We construct the rank-one, singular (point-like) perturbations of the d-dimensional fractional Laplacian in the physically meaningful norm-resolvent limit of fractional Schr{\"o}dinger operators with regular potentials centred around the perturbation point and shrinking to a delta-like shape. We analyse both possible regimes, the resonance-driven and the resonance-independent limit, depending on the power of the fractional Laplacian and the spatial dimension. To this aim, we also qualify the notion of zero-energy resonance for Schr{\"o}dinger operators formed by a fractional Laplacian and a regular potential.

}, issn = {1661-8262}, doi = {10.1007/s11785-019-00927-w}, url = {https://doi.org/10.1007/s11785-019-00927-w}, author = {Alessandro Michelangeli and Raffaele Scandone} } @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 {2018, title = {Peristaltic Waves as Optimal Gaits in Metameric Bio-Inspired Robots}, journal = {Frontiers in Robotics and AI}, volume = {5}, year = {2018}, month = {09/2018}, abstract = {*Peristalsis*, i.e., a motion pattern arising from the propagation of muscle contraction and expansion waves along the body, is a common locomotion strategy for limbless animals. Mimicking peristalsis in bio-inspired robots has attracted considerable attention in the literature. It has recently been observed that maximal velocity in a metameric earthworm-like robot is achieved by actuating the segments using a {\textquotedblleft}phase coordination{\textquotedblright} principle. This paper shows that, in fact, peristalsis (which requires not only phase coordination, but also that all segments oscillate at same frequency and amplitude) emerges from optimization principles. More precisely, basing our analysis on the assumption of small deformations, we show that peristaltic waves provide the optimal actuation solution in the ideal case of a periodic infinite system, and that this is approximately true, modulo edge effects, for the real, finite length system. Therefore, this paper confirms the effectiveness of mimicking peristalsis in bio-inspired robots, at least in the small-deformation regime. Further research will be required to test the effectiveness of this strategy if large deformations are allowed.

We study the periodic boundary value problem associated with the second order nonlinear equation u{\textquoteright}{\textquoteright}+(λa+(t)-μa-(t))g(u)=0, where g(u) has superlinear growth at zero and sublinear growth at infinity. For λ,μ positive and large, we prove the existence of 3^m-1 positive T-periodic solutions when the weight function a(t) has m positive humps separated by m negative ones (in a T-periodicity interval). As a byproduct of our approach we also provide abundance of positive subharmonic solutions and symbolic dynamics. The proof is based on coincidence degree theory for locally compact operators on open unbounded sets and also applies to Neumann and Dirichlet boundary conditions. Finally, we deal with radially symmetric positive solutions for the Neumann and the Dirichlet problems associated with elliptic PDEs.

}, url = {http://urania.sissa.it/xmlui/handle/1963/35264}, author = {Alberto Boscaggin and Guglielmo Feltrin and Fabio Zanolin} } @article {doi:10.1142/S0219199717500213, title = {Positive subharmonic solutions to nonlinear ODEs with indefinite weight}, journal = {Communications in Contemporary Mathematics}, volume = {20}, number = {01}, year = {2018}, pages = {1750021}, abstract = {We prove that the superlinear indefinite equation u" + a(t)up = 0, where p \> 1 and a(t) is a T-periodic sign-changing function satisfying the (sharp) mean value condition ∫0Ta(t)dt \< 0, has positive subharmonic solutions of order k for any large integer k, thus providing a further contribution to a problem raised by Butler in its pioneering paper [Rapid oscillation, nonextendability, and the existence of periodic solutions to second order nonlinear ordinary differential equations, J. Differential Equations 22 (1976) 467{\textendash}477]. The proof, which applies to a larger class of indefinite equations, combines coincidence degree theory (yielding a positive harmonic solution) with the Poincar{\'e}{\textendash}Birkhoff fixed point theorem (giving subharmonic solutions oscillating around it).

