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 Schr{\"o}dinger problem) admits:A threefold dynamical variational representation, in the spirit of the Benamou{\textendash}Brenier formula for the Wasserstein distance; A Hamilton{\textendash}Jacobi{\textendash}Bellman dual representation, in line with Bobkov{\textendash}Gentil{\textendash}Ledoux and Otto{\textendash}Villani results on the duality between Hamilton{\textendash}Jacobi and continuity equation for optimal transport;A Kantorovich-type duality formula, where the Hopf{\textendash}Lax semigroup is replaced by a suitable {\textquoteleft}entropic{\textquoteright} counterpart.We thus provide a complete and unifying picture of the equivalent variational representations of the Schr{\"o}dinger 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.

}, issn = {1432-2064}, doi = {10.1007/s00440-019-00909-1}, url = {https://doi.org/10.1007/s00440-019-00909-1}, author = {Nicola Gigli and Luca Tamanini} } @article {GIULIANI2019324, title = {BlackNUFFT: Modular customizable black box hybrid parallelization of type 3 NUFFT in 3D}, journal = {Computer Physics Communications}, volume = {235}, year = {2019}, pages = {324 - 335}, abstract = {Many 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.

}, keywords = {C++, Extensibility, FFT, Modularity, MPI, MRI image processing, NUFFT type 3, TBB}, issn = {0010-4655}, doi = {https://doi.org/10.1016/j.cpc.2018.10.005}, url = {http://www.sciencedirect.com/science/article/pii/S0010465518303539}, author = {Nicola Giuliani} } @article {GIGLI2019, title = {Differential structure associated to axiomatic Sobolev spaces}, journal = {Expositiones Mathematicae}, year = {2019}, abstract = {The aim of this note is to explain in which sense an axiomatic Sobolev space over a general metric measure space ({\`a} la Gol{\textquoteright}dshtein{\textendash}Troyanov) induces {\textendash} under suitable locality assumptions {\textendash} a first-order differential structure.

}, keywords = {Axiomatic Sobolev space, Cotangent module, Locality of differentials}, issn = {0723-0869}, doi = {https://doi.org/10.1016/j.exmath.2019.01.002}, url = {http://www.sciencedirect.com/science/article/pii/S0723086918300975}, author = {Nicola Gigli and Enrico Pasqualetto} } @conference {2019, title = {Efficient Reduction in Shape Parameter Space Dimension for Ship Propeller Blade Design}, booktitle = {VIII International Conference on Computational Methods in Marine Engineering}, year = {2019}, abstract = {In 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.

}, author = {Andrea Mola and Marco Tezzele and Mahmoud Gadalla and Valdenazzi, Federica and Grassi, Davide and Padovan, Roberta and Gianluigi Rozza} } @article {GENTIL2019, title = {An entropic interpolation proof of the HWI inequality}, journal = {Stochastic Processes and their Applications}, year = {2019}, abstract = {The HWI inequality is an {\textquotedblleft}interpolation{\textquotedblright}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 Schr{\"o}dinger problem. Our approach consists in making rigorous the Otto{\textendash}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.

}, keywords = {Entropic interpolations, Fisher information, Relative entropy, Schr{\"o}dinger problem, Wasserstein distance}, issn = {0304-4149}, doi = {https://doi.org/10.1016/j.spa.2019.04.002}, url = {http://www.sciencedirect.com/science/article/pii/S0304414918303454}, author = {Ivan Gentil and Christian L{\'e}onard and Luigia Ripani and Luca Tamanini} } @article {cotti2019, title = {Isomonodromy deformations at an irregular singularity with coalescing eigenvalues}, journal = {Duke Math. J.}, volume = {168}, number = {6}, year = {2019}, month = {04}, pages = {967{\textendash}1108}, publisher = {Duke University Press}, abstract = {We consider an n{\texttimes}n linear system of ODEs with an irregular singularity of Poincar\'e rank 1 at z=$\infty$, holomorphically depending on parameter t within a polydisc in Cn centred at t=0. The eigenvalues of the leading matrix at z=$\infty$ coalesce along a locus Δ contained in the polydisc, passing through t=0. Namely, z=$\infty$ 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.

}, doi = {10.1215/00127094-2018-0059}, url = {https://doi.org/10.1215/00127094-2018-0059}, author = {Giordano Cotti and Boris Dubrovin and Davide Guzzetti} } @article {gigli_rigoni_2019, title = {A Note About the Strong Maximum Principle on RCD Spaces}, journal = {Canadian Mathematical Bulletin}, volume = {62}, number = {2}, year = {2019}, pages = {259{\textendash}266}, publisher = {Canadian Mathematical Society}, abstract = {We give a direct proof of the strong maximum principle on finite dimensional RCD spaces based on the Laplacian comparison of the squared distance.

}, doi = {10.4153/CMB-2018-022-9}, author = {Nicola Gigli and Chiara Rigoni} } @article {1903.04302, title = {Quasi-continuous vector fields on RCD spaces}, year = {2019}, author = {Cl{\'e}ment Debin and Nicola Gigli and Enrico Pasqualetto} } @article {FEOLA2019932, title = {Reducibility of first order linear operators on tori via Moser{\textquoteright}s theorem}, journal = {Journal of Functional Analysis}, volume = {276}, number = {3}, year = {2019}, pages = {932 - 970}, abstract = {In 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{\textquoteright}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.

}, keywords = {Hyperbolic PDEs, KAM theory, Nash{\textendash}Moser, Reducibility}, issn = {0022-1236}, doi = {https://doi.org/10.1016/j.jfa.2018.10.009}, url = {http://www.sciencedirect.com/science/article/pii/S0022123618303793}, author = {Roberto Feola and Filippo Giuliani and Riccardo Montalto and Michela Procesi} } @article {giantesio2017comparison, title = {A Comparison Between Active Strain and Active Stress in Transversely Isotropic Hyperelastic Materials}, journal = {J. Elast.}, year = {2018}, publisher = {Springer Nature}, author = {Giulia Giantesio and Alessandro Musesti and Davide Riccobelli} } @article {20.500.11767_83906, title = {deal2lkit: A toolkit library for high performance programming in deal.II}, journal = {SOFTWAREX}, volume = {7}, year = {2018}, pages = {318{\textendash}327}, doi = {10.1016/j.softx.2018.09.004}, author = {Alberto Sartori and Nicola Giuliani and Mauro Bardelloni and Luca Heltai} } @article {20.500.11767_81694, title = {The deal.II Library, Version 9.0}, journal = {JOURNAL OF NUMERICAL MATHEMATICS}, year = {2018}, doi = {10.1515/jnma-2018-0054}, url = {https://doi.org/10.1515/jnma-2018-0054}, author = {Giovanni Alzetta and Arndt, Daniel and W. Bangerth and Boddu, Vishal and Brands, Benjamin and Denis Davydov and Gassm{\"o}ller, Rene and Timo Heister and Luca Heltai and Kormann, Katharina and Martin Kronbichler and Matthias Maier and Pelteret, Jean-Paul and B. Turcksin and David Wells} } @article {1807.10063, title = {Differential of metric valued Sobolev maps}, year = {2018}, author = {Nicola Gigli and Enrico Pasqualetto and Elefterios Soultanis} } @inbook {20.500.11767_86398, title = {A distributed lagrange formulation of the finite element immersed boundary method for fluids interacting with compressible solids}, booktitle = {Mathematical and Numerical Modeling of the Cardiovascular System and Applications}, volume = {16}, year = {2018}, pages = {1{\textendash}21}, publisher = {Springer International Publishing}, organization = {Springer International Publishing}, address = {Cham}, doi = {10.1007/978-3-319-96649-6_1}, url = {https://arxiv.org/abs/1712.02545v1}, author = {Boffi, Daniele and Gastaldi, Lucia and Luca Heltai} } @article {GEORGIEV20181551, title = {On fractional powers of singular perturbations of the Laplacian}, journal = {Journal of Functional Analysis}, volume = {275}, number = {6}, year = {2018}, pages = {1551 - 1602}, abstract = {We qualify a relevant range of fractional powers of the so-called Hamiltonian of point interaction in three dimensions, namely the singular perturbation of the negative Laplacian with a contact interaction supported at the origin. In particular we provide an explicit control of the domain of such a fractional operator and of its decomposition into regular and singular parts. We also qualify the norms of the resulting singular fractional Sobolev spaces and their mutual control with the corresponding classical Sobolev norms.

}, keywords = {Point interactions, Regular and singular component of a point-interaction operator, Singular perturbations of the Laplacian}, issn = {0022-1236}, doi = {https://doi.org/10.1016/j.jfa.2018.03.007}, url = {http://www.sciencedirect.com/science/article/pii/S0022123618301046}, author = {Vladimir Georgiev and Alessandro Michelangeli and Raffaele Scandone} } @article {2018, title = {On Geometric Quantum Confinement in Grushin-Like Manifolds}, number = {SISSA;36/2018/MATE}, year = {2018}, note = {16 pages}, abstract = {We 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{\textquoteright}s analysis in each fibre, thus fully characterising the regimes of presence and absence of essential self-adjointness of the associated Laplace-Beltrami operator.}, url = {http://preprints.sissa.it/handle/1963/35322}, author = {Matteo Gallone and Alessandro Michelangeli and Eugenio Pozzoli} } @article {2018, title = {Hydrogenoid Spectra with Central Perturbations}, number = {SISSA;34/2018/MATE}, year = {2018}, note = {Mathematics Subject Classification (2010) 34L10 . 34L15 . 34L16 . 47B15 . 47B25 . 47N20 . 81Q10 . 81Q80}, abstract = {Through the Krein-Vi{\v s}ik-Birman extension scheme, unlike the previous classical analysis based on von Neumann{\textquoteright}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 Sch{\"o}rdinger 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 Krein-Vi{\v s}ik-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.}, url = {http://preprints.sissa.it/handle/1963/35321}, author = {Matteo Gallone and Alessandro Michelangeli} } @article {2018, title = {Local moduli of semisimple Frobenius coalescent structures}, number = {arXiv;1712.08575}, year = {2018}, institution = {SISSA}, abstract = {There 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.

