible-vector-elds-2d-chain-rule-and-renormalization01071nas a2200145 4500008004100000245006700041210006600108260001300174520054900187100002200736700002400758700002200782700002000804856010100824 2013 en d00aCrawlers in viscous environments: linear vs nonlinear rheology0 aCrawlers in viscous environments linear vs nonlinear rheology bElsevier3 aWe study model self-propelled crawlers which derive their propulsive capabilities from the tangential resistance to motion offered by the environment. Two types of relationships between tangential forces and slip velocities are considered: a linear, Newtonian one and a nonlinear one of Bingham-type. Different behaviors result from the two different rheologies. These differences and their implications in terms of motility performance are discussed. Our aim is to develop new tools and insight for future studies of cell motility by crawling.1 aDeSimone, Antonio1 aGuarnieri, Federica1 aNoselli, Giovanni1 aTatone, Amabile uhttps://www.math.sissa.it/publication/crawlers-viscous-environments-linear-vs-nonlinear-rheology01219nas a2200145 4500008004100000245008300041210006900124260001000193520068100203100002000884700001800904700002100922700001800943856011200961 2013 en d00aOn critical behaviour in systems of Hamiltonian partial differential equations0 acritical behaviour in systems of Hamiltonian partial differentia bSISSA3 aWe study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlev\'e-I (P$_I$) equation or its fourth order analogue P$_I^2$. As concrete examples we discuss nonlinear Schr\"odinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture.

1 aDubrovin, Boris1 aGrava, Tamara1 aKlein, Christian1 aMoro, Antonio uhttps://www.math.sissa.it/publication/critical-behaviour-systems-hamiltonian-partial-differential-equations01298nas a2200145 4500008004100000245006100041210006100102260001000163520087100173100001601044700002101060700001101081700002401092856003601116 2013 en d00aDefect annihilation and proliferation in active nematics0 aDefect annihilation and proliferation in active nematics bSISSA3 aLiquid crystals inevitably possess topological defect excitations generated\r\nthrough boundary conditions, applied fields or in quenches to the ordered\r\nphase. In equilibrium pairs of defects coarsen and annihilate as the uniform\r\nground state is approached. Here we show that defects in active liquid crystals\r\nexhibit profoundly different behavior, depending on the degree of activity and\r\nits contractile or extensile character. While contractile systems enhance the\r\nannihilation dynamics of passive systems, extensile systems act to drive\r\ndefects apart so that they swarm around in the manner of topologically\r\nwell-characterized self-propelled particles. We develop a simple analytical\r\nmodel for the defect dynamics which reproduces the key features of both the\r\nnumerical solutions and recent experiments on microtuble-kinesin assemblies.1 aGiomi, Luca1 aBowick, Mark, J.1 aMa, Xu1 aMarchetti, Cristina uhttp://hdl.handle.net/1963/656601258nas a2200145 4500008004100000245010200041210006900143260008500212300001400297490000700311520069200318100002201010700002301032856005701055 2013 eng d00aGeneralized Sturm-Liouville boundary conditions for first order differential systems in the plane0 aGeneralized SturmLiouville boundary conditions for first order d bNicolaus Copernicus University, Juliusz P. Schauder Centre for Nonlinear Studies a293–3250 v423 aWe study asymptotically positively homogeneous first order systems in the plane, with boundary conditions which are positively homogeneous, as well. Defining a generalized concept of Fučík spectrum which extends the usual one for the scalar second order equation, we prove existence and multiplicity of solutions. In this way, on one hand we extend to the plane some known results for scalar second order equations (with Dirichlet, Neumann or Sturm-Liouville boundary conditions), while, on the other hand, we investigate some other kinds of boundary value problems, where the boundary points are chosen on a polygonal line, or in a cone. Our proofs rely on the shooting method.

1 aFonda, Alessandro1 aGarrione, Maurizio uhttps://projecteuclid.org:443/euclid.tmna/146124898101360nas a2200181 4500008004100000022001400041245008900055210006900144260000800213300001400221490000700235520080900242100001701051700002301068700002001091700002101111856004601132 2013 eng d a1559-002X00aLipschitz Classification of Almost-Riemannian Distances on Compact Oriented Surfaces0 aLipschitz Classification of AlmostRiemannian Distances on Compac cJan a438–4550 v233 aTwo-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. We consider the Carnot–Carathéodory distance canonically associated with an almost-Riemannian structure and study the problem of Lipschitz equivalence between two such distances on the same compact oriented surface. We analyze the generic case, allowing in particular for the presence of tangency points, i.e., points where two generators of the distribution and their Lie bracket are linearly dependent. The main result of the paper provides a characterization of the Lipschitz equivalence class of an almost-Riemannian distance in terms of a labeled graph associated with it.

1 aBoscain, Ugo1 aCharlot, Grégoire1 aGhezzi, Roberta1 aSigalotti, Mario uhttps://doi.org/10.1007/s12220-011-9262-401133nas a2200157 4500008004100000022001400041245008000055210007300135260000800208300001400216490000700230520064600237100002300883700002300906856004600929 2013 eng d a1420-900400aPlanar Hamiltonian systems at resonance: the Ahmad–Lazer–Paul condition0 aPlanar Hamiltonian systems at resonance the Ahmad–Lazer–Paul con cJun a825–8430 v203 aWe consider the planar Hamiltonian system\$\$Ju^{\backslashprime} = \backslashnabla F(u) + \backslashnabla_u R(t,u), \backslashquad t \backslashin [0,T], \backslash,u \backslashin \backslashmathbb{R}^2,\$\$with F(u) positive and positively 2-homogeneous and \$\${\backslashnabla_{u}R(t, u)}\$\$sublinear in u. By means of an Ahmad-Lazer-Paul type condition, we prove the existence of a T-periodic solution when the system is at resonance. The proof exploits a symplectic change of coordinates which transforms the problem into a perturbation of a linear one. The relationship with the Landesman–Lazer condition is analyzed, as well.

1 aBoscaggin, Alberto1 aGarrione, Maurizio uhttps://doi.org/10.1007/s00030-012-0181-201064nas a2200109 4500008004100000245002900041210002900070260001000099520079300109100001600902856003600918 2013 en d00aSoftly Constrained Films0 aSoftly Constrained Films bSISSA3 aThe shape of materials is often subject to a number of geometric constraints\r\nthat limit the size of the system or fix the structure of its boundary. In soft\r\nand biological materials, however, these constraints are not always hard, but\r\nare due to other physical mechanisms that affect the overall force balance. A\r\ncapillary film spanning a flexible piece of wire or a cell anchored to a\r\ncompliant substrate by mean of adhesive contacts are examples of these softly\r\nconstrained systems in the macroscopic and microscopic world. In this article I\r\nreview some of the important mathematical and physical developments that\r\ncontributed to our understanding of shape formation in softly constrained films\r\nand their recent application to the mechanics of adherent cells.1 aGiomi, Luca uhttp://hdl.handle.net/1963/656300744nas a2200097 4500008004100000245004800041210004300089520044800132100001800580856004800598 2013 en d00aThe splitting theorem in non-smooth context0 asplitting theorem in nonsmooth context3 aWe prove that an infinitesimally Hilbertian $CD(0,N)$ space containing a line splits as the product of $R$ and an infinitesimally Hilbertian $CD(0,N −1)$ space. By ‘infinitesimally Hilbertian’ we mean that the Sobolev space $W^{1,2}(X,d,m)$, which in general is a Banach space, is an Hilbert space. When coupled with a curvature-dimension bound, this condition is known to be stable with respect to measured Gromov-Hausdorff convergence.1 aGigli, Nicola uhttp://preprints.sissa.it/handle/1963/3530600584nas a2200145 4500008004100000022001400041245010700055210006900162300001500231490000700246100001900253700001700272700002000289856012900309 2013 eng d a0022-248800aStrong asymptotics for Cauchy biorthogonal polynomials with application to the Cauchy two-matrix model0 aStrong asymptotics for Cauchy biorthogonal polynomials with appl a043517, 250 v541 aBertola, Marco1 aGekhtman, M.1 aSzmigielski, J. uhttps://www.math.sissa.it/publication/strong-asymptotics-cauchy-biorthogonal-polynomials-application-cauchy-two-matrix-model01015nas a2200133 4500008004100000245004500041210003400086260001000120520062000130100001800750700001900768700002100787856007300808 2013 en d00aOn the tritronquée solutions of P$_I^2$0 atritronquée solutions of PI2 bSISSA3 aFor equation P$_I^2$, the second member in the P$_I$ hierarchy, we prove existence of various degenerate solutions depending on the complex parameter $t$ and evaluate the asymptotics in the complex $x$ plane for $|x|\to\infty$ and $t=o(x^{2/3})$. Using this result, we identify the most degenerate solutions $u^{(m)}(x,t)$, $\hat u^{(m)}(x,t)$, $m=0,\dots,6$, called {\em tritronqu\'ee}, describe the quasi-linear Stokes phenomenon and find the large $n$ asymptotics of the coefficients in a formal expansion of these solutions. We supplement our findings by a numerical study of the tritronqu\'ee solutions.

1 aGrava, Tamara1 aKapaev, Andrey1 aKlein, Christian uhttps://www.math.sissa.it/publication/tritronqu%C3%A9e-solutions-pi200934nas a2200133 4500008004100000245005700041210005100098260005100149520050200200100002100702700001700723700002400740856003600764 2012 en d00aOn 2-step, corank 2 nilpotent sub-Riemannian metrics0 a2step corank 2 nilpotent subRiemannian metrics bSociety for Industrial and Applied Mathematics3 aIn this paper we study the nilpotent 2-step, corank 2 sub-Riemannian metrics\\r\\nthat are nilpotent approximations of general sub-Riemannian metrics. We exhibit optimal syntheses for these problems. It turns out that in general the cut time is not equal to the first conjugate time but has a simple explicit expression. As a byproduct of this study we get some smoothness properties of the spherical Hausdorff measure in the case of a generic 6 dimensional, 2-step corank 2 sub-Riemannian metric.1 aBarilari, Davide1 aBoscain, Ugo1 aGauthier, Jean-Paul uhttp://hdl.handle.net/1963/606501113nas a2200145 4500008004100000245009600041210006900137260001300206300001200219490000700231520057100238100002000809700002400829856011400853 2012 en d00aOn a class of vector fields with discontinuity of divide-by-zero type and its applications0 aclass of vector fields with discontinuity of dividebyzero type a bSpringer a135-1580 v183 aWe study phase portraits and singular points of vector fields of a special type, that is, vector fields whose components are fractions with a common denominator vanishing on a smooth regular hypersurface in the phase space. We assume also some additional conditions, which are fulfilled, for instance, if the vector field is divergence-free. This problem is motivated by a large number of applications. In this paper, we consider three of them in the framework of differential geometry: singularities of geodesic flows in various singular metrics on surfaces.