}, doi = {10.1142/S0219199717500213}, url = {https://doi.org/10.1142/S0219199717500213}, author = {Alberto Boscaggin and Guglielmo Feltrin} } @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 {demo2018pydmd, title = {PyDMD: Python Dynamic Mode Decomposition}, journal = {The Journal of Open Source Software}, volume = {3}, number = {22}, year = {2018}, pages = {530}, doi = {10.21105/joss.00530}, url = {https://joss.theoj.org/papers/734e4326edd5062c6e8ee98d03df9e1d}, author = {Nicola Demo and Marco Tezzele and Gianluigi Rozza} } @article {Stabile2017, title = {POD-Galerkin reduced order methods for CFD using Finite Volume Discretisation: vortex shedding around a circular cylinder}, journal = {Communications in Applied and Industrial Mathematics}, volume = {8}, number = {1}, year = {2017}, month = {dec}, pages = {210{\textendash}236}, publisher = {Walter de Gruyter {GmbH}}, doi = {10.1515/caim-2017-0011}, url = {https://doi.org/10.1515/caim-2017-0011}, author = {Giovanni Stabile and Saddam Hijazi and Andrea Mola and Stefano Lorenzi and Gianluigi Rozza} } @article {2016, title = {Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super-sublinear case}, journal = {Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), 449{\textendash}474.}, number = {Proceedings of the Royal Society of Edinburgh. Section A. Mathematics;volume 146; issue 3; pages 449-474;}, year = {2016}, note = {AMS Subject Classification: Primary 34B18; 34C25; Secondary 34B15; 47H11;}, publisher = {Cambridge University Press}, abstract = {We study the periodic and Neumann boundary value problems associated with the second order nonlinear differential equation u{\textquoteright}{\textquoteright}+cu{\textquoteright}+λa(t)g(u)=0, where g:[0,+$\infty$[{\textrightarrow}[0,+$\infty$[ is a sublinear function at infinity having superlinear growth at zero. We prove the existence of two positive solutions when ∫a(t)dt 0 is sufficiently large. Our approach is based on Mawhin{\textquoteright}s coincidence degree theory and index computations.

}, doi = {10.1017/S0308210515000621}, url = {http://urania.sissa.it/xmlui/handle/1963/35262}, author = {Alberto Boscaggin and Guglielmo Feltrin and Fabio Zanolin} } @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 {arici2016pimsner, title = {Pimsner algebras and Gysin sequences from principal circle actions}, journal = {Journal of Noncommutative Geometry}, volume = {10}, year = {2016}, pages = {29{\textendash}64}, issn = {1661-6952}, doi = {10.4171/jncg/228}, url = {http://hdl.handle.net/2066/162951}, author = {Francesca Arici and Jens Kaad and Giovanni Landi} } @article {2016, title = {POD-Galerkin Method for Finite Volume Approximation of Navier-Stokes and RANS Equations}, year = {2016}, publisher = {Computer Methods in Applied Mechanics and Engineering, Elsevier}, abstract = {Numerical simulation of fluid flows requires important computational efforts but it is essential in engineering applications. Reduced Order Model (ROM) can be employed whenever fast simulations are required, or in general, whenever a trade-off between computational cost and solution accuracy is a preeminent issue as in process optimization and control. In this work, the efforts have been put to develop a ROM for Computational Fluid Dynamics (CFD) application based on Finite Volume approximation, starting from the results available in turbulent Reynold-Averaged Navier Stokes simulations in order to enlarge the application field of Proper Orthogonal Decomposition {\textendash} Reduced Order Model (POD {\textendash} ROM) technique to more industrial fields. The approach is tested in the classic benchmark of the numerical simulation of the 2D lid-driven cavity. In particular, two simulations at Re = 103 and Re = 105 have been considered in order to assess both a laminar and turbulent case. Some quantities have been compared with the Full Order Model in order to assess the performance of the proposed ROM procedure i.e., the kinetic energy of the system and the reconstructed quantities of interest (velocity, pressure and turbulent viscosity). In addition, for the laminar case, the comparison between the ROM steady-state solution and the data available in literature has been presented. The results have turned out to be very satisfactory both for the accuracy and the computational times. As a major outcome, the approach turns out not to be affected by the energy blow up issue characterizing the results obtained by classic turbulent POD-Galerkin methods.