}, url = {http://preprints.sissa.it/handle/1963/35304}, author = {Giordano Cotti and Boris Dubrovin and Davide Guzzetti} } @article {Max_arXiv, title = {A Localized Reduced-Order Modeling Approach for PDEs with Bifurcating Solutions}, journal = {ArXiv e-prints}, year = {2018}, keywords = {Mathematics - Numerical Analysis}, author = {Martin W. Hess and A. Alla and Annalisa Quaini and Gianluigi Rozza and M. Gunzburger} } @conference {tezzele2018model, title = {Model Order Reduction by means of Active Subspaces and Dynamic Mode Decomposition for Parametric Hull Shape Design Hydrodynamics}, booktitle = {Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship \& Maritime Research}, year = {2018}, publisher = {IOS Press}, organization = {IOS Press}, address = {Trieste, Italy}, abstract = {We 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.}, doi = {10.3233/978-1-61499-870-9-569}, url = {http://ebooks.iospress.nl/publication/49270}, author = {Marco Tezzele and Nicola Demo and Mahmoud Gadalla and Andrea Mola and Gianluigi Rozza} } @article {1803.05374, title = {On the notion of parallel transport on RCD spaces}, year = {2018}, author = {Nicola Gigli and Enrico Pasqualetto} } @article {doi:10.1098/rspa.2017.0458, title = {Numerical study of the Kadomtsev-Petviashvili equation and dispersive shock waves}, journal = {Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences}, volume = {474}, number = {2210}, year = {2018}, pages = {20170458}, abstract = {A detailed numerical study of the long time behaviour of dispersive shock waves in solutions to the Kadomtsev{\textendash}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 Schr{\"o}dinger equation in the semiclassical limit.

}, doi = {10.1098/rspa.2017.0458}, url = {https://royalsocietypublishing.org/doi/abs/10.1098/rspa.2017.0458}, author = {Tamara Grava and Christian Klein and Giuseppe Pitton} } @article {bertola2018painleve, title = {Painlev{\'e} IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane}, journal = {Symmetry, Integrability and Geometry. Methods and Applications}, volume = {14}, year = {2018}, publisher = {National Academy of Sciences of Ukraine}, abstract = {We 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{\textasciiacute}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{\textasciiacute}e transcendent is pole-free on a semiaxis.

}, doi = {10.3842/SIGMA.2018.091}, author = {Marco Bertola and Jos{\'e} Gustavo Elias Rebelo and Tamara Grava} } @article {20.500.11767_81735, title = {Predicting and Optimizing Microswimmer Performance from the Hydrodynamics of Its Components: The Relevance of Interactions}, journal = {SOFT ROBOTICS}, volume = {5}, year = {2018}, pages = {410{\textendash}424}, doi = {10.1089/soro.2017.0099}, url = {https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6094362/}, author = {Nicola Giuliani and Luca Heltai and Antonio DeSimone} } @article {Gigli2018, title = {Recognizing the flat torus among RCD*(0,N) spaces via the study of the first cohomology group}, journal = {Calculus of Variations and Partial Differential Equations}, volume = {57}, number = {4}, year = {2018}, month = {Jun}, pages = {104}, abstract = {We 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.

}, issn = {1432-0835}, doi = {10.1007/s00526-018-1377-z}, url = {https://doi.org/10.1007/s00526-018-1377-z}, author = {Nicola Gigli and Chiara Rigoni} } @article {1806.06604, title = {Reducibility for a class of weakly dispersive linear operators arising from the Degasperis Procesi equation}, year = {2018}, author = {Roberto Feola and Filippo Giuliani and Michela Procesi} } @article {gigli2018second, title = {Second order differentiation formula on RCD(K, N) spaces}, journal = {Rendiconti Lincei-Matematica e Applicazioni}, volume = {29}, number = {2}, year = {2018}, pages = {377{\textendash}386}, author = {Nicola Gigli and Luca Tamanini} } @article {1802.02463, title = {Second order differentiation formula on RCD*(K,N) spaces}, year = {2018}, author = {Nicola Gigli and Luca Tamanini} } @conference {demo2018shape, title = {Shape Optimization by means of Proper Orthogonal Decomposition and Dynamic Mode Decomposition}, booktitle = {Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship \& Maritime Research}, year = {2018}, publisher = {IOS Press}, organization = {IOS Press}, chapter = {212}, address = {Trieste, Italy}, abstract = {Shape 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.}, doi = {10.3233/978-1-61499-870-9-212}, url = {http://ebooks.iospress.nl/publication/49229}, author = {Nicola Demo and Marco Tezzele and Gianluca Gustin and Gianpiero Lavini and Gianluigi Rozza} } @article {doi:10.1098/rsta.2017.0424, title = {Symplectic invariants for parabolic orbits and cusp singularities of integrable systems}, journal = {Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences}, volume = {376}, number = {2131}, year = {2018}, pages = {20170424}, abstract = {We 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 {\textquoteleft}Finite dimensional integrable systems: new trends and methods{\textquoteright}.

}, doi = {10.1098/rsta.2017.0424}, url = {https://royalsocietypublishing.org/doi/abs/10.1098/rsta.2017.0424}, author = {Alexey Bolsinov and Lorenzo Guglielmi and Elena Kudryavtseva} } @article {giulianiEtAl2018, title = {π-BEM : A flexible parallel implementation for adaptive , geometry aware , and high order boundary element methods}, journal = {Advances in Engineering Software}, volume = {121}, year = {2018}, pages = {39{\textendash}58}, author = {Nicola Giuliani and Andrea Mola and Luca Heltai} } @article {doi:10.1142/S2010326317400044, title = {Analytic geometry of semisimple coalescent Frobenius structures}, journal = {Random Matrices: Theory and Applications}, volume = {06}, number = {04}, year = {2017}, pages = {1740004}, abstract = {We present some results of a joint paper with Dubrovin (see references), as exposed at the Workshop {\textquotedblleft}Asymptotic and Computational Aspects of Complex Differential Equations{\textquotedblright} 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.

}, doi = {10.1142/S2010326317400044}, url = {https://doi.org/10.1142/S2010326317400044}, author = {Giordano Cotti and Davide Guzzetti} } @article {FONDA20171064, title = {An avoiding cones condition for the Poincar{\'e}{\textendash}Birkhoff Theorem}, journal = {Journal of Differential Equations}, volume = {262}, number = {2}, year = {2017}, pages = {1064 - 1084}, abstract = {We provide a geometric assumption which unifies and generalizes the conditions proposed in [11], [12], so to obtain a higher dimensional version of the Poincar{\'e}{\textendash}Birkhoff fixed point Theorem for Poincar{\'e} maps of Hamiltonian systems.

}, keywords = {Avoiding cones condition, Hamiltonian systems, Periodic solutions, Poincar{\'e}{\textendash}Birkhoff theorem}, issn = {0022-0396}, doi = {https://doi.org/10.1016/j.jde.2016.10.002}, url = {http://www.sciencedirect.com/science/article/pii/S0022039616303278}, author = {Alessandro Fonda and Paolo Gidoni} } @article {2017, title = {Discrete spectra for critical Dirac-Coulomb Hamiltonians}, number = {SISSA;44/2017/MATE}, year = {2017}, abstract = {The 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{\textquoteright}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{\textquoteright}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.}, url = {http://preprints.sissa.it/handle/1963/35300}, author = {Matteo Gallone and Alessandro Michelangeli} } @article { refId0, title = {On the genesis of directional friction through bristle-like mediating elements}, journal = {ESAIM: COCV}, volume = {23}, number = {3}, year = {2017}, pages = {1023-1046}, abstract = {We 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{\textendash}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.

}, doi = {10.1051/cocv/2017030}, url = {https://doi.org/10.1051/cocv/2017030}, author = {Paolo Gidoni and Antonio DeSimone} } @article {2017arXiv170707595A, title = {The injectivity radius of Lie manifolds}, journal = {ArXiv e-prints}, year = {2017}, abstract = {We prove in a direct, geometric way that for any compatible Riemannian metric on a Lie manifold the injectivity radius is positive

}, keywords = {(58J40), 53C21, Mathematics - Differential Geometry}, url = {https://arxiv.org/pdf/1707.07595.pdf}, author = {Paolo Antonini and Guido De Philippis and Nicola Gigli} } @article {2017, title = {Krein-Visik-Birman self-adjoint extension theory revisited}, number = {SISSA;25/2017/MATE}, year = {2017}, abstract = {The 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.}, url = {http://preprints.sissa.it/handle/1963/35286}, author = {Matteo Gallone and Alessandro Michelangeli and Andrea Ottolini} } @article {GIULIANI20175052, title = {Quasi-periodic solutions for quasi-linear generalized KdV equations}, journal = {Journal of Differential Equations}, volume = {262}, number = {10}, year = {2017}, pages = {5052 - 5132}, abstract = {We 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{\textendash}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.

}, keywords = {KAM for PDE{\textquoteright}s, KdV, Nash{\textendash}Moser theory, Quasi-linear PDE{\textquoteright}s, Quasi-periodic solutions}, issn = {0022-0396}, doi = {https://doi.org/10.1016/j.jde.2017.01.021}, url = {http://www.sciencedirect.com/science/article/pii/S0022039617300487}, author = {Filippo Giuliani} } @article {1701.03932, title = {Second order differentiation formula on compact RCD*(K,N) spaces}, year = {2017}, author = {Nicola Gigli and Luca Tamanini} } @article {2017, title = {Self-Adjoint Extensions of Dirac Operator with Coulomb Potential}, number = {SISSA;09/2017/MATE}, year = {2017}, institution = {SISSA}, abstract = {In 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) = {\O}(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.}, url = {http://urania.sissa.it/xmlui/handle/1963/35273}, author = {Matteo Gallone} } @article {2017, title = {Self-adjoint realisations of the Dirac-Coulomb Hamiltonian for heavy nuclei}, number = {SISSA;26/2017/MATE}, year = {2017}, abstract = {We 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 KreIn-Vi{\v s}ik- Birman extension scheme, or also on Grubb{\textquoteright}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.}, url = {http://preprints.sissa.it/handle/1963/35287}, author = {Matteo Gallone and Alessandro Michelangeli} } @article {Gidoni2017, title = {Stasis domains and slip surfaces in the locomotion of a bio-inspired two-segment crawler}, journal = {Meccanica}, volume = {52}, number = {3}, year = {2017}, month = {Feb}, pages = {587{\textendash}601}, abstract = {We 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.

}, issn = {1572-9648}, doi = {10.1007/s11012-016-0408-0}, url = {https://doi.org/10.1007/s11012-016-0408-0}, author = {Paolo Gidoni and Antonio DeSimone} } @article {PhysRevLett.119.033901, title = {Universality of the Peregrine Soliton in the Focusing Dynamics of the Cubic Nonlinear Schr{\"o}dinger Equation}, journal = {Phys. Rev. Lett.}, volume = {119}, year = {2017}, month = {Jul}, pages = {033901}, publisher = {American Physical Society}, doi = {10.1103/PhysRevLett.119.033901}, url = {https://link.aps.org/doi/10.1103/PhysRevLett.119.033901}, author = {Tikan, Alexey and Billet, Cyril and Gennady El and Alexander Tovbis and Marco Bertola and Sylvestre, Thibaut and Gustave, Francois and Randoux, Stephane and Genty, Go{\"e}ry and Suret, Pierre and Dudley, John M.} } @article {1607.05188, title = {Behaviour of the reference measure on RCD spaces under charts}, year = {2016}, author = {Nicola Gigli and Enrico Pasqualetto} } @article {1611.09645, title = {Equivalence of two different notions of tangent bundle on rectifiable metric measure spaces}, year = {2016}, author = {Nicola Gigli and Enrico Pasqualetto} } @article {Fonda2016, title = {Generalizing the Poincar{\'e}{\textendash}Miranda theorem: the avoiding cones condition}, journal = {Annali di Matematica Pura ed Applicata (1923 -)}, volume = {195}, number = {4}, year = {2016}, month = {Aug}, pages = {1347{\textendash}1371}, abstract = {After proposing a variant of the Poincar{\'e}{\textendash}Bohl theorem, we extend the Poincar{\'e}{\textendash}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 \$\${\textpm}1. An illustrative application is provided for the study of functionals having degenerate multi-saddle points.