1 aGhezzi, Roberta1 aRemizov, Alexey, O. uhttps://www.math.sissa.it/publication/class-vector-fields-discontinuity-divide-zero-type-and-its-applications01113nas a2200133 4500008004100000245006400041210005900105260002800164520069000192653002700882100001600909700001800925856003600943 2012 en d00aThe KdV hierarchy: universality and a Painleve transcendent0 aKdV hierarchy universality and a Painleve transcendent bOxford University Press3 aWe study the Cauchy problem for the Korteweg-de Vries (KdV) hierarchy in the small dispersion limit where $\e\to 0$. For negative analytic initial data with a single negative hump, we prove that for small times, the solution is approximated by the solution to the hyperbolic transport equation which corresponds to $\e=0$. Near the time of gradient catastrophe for the transport equation, we show that the solution to the KdV hierarchy is approximated by a particular Painlev\'e transcendent. This supports Dubrovins universality conjecture concerning the critical behavior of Hamiltonian perturbations of hyperbolic equations. We use the Riemann-Hilbert approach to prove our results.10aSmall-Dispersion limit1 aClaeys, Tom1 aGrava, Tamara uhttp://hdl.handle.net/1963/692101662nas a2200121 4500008004100000245009600041210006900137260001300206520124300219100002001462700002201482856003601504 2012 en d00aNon-uniqueness results for critical metrics of regularized determinants in four dimensions0 aNonuniqueness results for critical metrics of regularized determ bSpringer3 aThe regularized determinant of the Paneitz operator arises in quantum gravity (see Connes 1994, IV.4.$\gamma$). An explicit formula for the relative determinant of two conformally related metrics was computed by Branson in Branson (1996). A similar formula holds for Cheeger's half-torsion, which plays a role in self-dual field theory (see Juhl, 2009), and is defined in terms of regularized determinants of the Hodge laplacian on $p$-forms ($p < n/2$). In this article we show that the corresponding actions are unbounded (above and below) on any conformal four-manifold. We also show that the conformal class of the round sphere admits a second solution which is not given by the pull-back of the round metric by a conformal map, thus violating uniqueness up to gauge equivalence. These results differ from the properties of the determinant of the conformal Laplacian established in Chang and Yang (1995), Branson, Chang, and Yang (1992), and Gursky (1997). We also study entire solutions of the Euler-Lagrange equation of $\log \det P$ and the half-torsion $\tau_h$ on $\mathbb{R}^4 \setminus {0}$, and show the existence of two families of periodic solutions. One of these families includes Delaunay-type solutions.1 aGursky, Matthew1 aMalchiodi, Andrea uhttp://hdl.handle.net/1963/655900918nas a2200133 4500008004100000245011000041210006900151260001300220520035800233653003100591100001800622700002100640856012300661 2012 en d00aNumerical study of the small dispersion limit of the Korteweg-de Vries equation and asymptotic solutions0 aNumerical study of the small dispersion limit of the Kortewegde bElsevier3 aWe study numerically the small dispersion limit for the Korteweg-de Vries (KdV) equation $u_t+6uu_x+\epsilon^{2}u_{xxx}=0$ for $\epsilon\ll1$ and give a quantitative comparison of the numerical solution with various asymptotic formulae for small $\epsilon$ in the whole $(x,t)$-plane. The matching of the asymptotic solutions is studied numerically.10aKorteweg-de Vries equation1 aGrava, Tamara1 aKlein, Christian uhttps://www.math.sissa.it/publication/numerical-study-small-dispersion-limit-korteweg-de-vries-equation-and-asymptotic00782nas a2200121 4500008004100000245008600041210006900127260001300196520037000209653002400579100002100603856003600624 2012 en d00aPoles Distribution of PVI Transcendents close to a Critical Point (summer 2011)0 aPoles Distribution of PVI Transcendents close to a Critical Poin bElsevier3 aThe distribution of the poles of Painlevé VI transcendents associated to semi-simple Frobenius manifolds is determined close to a critical point. It is shown that the poles accumulate at the critical point,asymptotically along two rays. As an example, the Frobenius manifold given by the quantum cohomology of CP2 is considered. The general PVI is also considered.10aPainleve' equations1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/652600498nas a2200121 4500008004100000245013300041210006900174260003300243300001400276490000700290100002300297856005600320 2012 eng d00aResonance at the first eigenvalue for first-order systems in the plane: vanishing Hamiltonians and the Landesman-Lazer condition0 aResonance at the first eigenvalue for firstorder systems in the bKhayyam Publishing, Inc.c05 a505–5260 v251 aGarrione, Maurizio uhttps://projecteuclid.org:443/euclid.die/135601267600581nas a2200121 4500008004100000245004500041210004300086260001000129520024000139653002300379100002100402856003600423 2012 en d00aA Review on The Sixth Painlevé Equation0 aReview on The Sixth Painlevé Equation bSISSA3 aFor the Painlev\\\'e 6 transcendents, we provide a unitary description of the\r\ncritical behaviours, the connection formulae, their complete tabulation, and\r\nthe asymptotic distribution of the poles close to a critical point.

10aPainlevé equation1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/652501270nas a2200109 4500008004100000245012700041210007000168260002800238520083700266100002101103856003601124 2012 en d00aSolving the Sixth Painlevé Equation: Towards the Classification of all the Critical Behaviors and the Connection Formulae0 aSolving the Sixth Painlevé Equation Towards the Classification o bOxford University Press3 aThe critical behavior of a three real parameter class of solutions of the\\r\\nsixth Painlev\\\\\\\'e equation is computed, and parametrized in terms of monodromy\\r\\ndata of the associated $2\\\\times 2$ matrix linear Fuchsian system of ODE. The\\r\\nclass may contain solutions with poles accumulating at the critical point. The\\r\\nstudy of this class closes a gap in the description of the transcendents in one\\r\\nto one correspondence with the monodromy data. These transcendents are reviewed in the paper. Some formulas that relate the monodromy data to the critical behaviors of the four real (two complex) parameter class of solutions are\\r\\nmissing in the literature, so they are computed here. A computational procedure to write the full expansion of the four and three real parameter class of solutions is proposed.1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/609300562nas a2200109 4500008004100000245004400041210004400085260001900129520024700148100002100395856003600416 2012 en d00aTabulation of Painlevé 6 transcendents0 aTabulation of Painlevé 6 transcendents bIOP Publishing3 aThe critical and asymptotic behaviors of solutions of the sixth Painlev'e equation PVI, obtained in the framework of the monodromy preserving deformation method, and their explicit parametrization in terms of monodromy data, are tabulated.1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/652000665nas a2200109 4500008004100000245007400041210007100115260001900186520029300205100002100498856003600519 2011 en d00aAn asymptotic reduction of a Painlevé VI equation to a Painlevé III0 aasymptotic reduction of a Painlevé VI equation to a Painlevé III bIOP Publishing3 aWhen the independent variable is close to a critical point, it is shown that\\r\\nPVI can be asymptotically reduced to PIII. In this way, it is possible to\\r\\ncompute the leading term of the critical behaviors of PVI transcendents\\r\\nstarting from the behaviors of PIII transcendents.1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/512400772nas a2200157 4500008004300000245008600043210007000129260003400199520023700233653003600470100001900506700002200525700001600547700001500563856003600578 2011 en_Ud 00aAxial symmetry of some steady state solutions to nonlinear Schrödinger equations0 aAxial symmetry of some steady state solutions to nonlinear Schrö bAmerican Mathematical Society3 aIn this note, we show the axial symmetry of steady state solutions of nonlinear Schrodinger equations when the exponent of the nonlinearity is between the critical Sobolev exponent of n dimensional space and n - 1 dimensional space.10aNonlinear Schrödinger equation1 aGui, Changfeng1 aMalchiodi, Andrea1 aXu, Haoyuan1 aYang, Paul uhttp://hdl.handle.net/1963/410001344nas a2200181 4500008004100000022001400041245010900055210007100164300001600235490000800251520070400259653002100963653003300984653002901017100002201046700002301068856007101091 2011 eng d a0022-039600aDouble resonance with Landesman–Lazer conditions for planar systems of ordinary differential equations0 aDouble resonance with Landesman–Lazer conditions for planar syst a1052 - 10820 v2503 aWe prove the existence of periodic solutions for first order planar systems at resonance. The nonlinearity is indeed allowed to interact with two positively homogeneous Hamiltonians, both at resonance, and some kind of Landesman–Lazer conditions are assumed at both sides. We are thus able to obtain, as particular cases, the existence results proposed in the pioneering papers by Lazer and Leach (1969) [27], and by Frederickson and Lazer (1969) [18]. Our theorem also applies in the case of asymptotically piecewise linear systems, and in particular generalizes Fabry's results in Fabry (1995) [10], for scalar equations with double resonance with respect to the Dancer–Fučik spectrum.

10aDouble resonance10aLandesman–Lazer conditions10aNonlinear planar systems1 aFonda, Alessandro1 aGarrione, Maurizio uhttp://www.sciencedirect.com/science/article/pii/S002203961000290100369nas a2200109 4500008004300000245006000043210005700103260002100160100002300181700001900204856003600223 2011 en_Ud 00aAn Estimate on the Flow Generated by Monotone Operators0 aEstimate on the Flow Generated by Monotone Operators bTaylor & Francis1 aBianchini, Stefano1 aGloyer, Matteo uhttp://hdl.handle.net/1963/364600442nas a2200109 4500008004100000245003800041210003400079520013500113100002500248700002300273856003600296 2011 en d00aThe geometry of Maximum Principle0 ageometry of Maximum Principle3 aAn invariant formulation of the maximum principle in optimal control is presented, and some second-order invariants are discussed.1 aAgrachev, Andrei, A.1 aGamkrelidze, Revaz uhttp://hdl.handle.net/1963/645600490nas a2200145 4500008004100000245008500041210006900126260002500195300001200220490000700232100001700239700001900256700002300275856004600298 2011 eng d00aMulti-physics modelling and sensitivity analysis of olympic rowing boat dynamics0 aMultiphysics modelling and sensitivity analysis of olympic rowin bSpringer Naturecnov a85–940 v141 aMola, Andrea1 aGhommem, Mehdi1 aHajj, Muhammad, R. uhttps://doi.org/10.1007/s12283-011-0075-200992nas a2200145 4500008004100000245009400041210006900135260003700204300001400241490000700255520041000262100002200672700002300694856012900717 2011 eng d00aNonlinear resonance: a comparison between Landesman-Lazer and Ahmad-Lazer-Paul conditions0 aNonlinear resonance a comparison between LandesmanLazer and Ahma bAdvanced Nonlinear Studies, Inc. a391–4040 v113 aWe show that the Ahmad-Lazer-Paul condition for resonant problems is more general than the Landesman-Lazer one, discussing some relations with other existence conditions, as well. As a consequence, such a relation holds, for example, when considering resonant boundary value problems associated with linear elliptic operators, the p-Laplacian and, in the scalar case, with an asymmetric oscillator.