}, author = {Stefano Lorenzi and Antonio Cammi and Lelio Luzzi and Gianluigi Rozza} } @article {11943, title = {POD{\textendash}Galerkin monolithic reduced order models for parametrized fluid-structure interaction problems}, journal = {International Journal Numerical Methods for Fluids}, year = {2016}, publisher = {Wiley}, abstract = {In this paper we propose a monolithic approach for reduced order modelling of parametrized fluid-structure interaction problems based on a proper orthogonal decomposition (POD){\textendash}Galerkin method. Parameters of the problem are related to constitutive properties of the fluid or structural problem, or to geometrical parameters related to the domain configuration at the initial time. We provide a detailed description of the parametrized formulation of the multiphysics problem in its components, together with some insights on how to obtain an offline-online efficient computational procedure through the approximation of parametrized nonlinear tensors. Then, we present the monolithic POD{\textendash}Galerkin method for the online computation of the global structural displacement, fluid velocity and pressure of the coupled problem. Finally, we show some numerical results to highlight the capabilities of the proposed reduced order method and its computational performances}, doi = {10.1002/fld.4252}, author = {Francesco Ballarin and Gianluigi Rozza} } @article {2016, title = {On point interactions realised as Ter-Martirosyan-Skornyakov Hamiltonians}, number = {SISSA;11/2016/MATE}, year = {2016}, abstract = {For quantum systems of zero-range interaction we discuss the mathematical scheme within which modelling the two-body interaction by means of the physically relevant ultra-violet asymptotics known as the {\textquoteleft}{\textquoteleft}Ter-Martirosyan--Skornyakov condition{\textquoteright}{\textquoteright} gives rise to a self-adjoint realisation of the corresponding Hamiltonian. This is done within the self-adjoint extension scheme of Krein, Visik, and Birman. We show that the Ter-Martirosyan--Skornyakov asymptotics is a condition of self-adjointness only when is imposed in suitable functional spaces, and not just as a point-wise asymptotics, and we discuss the consequences of this fact on a model of two identical fermions and a third particle of different nature.}, url = {http://urania.sissa.it/xmlui/handle/1963/35195}, author = {Alessandro Michelangeli and Andrea Ottolini} } @mastersthesis {2016, title = {Positive solutions to indefinite problems: a topological approach}, year = {2016}, note = {The research work described in this Ph.D. thesis has produced 10 papers.}, school = {SISSA}, abstract = {The present Ph.D. thesis is devoted to the study of positive solutions to indefinite problems. In particular, we deal with the second order nonlinear differential equation u{\textquoteright}{\textquoteright} + a(t) g(u) = 0, where g : [0,+$\infty$[{\textrightarrow}[0,+$\infty$[ is a continuous nonlinearity and a : [0,T]{\textrightarrow}R is a Lebesgue integrable sign-changing weight. We analyze the Dirichlet, Neumann and periodic boundary value problems on [0,T] associated with the equation and we provide existence, nonexistence and multiplicity results for positive solutions. In the first part of the manuscript, we investigate nonlinearities g(u) with a superlinear growth at zero and at infinity (including the classical superlinear case g(u)=u^p, with p>1). In particular, we prove that there exist 2^m-1 positive solutions when a(t) has m positive humps separated by negative ones and the negative part of a(t) is sufficiently large. Then, for the Dirichlet problem, we solve a conjecture by G{\'o}mez-Re{\~n}asco and L{\'o}pez-G{\'o}mez (JDE, 2000) and, for the periodic problem, we give a complete answer to a question raised by Butler (JDE, 1976). In the second part, we study the super-sublinear case (i.e. g(u) is superlinear at zero and sublinear at infinity). If a(t) has m positive humps separated by negative ones, we obtain the existence of 3^m-1 positive solutions of the boundary value problems associated with the parameter-dependent equation u{\textquoteright}{\textquoteright} + λ a(t) g(u) = 0, when both λ>0 and the negative part of a(t) are sufficiently large. We