}, issn = {1618-1891}, doi = {10.1007/s10231-015-0519-6}, url = {https://doi.org/10.1007/s10231-015-0519-6}, author = {Alessandro Fonda and Paolo Gidoni} } @article {fonda2016periodic, title = {Periodic perturbations of Hamiltonian systems}, journal = {Advances in Nonlinear Analysis}, volume = {5}, number = {4}, year = {2016}, pages = {367{\textendash}382}, publisher = {De Gruyter}, abstract = {We prove existence and multiplicity results for periodic solutions of Hamiltonian systems, by the use of a higher dimensional version of the Poincar{\'e}{\textendash}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.

}, doi = {10.1515/anona-2015-0122}, author = {Alessandro Fonda and Maurizio Garrione and Paolo Gidoni} } @inbook {Arici2016, title = {Pimsner Algebras and Circle Bundles}, booktitle = {Noncommutative Analysis, Operator Theory and Applications}, year = {2016}, pages = {1{\textendash}25}, publisher = {Springer International Publishing}, organization = {Springer International Publishing}, address = {Cham}, abstract = {We 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.

}, isbn = {978-3-319-29116-1}, doi = {10.1007/978-3-319-29116-1_1}, url = {https://doi.org/10.1007/978-3-319-29116-1_1}, author = {Francesca Arici and Francesco D{\textquoteright}Andrea and Giovanni Landi}, editor = {Alpay, Daniel and Cipriani, Fabio and Colombo, Fabrizio and Guido, Daniele and Sabadini, Irene and Sauvageot, Jean-Luc} } @article {gottsche2016refined, title = {Refined node polynomials via long edge graphs}, journal = {Communications in Number Theory and Physics}, volume = {10}, number = {2}, year = {2016}, pages = {193{\textendash}234}, publisher = {International Press of Boston}, abstract = {The 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.

}, doi = {10.4310/CNTP.2016.v10.n2.a2}, url = {http://dx.doi.org/10.4310/CNTP.2016.v10.n2.a2}, author = {Lothar G{\"o}ttsche and Benjamin Kipkirui Kikwai} } @article {doi:10.1137/15M1007380, title = {Renormalization for Autonomous Nearly Incompressible BV Vector Fields in Two Dimensions}, journal = {SIAM Journal on Mathematical Analysis}, volume = {48}, number = {1}, year = {2016}, pages = {1-33}, abstract = {Given 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{\textquoteright}s superposition principle [4].

}, doi = {10.1137/15M1007380}, url = {https://doi.org/10.1137/15M1007380}, author = {Stefano Bianchini and Paolo Bonicatto and N.A. Gusev} } @mastersthesis {2016, title = {Two explorations in Dynamical Systems and Mechanics}, year = {2016}, school = {SISSA}, abstract = {This 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".}, keywords = {Poincar{\'e}-Birkhoff Theorem}, author = {Paolo Gidoni} } @article {2015, title = {Deal2lkit: a Toolkit Library for High Performance Programming in deal.II}, year = {2015}, publisher = {SISSA}, abstract = {We 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.}, url = {http://urania.sissa.it/xmlui/handle/1963/35006}, author = {Alberto Sartori and Nicola Giuliani and Mauro Bardelloni and Luca Heltai} } @article {MR3369874, title = {A degeneration of two-phase solutions of the focusing nonlinear Schr{\"o}dinger equation via Riemann-Hilbert problems}, journal = {J. Math. Phys.}, volume = {56}, number = {6}, year = {2015}, pages = {061507, 17}, issn = {0022-2488}, doi = {10.1063/1.4922362}, url = {http://dx.doi.org/10.1063/1.4922362}, author = {Marco Bertola and Giavedoni, Pietro} } @article {2015, title = {FEM SUPG stabilisation of mixed isoparametric BEMs: application to linearised free surface flows}, journal = {Engineering Analysis with Boundary Elements 59 (2015), pp. 8-22}, year = {2015}, abstract = {In finite element formulations, transport dominated problems are often stabilised through the Streamline-Upwind-Petrov{\textendash}Galerkin (SUPG) method. Its application is straightforward when the problem at hand is solved using Galerkin methods. Applications of boundary integral formulations often resort to collocation techniques which are computationally more tractable. In this framework, the Galerkin method and the stabilisation may still be used to successfully apply boundary conditions and resolve instabilities that are frequently observed in transport dominated problems. We apply this technique to an adaptive collocation boundary element method for the solution of stationary potential flows, where we solve a mixed Poisson problem in boundary integral form, with the addition of linearised free surface boundary conditions. We use a mixed boundary element formulation to allow for different finite dimensional spaces describing the flow potential and its normal derivative, and we validate our method simulating the flow around both a submerged body and a surface piercing body. The coupling of mixed surface finite elements and strongly consistent stabilisation techniques with boundary elements opens up the possibility to use non conformal unstructured grids with local refinement, without introducing the inconsistencies of other stabilisation techniques based on up-winding and finite difference schemes.

}, doi = {10.1016/j.enganabound.2015.04.006}, url = {http://urania.sissa.it/xmlui/handle/1963/34466}, author = {Nicola Giuliani and Andrea Mola and Luca Heltai and L. Formaggia} } @article {agrachev2015geodesics, title = {Geodesics and horizontal-path spaces in Carnot groups}, journal = {Geometry \& Topology}, volume = {19}, number = {3}, year = {2015}, pages = {1569{\textendash}1630}, publisher = {Mathematical Sciences Publishers}, abstract = {We 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.

}, doi = {10.2140/gt.2015.19.1569}, author = {Andrei A. Agrachev and Alessandro Gentile and Antonio Lerario} } @article {DESIMONE2015254, title = {Liquid crystal elastomer strips as soft crawlers}, journal = {Journal of the Mechanics and Physics of Solids}, volume = {84}, year = {2015}, pages = {254 - 272}, abstract = {In 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, {\textquoteleft}breathing-like{\textquoteright} 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.

}, keywords = {Crawling motility, Directional surfaces, Frictional interactions, Liquid crystal elastomers, Soft biomimetic robots}, issn = {0022-5096}, doi = {https://doi.org/10.1016/j.jmps.2015.07.017}, url = {http://www.sciencedirect.com/science/article/pii/S0022509615300430}, author = {Antonio DeSimone and Paolo Gidoni and Giovanni Noselli} } @article {FONDA201573, title = {A permanence theorem for local dynamical systems}, journal = {Nonlinear Analysis: Theory, Methods \& Applications}, volume = {121}, year = {2015}, note = {Nonlinear Partial Differential Equations, in honor of Enzo Mitidieri for his 60th birthday}, pages = {73 - 81}, abstract = {We 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{\textendash}Volterra predator{\textendash}prey model with intraspecific competition.

}, keywords = {Lotka{\textendash}Volterra, permanence, Predator{\textendash}prey, Uniform persistence}, issn = {0362-546X}, doi = {https://doi.org/10.1016/j.na.2014.10.011}, url = {http://www.sciencedirect.com/science/article/pii/S0362546X14003332}, author = {Alessandro Fonda and Paolo Gidoni} } @article {2014, title = {Approximate Hermitian{\textendash}Yang{\textendash}Mills structures on semistable principal Higgs bundles}, number = {Annals of global analysis and geometry;volume 47; issue 1; pp 1-11}, year = {2014}, publisher = {Springer}, abstract = {We 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.}, doi = {10.1007/s10455-014-9433-1}, url = {http://urania.sissa.it/xmlui/handle/1963/34645}, author = {Ugo Bruzzo and Beatriz Grana-Otero} } @article {MR3162486, title = {Cauchy-Laguerre two-matrix model and the Meijer-G random point field}, journal = {Comm. Math. Phys.}, volume = {326}, number = {1}, year = {2014}, pages = {111{\textendash}144}, issn = {0010-3616}, doi = {10.1007/s00220-013-1833-8}, url = {http://dx.doi.org/10.1007/s00220-013-1833-8}, author = {Marco Bertola and Gekhtman, M. and Szmigielski, J.} } @article {2014, title = {Conformal invariants from nodal sets. I. negative eigenvalues and curvature prescription}, number = {International Mathematics Research Notices;volume 2014; issue 9; pages 2356-2400;}, year = {2014}, publisher = {Oxford University Press}, abstract = {In 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{\textquoteright}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.}, doi = {10.1093/imrn/rns295}, url = {http://urania.sissa.it/xmlui/handle/1963/35128}, author = {Rod R. Gover and Yaiza Canzani and Dmitry Jakobson and Rapha{\"e}l Ponge and Andrea Malchiodi} } @article {GIDONI201465, title = {Crawling on directional surfaces}, journal = {International Journal of Non-Linear Mechanics}, volume = {61}, year = {2014}, pages = {65 - 73}, abstract = {In 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 {\textquoteleft}along the grain{\textquoteright}, and high resistance when sliding {\textquoteleft}against the grain{\textquoteright}. 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.