1 aFonda, Alessandro1 aGarrione, Maurizio uhttps://www.math.sissa.it/publication/nonlinear-resonance-comparison-between-landesman-lazer-and-ahmad-lazer-paul-conditions01547nas a2200133 4500008004100000245008700041210006900128260000900197520111200206100002001318700001801338700002101356856003601377 2011 en d00aNumerical Study of breakup in generalized Korteweg-de Vries and Kawahara equations0 aNumerical Study of breakup in generalized Kortewegde Vries and K bSIAM3 aThis article is concerned with a conjecture in [B. Dubrovin, Comm. Math. Phys., 267 (2006), pp. 117–139] on the formation of dispersive shocks in a class of Hamiltonian dispersive regularizations of the quasi-linear transport equation. The regularizations are characterized by two arbitrary functions of one variable, where the condition of integrability implies that one of these functions must not vanish. It is shown numerically for a large class of equations that the local behavior of their solution near the point of gradient catastrophe for the transport equation is described by a special solution of a Painlevé-type equation. This local description holds also for solutions to equations where blowup can occur in finite time. Furthermore, it is shown that a solution of the dispersive equations away from the point of gradient catastrophe is approximated by a solution of the transport equation with the same initial data, modulo terms of order $\\\\epsilon^2$, where $\\\\epsilon^2$ is the small dispersion parameter. Corrections up to order $\\\\epsilon^4$ are obtained and tested numerically.1 aDubrovin, Boris1 aGrava, Tamara1 aKlein, Christian uhttp://hdl.handle.net/1963/495100609nas a2200121 4500008004100000245003000041210002900071260001000100520030300110100001600413700002200429856003600451 2011 en d00aQ-factorial Laurent rings0 aQfactorial Laurent rings bSISSA3 aDolgachev proved that, for any field k, the ring naturally associated to a\\r\\ngeneric Laurent polynomial in d variables, $d \\\\geq 4$, is factorial. We prove a\\r\\nsufficient condition for the ring associated to a very general complex Laurent\\r\\npolynomial in d=3 variables to be Q-factorial.1 aBruzzo, Ugo1 aGrassi, Antonella uhttp://hdl.handle.net/1963/418301079nas a2200121 4500008004100000245007600041210006900117300001400186490000700200520062100207100002300828856010600851 2011 eng d00aResonance and Landesman-Lazer conditions for first order systems in R^20 aResonance and LandesmanLazer conditions for first order systems a153–1600 v663 aThe first part of the paper surveys the concept of resonance for $T$-periodic nonlinear problems. In the second part, some new results about existence conditions for nonlinear planar systems are presented. In particular, the Landesman-Lazer conditions are generalized to systems in $\mathbbR^2$ where the nonlinearity interacts with two resonant Hamiltonians. Such results apply to second order equations, generalizing previous theorems by Fabry [4] (for the undamped case), and Frederickson-Lazer [9] (for the case with friction). The results have been obtained with A. Fonda, and have been published in [8].

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

10aMultiple periodic solutions10aPoincaré–Birkhoff theorem10aResonance10aRotation number1 aBoscaggin, Alberto1 aGarrione, Maurizio uhttp://www.sciencedirect.com/science/article/pii/S0362546X1100181701465nas a2200121 4500008004300000245006700043210006500110260001300175520107800188100001601266700002501282856003601307 2011 en_Ud 00aSemistable and numerically effective principal (Higgs) bundles0 aSemistable and numerically effective principal Higgs bundles bElsevier3 aWe study Miyaoka-type semistability criteria for principal Higgs G-bundles E on complex projective manifolds of any dimension. We prove that E has the property of being semistable after pullback to any projective curve if and only if certain line bundles, obtained from some characters of the parabolic subgroups of G, are numerically effective. One also proves that these conditions are met for semistable principal Higgs bundles whose adjoint bundle has vanishing second Chern class.\\r\\n\\r\\nIn a second part of the paper, we introduce notions of numerical effectiveness and numerical flatness for principal (Higgs) bundles, discussing their main properties. For (non-Higgs) principal bundles, we show that a numerically flat principal bundle admits a reduction to a Levi factor which has a flat Hermitian–Yang–Mills connection, and, as a consequence, that the cohomology ring of a numerically flat principal bundle with coefficients in R is trivial. To our knowledge this notion of numerical effectiveness is new even in the case of (non-Higgs) principal bundles.1 aBruzzo, Ugo1 aGrana-Otero, Beatriz uhttp://hdl.handle.net/1963/363801019nas a2200133 4500008004100000020001800041245004500059210004500104260001000149520064400159653002500803100002100828856003600849 2011 en d a978311027558200aSolving PVI by Isomonodromy Deformations0 aSolving PVI by Isomonodromy Deformations bSISSA3 aThe critical and asymptotic behaviors of solutions of the sixth Painlev\\\'e\r\nequation, an their parametrization in terms of monodromy data, are\r\nsynthetically reviewed. The explicit formulas are given. This paper has been\r\nwithdrawn by the author himself, because some improvements are necessary.\r\nThis is a proceedings of the international conference \"Painlevé Equations and Related Topics\" which was taking place at the Euler International Mathematical Institute, a branch of the Saint Petersburg Department of the SteklovInstitute of Mathematicsof theRussian Academy of Sciences, in Saint Petersburg on June 17 to 23, 2011.10aPainlevé Equations1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/652200991nas a2200169 4500008004100000245009900041210006900140260001300209300001200222490000800234520046000242100002100702700002300723700002000746700001900766856003600785 2011 en d00aThe sphere and the cut locus at a tangency point in two-dimensional almost-Riemannian geometry0 asphere and the cut locus at a tangency point in twodimensional a bSpringer a141-1610 v17 3 aWe study the tangential case in 2-dimensional almost-Riemannian geometry. We\\r\\nanalyse the connection with the Martinet case in sub-Riemannian geometry. We\\r\\ncompute estimations of the exponential map which allow us to describe the\\r\\nconjugate locus and the cut locus at a tangency point. We prove that this last\\r\\none generically accumulates at the tangency point as an asymmetric cusp whose branches are separated by the singular set.

1 aBonnard, Bernard1 aCharlot, Grégoire1 aGhezzi, Roberta1 aJanin, Gabriel uhttp://hdl.handle.net/1963/491400340nas a2200097 4500008004100000245006800041210006700109260001000176100002000186856003600206 2010 en d00aAlmost-Riemannian Geometry from a Control Theoretical Viewpoint0 aAlmostRiemannian Geometry from a Control Theoretical Viewpoint bSISSA1 aGhezzi, Roberta uhttp://hdl.handle.net/1963/470500422nas a2200145 4500008004100000022001400041245003600055210003600091300001400127490000800141100001900149700001700168700002000185856007100205 2010 eng d a0021-904500aCauchy biorthogonal polynomials0 aCauchy biorthogonal polynomials a832–8670 v1621 aBertola, Marco1 aGekhtman, M.1 aSzmigielski, J. uhttp://0-dx.doi.org.mercury.concordia.ca/10.1016/j.jat.2009.09.00801271nas a2200133 4500008004300000245007300043210006800116520083300184100001801017700001901035700002301054700002401077856003601101 2010 en_Ud 00aChern-Simons theory on L(p,q) lens spaces and Gopakumar-Vafa duality0 aChernSimons theory on Lpq lens spaces and GopakumarVafa duality3 aWe consider aspects of Chern-Simons theory on L(p,q) lens spaces and its relation with matrix models and topological string theory on Calabi-Yau threefolds, searching for possible new large N dualities via geometric transition for non-SU(2) cyclic quotients of the conifold. To this aim we find, on one hand, some novel matrix integral representations of the SU(N) CS partition function in a generic flat background for the whole L(p,q) family and provide a solution for its large N dynamics; on the other, we perform in full detail the construction of a family of would-be dual closed string backgrounds via conifold geometric transition from T^*L(p,q). We can then explicitly prove that Gopakumar-Vafa duality in a fixed vacuum fails in the case q>1, and briefly discuss how it could be restored in a non-perturbative setting.1 aBrini, Andrea1 aGriguolo, Luca1 aSeminara, Domenico1 aTanzini, Alessandro uhttp://hdl.handle.net/1963/293800912nas a2200133 4500008004300000245011700043210006900160520042800229100002600657700002200683700001900705700001800724856003600742 2010 en_Ud 00aConcentration of solutions for some singularly perturbed mixed problems: Asymptotics of minimal energy solutions0 aConcentration of solutions for some singularly perturbed mixed p3 aIn this paper we carry on the study of asymptotic behavior of some solutions to a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions, started in the first paper. Here we are mainly interested in the analysis of the location and shape of least energy solutions when the singular perturbation parameter tends to zero. We show that in many cases they coincide with the new solutions produced in.1 aGarcia Azorero, Jesus1 aMalchiodi, Andrea1 aMontoro, Luigi1 aPeral, Ireneo uhttp://hdl.handle.net/1963/340900856nas a2200133 4500008004300000245010300043210006900146520038600215100002600601700002200627700001900649700001800668856003600686 2010 en_Ud 00aConcentration of solutions for some singularly perturbed mixed problems. Part I: existence results0 aConcentration of solutions for some singularly perturbed mixed p3 aIn this paper we study the asymptotic behavior of some solutions to a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions. We prove that, under suitable geometric conditions on the boundary of the domain, there exist solutions which approach the intersection of the Neumann and the Dirichlet parts as the singular perturbation parameter tends to zero.1 aGarcia Azorero, Jesus1 aMalchiodi, Andrea1 aMontoro, Luigi1 aPeral, Ireneo uhttp://hdl.handle.net/1963/340600371nas a2200097 4500008004300000245008300043210006900126100002300195700001900218856003600237 2010 en_Ud 00aOn the Euler-Lagrange equation for a variational problem : the general case II0 aEulerLagrange equation for a variational problem the general cas1 aBianchini, Stefano1 aGloyer, Matteo uhttp://hdl.handle.net/1963/255102590nas a2200265 4500008004100000245013200041210006900173260001000242520175000252100001702002700002402019700002002043700001902063700002102082700001802103700003002121700001802151700001702169700001702186700002002203700002202223700002402245700001902269856003602288 2010 en d00aGene expression analysis of the emergence of epileptiform activity after focal injection of kainic acid into mouse hippocampus.0 aGene expression analysis of the emergence of epileptiform activi bWiley3 aWe report gene profiling data on genomic processes underlying the progression towards recurrent seizures after injection of kainic acid (KA) into the mouse hippocampus. Focal injection enabled us to separate the effects of proepileptic stimuli initiated by KA injection. Both the injected and contralateral hippocampus participated in the status epilepticus. However, neuronal death induced by KA treatment was restricted to the injected hippocampus, although there was some contralateral axonal degeneration. We profiled gene expression changes in dorsal and ventral regions of both the injected and contralateral hippocampus. Changes were detected in the expression of 1526 transcripts in samples from three time-points: (i) during the KA-induced status epilepticus, (ii) at 2 weeks, before recurrent seizures emerged, and (iii) at 6 months after seizures emerged. Grouping genes with similar spatio-temporal changes revealed an early transcriptional response, strong immune, cell death and growth responses at 2 weeks and an activation of immune and extracellular matrix genes persisting at 6 months. Immunostaining for proteins coded by genes identified from array studies provided evidence for gliogenesis and suggested that the proteoglycan biglycan is synthesized by astrocytes and contributes to a glial scar. Gene changes at 6 months after KA injection were largely restricted to tissue from the injection site. This suggests that either recurrent seizures might depend on maintained processes including immune responses and changes in extracellular matrix proteins near the injection site or alternatively might result from processes, such as growth, distant from the injection site and terminated while seizures are maintained.