propose a new approach based on topological degree theory for locally compact operators on open possibly unbounded sets, which applies for Dirichlet, Neumann and periodic boundary conditions. As a byproduct of our method, we obtain infinitely many subharmonic solutions and globally defined positive solutions with complex behavior, and we deal with chaotic dynamics. Moreover, we study positive radially symmetric solutions to the Dirichlet and Neumann problems associated with elliptic PDEs on annular domains. Furthermore, this innovative technique has the potential and the generality needed to deal with indefinite problems with more general differential operators. Indeed, our approach apply also for the non-Hamiltonian equation u{\textquoteright}{\textquoteright} + cu{\textquoteright} + a(t) g(u) = 0. Meanwhile, more general operators in the one-dimensional case and problems involving PDEs will be subjects of future investigations.}, keywords = {positive solutions}, author = {Guglielmo Feltrin} } @article {MR3342766, title = {The partition function of the extended $r$-reduced Kadomtsev-Petviashvili hierarchy}, journal = {J. Phys. A}, volume = {48}, number = {19}, year = {2015}, pages = {195205, 20}, issn = {1751-8113}, doi = {10.1088/1751-8113/48/19/195205}, url = {http://dx.doi.org/10.1088/1751-8113/48/19/195205}, author = {Marco Bertola and Di Yang} } @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 {2015, title = {The phototransduction machinery in the rod outer segment has a strong efficacy gradient}, number = {Proceedings of the National Academy of Sciences of the United States of America;Volume 112, issue 20; pp. E2715-E2724}, year = {2015}, note = {Open Access article}, publisher = {National Academy of Sciences}, doi = {10.1073/pnas.1423162112}, url = {http://urania.sissa.it/xmlui/handle/1963/35157}, author = {Monica Mazzolini and Giuseppe Facchetti and L. Andolfi and R. Proietti Zaccaria and S. Tuccio and J. Treud and Claudio Altafini and Enzo M. Di Fabrizio and Marco Lazzarino and G. Rapp and Vincent Torre} } @article {2015, title = {Poisson cohomology of scalar multidimensional Dubrovin-Novikov brackets}, year = {2015}, abstract = {We compute the Poisson cohomology of a scalar Poisson bracket of Dubrovin-Novikov type with D independent variables. We find that the second and third cohomology groups are generically non-vanishing in D>1. Hence, in contrast with the D=1 case, the deformation theory in the multivariable case is non-trivial.}, author = {Guido Carlet and Matteo Casati and Sergey Shadrin} } @mastersthesis {2015, title = {Principal circle bundles, Pimsner algebras and Gysin sequences}, year = {2015}, school = {SISSA}, abstract = {Principal circle bundles and Gysin sequences play a crucial role in mathematical physics, in particular in Chern-Simons theories and T-duality. This works focuses on the noncommutative topology of principal circle bundles: we investigate the connections between noncommutative principal circle bundles, Pimsner algebras and strongly graded algebras. At the C*-algebraic level, we start from a self-Morita equivalence bimodule E for a C*-algebra B which we think of as a non commutative line bundle over the {\textquoteleft}base space{\textquoteright} algebra B. The corresponding Pimsner algebra O_E, is then the total space algebra of an associated circle bundle. A natural six term exact sequence, an analogue of the Gysin sequence for circle bundles, relates the KK-theories of O_E and of the base space B. We illustrate several results with the examples of quantum weighted projective and lens spaces.}, author = {Francesca Arici} } @article {2014, title = {Pfaffian representations of cubic surfaces}, number = {Geometriae dedicata;volume 168; issue 1; pages 69-86;}, year = {2014}, publisher = {Springer}, abstract = {Let K be a field of characteristic zero. We describe an algorithm which requires a homogeneous polynomial F of degree three in K[x0,x1,x2,x3] and a zero a of F in P3 K and ensures a linear Pfaffian representation of V(F) with entries in K[x0,x1,x2,x3], under mild assumptions on F and a. We use this result to give an explicit construction of (and to prove the existence of) a linear Pfaffian representation of V (F), with entries in K'[x0,x1,x2,x3], being K' an algebraic extension of K of degree at most six. An explicit example of such a construction is given.