}, keywords = {Bio-mimetic micro-robots, Cell migration, Crawling motility, Directional surfaces, Self-propulsion}, issn = {0020-7462}, doi = {https://doi.org/10.1016/j.ijnonlinmec.2014.01.012}, url = {http://www.sciencedirect.com/science/article/pii/S0020746214000213}, author = {Paolo Gidoni and Giovanni Noselli and Antonio DeSimone} } @article {2014, title = {An effective model for nematic liquid crystal composites with ferromagnetic inclusions}, number = {SIAM Journal on Applied Mathematics;volume 74; issue 2; pages 237-262;}, year = {2014}, publisher = {Society for Industrial and Applied Mathematics Publications}, abstract = {Molecules 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.}, doi = {10.1137/130910348}, url = {http://urania.sissa.it/xmlui/handle/1963/34940}, author = {Maria Carme Calderer and Antonio DeSimone and Dmitry Golovaty and Alexander Panchenko} } @article {2014, title = {A Review of the Sixth Painlev{\'e} Equation}, number = {Constructive approximation;volume 41; issue 3; pages 495-527;}, year = {2014}, publisher = {Springer}, abstract = {For the Painlev{\'e} 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.}, doi = {10.1007/s00365-014-9250-6}, url = {http://urania.sissa.it/xmlui/handle/1963/34658}, author = {Davide Guzzetti} } @article {2014, title = {Spontaneous division and motility in active nematic droplets}, number = {Physical review letters;volume 112; issue 14; article number 147802;}, year = {2014}, publisher = {American Physical Society}, abstract = {We 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.}, doi = {10.1103/PhysRevLett.112.147802}, url = {http://urania.sissa.it/xmlui/handle/1963/34902}, author = {Luca Giomi and Antonio DeSimone} } @article {2014, title = {Steady nearly incompressible vector elds in 2D: chain rule and renormalization}, year = {2014}, institution = {SISSA}, author = {Stefano Bianchini and N.A. Gusev} } @article {2013, title = {Crawlers in viscous environments: linear vs nonlinear rheology}, journal = {International Journal of Non-Linear Mechanics 56, 142-147 (2013)}, year = {2013}, publisher = {Elsevier}, abstract = {We 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.}, doi = {10.1016/j.ijnonlinmec.2013.02.007}, author = {Antonio DeSimone and Federica Guarnieri and Giovanni Noselli and Amabile Tatone} } @article {10978, title = {On critical behaviour in systems of Hamiltonian partial differential equations}, year = {2013}, institution = {SISSA}, abstract = {We 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.

}, author = {Boris Dubrovin and Tamara Grava and Christian Klein and Antonio Moro} } @article {2013, title = {Defect annihilation and proliferation in active nematics}, number = {arXiv:1303.4720;}, year = {2013}, note = {5 pages, 4 figures}, institution = {SISSA}, abstract = {Liquid 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.}, url = {http://hdl.handle.net/1963/6566}, author = {Luca Giomi and Mark J. Bowick and Xu Ma and M. Cristina Marchetti} } @article {fonda2013, title = {Generalized Sturm-Liouville boundary conditions for first order differential systems in the plane}, journal = {Topol. Methods Nonlinear Anal.}, volume = {42}, number = {2}, year = {2013}, pages = {293{\textendash}325}, publisher = {Nicolaus Copernicus University, Juliusz P. Schauder Centre for Nonlinear Studies}, abstract = {We study asymptotically positively homogeneous first order systems in the plane, with boundary conditions which are positively homogeneous, as well. Defining a generalized concept of Fu{\v c}{\'\i}k 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.

}, url = {https://projecteuclid.org:443/euclid.tmna/1461248981}, author = {Alessandro Fonda and Maurizio Garrione} } @article {Boscain2013, title = {Lipschitz Classification of Almost-Riemannian Distances on Compact Oriented Surfaces}, journal = {Journal of Geometric Analysis}, volume = {23}, number = {1}, year = {2013}, month = {Jan}, pages = {438{\textendash}455}, abstract = {Two-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{\textendash}Carath{\'e}odory 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.

}, issn = {1559-002X}, doi = {10.1007/s12220-011-9262-4}, url = {https://doi.org/10.1007/s12220-011-9262-4}, author = {Ugo Boscain and Gr{\'e}goire Charlot and Roberta Ghezzi and Mario Sigalotti} } @article {Boscaggin2013, title = {Planar Hamiltonian systems at resonance: the Ahmad{\textendash}Lazer{\textendash}Paul condition}, journal = {Nonlinear Differential Equations and Applications NoDEA}, volume = {20}, number = {3}, year = {2013}, month = {Jun}, pages = {825{\textendash}843}, abstract = {We 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{\textendash}Lazer condition is analyzed, as well.

}, issn = {1420-9004}, doi = {10.1007/s00030-012-0181-2}, url = {https://doi.org/10.1007/s00030-012-0181-2}, author = {Alberto Boscaggin and Maurizio Garrione} } @article {2013, title = {Softly Constrained Films}, number = {arXiv:1304.1077;}, year = {2013}, note = {Review article, 21 pages, 16 figures, submitted to Soft Matter}, publisher = {SISSA}, abstract = {The 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.}, url = {http://hdl.handle.net/1963/6563}, author = {Luca Giomi} } @article {2013, title = {The splitting theorem in non-smooth context}, year = {2013}, abstract = {We 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 {\textquoteleft}infinitesimally Hilbertian{\textquoteright} 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.}, url = {http://preprints.sissa.it/handle/1963/35306}, author = {Nicola Gigli} } @article {MR3088819, title = {Strong asymptotics for Cauchy biorthogonal polynomials with application to the Cauchy two-matrix model}, journal = {J. Math. Phys.}, volume = {54}, number = {4}, year = {2013}, pages = {043517, 25}, issn = {0022-2488}, author = {Marco Bertola and Gekhtman, M. and Szmigielski, J.} } @article {10979, title = {On the tritronqu{\'e}e solutions of P$_I^2$}, year = {2013}, institution = {SISSA}, abstract = {For 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.

}, author = {Tamara Grava and Andrey Kapaev and Christian Klein} } @article {2012, title = {On 2-step, corank 2 nilpotent sub-Riemannian metrics}, journal = {SIAM J. Control Optim., 50 (2012) 559{\textendash}582}, number = {arXiv:1105.5766;}, year = {2012}, publisher = {Society for Industrial and Applied Mathematics}, abstract = {In 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.}, doi = {10.1137/110835700}, url = {http://hdl.handle.net/1963/6065}, author = {Davide Barilari and Ugo Boscain and Jean-Paul Gauthier} } @article {2012, title = {On a class of vector fields with discontinuity of divide-by-zero type and its applications}, journal = {Journal of dynamical and control systems }, volume = {18}, number = {arXiv:1007.0912;}, year = {2012}, pages = {135-158}, publisher = {Springer}, abstract = {We 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.

}, doi = {10.1007/s10883-012-9137-4}, author = {Roberta Ghezzi and Alexey O. Remizov} } @article {2012, title = {The KdV hierarchy: universality and a Painleve transcendent}, journal = {International Mathematics Research Notices, vol. 22 (2012) , page 5063-5099}, number = {arXiv:1101.2602;}, year = {2012}, note = {This article was published in "International Mathematics Research Notices, vol. 22 (2012) , page 5063-5099}, publisher = {Oxford University Press}, abstract = {We 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.}, keywords = {Small-Dispersion limit}, url = {http://hdl.handle.net/1963/6921}, author = {Tom Claeys and Tamara Grava} } @article {2012, title = {Non-uniqueness results for critical metrics of regularized determinants in four dimensions}, journal = {Communications in Mathematical Physics, Volume 315, Issue 1, September 2012, Pages 1-37}, number = {arXiv:1105.3762;}, year = {2012}, note = {35 pages, title changed, added determinant of half-torsion, references added. Comm. Math. Phys., to appear}, publisher = {Springer}, abstract = {The 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{\textquoteright}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.}, doi = {10.1007/s00220-012-1535-7}, url = {http://hdl.handle.net/1963/6559}, author = {Matthew Gursky and Andrea Malchiodi} } @article {2012, title = {Numerical study of the small dispersion limit of the Korteweg-de Vries equation and asymptotic solutions}, journal = {Physica D 241, nr. 23-24 (2012): 2246-2264}, number = {arXiv:1202.0962;}, year = {2012}, publisher = {Elsevier}, abstract = {We 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.}, keywords = {Korteweg-de Vries equation}, doi = {10.1016/j.physd.2012.04.001}, author = {Tamara Grava and Christian Klein} } @article {2012, title = {Poles Distribution of PVI Transcendents close to a Critical Point (summer 2011)}, journal = {Physica D: Nonlinear Phenomena, Volume 241, Issue 23-24, 1 December 2012, Pages 2188-2203}, number = {arXiv:1104.5066;}, year = {2012}, publisher = {Elsevier}, abstract = {The distribution of the poles of Painlev{\'e} 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.}, keywords = {Painleve{\textquoteright} equations}, doi = {doi:10.1016/j.physd.2012.02.015}, url = {http://hdl.handle.net/1963/6526}, author = {Davide Guzzetti} } @article {garrione2012, title = {Resonance at the first eigenvalue for first-order systems in the plane: vanishing Hamiltonians and the Landesman-Lazer condition}, journal = {Differential Integral Equations}, volume = {25}, number = {5/6}, year = {2012}, month = {05}, pages = {505{\textendash}526}, publisher = {Khayyam Publishing, Inc.}, url = {https://projecteuclid.org:443/euclid.die/1356012676}, author = {Maurizio Garrione} } @article {2012, title = {A Review on The Sixth Painlev{\'e} Equation}, number = {arXiv:1210.0311;}, year = {2012}, note = {31 pages, 10 figures}, publisher = {SISSA}, abstract = {For 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.

}, keywords = {Painlev{\'e} equation}, url = {http://hdl.handle.net/1963/6525}, author = {Davide Guzzetti} } @article {2012, title = {Solving the Sixth Painlev{\'e} Equation: Towards the Classification of all the Critical Behaviors and the Connection Formulae}, journal = {Int Math Res Notices (2012) 2012 (6): 1352-1413}, number = {arXiv:1010.1895;}, year = {2012}, note = {53 pages, 2 figures}, publisher = {Oxford University Press}, abstract = {The 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.}, doi = {10.1093/imrn/rnr071}, url = {http://hdl.handle.net/1963/6093}, author = {Davide Guzzetti} } @article {2012, title = {Tabulation of Painlev{\'e} 6 transcendents}, journal = {Nonlinearity, Volume 25, Issue 12, December 2012, Pages 3235-3276}, number = {arXiv:1108.3401;}, year = {2012}, note = {30 pages, 1 figure; this article was published in "Nonlinearity" in 2012}, publisher = {IOP Publishing}, abstract = {The critical and asymptotic behaviors of solutions of the sixth Painlev{\textquoteright}e equation PVI, obtained in the framework of the monodromy preserving deformation method, and their explicit parametrization in terms of monodromy data, are tabulated.}, doi = {10.1088/0951-7715/25/12/3235}, url = {http://hdl.handle.net/1963/6520}, author = {Davide Guzzetti} } @article {2011, title = {An asymptotic reduction of a Painlev{\'e} VI equation to a Painlev{\'e} III}, journal = {J.Phys.A: Math.Theor. 44 (2011) 215203}, number = {arXiv:1101.4705;}, year = {2011}, publisher = {IOP Publishing}, abstract = {When 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.}, doi = {10.1088/1751-8113/44/21/215203}, url = {http://hdl.handle.net/1963/5124}, author = {Davide Guzzetti} } @article {2011, title = {Axial symmetry of some steady state solutions to nonlinear Schr{\"o}dinger equations}, journal = {Proc. Amer. Math. Soc. 139 (2011), 1023-1032}, number = {SISSA;75/2010/M}, year = {2011}, publisher = {American Mathematical Society}, abstract = {In 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.}, keywords = {Nonlinear Schr{\"o}dinger equation}, doi = {10.1090/S0002-9939-2010-10638-X}, url = {http://hdl.handle.net/1963/4100}, author = {Changfeng Gui and Andrea Malchiodi and Haoyuan Xu and Paul Yang} } @article {FONDA20111052, title = {Double resonance with Landesman{\textendash}Lazer conditions for planar systems of ordinary differential equations}, journal = {Journal of Differential Equations}, volume = {250}, number = {2}, year = {2011}, pages = {1052 - 1082}, abstract = {We 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{\textendash}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{\textquoteright}s results in Fabry (1995) [10], for scalar equations with double resonance with respect to the Dancer{\textendash}Fu{\v c}ik spectrum.