1 aMotti, Dario1 aLe Duigou, Caroline1 aChemaly, Nicole1 aWittner, Lucia1 aLazarevic, Dejan1 aKrmac, Helena1 aMarstrand, Troels, Torben1 aValen, Eivind1 aSanges, Remo1 aStupka, Elia1 aSandelin, Albin1 aCherubini, Enrico1 aGustincich, Stefano1 aMiles, Richard uhttp://hdl.handle.net/1963/448001051nas a2200169 4500008004300000245007900043210006900122260003000191520049800221100001800719700002600737700002300763700001800786700002200804700001900826856003600845 2010 en_Ud 00aHomogeneous binary trees as ground states of quantum critical Hamiltonians0 aHomogeneous binary trees as ground states of quantum critical Ha bAmerican Physical Society3 aMany-body states whose wave-function admits a representation in terms of a uniform binary-tree tensor decomposition are shown to obey to power-law two-body correlations functions. Any such state can be associated with the ground state of a translational invariant Hamiltonian which, depending on the dimension of the systems sites, involve at most couplings between third-neighboring sites. A detailed analysis of their spectra shows that they admit an exponentially large ground space.

1 aSilvi, Pietro1 aGiovannetti, Vittorio1 aMontangero, Simone1 aRizzi, Matteo1 aCirac, J. Ignacio1 aFazio, Rosario uhttp://hdl.handle.net/1963/390901188nas a2200157 4500008004300000245010800043210006900151260001900220520065100239100001800890700002300908700001800931700002600949700001900975856003600994 2010 en_Ud 00aHomogeneous multiscale entanglement renormalization ansatz tensor networks for quantum critical systems0 aHomogeneous multiscale entanglement renormalization ansatz tenso bIOP Publishing3 aIn this paper, we review the properties of homogeneous multiscale entanglement renormalization ansatz (MERA) to describe quantum critical systems.We discuss in more detail our results for one-dimensional (1D) systems (the Ising and Heisenberg models) and present new data for the 2D Ising model. Together with the results for the critical exponents, we provide a detailed description of the numerical algorithm and a discussion of new optimization\\nstrategies. The relation between the critical properties of the system and the tensor structure of the MERA is expressed using the formalism of quantum channels, which we review and extend.

1 aRizzi, Matteo1 aMontangero, Simone1 aSilvi, Pietro1 aGiovannetti, Vittorio1 aFazio, Rosario uhttp://hdl.handle.net/1963/406700901nas a2200121 4500008004300000245004400043210004300087520054400130100002200674700002200696700002500718856003600743 2010 en_Ud 00aLorentz Covariant k-Minkowski Spacetime0 aLorentz Covariant kMinkowski Spacetime3 aIn recent years, different views on the interpretation of Lorentz covariance of non commuting coordinates were discussed. Here, by a general procedure, we construct the minimal canonical central covariantisation of the k-Minkowski spacetime. We then show that, though the usual k-Minkowski spacetime is covariant under deformed (or twisted) Lorentz action, the resulting framework is equivalent to taking a non covariant restriction of the covariantised model. We conclude with some general comments on the approach of deformed covariance.1 aDabrowski, Ludwik1 aGodlinski, Michal1 aPiacitelli, Gherardo uhttp://hdl.handle.net/1963/382900493nas a2200109 4500008004100000245009300041210006900134100001700203700002300220700002000243856012000263 2010 eng d00aA normal form for generic 2-dimensional almost-Riemannian structures at a tangency point0 anormal form for generic 2dimensional almostRiemannian structures1 aBoscain, Ugo1 aCharlot, Grégoire1 aGhezzi, Roberta uhttps://www.math.sissa.it/publication/normal-form-generic-2-dimensional-almost-riemannian-structures-tangency-point01048nas a2200121 4500008004300000245012100043210006900164520059600233100002200829700001800851700002100869856003600890 2010 en_Ud 00aNumerical Solution of the Small Dispersion Limit of the Camassa-Holm and Whitham Equations and Multiscale Expansions0 aNumerical Solution of the Small Dispersion Limit of the CamassaH3 aThe small dispersion limit of solutions to the Camassa-Holm (CH) equation is characterized by the appearance of a zone of rapid modulated oscillations. An asymptotic description of these oscillations is given, for short times, by the one-phase solution to the CH equation, where the branch points of the corresponding elliptic curve depend on the physical coordinates via the Whitham equations. We present a conjecture for the phase of the asymptotic solution. A numerical study of this limit for smooth hump-like initial data provides strong evidence for the validity of this conjecture....1 aAbenda, Simonetta1 aGrava, Tamara1 aKlein, Christian uhttp://hdl.handle.net/1963/384000900nas a2200121 4500008004300000245014000043210007000183260001000253520044500263100001600708700001800724856003600742 2010 en_Ud 00aPainlevé II asymptotics near the leading edge of the oscillatory zone for the Korteweg-de Vries equation in the small-dispersion limit0 aPainlevé II asymptotics near the leading edge of the oscillatory bWiley3 aIn the small dispersion limit, solutions to the Korteweg-de Vries equation develop an interval of fast oscillations after a certain time. We obtain a universal asymptotic expansion for the Korteweg-de Vries solution near the leading edge of the oscillatory zone up to second order corrections. This expansion involves the Hastings-McLeod solution of the Painlev\\\\\\\'e II equation. We prove our results using the Riemann-Hilbert approach.1 aClaeys, Tom1 aGrava, Tamara uhttp://hdl.handle.net/1963/379900671nas a2200109 4500008004300000245005300043210005300096520033800149100001600487700002200503856003600525 2010 en_Ud 00aPicard group of hypersurfaces in toric varieties0 aPicard group of hypersurfaces in toric varieties3 aWe show that the usual sufficient criterion for a generic hypersurface in a smooth projective manifold to have the same Picard number as the ambient variety can be generalized to hypersurfaces in complete simplicial toric varieties. This sufficient condition is always satisfied by generic K3 surfaces embedded in Fano toric 3-folds.1 aBruzzo, Ugo1 aGrassi, Antonella uhttp://hdl.handle.net/1963/410301121nas a2200121 4500008004300000245007400043210006900117260003700186520069600223100002200919700002200941856003600963 2010 en_Ud 00aRiemann-Roch theorems and elliptic genus for virtually smooth schemes0 aRiemannRoch theorems and elliptic genus for virtually smooth sch bMathematical Sciences Publishers3 aFor a proper scheme X with a fixed 1-perfect obstruction theory, we define virtual versions of holomorphic Euler characteristic, chi y-genus, and elliptic genus; they are deformation invariant, and extend the usual definition in the smooth case. We prove virtual versions of the Grothendieck-Riemann-Roch and Hirzebruch-Riemann-Roch theorems. We show that the virtual chi y-genus is a polynomial, and use this to define a virtual topological Euler characteristic. We prove that the virtual elliptic genus satisfies a Jacobi modularity property; we state and prove a localization theorem in the toric equivariant case. We show how some of our results apply to moduli spaces of stable sheaves.1 aFantechi, Barbara1 aGöttsche, Lothar uhttp://hdl.handle.net/1963/388800906nas a2200109 4500008004300000245009100043210006900134520052300203100001800726700001600744856003600760 2010 en_Ud 00aSolitonic asymptotics for the Korteweg-de Vries equation in the small dispersion limit0 aSolitonic asymptotics for the Kortewegde Vries equation in the s3 aWe study the small dispersion limit for the Korteweg-de Vries (KdV) equation $u_t+6uu_x+\\\\epsilon^{2}u_{xxx}=0$ in a critical scaling regime where $x$ approaches the trailing edge of the region where the KdV solution shows oscillatory behavior. Using the Riemann-Hilbert approach, we obtain an asymptotic expansion for the KdV solution in a double scaling limit, which shows that the oscillations degenerate to sharp pulses near the trailing edge. Locally those pulses resemble soliton solutions of the KdV equation.1 aGrava, Tamara1 aClaeys, Tom uhttp://hdl.handle.net/1963/383901439nas a2200181 4500008004300000245007000043210006800113260001300181300001200194490000700206520090200213100002501115700001701140700002301157700002001180700002101200856003601221 2010 en_Ud 00aTwo-dimensional almost-Riemannian structures with tangency points0 aTwodimensional almostRiemannian structures with tangency points bElsevier a793-8070 v273 aTwo-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. We study the relation between the topological invariants of an almost-Riemannian structure on a compact oriented surface and the rank-two vector bundle over the surface which defines the structure. We analyse the generic case including the presence of tangency points, i.e. points where two generators of the distribution and their Lie bracket are linearly dependent. The main result of the paper provides a classification of oriented almost-Riemannian structures on compact oriented surfaces in terms of the Euler number of the vector bundle corresponding to the structure. Moreover, we present a Gauss-Bonnet formula for almost-Riemannian structures with tangency points.