}, doi = {10.1007/s10711-012-9818-x}, url = {http://urania.sissa.it/xmlui/handle/1963/34688}, author = {Fabio Tanturri} } @conference {mola2014ship, title = {Potential Model for Ship Hydrodynamics Simulations Directly Interfaced with CAD Data Structures}, booktitle = {The 24th International Ocean and Polar Engineering Conference}, volume = {4}, year = {2014}, pages = {815{\textendash}822}, publisher = {International Society of Offshore and Polar Engineers}, organization = {International Society of Offshore and Polar Engineers}, author = {Andrea Mola and Luca Heltai and Antonio DeSimone and Massimiliano Berti} } @article {2014, title = {Pseudo-automorphisms of positive entropy on the blowups of products of projective spaces}, number = {Mathematische annalen;volume 359; issue 1-2; pages 189-209;}, year = {2014}, publisher = {Springer}, abstract = {We use a concise method to construct pseudo-automorphisms fn of the first dynamical degree d1(fn) > 1 on the blowups of the projective n-space for all n >= 2 and more generally on the blowups of products of projective spaces. These fn, for n=3 have positive entropy, and for n>= 4 seem to be the first examples of pseudo-automorphisms with d1(fn) > 1 (and of non-product type) on rational varieties of higher dimensions.}, doi = {10.1007/s00208-013-0992-4}, url = {http://urania.sissa.it/xmlui/handle/1963/34714}, author = {Fabio Perroni and Deqi Zhang} } @article {boscaggin2013pairs, title = {Pairs of nodal solutions for a class of nonlinear problems with one-sided growth conditions}, journal = {Advanced Nonlinear Studies}, volume = {13}, number = {1}, year = {2013}, pages = {13{\textendash}53}, publisher = {Advanced Nonlinear Studies, Inc.}, doi = {10.1515/ans-2013-0103}, author = {Alberto Boscaggin and Fabio Zanolin} } @article {fonda2013periodic, title = {Periodic bouncing solutions for nonlinear impact oscillators}, journal = {Advanced Nonlinear Studies}, volume = {13}, number = {1}, year = {2013}, pages = {179{\textendash}189}, publisher = {Advanced Nonlinear Studies, Inc.}, doi = {10.1515/ans-2013-0110}, author = {Alessandro Fonda and Andrea Sfecci} } @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 {BOSCAGGIN20122900, title = {Pairs of positive periodic solutions of second order nonlinear equations with indefinite weight}, journal = {Journal of Differential Equations}, volume = {252}, number = {3}, year = {2012}, pages = {2900 - 2921}, abstract = {We study the problem of the existence and multiplicity of positive periodic solutions to the scalar ODEu"+λa(t)g(u)=0,λ\>0, where g(x) is a positive function on R+, superlinear at zero and sublinear at infinity, and a(t) is a T-periodic and sign indefinite weight with negative mean value. We first show the nonexistence of solutions for some classes of nonlinearities g(x) when λ is small. Then, using critical point theory, we prove the existence of at least two positive T-periodic solutions for λ large. Some examples are also provided.

}, keywords = {Critical points, Necessary conditions, Pairs of positive solutions, Periodic solutions}, issn = {0022-0396}, doi = {https://doi.org/10.1016/j.jde.2011.09.011}, url = {http://www.sciencedirect.com/science/article/pii/S0022039611003895}, author = {Alberto Boscaggin and Fabio Zanolin} } @article {fonda2012, title = {Periodic solutions of a system of coupled oscillators with one-sided superlinear retraction forces}, journal = {Differential Integral Equations}, volume = {25}, number = {11/12}, year = {2012}, month = {11}, pages = {993{\textendash}1010}, publisher = {Khayyam Publishing, Inc.}, url = {https://projecteuclid.org:443/euclid.die/1356012248}, author = {Alessandro Fonda and Andrea Sfecci} } @article {boscaggin2012periodic, title = {Periodic solutions to superlinear planar Hamiltonian systems}, journal = {Portugaliae Mathematica}, volume = {69}, number = {2}, year = {2012}, pages = {127{\textendash}141}, publisher = {European Mathematical Society Publishing House}, abstract = {We prove the existence of infinitely many periodic (harmonic and subharmonic) solutions to planar Hamiltonian systems satisfying a suitable superlinearity condition at infinity. The proof relies on the Poincare-Birkhoff fixed point theorem.

}, author = {Alberto Boscaggin} } @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 {BOSCAGGIN20122922, title = {Positive periodic solutions of second order nonlinear equations with indefinite weight: Multiplicity results and complex dynamics}, journal = {Journal of Differential Equations}, volume = {252}, number = {3}, year = {2012}, pages = {2922 - 2950}, abstract = {We prove the existence of a pair of positive T-periodic solutions as well as the existence of positive subharmonic solutions of any order and the presence of chaotic-like dynamics for the scalar second order ODEu"+aλ,μ(t)g(u)=0, where g(x) is a positive function on R+, superlinear at zero and sublinear at infinity, and aλ,μ(t) is a T-periodic and sign indefinite weight of the form λa+(t)-μa-(t), with λ,μ\>0 and large.