}, keywords = {Double resonance, Landesman{\textendash}Lazer conditions, Nonlinear planar systems}, issn = {0022-0396}, doi = {https://doi.org/10.1016/j.jde.2010.08.006}, url = {http://www.sciencedirect.com/science/article/pii/S0022039610002901}, author = {Alessandro Fonda and Maurizio Garrione} } @article {2011, title = {An Estimate on the Flow Generated by Monotone Operators}, journal = {Communications in Partial Differential Equations 36 (2011) 777-796}, number = {SISSA;29/2009/M}, year = {2011}, publisher = {Taylor \& Francis}, doi = {10.1080/03605302.2010.534224}, url = {http://hdl.handle.net/1963/3646}, author = {Stefano Bianchini and Matteo Gloyer} } @article {2011, title = {The geometry of Maximum Principle}, journal = {Proceedings of the Steklov Institute of mathematics. vol. 273 (2011), page: 5-27 ; ISSN: 0081-5438}, number = {Proceedings of the Steklov Institute of mathematics;v.273}, year = {2011}, abstract = {An invariant formulation of the maximum principle in optimal control is presented, and some second-order invariants are discussed.}, url = {http://hdl.handle.net/1963/6456}, author = {Andrei A. Agrachev and Revaz Gamkrelidze} } @article {Mola2011, title = {Multi-physics modelling and sensitivity analysis of olympic rowing boat dynamics}, journal = {Sports Engineering}, volume = {14}, number = {2-4}, year = {2011}, month = {nov}, pages = {85{\textendash}94}, publisher = {Springer Nature}, doi = {10.1007/s12283-011-0075-2}, url = {https://doi.org/10.1007/s12283-011-0075-2}, author = {Andrea Mola and Mehdi Ghommem and Muhammad R. Hajj} } @article {fonda2011nonlinear, title = {Nonlinear resonance: a comparison between Landesman-Lazer and Ahmad-Lazer-Paul conditions}, journal = {Advanced Nonlinear Studies}, volume = {11}, number = {2}, year = {2011}, pages = {391{\textendash}404}, publisher = {Advanced Nonlinear Studies, Inc.}, abstract = {We 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.

}, doi = {10.1515/ans-2011-0209}, author = {Alessandro Fonda and Maurizio Garrione} } @article {2011, title = {Numerical Study of breakup in generalized Korteweg-de Vries and Kawahara equations}, journal = {SIAM J. Appl. Math. 71 (2011) 983-1008}, number = {arXiv:1101.0268;}, year = {2011}, publisher = {SIAM}, abstract = {This article is concerned with a conjecture in [B. Dubrovin, Comm. Math. Phys., 267 (2006), pp. 117{\textendash}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 Painlev{\'e}-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.}, doi = {10.1137/100819783}, url = {http://hdl.handle.net/1963/4951}, author = {Boris Dubrovin and Tamara Grava and Christian Klein} } @article {2011, title = {Q-factorial Laurent rings}, number = {arXiv:1108.4116v1;}, year = {2011}, note = {5 pages}, institution = {SISSA}, abstract = {Dolgachev 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.}, url = {http://hdl.handle.net/1963/4183}, author = {Ugo Bruzzo and Antonella Grassi} } @article {garrione2011resonance, title = {Resonance and Landesman-Lazer conditions for first order systems in R^2}, journal = {Le Matematiche}, volume = {66}, number = {1}, year = {2011}, pages = {153{\textendash}160}, abstract = {The 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].

}, author = {Maurizio Garrione} } @article {BOSCAGGIN20114166, title = {Resonance and rotation numbers for planar Hamiltonian systems: Multiplicity results via the Poincar{\'e}{\textendash}Birkhoff theorem}, journal = {Nonlinear Analysis: Theory, Methods \& Applications}, volume = {74}, number = {12}, year = {2011}, pages = {4166 - 4185}, abstract = {In the general setting of a planar first order system (0.1)u'=G(t,u),u∈R2, with G:[0,T]{\texttimes}R2{\textrightarrow}R2, we study the relationships between some classical nonresonance conditions (including the Landesman{\textendash}Lazer one) {\textemdash} at infinity and, in the unforced case, i.e. G(t,0)=0, at zero {\textemdash} and the rotation numbers of {\textquotedblleft}large{\textquotedblright} and {\textquotedblleft}small{\textquotedblright} solutions of (0.1), respectively. Such estimates are then used to establish, via the Poincar{\'e}{\textendash}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{\textendash}Lazer condition, we obtain sharp conclusions when the system is resonant at infinity.

}, keywords = {Multiple periodic solutions, Poincar{\'e}{\textendash}Birkhoff theorem, Resonance, Rotation number}, issn = {0362-546X}, doi = {https://doi.org/10.1016/j.na.2011.03.051}, url = {http://www.sciencedirect.com/science/article/pii/S0362546X11001817}, author = {Alberto Boscaggin and Maurizio Garrione} } @article {2011, title = {Semistable and numerically effective principal (Higgs) bundles}, journal = {Advances in Mathematics 226 (2011) 3655-3676}, number = {SISSA;27/2009/FM}, year = {2011}, publisher = {Elsevier}, abstract = {We 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{\textendash}Yang{\textendash}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.}, doi = {10.1016/j.aim.2010.10.026}, url = {http://hdl.handle.net/1963/3638}, author = {Ugo Bruzzo and Beatriz Grana-Otero} } @inbook {2011, title = {Solving PVI by Isomonodromy Deformations}, booktitle = {Painlevé equations and related topics : proceedings of the international conference, Saint Petersburg, Russia, June 17-23, 2011 / Aleksandr Dmitrievich Briuno; Alexander B Batkhin. - Berlin : De Gruyter, [2012]. - p. 101-105}, number = {arXiv:1106.2636;}, year = {2011}, note = {12 pages, 1 figurethis paper has been}, publisher = {SISSA}, organization = {SISSA}, abstract = {The 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.}, keywords = {Painlevé Equations}, isbn = {9783110275582}, url = {http://hdl.handle.net/1963/6522}, author = {Davide Guzzetti} } @article {2011, title = {The sphere and the cut locus at a tangency point in two-dimensional almost-Riemannian geometry}, journal = {Journal of Dynamical and Control Systems }, volume = {17 }, number = {arXiv:1009.2612;}, year = {2011}, pages = {141-161}, publisher = {Springer}, abstract = {We 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.

}, doi = {10.1007/s10883-011-9113-4}, url = {http://hdl.handle.net/1963/4914}, author = {Bernard Bonnard and Gr{\'e}goire Charlot and Roberta Ghezzi and Gabriel Janin} } @mastersthesis {2010, title = {Almost-Riemannian Geometry from a Control Theoretical Viewpoint}, year = {2010}, school = {SISSA}, url = {http://hdl.handle.net/1963/4705}, author = {Roberta Ghezzi} } @article {Bertola:CBOPs, title = {Cauchy biorthogonal polynomials}, journal = {J. Approx. Theory}, volume = {162}, number = {4}, year = {2010}, pages = {832{\textendash}867}, issn = {0021-9045}, doi = {10.1016/j.jat.2009.09.008}, url = {http://0-dx.doi.org.mercury.concordia.ca/10.1016/j.jat.2009.09.008}, author = {Marco Bertola and Gekhtman, M. and Szmigielski, J.} } @article {2010, title = {Chern-Simons theory on L(p,q) lens spaces and Gopakumar-Vafa duality}, journal = {J. Geom. Phys. 60 (2010) 417-429}, number = {SISSA;56/2008/FM}, year = {2010}, abstract = {We 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.}, doi = {10.1016/j.geomphys.2009.11.006}, url = {http://hdl.handle.net/1963/2938}, author = {Andrea Brini and Luca Griguolo and Domenico Seminara and Alessandro Tanzini} } @article {2010, title = {Concentration of solutions for some singularly perturbed mixed problems: Asymptotics of minimal energy solutions}, journal = {Ann. Inst. H. Poincare Anal. Non Lineaire 27 (2010) 37-56}, number = {SISSA;03/2009/M}, year = {2010}, abstract = {In 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.}, doi = {10.1016/j.anihpc.2009.06.005}, url = {http://hdl.handle.net/1963/3409}, author = {Jesus Garcia Azorero and Andrea Malchiodi and Luigi Montoro and Ireneo Peral} } @article {2010, title = {Concentration of solutions for some singularly perturbed mixed problems. Part I: existence results}, journal = {Arch. Ration. Mech. Anal. 196 (2010) 907-950}, number = {SISSA;02/2009/M}, year = {2010}, abstract = {In 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.}, doi = {10.1007/s00205-009-0259-0}, url = {http://hdl.handle.net/1963/3406}, author = {Jesus Garcia Azorero and Andrea Malchiodi and Luigi Montoro and Ireneo Peral} } @article {2010, title = {On the Euler-Lagrange equation for a variational problem : the general case II}, journal = {Math. Z. 265 (2010) 889-923}, number = {SISSA;75/2007/M}, year = {2010}, doi = {10.1007/s00209-009-0547-2}, url = {http://hdl.handle.net/1963/2551}, author = {Stefano Bianchini and Matteo Gloyer} } @article {2010, title = {Gene expression analysis of the emergence of epileptiform activity after focal injection of kainic acid into mouse hippocampus.}, journal = {The European journal of neuroscience. 2010 Oct; 32(8):1364-79}, number = {PMID:20950280;}, year = {2010}, publisher = {Wiley}, abstract = {We 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.

}, doi = {10.1111/j.1460-9568.2010.07403.x}, url = {http://hdl.handle.net/1963/4480}, author = {Dario Motti and Caroline Le Duigou and Nicole Chemaly and Lucia Wittner and Dejan Lazarevic and Helena Krmac and Troels Torben Marstrand and Eivind Valen and Remo Sanges and Elia Stupka and Albin Sandelin and Enrico Cherubini and Stefano Gustincich and Richard Miles} } @article {2010, title = {Homogeneous binary trees as ground states of quantum critical Hamiltonians}, journal = {Phys. Rev. A 81 (2010) 062335}, number = {arXiv.org;0912.0466v2}, year = {2010}, publisher = {American Physical Society}, abstract = {Many-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.