1 aAgrachev, Andrei, A.1 aBoscain, Ugo1 aCharlot, Grégoire1 aGhezzi, Roberta1 aSigalotti, Mario uhttp://hdl.handle.net/1963/387000390nas a2200133 4500008004100000245003400041210003000075300001500105490000800120100001900128700001600147700001900163856007400182 2009 eng d00aThe Cauchy two–matrix model0 aCauchy two–matrix model a983–10140 v2871 aBertola, Marco1 aGekhtman, M1 aSzmigielski, J uhttps://www.math.sissa.it/publication/cauchy-two%E2%80%93matrix-model00501nas a2200145 4500008004100000022001400041245007700055210006900132300001500201490000700216100001900223700001700242700002000259856007600279 2009 eng d a1751-811300aCubic string boundary value problems and Cauchy biorthogonal polynomials0 aCubic string boundary value problems and Cauchy biorthogonal pol a454006, 130 v421 aBertola, Marco1 aGekhtman, M.1 aSzmigielski, J. uhttp://0-dx.doi.org.mercury.concordia.ca/10.1088/1751-8113/42/45/45400600381nas a2200097 4500008004300000245009500043210006900138100002000207700002000227856003600247 2009 en_Ud 00aHardy-Sobolev-Maz\\\'ja inequalities: symmetry and breaking symmetry of extremal functions0 aHardySobolevMazja inequalities symmetry and breaking symmetry of1 aGazzini, Marita1 aMusina, Roberta uhttp://hdl.handle.net/1963/256900963nas a2200133 4500008004300000245008100043210006900124260001300193520052900206100001800735700002200753700001800775856003600793 2009 en_Ud 00aInitial value problem of the Whitham equations for the Camassa-Holm equation0 aInitial value problem of the Whitham equations for the CamassaHo bElsevier3 aWe study the Whitham equations for the Camassa-Holm equation. The equations are neither strictly hyperbolic nor genuinely nonlinear. We are interested in the initial value problem of the Whitham equations. When the initial values are given by a step function, the Whitham solution is self-similar. When the initial values are given by a smooth function, the Whitham solution exists within a cusp in the x-t plane. On the boundary of the cusp, the Whitham equation matches the Burgers solution, which exists outside the cusp.1 aGrava, Tamara1 aPierce, Virgil U.1 aTian, Fei-Ran uhttp://hdl.handle.net/1963/342901520nas a2200133 4500008004300000245008600043210006900129520106500198100002501263700001701288700002401305700002101329856003601350 2009 en_Ud 00aThe intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups0 aintrinsic hypoelliptic Laplacian and its heat kernel on unimodul3 aWe present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector, using the Popp\\\'s volume form introduced by Montgomery. This definition generalizes the one of the Laplace-Beltrami operator in Riemannian geometry. In the case of left-invariant problems on unimodular Lie groups we prove that it coincides with the usual sum of squares.\\nWe then extend a method (first used by Hulanicki on the Heisenberg group) to compute explicitly the kernel of the hypoelliptic heat equation on any unimodular Lie group of type I. The main tool is the noncommutative Fourier transform. We then study some relevant cases: SU(2), SO(3), SL(2) (with the metrics inherited by the Killing form), and the group SE(2) of rototranslations of the plane.\\nOur study is motivated by some recent results about the cut and conjugate loci on these sub-Riemannian manifolds. The perspective is to understand how singularities of the sub-Riemannian distance reflect on the kernel of the corresponding hypoelliptic heat equation.1 aAgrachev, Andrei, A.1 aBoscain, Ugo1 aGauthier, Jean-Paul1 aRossi, Francesco uhttp://hdl.handle.net/1963/266900362nas a2200097 4500008004300000245007600043210006900119100002000188700002000208856003600228 2009 en_Ud 00aOn a Sobolev type inequality related to the weighted p-Laplace operator0 aSobolev type inequality related to the weighted pLaplace operato1 aGazzini, Marita1 aMusina, Roberta uhttp://hdl.handle.net/1963/261300978nas a2200121 4500008004300000245018700043210006900230520046200299100002000761700001800781700002100799856003600820 2009 en_Ud 00aOn universality of critical behaviour in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the {\\\\it tritronquée} solution to the Painlevé-I equation0 auniversality of critical behaviour in the focusing nonlinear Sch3 aWe argue that the critical behaviour near the point of ``gradient catastrophe\\\" of the solution to the Cauchy problem for the focusing nonlinear Schr\\\\\\\"odinger equation $ i\\\\epsilon \\\\psi_t +\\\\frac{\\\\epsilon^2}2\\\\psi_{xx}+ |\\\\psi|^2 \\\\psi =0$ with analytic initial data of the form $\\\\psi(x,0;\\\\epsilon) =A(x) e^{\\\\frac{i}{\\\\epsilon} S(x)}$ is approximately described by a particular solution to the Painlev\\\\\\\'e-I equation.1 aDubrovin, Boris1 aGrava, Tamara1 aKlein, Christian uhttp://hdl.handle.net/1963/252501175nas a2200109 4500008004300000245012700043210006900170520075600239100001800995700001601013856003601029 2009 en_Ud 00aUniversality of the break-up profile for the KdV equation in the small dispersion limit using the Riemann-Hilbert approach0 aUniversality of the breakup profile for the KdV equation in the 3 aWe obtain an asymptotic expansion for the solution of the Cauchy problem for the Korteweg-de Vries (KdV) equation in the small dispersion limit near the point of gradient catastrophe (x_c,t_c) for the solution of the dispersionless equation.\\nThe sub-leading term in this expansion is described by the smooth solution of a fourth order ODE, which is a higher order analogue to the Painleve I equation. This is in accordance with a conjecture of Dubrovin, suggesting that this is a universal phenomenon for any Hamiltonian perturbation of a hyperbolic equation. Using the Deift/Zhou steepest descent method applied on the Riemann-Hilbert problem for the KdV equation, we are able to prove the asymptotic expansion rigorously in a double scaling limit.1 aGrava, Tamara1 aClaeys, Tom uhttp://hdl.handle.net/1963/263600454nas a2200109 4500008004300000245012300043210006900166100002100235700002600256700002600282856003600308 2009 en_Ud 00aA variational model for quasistatic crack growth in nonlinear elasticity: some qualitative properties of the solutions0 avariational model for quasistatic crack growth in nonlinear elas1 aDal Maso, Gianni1 aGiacomini, Alessandro1 aPonsiglione, Marcello uhttp://hdl.handle.net/1963/267500681nas a2200109 4500008004300000245012000043210006900163520026100232100002100493700002100514856003600535 2008 en_Ud 00aGradient bounds for minimizers of free discontinuity problems related to cohesive zone models in fracture mechanics0 aGradient bounds for minimizers of free discontinuity problems re3 aIn this note we consider a free discontinuity problem for a scalar function, whose energy depends also on the size of the jump. We prove that the gradient of every smooth local minimizer never exceeds a constant, determined only by the data of the problem.1 aDal Maso, Gianni1 aGarroni, Adriana uhttp://hdl.handle.net/1963/172300688nas a2200109 4500008004100000245008400041210006900125260001000194520031700204100002100521856003600542 2008 en d00aOn the Logarithmic Asymptotics of the Sixth Painleve\' Equation (Summer 2007)0 aLogarithmic Asymptotics of the Sixth Painleve Equation Summer 20 bSISSA3 aWe study the solutions of the sixth Painlev\'e equation with a logarithmic\r\nasymptotic behavior at a critical point. We compute the monodromy group\r\nassociated to the solutions by the method of monodromy preserving deformations\r\nand we characterize the asymptotic behavior in terms of the monodromy itself.1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/652101399nas a2200109 4500008004300000245010600043210006900149520099600218100001801214700002101232856003601253 2008 en_Ud 00aNumerical study of a multiscale expansion of the Korteweg-de Vries equation and Painlevé-II equation0 aNumerical study of a multiscale expansion of the Kortewegde Vrie3 aThe Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order $\\\\e^2$, $\\\\e\\\\ll 1$, is characterized by the appearance of a zone of rapid modulated oscillations. These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wave-number and frequency are not constant but evolve according to the Whitham equations. Whereas the difference between the KdV and the asymptotic solution decreases as $\\\\epsilon$ in the interior of the Whitham oscillatory zone, it is known to be only of order $\\\\epsilon^{1/3}$ near the leading edge of this zone. To obtain a more accurate description near the leading edge of the oscillatory zone we present a multiscale expansion of the solution of KdV in terms of the Hastings-McLeod solution of the Painlev\\\\\\\'e-II equation. We show numerically that the resulting multiscale solution approximates the KdV solution, in the small dispersion limit, to the order $\\\\epsilon^{2/3}$.1 aGrava, Tamara1 aKlein, Christian uhttp://hdl.handle.net/1963/259200781nas a2200133 4500008004100000020002200041245006700063210006700130260001300197520035900210100002300569700001900592856003600611 2008 en d a978-3-642-21718-000aTransport Rays and Applications to Hamilton–Jacobi Equations0 aTransport Rays and Applications to Hamilton–Jacobi Equations bSpringer3 aThe aim of these notes is to introduce the readers to the use of the Disintegration Theorem for measures as an effective tool for reducing problems in transport equations to simpler ones. The basic idea is to partition Rd into one dimensional sets, on which the problem under consideration becomes one space dimensional (and thus much easier, hopefully).1 aBianchini, Stefano1 aGloyer, Matteo uhttp://hdl.handle.net/1963/546300925nas a2200121 4500008004100000245012500041210006900166260004700235520035300282100002100635700002100656856012600677 2007 en d00aThe Asymptotic Behaviour of the Fourier Transforms of Orthogonal Polynomials II: L.I.F.S. Measures and Quantum Mechanics0 aAsymptotic Behaviour of the Fourier Transforms of Orthogonal Pol b2007 Birkh¨auser Verlag Basel/Switzerland3 aWe study measures generated by systems of linear iterated functions,\r\ntheir Fourier transforms, and those of their orthogonal polynomials. We\r\ncharacterize the asymptotic behaviours of their discrete and continuous averages.\r\nFurther related quantities are analyzed, and relevance of this analysis\r\nto quantum mechanics is briefly discussed1 aGuzzetti, Davide1 aMantica, Giorgio uhttps://www.math.sissa.it/publication/asymptotic-behaviour-fourier-transforms-orthogonal-polynomials-ii-lifs-measures-and00554nas a2200133 4500008004100000022001400041245011300055210007000168300001400238490000700252100001900259700001700278856012500295 2007 eng d a0176-427600aBiorthogonal Laurent polynomials, Töplitz determinants, minimal Toda orbits and isomonodromic tau functions0 aBiorthogonal Laurent polynomials Töplitz determinants minimal To a383–4300 v261 aBertola, Marco1 aGekhtman, M. uhttps://www.math.sissa.it/publication/biorthogonal-laurent-polynomials-t%C3%B6plitz-determinants-minimal-toda-orbits-and01457nas a2200133 4500008004300000245010800043210006900151520097900220100001901199700002301218700002201241700002401263856003601287 2007 en_Ud 00aBlack Holes, Instanton Counting on Toric Singularities and q-Deformed Two-Dimensional Yang-Mills Theory0 aBlack Holes Instanton Counting on Toric Singularities and qDefor3 aWe study the relationship between instanton counting in N=4 Yang-Mills theory on a generic four-dimensional toric orbifold and the semi-classical expansion of q-deformed Yang-Mills theory on the blowups of the minimal resolution of the orbifold singularity, with an eye to clarifying the recent proposal of using two-dimensional gauge theories to count microstates of black holes in four dimensions. We describe explicitly the instanton contributions to the counting of D-brane bound states which are captured by the two-dimensional gauge theory. We derive an intimate relationship between the two-dimensional Yang-Mills theory and Chern-Simons theory on generic Lens spaces, and use it to show that the correct instanton counting is only reproduced when the Chern-Simons contributions are treated as non-dynamical boundary conditions in the D4-brane gauge theory. We also use this correspondence to discuss the counting of instantons on higher genus ruled Riemann surfaces.1 aGriguolo, Luca1 aSeminara, Domenico1 aSzabo, Richard J.1 aTanzini, Alessandro uhttp://hdl.handle.net/1963/188800503nas a2200121 4500008004100000022001400041245008200055210006900137300002300206100001900229700001700248856011600265 2007 eng d a1073-792800aEffective inverse spectral problem for rational Lax matrices and applications0 aEffective inverse spectral problem for rational Lax matrices and aArt. ID rnm103, 391 aBertola, Marco1 aGekhtman, M. uhttps://www.math.sissa.it/publication/effective-inverse-spectral-problem-rational-lax-matrices-and-applications00410nas a2200097 4500008004300000245012400043210006900167100002000236700002000256856003600276 2007 en_Ud 00aOn the Maz\\\'ya inequalities: existence and multiplicity results for an elliptic problem involving cylindrical weights0 aMazya inequalities existence and multiplicity results for an ell1 aGazzini, Marita1 aMusina, Roberta uhttp://hdl.handle.net/1963/252201055nas a2200109 4500008004300000245006600043210006600109520069300175100001600868700002500884856003600909 2007 en_Ud 00aMetrics on semistable and numerically effective Higgs bundles0 aMetrics on semistable and numerically effective Higgs bundles3 aWe consider fibre metrics on Higgs vector bundles on compact K\\\\\\\"ahler manifolds, providing notions of numerical effectiveness and numerical flatness in terms of such metrics. We prove several properties of bundles satisfying such conditions and in particular we show that numerically flat Higgs bundles have vanishing Chern classes, and that they admit filtrations whose quotients are stable flat Higgs bundles. We compare these definitions with those previously given in the case of projective varieties. Finally we study the relations between numerically effectiveness and semistability, providing semistability criteria for Higgs bundles on projective manifolds of any dimension.1 aBruzzo, Ugo1 aGrana-Otero, Beatriz uhttp://hdl.handle.net/1963/184001154nas a2200121 4500008004300000245004500043210004300088520080300131100002200934700002400956700001600980856003600996 2007 en_Ud 00aA new model for contact angle hysteresis0 anew model for contact angle hysteresis3 aWe present a model which explains several experimental observations relating contact angle hysteresis with surface roughness. The model is based on the balance between released energy and dissipation, and it describes the stick-slip behavior of drops on a rough surface using ideas similar to those employed in dry friction, elasto-plasticity and fracture mechanics. The main results of our analysis are formulas giving the interval of stable contact angles as a function of the surface roughness. These formulas show that the difference between advancing and receding angles is much larger for a drop in complete contact with the substrate (Wenzel drop) than for one whose cavities are filled with air (Cassie-Baxter drop). This fact is used as the key tool to interpret the experimental evidence.1 aDeSimone, Antonio1 aGruenewald, Natalie1 aOtto, Felix uhttp://hdl.handle.net/1963/184801358nas a2200109 4500008004300000245009600043210006900139520096500208100001801173700002101191856003601212 2007 en_Ud 00aNumerical solution of the small dispersion limit of Korteweg de Vries and Whitham equations0 aNumerical solution of the small dispersion limit of Korteweg de 3 aThe Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order $\\\\epsilon^2$, is characterized by the appearance of a zone of rapid modulated oscillations of wave-length of order $\\\\epsilon$. These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wave-number and frequency are not constant but evolve according to the Whitham equations. In this manuscript we give a quantitative analysis of the discrepancy between the numerical solution of the KdV equation in the small dispersion limit and the corresponding approximate solution for values of $\\\\epsilon$ between $10^{-1}$ and $10^{-3}$. The numerical results are compatible with a difference of order $\\\\epsilon$ within the `interior\\\' of the Whitham oscillatory zone, of order $\\\\epsilon^{1/3}$ at the left boundary outside the Whitham zone and of order $\\\\epsilon^{1/2}$ at the right boundary outside the Whitham zone.1 aGrava, Tamara1 aKlein, Christian uhttp://hdl.handle.net/1963/178801084nas a2200109 4500008004300000245007900043210006900122520070800191100001800899700002100917856003600938 2007 en_Ud 00aNumerical study of a multiscale expansion of KdV and Camassa-Holm equation0 aNumerical study of a multiscale expansion of KdV and CamassaHolm3 aWe study numerically solutions to the Korteweg-de Vries and Camassa-Holm equation close to the breakup of the corresponding solution to the dispersionless equation. The solutions are compared with the properly rescaled numerical solution to a fourth order ordinary differential equation, the second member of the Painlev\\\\\\\'e I hierarchy. It is shown that this solution gives a valid asymptotic description of the solutions close to breakup. We present a detailed analysis of the situation and compare the Korteweg-de Vries solution quantitatively with asymptotic solutions obtained via the solution of the Hopf and the Whitham equations. We give a qualitative analysis for the Camassa-Holm equation1 aGrava, Tamara1 aKlein, Christian uhttp://hdl.handle.net/1963/252700810nas a2200109 4500008004300000245004200043210004200085520049600127100001600623700002500639856003600664 2007 en_Ud 00aNumerically flat Higgs vector bundles0 aNumerically flat Higgs vector bundles3 aAfter providing a suitable definition of numerical effectiveness for Higgs bundles, and a related notion of numerical flatness, in this paper we prove, together with some side results, that all Chern classes of a Higgs-numerically flat Higgs bundle vanish, and that a Higgs bundle is Higgs-numerically flat if and only if it is has a filtration whose quotients are flat stable Higgs bundles. We also study the relation between these numerical properties of Higgs bundles and (semi)stability.1 aBruzzo, Ugo1 aGrana-Otero, Beatriz uhttp://hdl.handle.net/1963/175700973nas a2200109 4500008004300000245006600043210006600109520061200175100002200787700001800809856003600827 2007 en_Ud 00aReciprocal transformations and flat metrics on Hurwitz spaces0 aReciprocal transformations and flat metrics on Hurwitz spaces3 aWe consider hydrodynamic systems which possess a local Hamiltonian structure of Dubrovin-Novikov type. To such a system there are also associated an infinite number of nonlocal Hamiltonian structures. We give necessary and sufficient conditions so that, after a nonlinear transformation of the independent variables, the reciprocal system still possesses a local Hamiltonian structure of Dubrovin-Novikov type. We show that, under our hypotheses, bi-hamiltonicity is preserved by the reciprocal transformation. Finally we apply such results to reciprocal systems of genus g Whitham-KdV modulation equations.1 aAbenda, Simonetta1 aGrava, Tamara uhttp://hdl.handle.net/1963/221000296nas a2200097 4500008004300000245003900043210003900082100001600121700002500137856003600162 2007 en_Ud 00aSemistable principal Higgs bundles0 aSemistable principal Higgs bundles1 aBruzzo, Ugo1 aGrana-Otero, Beatriz uhttp://hdl.handle.net/1963/253302436nas a2200169 4500008004100000245007600041210006900117260007200186520184400258100002002102700002202122700001802144700002502162700001902187700002402206856003602230 2006 en d00aExperimental and modeling studies of desensitization