}, keywords = {Complex dynamics, Poincar{\'e} map, Positive periodic solutions, Subharmonics}, issn = {0022-0396}, doi = {https://doi.org/10.1016/j.jde.2011.09.010}, url = {http://www.sciencedirect.com/science/article/pii/S0022039611003883}, author = {Alberto Boscaggin and Fabio Zanolin} } @article {2012, title = {Predicting and characterizing selective multiple drug treatments for metabolic diseases and cancer.}, journal = {BMC Systems Biology. 29 August 2012, Page 115}, year = {2012}, publisher = {BioMed Central}, abstract = {Background: In the field of drug discovery, assessing the potential of multidrug therapies is a difficult task because of the combinatorial complexity (both theoretical and experimental) and because of the requirements on the selectivity of the therapy. To cope with this problem, we have developed a novel method for the systematic in silico investigation of synergistic effects of currently available drugs on genome-scale metabolic networks. The algorithm finds the optimal combination of drugs which guarantees the inhibition of an objective function, while minimizing the side effect on the overall network. Results: Two different applications are considered: finding drug synergisms for human metabolic diseases (like diabetes, obesity and hypertension) and finding antitumoral drug combinations with minimal side effect on the normal human metabolism.The results we obtain are consistent with some of the available therapeutic indications and predict some new multiple drug treatments.A cluster analysis on all possible interactions among the currently available drugs indicates a limited variety on the metabolic targets for the approved drugs. Conclusion: The in silico prediction of drug synergism can represent an important tool for the repurposing of drug in a realistic perspective which considers also the selectivty of the therapy. Moreover, for a more profitable exploitation of drug-drug interactions, also drugs which show a too low efficacy but which have a non-common mechanism of action, can be reconsider as potential ingredients of new multicompound therapeutic indications.Needless to say the clues provided by a computational study like ours need in any case to be thoroughly evaluated experimentally.}, doi = {doi:10.1186/1752-0509-6-115}, url = {http://hdl.handle.net/1963/6515}, author = {Giuseppe Facchetti and Claudio Altafini and Mattia Zampieri} } @article {1110.6124, title = {A planar bi-Lipschitz extension Theorem}, year = {2011}, url = {http://arxiv.org/abs/1110.6124}, author = {Sara Daneri and Aldo Pratelli} } @article {2011, title = {Planar loops with prescribed curvature: existence, multiplicity and uniqueness results}, journal = {Proceedings of the American Mathematical Society 139 (2011) 4445-4459}, number = {SISSA;08/2010/M}, year = {2011}, publisher = {American Mathematical Society}, keywords = {Plane curves}, doi = {10.1090/S0002-9939-2011-10915-8}, url = {http://hdl.handle.net/1963/3842}, author = {Roberta Musina} } @article {2011, title = {Poincar{\'e} covariance and κ-Minkowski spacetime}, journal = {Physics Letters A 375 (2011) 3496-3498}, number = {SISSA;43/2010/FM}, year = {2011}, publisher = {Elsevier}, abstract = {A fully Poincar{\'e} covariant model is constructed out of the k-Minkowski spacetime. Covariance is implemented by a unitary representation of the Poincar{\'e} group, and thus complies with the original Wigner approach to quantum symmetries. This provides yet another example (besides the DFR model), where Poincar{\'e} covariance is realised {\'a} la Wigner in the presence of two characteristic dimensionful parameters: the light speed and the Planck length. In other words, a Doubly Special Relativity (DSR) framework may well be realised without deforming the meaning of \\\"Poincar{\'e} covariance\\\".}, doi = {10.1016/j.physleta.2011.08.011}, url = {http://hdl.handle.net/1963/3893}, author = {Ludwik Dabrowski and Gherardo Piacitelli} } @article {2011, title = {Poincar{\'e} polynomial of moduli spaces of framed sheaves on (stacky) Hirzebruch surfaces}, journal = {Communications in Mathematical Physics 304 (2011) 395-409}, volume = {304}, number = {SISSA;56/2009/FM}, year = {2011}, month = {06/2011}, pages = {395-409}, publisher = {Springer}, abstract = {We perform a study of the moduli space of framed torsion-free sheaves on Hirzebruch surfaces by using localization techniques. We discuss some general properties of this moduli space by studying it in the framework of Huybrechts-Lehn theory of framed modules. We classify the fixed points under a toric action on the moduli space, and use this to compute the Poincare polynomial of the latter. This will imply that the moduli spaces we are considering are irreducible. We also consider fractional first Chern classes, which means that we are extending our computation to a stacky deformation of a Hirzebruch surface. From the physical viewpoint, our results provide the partition function of N=4 Vafa-Witten theory on total spaces of line bundles on P1, which is relevant in black hole entropy counting problems according to a conjecture due to Ooguri, Strominger and Vafa.