}, doi = {10.1103/PhysRevA.81.062335}, url = {http://hdl.handle.net/1963/3909}, author = {Pietro Silvi and Vittorio Giovannetti and Simone Montangero and Matteo Rizzi and J. Ignacio Cirac and Rosario Fazio} } @article {2010, title = {Homogeneous multiscale entanglement renormalization ansatz tensor networks for quantum critical systems}, journal = {New J. Phys. 12 (2010) 075018}, year = {2010}, publisher = {IOP Publishing}, abstract = {In 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.

}, doi = {10.1088/1367-2630/12/7/075018}, url = {http://hdl.handle.net/1963/4067}, author = {Matteo Rizzi and Simone Montangero and Pietro Silvi and Vittorio Giovannetti and Rosario Fazio} } @article {2010, title = {Lorentz Covariant k-Minkowski Spacetime}, journal = {Phys. Rev. D 81 (2010) 125024}, number = {SISSA;04/2010/FM}, year = {2010}, abstract = {In 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.}, doi = {10.1103/PhysRevD.81.125024}, url = {http://hdl.handle.net/1963/3829}, author = {Ludwik Dabrowski and Michal Godlinski and Gherardo Piacitelli} } @article {boscain2010normal, title = {A normal form for generic 2-dimensional almost-Riemannian structures at a tangency point}, journal = {arXiv preprint arXiv:1008.5036}, year = {2010}, author = {Ugo Boscain and Gr{\'e}goire Charlot and Roberta Ghezzi} } @article {2010, title = {Numerical Solution of the Small Dispersion Limit of the Camassa-Holm and Whitham Equations and Multiscale Expansions}, number = {SISSA;10/2010/FM}, year = {2010}, abstract = {The 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....}, url = {http://hdl.handle.net/1963/3840}, author = {Simonetta Abenda and Tamara Grava and Christian Klein} } @article {2010, title = {Painlev{\'e} II asymptotics near the leading edge of the oscillatory zone for the Korteweg-de Vries equation in the small-dispersion limit}, journal = {Comm. Pure Appl. Math. 63 (2010) 203-232}, number = {arXiv.org;0812.4142v1}, year = {2010}, publisher = {Wiley}, abstract = {In 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.}, doi = {10.1002/cpa.20277}, url = {http://hdl.handle.net/1963/3799}, author = {Tom Claeys and Tamara Grava} } @article {2010, title = {Picard group of hypersurfaces in toric varieties}, number = {SISSA;78/2010/FM}, year = {2010}, abstract = {We 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.}, url = {http://hdl.handle.net/1963/4103}, author = {Ugo Bruzzo and Antonella Grassi} } @article {2010, title = {Riemann-Roch theorems and elliptic genus for virtually smooth schemes}, journal = {Geom. Topol. 14 (2010) 83-115}, number = {arXiv.org;0706.0988v1}, year = {2010}, publisher = {Mathematical Sciences Publishers}, abstract = {For 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.}, doi = {10.2140/gt.2010.14.83}, url = {http://hdl.handle.net/1963/3888}, author = {Barbara Fantechi and Lothar G{\"o}ttsche} } @article {2010, title = {Solitonic asymptotics for the Korteweg-de Vries equation in the small dispersion limit}, journal = {SIAM J. Math. Anal. 42 (2010) 2132-2154}, number = {SISSA;09/2010/FM}, year = {2010}, abstract = {We 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.}, doi = {10.1137/090779103}, url = {http://hdl.handle.net/1963/3839}, author = {Tamara Grava and Tom Claeys} } @article {2010, title = {Two-dimensional almost-Riemannian structures with tangency points}, journal = {Ann. Inst. H. Poincare Anal. Non Lineaire }, volume = {27}, number = {arXiv.org;0908.2564v1}, year = {2010}, pages = {793-807}, publisher = {Elsevier}, abstract = {Two-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.