of P2X3 receptors.0 aExperimental and modeling studies of desensitization of P2X3 rec bthe American Society for Pharmacology and Experimental Therapeutics3 aThe function of ATP-activated P2X3 receptors involved in pain sensation is modulated by desensitization, a phenomenon poorly understood. The present study used patch-clamp recording from cultured rat or mouse sensory neurons and kinetic modeling to clarify the properties of P2X3 receptor desensitization. Two types of desensitization were observed, a fast process (t1/2 = 50 ms; 10 microM ATP) following the inward current evoked by micromolar agonist concentrations, and a slow process (t1/2 = 35 s; 10 nM ATP) that inhibited receptors without activating them. We termed the latter high-affinity desensitization (HAD). Recovery from fast desensitization or HAD was slow and agonist-dependent. When comparing several agonists, there was analogous ranking order for agonist potency, rate of desensitization and HAD effectiveness, with 2-methylthioadenosine triphosphate the strongest and beta,gamma-methylene-ATP the weakest. HAD was less developed with recombinant (ATP IC50 = 390 nM) than native P2X3 receptors (IC50 = 2.3 nM). HAD could also be induced by nanomolar ATP when receptors seemed to be nondesensitized, indicating that resting receptors could express high-affinity binding sites. Desensitization properties were well accounted for by a cyclic model in which receptors could be desensitized from either open or closed states. Recovery was assumed to be a multistate process with distinct kinetics dependent on the agonist-dependent dissociation rate from desensitized receptors. Thus, the combination of agonist-specific mechanisms such as desensitization onset, HAD, and resensitization could shape responsiveness of sensory neurons to P2X3 receptor agonists. By using subthreshold concentrations of an HAD-potent agonist, it might be possible to generate sustained inhibition of P2X3 receptors for controlling chronic pain.1 aSokolova, Elena1 aSkorinkin, Andrei1 aMoiseev, Igor1 aAgrachev, Andrei, A.1 aNistri, Andrea1 aGiniatullin, Rashid uhttp://hdl.handle.net/1963/497400940nas a2200109 4500008004300000245005300043210005300096520060900149100001800758700001800776856003600794 2006 en_Ud 00aLarge Parameter Behavior of Equilibrium Measures0 aLarge Parameter Behavior of Equilibrium Measures3 aWe study the equilibrium measure for a logarithmic potential in the presence of an external field V*(x) + tp(x), where t is a parameter, V*(x) is a smooth function and p(x) a monic polynomial. When p(x) is of an odd degree, the equilibrium measure is shown to be supported on a single interval as |t| is sufficiently large. When p(x) is of an even degree, the equilibrium measure is supported on two disjoint intervals as t is negatively large; it is supported on a single interval for convex p(x) as t is positively large and is likely to be supported on multiple disjoint intervals for non-convex p(x).1 aGrava, Tamara1 aTian, Fei-Ran uhttp://hdl.handle.net/1963/178900691nas a2200109 4500008004100000245006700041210006500108260001000173520034100183100002100524856003600545 2006 en d00aMatching Procedure for the Sixth Painlevé Equation (May 2006)0 aMatching Procedure for the Sixth Painlevé Equation May 2006 bSISSA3 aWe present a constructive procedure to obtain the critical behavior of\r\nPainleve\' VI transcendents and solve the connection problem. This procedure\r\nyields two and one parameter families of solutions, including trigonometric and\r\nlogarithmic behaviors, and three classes of solutions with Taylor expansion at\r\na critical point.1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/652400778nas a2200109 4500008004300000245004900043210004800092520045100140100002300591700001800614856003600632 2006 en_Ud 00aThomae type formulae for singular Z_N curves0 aThomae type formulae for singular ZN curves3 aWe give an elementary and rigorous proof of the Thomae type formula for singular $Z_N$ curves. To derive the Thomae formula we use the traditional variational method which goes back to Riemann, Thomae and Fuchs. An important step of the proof is the use of the Szego kernel computed explicitly in algebraic form for non-singular 1/N-periods. The proof inherits principal points of Nakayashiki\\\'s proof [31], obtained for non-singular ZN curves.1 aEnolski, Victor Z.1 aGrava, Tamara uhttp://hdl.handle.net/1963/212500679nas a2200121 4500008004100000245003100041210003100072260000900103520036500112100002100477700002300498856003600521 2005 en d00aHybrid necessary principle0 aHybrid necessary principle bSIAM3 aWe consider a hybrid control system and general optimal control problems for this system. We suppose that the switching strategy imposes restrictions on control sets and we provide necessary conditions for an optimal hybrid trajectory, stating a hybrid necessary principle (HNP). Our result generalizes various necessary principles available in the literature.1 aGaravello, Mauro1 aPiccoli, Benedetto uhttp://hdl.handle.net/1963/164101078nas a2200109 4500008004300000245007500043210006900118520070500187100002200892700001800914856003600932 2005 en_Ud 00aModulation of the Camassa-Holm equation and reciprocal transformations0 aModulation of the CamassaHolm equation and reciprocal transforma3 aWe derive the modulation equations or Whitham equations for the Camassa-Holm (CH) equation. We show that the modulation equations are hyperbolic and admit bi-Hamiltonian structure. Furthermore they are connected by a reciprocal transformation to the modulation equations of the first negative flow of the Korteweg de Vries (KdV) equation. The reciprocal transformation is generated by the Casimir of the second Poisson bracket of the KdV averaged flow. We show that the geometry of the bi-Hamiltonian structure of the KdV and CH modulation equations is quite different: indeed the KdV averaged bi-Hamiltonian structure can always be related to a semisimple Frobenius manifold while the CH one cannot.1 aAbenda, Simonetta1 aGrava, Tamara uhttp://hdl.handle.net/1963/230501069nas a2200133 4500008004100000245003500041210003500076260001800111520069800129100002800827700002300855700002100878856003600899 2005 en d00aTraffic flow on a road network0 aTraffic flow on a road network bSISSA Library3 aThis paper is concerned with a fluidodynamic model for traffic flow. More precisely, we consider a single conservation law, deduced from conservation of the number of cars,\\ndefined on a road network that is a collection of roads with junctions. The evolution problem is underdetermined at junctions, hence we choose to have some fixed rules for the distribution of traffic plus an optimization criteria for the flux. We prove existence, uniqueness and stability of solutions to the Cauchy problem. Our method is based on wave front tracking approach, see [6], and works also for boundary data and time dependent coefficients of traffic distribution at junctions, so including traffic lights.1 aCoclite, Giuseppe Maria1 aPiccoli, Benedetto1 aGaravello, Mauro uhttp://hdl.handle.net/1963/158400854nas a2200133 4500008004100000020002200041245006500063210006000128260003400188520041800222653002300640100002100663856003600684 2004 en d a978-2-85629-229-700aThe elliptic representation of the sixth Painlevé equation.0 aelliptic representation of the sixth Painlevé equation bSociete Matematique de France3 aWe find a class of solutions of the sixth Painlev´e equation corresponding\r\nto almost all the monodromy data of the associated linear system; actually, all data\r\nbut one point in the space of data. We describe the critical behavior close to the\r\ncritical points by means of the elliptic representation, and we find the relation among\r\nthe parameters at the different critical points (connection problem).10aPainlevé equation1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/652901974nas a2200109 4500008004300000245009900043210006900142520157600211100002301787700001801810856003601828 2004 en_Ud 00aSingular Z_N curves, Riemann-Hilbert problem and modular solutions of the Schlesinger equation0 aSingular ZN curves RiemannHilbert problem and modular solutions 3 aWe are solving the classical Riemann-Hilbert problem of rank N>1 on the extended complex plane punctured in 2m+2 points, for NxN quasi-permutation monodromy matrices. Following Korotkin we solve the Riemann-Hilbert problem in terms of the Szego kernel of certain Riemann surfaces branched over the given 2m+2 points. These Riemann surfaces are constructed from a permutation representation of the symmetric group S_N to which the quasi-permutation monodromy representation has been reduced. The permutation representation of our problem generates the cyclic subgroup Z_N. For this reason the corresponding Riemann surfaces of genus N(m-1) have Z_N symmetry. This fact enables us to write the matrix entries of the solution of the NxN Riemann-Hilbert problem as a product of an algebraic function and theta-function quotients. The algebraic function turns out to be related to the Szego kernel with zero characteristics. From the solution of the Riemann- Hilbert problem we automatically obtain a particular solution of the Schlesinger system. The tau-function of the Schlesinger system is computed explicitly. The rank 3 problem with four singular points (0,t,1,\\\\infty) is studied in detail. The corresponding solution of the Riemann-Hilbert problem and the Schlesinger system is given in terms of Jacobi\\\'s theta-function with modulus T=T(t), Im(T)>0. The function T=T(t) is invertible if it belongs to the Siegel upper half space modulo the subgroup \\\\Gamma_0(3) of the modular group. The inverse function t=t(T) generates a solution of a general Halphen system.1 aEnolski, Victor Z.1 aGrava, Tamara uhttp://hdl.handle.net/1963/254000759nas a2200121 4500008004100000245005300041210005300094260001800147520038500165100002800550700002300578856003600601 2004 en d00aSolitary waves for Maxwell Schrodinger equations0 aSolitary waves for Maxwell Schrodinger equations bSISSA Library3 aIn this paper we study solitary waves for the coupled system of Schrodinger-Maxwell equations in the three-dimensional space. We prove the existence of a sequence of radial solitary waves for these equations with a fixed L^2 norm. We study the asymptotic behavior and the smoothness of these solutions. We show also that the eigenvalues are negative and the first one is isolated.1 aCoclite, Giuseppe Maria1 aGeorgiev, Vladimir uhttp://hdl.handle.net/1963/158200868nas a2200145 4500008004300000245005900043210005800102260002300160520042200183100001800605700002100623700001900644700002300663856003600686 2003 en_Ud 00aHybrid optimal control: case study of a car with gears0 aHybrid optimal control case study of a car with gears bTaylor and Francis3 aThe purpose of this paper is to show the use of some analytical tools for hybrid optimal control. We illustrate both the hybrid maximum principle and the hybrid necessary principle at work on a simple example of a car with gears. The model is sufficiently rich to generate non-trivial optimization problems and the obtained results match with intuition. Finally, computer simulations confirm the theoretical analysis.1 aD'Apice, Ciro1 aGaravello, Mauro1 aManzo, Rosanna1 aPiccoli, Benedetto uhttp://hdl.handle.net/1963/302200651nas a2200109 4500008004100000245006800041210006400109260001000173520030100183100002100484856003600505 2002 en d00aThe Elliptic Representation of the General Painlevé 6 Equation0 aElliptic Representation of the General Painlevé 6 Equation bSISSA3 aWe study the analytic properties and the critical behavior of the elliptic\r\nrepresentation of solutions of the Painlev\\\'e 6 equation. We solve the\r\nconnection problem for elliptic representation in the generic case and in a\r\nnon-generic case equivalent to WDVV equations of associativity.1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/652300515nas a2200121 4500008004100000245005900041210005500100260006700155520009100222653002300313100002100336856003600357 2002 en d00aThe Elliptic Representation of the Painleve 6 Equation0 aElliptic Representation of the Painleve 6 Equation bKyoto University, Research Institute for Mathematical Sciences3 aWe review our results on the elliptic representation of the sixth Painleve’ equation10aPainleve equations1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/653000435nas a2200121 4500008004100000245008600041210006900127260001800196100001700214700002200231700002400253856003600277 2002 en d00aOn the K+P problem for a three-level quantum system: optimality implies resonance0 aKP problem for a threelevel quantum system optimality implies re bSISSA Library1 aBoscain, Ugo1 aChambrion, Thomas1 aGauthier, Jean-Paul uhttp://hdl.handle.net/1963/160100481nas a2200121 4500008004300000245011600043210006900159260003000228100002000258700002400278700002100302856003600323 2002 en_Ud 00aThe passage from nonconvex discrete systems to variational problems in Sobolev spaces: the one-dimensional case0 apassage from nonconvex discrete systems to variational problems bMAIK Nauka/Interperiodica1 aBraides, Andrea1 aGelli, Maria Stella1 aSigalotti, Mario uhttp://hdl.handle.net/1963/313001123nas a2200121 4500008004100000245011200041210006900153260003100222520054200253653005000795100002100845856013500866 2001 en d00aOn the Critical Behavior, the Connection Problem and the Elliptic Representation of a Painlevé VI Equation0 aCritical Behavior the Connection Problem and the Elliptic Repres bKluwer Academic Publishers3 aIn this paper we find a class of solutions of the sixth Painlevé equation appearing in\r\nthe theory of WDVV equations. This class covers almost all the monodromy data associated to\r\nthe equation, except one point in the space of the data. We describe the critical behavior close to\r\nthe critical points in terms of two parameters and we find the relation among the parameters at\r\nthe different critical points (connection problem). We also study the critical behavior of Painlevé\r\ntranscendents in the elliptic representation.10aPainleve Equations, Isomonodromy deformations1 aGuzzetti, Davide uhttps://www.math.sissa.it/publication/critical-behavior-connection-problem-and-elliptic-representation-painlev%C3%A9-vi-equation-000362nas a2200121 4500008004100000245004000041210003900081260001800120100002100138700001900159700002600178856003600204 2001 en d00aDieletric breakdown: optimal bounds0 aDieletric breakdown optimal bounds bSISSA Library1 aGarroni, Adriana1 aNesi, Vincenzo1 aPonsiglione, Marcello uhttp://hdl.handle.net/1963/156900385nas 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/1963/122801141nas a2200121 4500008004100000245008100041210006900122260002700191520059400218653007100812100002100883856011500904 2001 en d00aInverse Problem and Monodromy Data for Three-Dimensional Frobenius Manifolds0 aInverse Problem and Monodromy Data for ThreeDimensional Frobeniu bRIMS, Kyoto University3 aWe study the inverse problem for semi-simple Frobenius manifolds of dimension 3 and we\r\nexplicitly compute a parametric form of the solutions of theWDVV equations in terms of Painlevé VI\r\ntranscendents. We show that the solutions are labeled by a set of monodromy data. We use our parametric\r\nform to explicitly construct polynomial and algebraic solutions and to derive the generating\r\nfunction of Gromov–Witten invariants of the quantum cohomology of the two-dimensional projective\r\nspace. The procedure is a relevant application of the theory of isomonodromic deformations.10aFrobenius Manifolds, Painleve Equations, Isomonodromy deformations1 aGuzzetti, Davide uhttps://www.math.sissa.it/publication/inverse-problem-and-monodromy-data-three-dimensional-frobenius-manifolds00413nas a2200133 4500008004100000022001400041245004300055210004300098300001400141490000700155100001900162700001700181856008100198 2001 eng d a1120-718300aLie triple systems and warped products0 aLie triple systems and warped products a275–2930 v211 aBertola, Marco1 aGouthier, D. uhttps://www.math.sissa.it/publication/lie-triple-systems-and-warped-products00364nas a2200109 4500008004100000245005900041210005100100260001800151100002500169700002400194856003600218 2001 en d00aOn the subanalyticity of Carnot-Caratheodory distances0 asubanalyticity of CarnotCaratheodory distances bSISSA Library1 aAgrachev, Andrei, A.1 aGauthier, Jean-Paul uhttp://hdl.handle.net/1963/148300444nas a2200133 4500008004100000022001400041245005400055210005400109300001200163490000700175100001900182700002200201856008700223 2001 eng d a0100-356900aWarped products with special Riemannian curvature0 aWarped products with special Riemannian curvature a45–620 v321 aBertola, Marco1 aGouthier, Daniele uhttps://www.math.sissa.it/publication/warped-products-special-riemannian-curvature00587nas a2200169 4500008004100000245010300041210006900144260001800213100001900231700001700250700002400267700001800291700002700309700002300336700002200359856003600381 2000 en d00a3D superconformal theories from Sasakian seven-manifolds: new nontrivial evidences for AdS_4/CFT_30 a3D superconformal theories from Sasakian sevenmanifolds new nont bSISSA Library1 aFabbri, Davide1 aFré, Pietro1 aGualtieri, Leonardo1 aReina, Cesare1 aTomasiello, Alessandro1 aZaffaroni, Alberto1 aZampa, Alessandro uhttp://hdl.handle.net/1963/132700505nas a2200169 4500008004100000022001400041245004100055210004100096300001400137490000800151100001900159700001800178700002100196700001900217700002300236856007600259 2000 eng d a0550-321300aDecomposing quantum fields on branes0 aDecomposing quantum fields on branes a575–6030 v5811 aBertola, Marco1 aBros, Jacques1 aGorini, Vittorio1 aMoschella, Ugo1 aSchaeffer, Richard uhttps://www.math.sissa.it/publication/decomposing-quantum-fields-branes00420nas a2200121 4500008004100000245007100041210006400112260001800176100002400194700002600218700001800244856003600262 2000 en d00aElliptic variational problems in $ R\\\\sp N$ with critical growth0 aElliptic variational problems in Rsp N with critical growth bSISSA Library1 aAmbrosetti, Antonio1 aGarcia Azorero, Jesus1 aPeral, Ireneo uhttp://hdl.handle.net/1963/125800442nas a2200121 4500008004100000245008800041210006900129260001800198100002400216700002600240700001800266856003600284 2000 en d00aExistence and multiplicity results for some nonlinear elliptic equations: a survey.0 aExistence and multiplicity results for some nonlinear elliptic e bSISSA Library1 aAmbrosetti, Antonio1 aGarcia Azorero, Jesus1 aPeral, Ireneo uhttp://hdl.handle.net/1963/146200383nas a2200097 4500008004100000245010000041210006900141260001800210100002100228856003600249 2000 en d00aInverse problem for Semisimple Frobenius Manifolds Monodromy Data and the Painlevé VI Equation0 aInverse problem for Semisimple Frobenius Manifolds Monodromy Dat bSISSA Library1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/155700947nas a2200121 4500008004300000245005900043210005800102260002300160520056700183100002100750700001800771856003600789 2000 en_Ud 00aStability of L^infty Solutions of Temple Class Systems0 aStability of Linfty Solutions of Temple Class Systems bKhayyam Publishing3 aLet $u_t+f(u)_x=0$ be a strictly hyperbolic, genuinely nonlinear system of conservation laws of Temple class. In this paper, a continuous semigroup of solutions is constructed on a domain of $L^\infty$ functions, with possibly unbounded variation. Trajectories depend Lipschitz continuously on the initial data, in the $L^1$ distance. Moreover, we show that a weak solution of the Cauchy problem coincides with the corresponding semigroup trajectory if and only if it satisfies an entropy condition of Oleinik type, concerning the decay of positive waves.