}, doi = {10.1007/s00220-011-1231-z}, url = {http://hdl.handle.net/1963/3738}, author = {Ugo Bruzzo and Rubik Poghossian and Alessandro Tanzini} } @article {2011, title = {Product of real spectral triples}, journal = {International Journal of Geometric Methods in Modern Physics 8 (2011) 1833-1848}, number = {arXiv:1011.4456;}, year = {2011}, note = {Based on the talk given at the conference \\\"Noncommutative Geometry and Quantum Physics, Vietri sul Mare, Aug 31 - Sept 5, 2009\\\"}, publisher = {World Scientific}, abstract = {We construct the product of real spectral triples of arbitrary finite dimension (and arbitrary parity) taking into account the fact that in the even case there are two possible real structures, in the odd case there are two inequivalent representations of the gamma matrices (Clifford algebra), and in the even-even case there are two natural candidates for the Dirac operator of the product triple.}, doi = {10.1142/S021988781100597X}, url = {http://hdl.handle.net/1963/5510}, author = {Ludwik Dabrowski and Giacomo Dossena} } @article {2011, title = {A proof of Sudakov theorem with strictly convex norms}, journal = {Mathematische Zeitschrift 268 (2011) 371-407}, number = {SISSA;64/2008/M}, year = {2011}, publisher = {Springer}, abstract = {We establish a solution to the Monge problem in Rn, with an asymmetric, strictly convex norm cost function, when the initial measure is absolutely continuous. We focus on the strategy, based on disintegration of measures, initially proposed by Sudakov. As known, there is a gap to fill. The missing step is completed when the unit ball is strictly convex, but not necessarily differentiable nor uniformly convex. The key disintegration is achieved following a similar proof for a variational problem.}, doi = {10.1007/s00209-010-0677-6}, url = {http://hdl.handle.net/1963/2967}, author = {Laura Caravenna} } @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} } @conference {10.1007/978-90-481-9195-6_4, title = {A Phase Field Approach to Wetting and Contact Angle Hysteresis Phenomena}, booktitle = {IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials}, year = {2010}, pages = {51{\textendash}63}, publisher = {Springer Netherlands}, organization = {Springer Netherlands}, address = {Dordrecht}, abstract = {We discuss a phase field model for the numerical simulation of contact angle hysteresis phenomena in wetting. The performance of the model is assessed by comparing its predictions with experimental data on the critical size of drops that can stick on a vertical glass plate.

}, isbn = {978-90-481-9195-6}, author = {Antonio DeSimone and Livio Fedeli and Turco, Alessandro}, editor = {Hackl, Klaus} } @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 = {Poles of Integrale Tritronquee and Anharmonic Oscillators. Asymptotic localization from WKB analysis}, journal = {Nonlinearity. vol. 23, (2010), page 2501-2507}, number = {SISSA;11/2010/FM}, year = {2010}, abstract = {Poles of integrale tritronquee are in bijection with cubic oscillators that admit the simultaneous solutions of two quantization conditions. We show that the poles lie near the solutions of a pair of Bohr-Sommerfeld quantization conditions (the Bohr-Sommerfeld-Boutroux system): the distance between a pole and the corresponding solution of the Bohr-Sommerfeld-Boutroux system vanishes asymptotically.