}, doi = {10.1016/j.anihpc.2009.11.011}, url = {http://hdl.handle.net/1963/3870}, author = {Andrei A. Agrachev and Ugo Boscain and Gr{\'e}goire Charlot and Roberta Ghezzi and Mario Sigalotti} } @article {Bertola:CauchyMM, title = {The Cauchy two{\textendash}matrix model}, journal = {Comm. Math. Phys.}, volume = {287}, number = {3}, year = {2009}, pages = {983{\textendash}1014}, author = {Marco Bertola and M Gekhtman and J Szmigielski} } @article {Bertola:Cubic, title = {Cubic string boundary value problems and Cauchy biorthogonal polynomials}, journal = {J. Phys. A}, volume = {42}, number = {45}, year = {2009}, pages = {454006, 13}, issn = {1751-8113}, doi = {10.1088/1751-8113/42/45/454006}, url = {http://0-dx.doi.org.mercury.concordia.ca/10.1088/1751-8113/42/45/454006}, author = {Marco Bertola and Gekhtman, M. and Szmigielski, J.} } @article {2009, title = {Hardy-Sobolev-Maz\\\'ja inequalities: symmetry and breaking symmetry of extremal functions}, journal = {Commun. Contemp. Math. 11 (2009) 993-1007}, number = {SISSA;06/2008/M}, year = {2009}, doi = {10.1142/S0219199709003636}, url = {http://hdl.handle.net/1963/2569}, author = {Marita Gazzini and Roberta Musina} } @article {2009, title = {Initial value problem of the Whitham equations for the Camassa-Holm equation}, journal = {Physica D 238 (2009) 55-66}, number = {arXiv.org;0805.2558v1}, year = {2009}, publisher = {Elsevier}, abstract = {We 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.}, doi = {10.1016/j.physd.2008.08.016}, url = {http://hdl.handle.net/1963/3429}, author = {Tamara Grava and Virgil U. Pierce and Fei-Ran Tian} } @article {2009, title = {The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups}, journal = {J. Funct. Anal. 256 (2009) 2621-2655}, number = {SISSA;33/2008/M}, year = {2009}, abstract = {We 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.}, doi = {10.1016/j.jfa.2009.01.006}, url = {http://hdl.handle.net/1963/2669}, author = {Andrei A. Agrachev and Ugo Boscain and Jean-Paul Gauthier and Francesco Rossi} } @article {2009, title = {On a Sobolev type inequality related to the weighted p-Laplace operator}, journal = {J. Math. Anal. Appl. 352 (2009) 99-111}, number = {SISSA;18/2008/M}, year = {2009}, doi = {10.1016/j.jmaa.2008.06.021}, url = {http://hdl.handle.net/1963/2613}, author = {Marita Gazzini and Roberta Musina} } @article {2009, title = {On universality of critical behaviour in the focusing nonlinear Schr{\"o}dinger equation, elliptic umbilic catastrophe and the {\\\\it tritronqu{\'e}e} solution to the Painlev{\'e}-I equation}, journal = {J. Nonlinear Sci. 19 (2009) 57-94}, number = {arXiv.org;0704.0501}, year = {2009}, abstract = {We argue that the critical behaviour near the point of {\textquoteleft}{\textquoteleft}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.}, doi = {10.1007/s00332-008-9025-y}, url = {http://hdl.handle.net/1963/2525}, author = {Boris Dubrovin and Tamara Grava and Christian Klein} } @article {2009, title = {Universality of the break-up profile for the KdV equation in the small dispersion limit using the Riemann-Hilbert approach}, journal = {Comm. Math. Phys. 286 (2009) 979-1009}, number = {arXiv.org;0801.2326}, year = {2009}, abstract = {We 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.}, doi = {10.1007/s00220-008-0680-5}, url = {http://hdl.handle.net/1963/2636}, author = {Tamara Grava and Tom Claeys} } @article {2009, title = {A variational model for quasistatic crack growth in nonlinear elasticity: some qualitative properties of the solutions}, journal = {Boll. Unione Mat. Ital. (9) 2 (2009) 371-390}, number = {SISSA;41/2008/M}, year = {2009}, url = {http://hdl.handle.net/1963/2675}, author = {Gianni Dal Maso and Alessandro Giacomini and Marcello Ponsiglione} } @article {2008, title = {Gradient bounds for minimizers of free discontinuity problems related to cohesive zone models in fracture mechanics}, journal = {Calc. Var. Partial Differential Equations 31 (2008) 137-145}, number = {SISSA;50/2005/M}, year = {2008}, abstract = {In 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.}, doi = {10.1007/s00526-006-0084-3}, url = {http://hdl.handle.net/1963/1723}, author = {Gianni Dal Maso and Adriana Garroni} } @article {2008, title = {On the Logarithmic Asymptotics of the Sixth Painleve\' Equation (Summer 2007)}, journal = {J.Phys.A: Math.Theor. 41,(2008), 205201-205247}, number = {arXiv:0801.1157;}, year = {2008}, note = {This paper appeared as a preprint in August 2007. It is published in Journal of Physics A: Mathematical and Theoretical, Volume 41, Issue 20, 6 May 2008, p. 205201-205247. It was on the archive in January 2008 (arXiv:0801.1157). This version does not differ from the published one except for two facts: 1)the addition of subsection 8.2, which proves that tr(M0Mx) = -2 for solutions y(x) \~{} a (ln x)n , n = 1, 2, x {\textrightarrow} 0. 2). The title of the journal article is : The logarithmic asymptotics of the sixth Painlev{\'e} equation}, publisher = {SISSA}, abstract = {We 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.}, doi = {10.1088/1751-8113/41/20/205201}, url = {http://hdl.handle.net/1963/6521}, author = {Davide Guzzetti} } @article {2008, title = {Numerical study of a multiscale expansion of the Korteweg-de Vries equation and Painlev{\'e}-II equation}, journal = {Proc. R. Soc. A 464 (2008) 733-757}, number = {arXiv.org;0708.0638v3}, year = {2008}, abstract = {The 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 $\\\\epsilon^{2/3}$.}, doi = {10.1098/rspa.2007.0249}, url = {http://hdl.handle.net/1963/2592}, author = {Tamara Grava and Christian Klein} } @inbook {2008, title = {Transport Rays and Applications to Hamilton{\textendash}Jacobi Equations}, booktitle = {Nonlinear PDE{\textquoteright}s and Applications : C.I.M.E. Summer School, Cetraro, Italy 2008 / Stefano Bianchini, Eric A. Carlen, Alexander Mielke, C{\'e}dric Villani. Eds. Luigi Ambrosio, Giuseppe Savar{\'e}. - Berlin : Springer, 2011. - (Lecture Notes in Mathematics ; 20}, year = {2008}, note = {This volume collects the notes of the CIME course Nonlinear PDE{\textquoteright}s and\\r\\napplications held in Cetraro (Italy) on June 23{\textendash}28, 2008. The school consisted\\r\\nin 5 series of lectures, delivered by Stefano Bianchini (SISSA, Trieste), Eric A. Carlen (Rutgers University), Alexander Mielke (WIAS, Berlin), Felix Otto (Bonn University), Cedric Villani (Ecole Normale Superieure de Lyon).}, publisher = {Springer}, organization = {Springer}, abstract = {The 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).}, isbn = {978-3-642-21718-0}, doi = {10.1007/978-3-642-21861-3_1}, url = {http://hdl.handle.net/1963/5463}, author = {Stefano Bianchini and Matteo Gloyer} } @article {2007, title = {The Asymptotic Behaviour of the Fourier Transforms of Orthogonal Polynomials II: L.I.F.S. Measures and Quantum Mechanics}, journal = {Ann. Henri Poincar{\textasciiacute}e 8 (2007), 301{\textendash}336}, year = {2007}, publisher = {2007 Birkh{\textasciidieresis}auser Verlag Basel/Switzerland}, abstract = {We 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 discussed}, doi = {10.1007/s00023-006-0309-1}, author = {Davide Guzzetti and Giorgio Mantica} } @article {Bertola:LOPs, title = {Biorthogonal Laurent polynomials, T{\"o}plitz determinants, minimal Toda orbits and isomonodromic tau functions}, journal = {Constr. Approx.}, volume = {26}, number = {3}, year = {2007}, pages = {383{\textendash}430}, issn = {0176-4276}, author = {Marco Bertola and Gekhtman, M.} } @article {2007, title = {Black Holes, Instanton Counting on Toric Singularities and q-Deformed Two-Dimensional Yang-Mills Theory}, number = {SISSA;61/2006/FM}, year = {2007}, abstract = {We 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.}, doi = {10.1016/j.nuclphysb.2007.02.030}, url = {http://hdl.handle.net/1963/1888}, author = {Luca Griguolo and Domenico Seminara and Richard J. Szabo and Alessandro Tanzini} } @article {Bertola:EffectiveIMRN, title = {Effective inverse spectral problem for rational Lax matrices and applications}, journal = {Int. Math. Res. Not. IMRN}, number = {23}, year = {2007}, pages = {Art. ID rnm103, 39}, issn = {1073-7928}, author = {Marco Bertola and Gekhtman, M.} } @article {2007, title = {On the Maz\\\'ya inequalities: existence and multiplicity results for an elliptic problem involving cylindrical weights}, number = {SISSA;92/2007/M}, year = {2007}, url = {http://hdl.handle.net/1963/2522}, author = {Marita Gazzini and Roberta Musina} } @article {2007, title = {Metrics on semistable and numerically effective Higgs bundles}, journal = {J. Reine Angew. Math. 612 (2007) 59-79}, number = {SISSA;17/2006/FM}, year = {2007}, abstract = {We 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.}, doi = {10.1515/CRELLE.2007.084}, url = {http://hdl.handle.net/1963/1840}, author = {Ugo Bruzzo and Beatriz Grana-Otero} } @article {2007, title = {A new model for contact angle hysteresis}, number = {SISSA;37/2006/M}, year = {2007}, abstract = {We 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.}, url = {http://hdl.handle.net/1963/1848}, author = {Antonio DeSimone and Natalie Gruenewald and Felix Otto} } @article {2007, title = {Numerical solution of the small dispersion limit of Korteweg de Vries and Whitham equations}, number = {SISSA;91/2005/FM}, year = {2007}, abstract = {The 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 {\textquoteleft}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.}, doi = {10.1002/cpa.20183}, url = {http://hdl.handle.net/1963/1788}, author = {Tamara Grava and Christian Klein} } @article {2007, title = {Numerical study of a multiscale expansion of KdV and Camassa-Holm equation}, number = {arXiv.org;math-ph/0702038v1}, year = {2007}, abstract = {We 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 equation}, url = {http://hdl.handle.net/1963/2527}, author = {Tamara Grava and Christian Klein} } @article {2007, title = {Numerically flat Higgs vector bundles}, number = {SISSA;39/2005/FM}, year = {2007}, abstract = {After 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.}, doi = {10.1142/S0219199707002526}, url = {http://hdl.handle.net/1963/1757}, author = {Ugo Bruzzo and Beatriz Grana-Otero} } @article {2007, title = {Reciprocal transformations and flat metrics on Hurwitz spaces}, journal = {J. Phys. A 40 (2007) 10769-10790}, number = {arXiv.org;0704.1779v2}, year = {2007}, abstract = {We 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.}, doi = {10.1088/1751-8113/40/35/004}, url = {http://hdl.handle.net/1963/2210}, author = {Simonetta Abenda and Tamara Grava} } @article {2007, title = {Semistable principal Higgs bundles}, number = {SISSA;89/2007/MP}, year = {2007}, url = {http://hdl.handle.net/1963/2533}, author = {Ugo Bruzzo and Beatriz Grana-Otero} } @article {2006, title = {Experimental and modeling studies of desensitization of P2X3 receptors.}, journal = {Molecular pharmacology. 2006 Jul; 70(1):373-82}, number = {PMID:16627751;}, year = {2006}, publisher = {the American Society for Pharmacology and Experimental Therapeutics}, abstract = {The 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.}, doi = {10.1124/mol.106.023564}, url = {http://hdl.handle.net/1963/4974}, author = {Elena Sokolova and Andrei Skorinkin and Igor Moiseev and Andrei A. Agrachev and Andrea Nistri and Rashid Giniatullin} } @article {2006, title = {Large Parameter Behavior of Equilibrium Measures}, number = {SISSA;92/2005/FM}, year = {2006}, abstract = {We 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).}, url = {http://hdl.handle.net/1963/1789}, author = {Tamara Grava and Fei-Ran Tian} } @article {2006, title = {Matching Procedure for the Sixth Painlev{\'e} Equation (May 2006)}, journal = {Journal of Physics A: Mathematical and General, Volume 39, Issue 39, 29 September 2006, Article numberS02, Pages 11973-12031}, number = {arXiv:1010.1952;}, year = {2006}, note = {This paper appeared in May 2006. I put it on the archive now, with more that four years of delay, for completeness sake. The paper is published in J.Phys.A: Math.Gen. 39 (2006), 11973-12031, with some modifications.}, publisher = {SISSA}, abstract = {We 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.}, doi = {doi:10.1088/0305-4470/39/39/S02}, url = {http://hdl.handle.net/1963/6524}, author = {Davide Guzzetti} } @article {2006, title = {Thomae type formulae for singular Z_N curves}, journal = {Lett. Math. Phys. 76 (2006) 187-214}, number = {arXiv.org;math-ph/0602017v1}, year = {2006}, abstract = {We 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.}, doi = {10.1007/s11005-006-0073-7}, url = {http://hdl.handle.net/1963/2125}, author = {Victor Z. Enolski and Tamara Grava} } @article {2005, title = {Hybrid necessary principle}, journal = {SIAM J. Control Optim. 43 (2005) 1867-1887}, number = {SISSA;71/2002/M}, year = {2005}, note = {Proceedings of IFAC Conference on Analysis and Design of Hybrid Systems, Saint Malo, France}, publisher = {SIAM}, abstract = {We 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.}, doi = {10.1137/S0363012903416219}, url = {http://hdl.handle.net/1963/1641}, author = {Mauro Garavello and Benedetto Piccoli} } @article {2005, title = {Modulation of the Camassa-Holm equation and reciprocal transformations}, journal = {Ann. Inst. Fourier (Grenoble) 55 (2005) 1803-1834}, number = {SISSA;107/2004/FM}, year = {2005}, abstract = {We 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.}, url = {http://hdl.handle.net/1963/2305}, author = {Simonetta Abenda and Tamara Grava} } @article {2005, title = {Traffic flow on a road network}, journal = {SIAM J. Math. Anal. 36 (2005) 1862-1886}, number = {SISSA;13/2002/M}, year = {2005}, publisher = {SISSA Library}, abstract = {This 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.}, doi = {10.1137/S0036141004402683}, url = {http://hdl.handle.net/1963/1584}, author = {Giuseppe Maria Coclite and Benedetto Piccoli and Mauro Garavello} } @proceedings {2004, title = {The elliptic representation of the sixth Painlev{\'e} equation.}, year = {2004}, publisher = {Societe Matematique de France}, abstract = {We find a class of solutions of the sixth Painlev{\textasciiacute}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).}, keywords = {Painlev{\'e} equation}, isbn = {978-2-85629-229-7}, url = {http://hdl.handle.net/1963/6529}, author = {Davide Guzzetti} } @article {2004, title = {Singular Z_N curves, Riemann-Hilbert problem and modular solutions of the Schlesinger equation}, journal = {Int. Math. Res. Not. 2004, no. 32, 1619-1683}, number = {arXiv.org;math-ph/0306050}, year = {2004}, abstract = {We 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.}, doi = {10.1155/S1073792804132625}, url = {http://hdl.handle.net/1963/2540}, author = {Victor Z. Enolski and Tamara Grava} } @article {2004, title = {Solitary waves for Maxwell Schrodinger equations}, journal = {Electron. J. Differential Equations (2004) 94}, number = {SISSA;11/2002/M}, year = {2004}, publisher = {SISSA Library}, abstract = {In 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.}, url = {http://hdl.handle.net/1963/1582}, author = {Giuseppe Maria Coclite and Vladimir Georgiev} } @article {2003, title = {Hybrid optimal control: case study of a car with gears}, journal = {Int. J. Control 76 (2003) 1272-1284}, year = {2003}, publisher = {Taylor and Francis}, abstract = {The 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.}, doi = {10.1080/0020717031000147520}, url = {http://hdl.handle.net/1963/3022}, author = {Ciro D{\textquoteright}Apice and Mauro Garavello and Rosanna Manzo and Benedetto Piccoli} } @article {2002, title = {The Elliptic Representation of the General Painlev{\'e} 6 Equation}, journal = {Communications on Pure and Applied Mathematics, Volume 55, Issue 10, October 2002, Pages 1280-1363}, number = {arXiv:math/0108073;}, year = {2002}, note = {60 pages; Latex; 3 figures. The statements of theorems have been\r\n simplified}, publisher = {SISSA}, abstract = {We 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.}, doi = {10.1002/cpa.10045}, url = {http://hdl.handle.net/1963/6523}, author = {Davide Guzzetti} } @proceedings {2002, title = {The Elliptic Representation of the Painleve 6 Equation}, number = {RIMS Kokyuroku;vol. 1296}, year = {2002}, publisher = {Kyoto University, Research Institute for Mathematical Sciences}, abstract = {We review our results on the elliptic representation of the sixth Painleve{\textquoteright} equation}, keywords = {Painleve equations}, url = {http://hdl.handle.net/1963/6530}, author = {Davide Guzzetti} } @article {2002, title = {On the K+P problem for a three-level quantum system: optimality implies resonance}, journal = {J.Dynam. Control Systems 8 (2002),no.4, 547}, number = {SISSA;30/2002/M}, year = {2002}, publisher = {SISSA Library}, doi = {10.1023/A:1020767419671}, url = {http://hdl.handle.net/1963/1601}, author = {Ugo Boscain and Thomas Chambrion and Jean-Paul Gauthier} } @article {2002, title = {The passage from nonconvex discrete systems to variational problems in Sobolev spaces: the one-dimensional case}, journal = {Proc. Steklov Inst. Math. 236 (2002) 395-414}, number = {SISSA;11/2001/M}, year = {2002}, publisher = {MAIK Nauka/Interperiodica}, url = {http://hdl.handle.net/1963/3130}, author = {Andrea Braides and Maria Stella Gelli and Mario Sigalotti} } @article {2001, title = {On the Critical Behavior, the Connection Problem and the Elliptic Representation of a Painlev{\'e} VI Equation}, journal = {Mathematical Physics, Analysis and Geometry 4: 293{\textendash}377, 2001}, year = {2001}, publisher = {Kluwer Academic Publishers}, abstract = {In this paper we find a class of solutions of the sixth Painlev{\'e} 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 Painlev{\'e}\r\ntranscendents in the elliptic representation.}, keywords = {Painleve Equations, Isomonodromy deformations}, doi = {10.1023/A:1014265919008}, author = {Davide Guzzetti} } @article {2014, title = {Dieletric breakdown: optimal bounds}, journal = {Proc. of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences 457 (2001): p. 2317-2335, OCT. 8, 2001}, number = {SISSA;113/00/M}, year = {2001}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1569}, author = {Adriana Garroni and Vincenzo Nesi and Marcello Ponsiglione} } @article {2001, title = {Finite Difference Approximation of Free Discontinuity Problems}, journal = {Proc. Royal Soc. Edinb. Ser. A 131 (2001), no. 3, 567-595}, number = {SISSA;14/99/M}, year = {2001}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1228}, author = {Massimo Gobbino and Maria Giovanna Mora} } @article {2001, title = {Inverse Problem and Monodromy Data for Three-Dimensional Frobenius Manifolds}, journal = {Mathematical Physics, Analysis and Geometry 4: 245{\textendash}291, 2001}, year = {2001}, publisher = {RIMS, Kyoto University}, abstract = {We 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 Painlev{\'e} 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{\textendash}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.}, keywords = {Frobenius Manifolds, Painleve Equations, Isomonodromy deformations}, doi = {10.1023/A:1012933622521}, author = {Davide Guzzetti} } @article {Bertola:LietripleRMA, title = {Lie triple systems and warped products}, journal = {Rend. Mat. Appl. (7)}, volume = {21}, number = {1-4}, year = {2001}, pages = {275{\textendash}293}, issn = {1120-7183}, author = {Marco Bertola and Gouthier, D.} } @article {2001, title = {On the subanalyticity of Carnot-Caratheodory distances}, journal = {Ann. I. H. Poincare - An., 2001, 18, 359}, number = {SISSA;25/00/M}, year = {2001}, publisher = {SISSA Library}, doi = {10.1016/S0294-1449(00)00064-0}, url = {http://hdl.handle.net/1963/1483}, author = {Andrei A. Agrachev and Jean-Paul Gauthier} } @article {Bertola:Warped, title = {Warped products with special Riemannian curvature}, journal = {Bol. Soc. Brasil. Mat. (N.S.)}, volume = {32}, number = {1}, year = {2001}, pages = {45{\textendash}62}, issn = {0100-3569}, author = {Marco Bertola and Gouthier, Daniele} } @article {2000, title = {3D superconformal theories from Sasakian seven-manifolds: new nontrivial evidences for AdS_4/CFT_3}, journal = {Nucl.Phys. B577 (2000) 547-608}, number = {SISSA;113/99/FM}, year = {2000}, publisher = {SISSA Library}, doi = {10.1016/S0550-3213(00)00098-5}, url = {http://hdl.handle.net/1963/1327}, author = {Davide Fabbri and Pietro Fr{\'e} and Leonardo Gualtieri and Cesare Reina and Alessandro Tomasiello and Alberto Zaffaroni and Alessandro Zampa} } @article {Bertola:DecomposingNPB, title = {Decomposing quantum fields on branes}, journal = {Nuclear Phys. B}, volume = {581}, number = {1-2}, year = {2000}, pages = {575{\textendash}603}, issn = {0550-3213}, author = {Marco Bertola and Bros, Jacques and Gorini, Vittorio and Moschella, Ugo and Schaeffer, Richard} } @article {2000, title = {Elliptic variational problems in $ R\\\\sp N$ with critical growth}, journal = {J. Differential Equations 168 (2000), no. 1, 10--32}, number = {SISSA;44/99/M}, year = {2000}, publisher = {SISSA Library}, doi = {10.1006/jdeq.2000.3875}, url = {http://hdl.handle.net/1963/1258}, author = {Antonio Ambrosetti and Jesus Garcia Azorero and Ireneo Peral} } @article {2000, title = {Existence and multiplicity results for some nonlinear elliptic equations: a survey.}, journal = {Rend. Mat. Appl., 2000, 20, 167}, number = {SISSA;4/00/M}, year = {2000}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1462}, author = {Antonio Ambrosetti and Jesus Garcia Azorero and Ireneo Peral} } @article {2000, title = {Inverse problem for Semisimple Frobenius Manifolds Monodromy Data and the Painlev{\'e} VI Equation}, number = {SISSA;101/00/FM}, year = {2000}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1557}, author = {Davide Guzzetti} } @article {2000, title = {Stability of L^infty Solutions of Temple Class Systems}, journal = {Differential Integral Equations 13 (2000) 1503-1528}, number = {SISSA;134/98/M}, year = {2000}, publisher = {Khayyam Publishing}, abstract = {Let $u_t+f(u)_x=0$ 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.