1 aBressan, Alberto1 aGoatin, Paola uhttp://hdl.handle.net/1963/325600604nas a2200133 4500008004100000020001800041245007300059210007000132260002700202520016100229653002300390100002100413856003600434 2000 en d a4-907719-07-800aStokes Matrices for Frobenius Manifolds and the 6 Painlevé Equation0 aStokes Matrices for Frobenius Manifolds and the 6 Painlevé Equat bKobe University, Japan3 aThese notes are a short review on the theory of Frobenius manifolds and its connection to problems of isomonodromy deformations and to Painlev'e equations.10aPainlevé equation1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/654600568nas a2200157 4500008004100000022001400041245007200055210006900127300001400196490000800210100001900218700002100237700001900258700002300277856011000300 1999 eng d a0370-269300aCorrespondence between Minkowski and de Sitter quantum field theory0 aCorrespondence between Minkowski and de Sitter quantum field the a249–2530 v4621 aBertola, Marco1 aGorini, Vittorio1 aMoschella, Ugo1 aSchaeffer, Richard uhttps://www.math.sissa.it/publication/correspondence-between-minkowski-and-de-sitter-quantum-field-theory00817nas a2200133 4500008004100000245009500041210006900136260001000205520036300215100002100578700002700599700002100626856003600647 1999 en d00aA Lipschitz selection from the set of minimizers of a nonconvex functional of the gradient0 aLipschitz selection from the set of minimizers of a nonconvex fu bSISSA3 aA constructive and improved version of the proof that there exist a continuous map that solves the convexified problem is presented. A Lipschitz continuous map is analyzed such that a map vector minimizes the functional at each vector satisfying Cellina\\\'s condition of existence of minimum. This map is explicitly given by a direct constructive algorithm.1 aDal Maso, Gianni1 aGoncharov, Vladimir V.1 aOrnelas, Antonio uhttp://hdl.handle.net/1963/643900990nas a2200121 4500008004300000245007000043210006900113260001300182520059800195100002100793700001800814856003600832 1999 en_Ud 00aOleinik type estimates and uniqueness for n x n conservation laws0 aOleinik type estimates and uniqueness for n x n conservation law bElsevier3 aLet $u_t+f(u)_x=0$ be a strictly hyperbolic $n\\\\times n$ system of conservation laws in one space dimension. Relying on the existence of a semigroup of solutions, we first establish the uniqueness of entropy admissible weak solutions to the Cauchy problem, under a mild assumption on the local oscillation of $u$ in a forward neighborhood of each point in the $t\\\\text{-}x$ plane. In turn, this yields the uniqueness of weak solutions which satisfy a decay estimate on positive waves of genuinely nonlinear families, thus extending a classical result proved by Oleĭnik in the scalar case.1 aBressan, Alberto1 aGoatin, Paola uhttp://hdl.handle.net/1963/337500698nas a2200133 4500008004300000245010800043210006900151260001300220520022700233100002400460700002600484700001800510856003600528 1999 en_Ud 00aPerturbation of $\Delta u+u^{(N+2)/(N-2)}=0$, the scalar curvature problem in $R^N$, and related topics0 aPerturbation of Delta uu N2N2 0 the scalar curvature problem in bElsevier3 aSome nonlinear elliptic equations on $R^N$ which arise perturbing the problem with the critical Sobolev exponent are studied. In particular, some results dealing with the scalar curvature problem in $R^N$ are given.