}, url = {http://hdl.handle.net/1963/3256}, author = {Alberto Bressan and Paola Goatin} } @inbook {2000, title = {Stokes Matrices for Frobenius Manifolds and the 6 Painlev{\'e} Equation}, booktitle = {Rokko Lectures in Mathematics, Vol 7 [Issue title: Perspective of Painleve equations], (2000), pages : 101-109}, number = {Rokko lectures in mathematics;vol. 7}, year = {2000}, publisher = {Kobe University, Japan}, organization = {Kobe University, Japan}, abstract = {These notes are a short review on the theory of Frobenius manifolds and its connection to problems of isomonodromy deformations and to Painlev{\textquoteright}e equations.}, keywords = {Painlev{\'e} equation}, isbn = {4-907719-07-8}, url = {http://hdl.handle.net/1963/6546}, author = {Davide Guzzetti} } @article {Bertola:CorrespondencePLB, title = {Correspondence between Minkowski and de Sitter quantum field theory}, journal = {Phys. Lett. B}, volume = {462}, number = {3-4}, year = {1999}, pages = {249{\textendash}253}, issn = {0370-2693}, author = {Marco Bertola and Gorini, Vittorio and Moschella, Ugo and Schaeffer, Richard} } @article {1999, title = {A Lipschitz selection from the set of minimizers of a nonconvex functional of the gradient}, journal = {Nonlinear Analysis, Theory, Methods and Applications. Volume 37, Issue 6, September 1999, Pages 707-717}, year = {1999}, publisher = {SISSA}, abstract = {A 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.}, doi = {10.1016/S0362-546X(98)00067-4}, url = {http://hdl.handle.net/1963/6439}, author = {Gianni Dal Maso and Vladimir V. Goncharov and Antonio Ornelas} } @article {1999, title = {Oleinik type estimates and uniqueness for n x n conservation laws}, journal = {J. Differential Equations 156 (1999), no. 1, 26--49}, number = {SISSA;150/97/M}, year = {1999}, publisher = {Elsevier}, abstract = {Let $u_t+f(u)_x=0$ 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 Oleinik in the scalar case.}, doi = {10.1006/jdeq.1998.3606}, url = {http://hdl.handle.net/1963/3375}, author = {Alberto Bressan and Paola Goatin} } @article {1999, title = {Perturbation of $\Delta u+u^{(N+2)/(N-2)}=0$, the scalar curvature problem in $R^N$, and related topics}, journal = {J. Funct. Anal. 165 (1999) 117-149}, number = {SISSA;141/98/M}, year = {1999}, publisher = {Elsevier}, abstract = {Some 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.

}, doi = {10.1006/jfan.1999.3390}, url = {http://hdl.handle.net/1963/3255}, author = {Antonio Ambrosetti and Jesus Garcia Azorero and Ireneo Peral} } @article {1999, title = {Stokes matrices and monodromy of the quantum cohomology of projective spaces}, journal = {Comm. Math. Phys. 207 (1999) 341-383}, number = {arXiv.org;math/9904099v1}, year = {1999}, publisher = {Springer}, abstract = {n 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.}, doi = {10.1007/s002200050729}, url = {http://hdl.handle.net/1963/3475}, author = {Davide Guzzetti} } @article {1999, title = {Variational formulation of softening phenomena in fracture mechanics. The one-dimensional case}, journal = {Arch. Ration. Mech. Anal. 146 (1999), no. 1, 23--58}, number = {SISSA;160/97/M}, year = {1999}, publisher = {Springer}, abstract = {Starting 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.}, doi = {10.1007/s002050050135}, url = {http://hdl.handle.net/1963/3371}, author = {Andrea Braides and Gianni Dal Maso and Adriana Garroni} } @mastersthesis {1998, title = {On the Cauchy Problem for the Whitham Equations}, year = {1998}, school = {SISSA}, keywords = {Korteweg de Vries equation}, url = {http://hdl.handle.net/1963/5555}, author = {Tamara Grava} } @article {Bertola:GenerationGC, title = {Generation of primordial fluctuations in curved spaces}, journal = {Gravit. Cosmol.}, volume = {4}, number = {2}, year = {1998}, pages = {121{\textendash}127}, issn = {0202-2893}, author = {Schaeffer, Richard and Moschella, Ugo and Marco Bertola and Gorini, Vittorio} } @article {1998, title = {Special functions with bounded variation and with weakly differentiable traces on the jump set}, journal = {NoDEA Nonlinear Differential Equations Appl. 5 (1998), no. 2, 219--243}, number = {SISSA;123/95/M}, year = {1998}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1025}, author = {Luigi Ambrosio and Andrea Braides and Adriana Garroni} } @article {1997, title = {Shift-differentiability of the flow generated by a conservation law}, journal = {Discrete Contin. Dynam. Systems 3 (1997), no. 1, 35--58.}, number = {SISSA;131/95/M}, year = {1997}, publisher = {SISSA Library}, abstract = {The 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.}, url = {http://hdl.handle.net/1963/1033}, author = {Alberto Bressan and Graziano Guerra} } @mastersthesis {1994, title = {Asymptotic Behaviour of Dirichlet Problems in Perforated Domains}, year = {1994}, school = {SISSA}, keywords = {Dirichlet problems}, url = {http://hdl.handle.net/1963/5714}, author = {Adriana Garroni} }