1 aAmbrosetti, Antonio1 aGarcia Azorero, Jesus1 aPeral, Ireneo uhttp://hdl.handle.net/1963/325500949nas a2200109 4500008004300000245008100043210006900124260001300193520057600206100002100782856003600803 1999 en_Ud 00aStokes matrices and monodromy of the quantum cohomology of projective spaces0 aStokes matrices and monodromy of the quantum cohomology of proje bSpringer3 an this paper we compute Stokes matrices and monodromy of the quantum cohomology of projective spaces. This problem can be formulated in a \\\"classical\\\" framework, as the problem of computation of Stokes matrices and monodromy of differential equations with regular and irregular singularities. We prove that the Stokes\\\' matrix of the quantum cohomology coincides with the Gram matrix in the theory of derived categories of coherent sheaves. We also study the monodromy group of the quantum cohomology and we show that it is related to hyperbolic triangular groups.1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/347501405nas a2200133 4500008004300000245009900043210006900142260001300211520094900224100002001173700002101193700002101214856003601235 1999 en_Ud 00aVariational formulation of softening phenomena in fracture mechanics. The one-dimensional case0 aVariational formulation of softening phenomena in fracture mecha bSpringer3 aStarting from experimental evidence, the authors justify a variational model for softening phenomena in fracture of one-dimensional bars where the energy is given by the contribution and interaction of two terms: a typical bulk energy term depending on elastic strain and a discrete part that depends upon the jump discontinuities that occur in fracture. A more formal, rigorous derivation of the model is presented by examining the $\\\\Gamma$-convergence of discrete energy functionals associated to an array of masses and springs. Close attention is paid to the softening and fracture regimes. \\nOnce the continuous model is derived, it is fully analyzed without losing sight of its discrete counterpart. In particular, the associated boundary value problem is studied and a detailed analysis of the stationary points under the presence of a dead load is performed. A final, interesting section on the scale effect on the model is included.1 aBraides, Andrea1 aDal Maso, Gianni1 aGarroni, Adriana uhttp://hdl.handle.net/1963/337100343nas a2200109 4500008004100000245005200041210004500093260001000138653003100148100001800179856003600197 1998 en d00aOn the Cauchy Problem for the Whitham Equations0 aCauchy Problem for the Whitham Equations bSISSA10aKorteweg de Vries equation1 aGrava, Tamara uhttp://hdl.handle.net/1963/555500524nas a2200157 4500008004100000022001400041245005900055210005900114300001400173490000600187100002300193700001900216700001900235700002100254856009100275 1998 eng d a0202-289300aGeneration of primordial fluctuations in curved spaces0 aGeneration of primordial fluctuations in curved spaces a121–1270 v41 aSchaeffer, Richard1 aMoschella, Ugo1 aBertola, Marco1 aGorini, Vittorio uhttps://www.math.sissa.it/publication/generation-primordial-fluctuations-curved-spaces00446nas a2200121 4500008004100000245009900041210006900140260001800209100002000227700002000247700002100267856003600288 1998 en d00aSpecial functions with bounded variation and with weakly differentiable traces on the jump set0 aSpecial functions with bounded variation and with weakly differe bSISSA Library1 aAmbrosio, Luigi1 aBraides, Andrea1 aGarroni, Adriana uhttp://hdl.handle.net/1963/102500768nas a2200121 4500008004100000245007200041210006900113260001800182520036800200100002100568700002100589856003600610 1997 en d00aShift-differentiability of the flow generated by a conservation law0 aShiftdifferentiability of the flow generated by a conservation l bSISSA Library3 aThe paper introduces a notion of \\\"shift-differentials\\\" for maps with values in the space BV. These differentials describe first order variations of a given functin $u$, obtained by horizontal shifts of the points of its graph. The flow generated by a scalar conservation law is proved to be generically shift-differentiable, according to the new definition.1 aBressan, Alberto1 aGuerra, Graziano uhttp://hdl.handle.net/1963/103300379nas a2200109 4500008004100000245006900041210006900110260001000179653002300189100002100212856003600233 1994 en d00aAsymptotic Behaviour of Dirichlet Problems in Perforated Domains0 aAsymptotic Behaviour of Dirichlet Problems in Perforated Domains bSISSA10aDirichlet problems1 aGarroni, Adriana uhttp://hdl.handle.net/1963/5714