}, doi = {10.1088/0951-7715/23/10/008}, url = {http://hdl.handle.net/1963/3841}, author = {Davide Masoero} } @article {cerami2010positive, title = {Positive solutions for some non-autonomous Schr{\"o}dinger{\textendash}Poisson systems}, journal = {Journal of Differential Equations}, volume = {248}, number = {3}, year = {2010}, pages = {521{\textendash}543}, publisher = {Academic Press}, author = {Giovanna Cerami and Giusi Vaira} } @article {2010, title = {Projective Reeds-Shepp car on $S^2$ with quadratic cost}, journal = {ESAIM COCV 16 (2010) 275-297}, number = {SISSA;34/2008/M}, year = {2010}, abstract = {Fix two points $x,\\\\bar{x}\\\\in S^2$ and two directions (without orientation) $\\\\eta,\\\\bar\\\\eta$ of the velocities in these points. In this paper we are interested to the problem of minimizing the cost $$ J[\\\\gamma]=\\\\int_0^T g_{\\\\gamma(t)}(\\\\dot\\\\gamma(t),\\\\dot\\\\gamma(t))+\\nK^2_{\\\\gamma(t)}g_{\\\\gamma(t)}(\\\\dot\\\\gamma(t),\\\\dot\\\\gamma(t)) ~dt$$ along all smooth curves starting from $x$ with direction $\\\\eta$ and ending in $\\\\bar{x}$ with direction $\\\\bar\\\\eta$. Here $g$ is the standard Riemannian metric on $S^2$ and $K_\\\\gamma$ is the corresponding geodesic curvature.\\nThe interest of this problem comes from mechanics and geometry of vision. It can be formulated as a sub-Riemannian problem on the lens space L(4,1).\\nWe compute the global solution for this problem: an interesting feature is that some optimal geodesics present cusps. The cut locus is a stratification with non trivial topology.}, doi = {10.1051/cocv:2008075}, url = {http://hdl.handle.net/1963/2668}, author = {Ugo Boscain and Francesco Rossi} } @article {Bertola:IZtau, title = {The partition function of the two-matrix model as an isomonodromic τ function}, journal = {J. Math. Phys.}, volume = {50}, number = {1}, year = {2009}, pages = {013529, 17}, issn = {0022-2488}, doi = {10.1063/1.3054865}, url = {http://0-dx.doi.org.mercury.concordia.ca/10.1063/1.3054865}, author = {Marco Bertola and Marchal, O.} } @article {Berti2008601, title = {On periodic elliptic equations with gradient dependence}, journal = {Communications on Pure and Applied Analysis}, volume = {7}, number = {3}, year = {2008}, note = {cited By (since 1996)1}, pages = {601-615}, abstract = {We construct entire solutions of Δu = f(x, u, ∇u) which are superpositions of odd, periodic functions and linear ones, with prescribed integer or rational slope.}, issn = {15340392}, author = {Massimiliano Berti and Matzeu, M and Enrico Valdinoci} } @article {mercuri2008positive, title = {Positive solutions of nonlinear Schr{\"o}dinger-Poisson systems with radial potentials vanishing at infinity}, journal = {Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl}, volume = {19}, number = {3}, year = {2008}, pages = {211{\textendash}227}, publisher = {Citeseer}, abstract = {We deal with a weighted nonlinear Schr{\textasciidieresis}odinger-Poisson system, allowing the potentials to vanish at infinity.

}, doi = {10.1.1.510.3635}, url = {http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.510.3635\&rep=rep1\&type=pdf}, author = {Mercuri, Carlo} } @article {2007, title = {Parametrized curves in Lagrange Grassmannians}, journal = {C. R. Math. 345 (2007) 647-652}, number = {arXiv.org;0708.1100v1}, year = {2007}, doi = {10.1016/j.crma.2007.10.034}, url = {http://hdl.handle.net/1963/2560}, author = {Igor Zelenko and Li Chengbo} } @article {2007, title = {Perturbation techniques applied to the real vanishing viscosity approximation of an initial boundary value problem}, year = {2007}, institution = {SISSA}, url = {http://preprints.sissa.it/handle/1963/35315}, author = {Stefano Bianchini} } @article {2006, title = {On Palais-Smale sequences for H-systems: some examples}, journal = {Adv. Differential Equations 11 (2006) 931-960}, number = {SISSA;32/2005/M}, year = {2006}, abstract = {We exhibit a series of examples of Palais-Smale sequences for the Dirichlet problem associated to the mean curvature equation with null boundary condition, and we show that in the case of nonconstant mean curvature functions different kinds of blow up phenomena appear and Palais-Smale sequences may have quite wild behaviour.}, url = {http://hdl.handle.net/1963/2157}, author = {Paolo Caldiroli and Roberta Musina} } @article {Bertola:PDEBOPs, title = {The PDEs of biorthogonal polynomials arising in the two-matrix model}, journal = {Math. Phys. Anal. Geom.}, volume = {9}, number = {1}, year = {2006}, pages = {23{\textendash}52}, issn = {1385-0172}, author = {Marco Bertola and B. Eynard} } @article {Baldi2006257, title = {Periodic solutions of nonlinear wave equations for asymptotically full measure sets of frequencies}, journal = {Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni}, volume = {17}, number = {3}, year = {2006}, note = {cited By (since 1996)5}, pages = {257-277}, abstract = {We prove existence and multiplicity of small amplitude periodic solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for asymptotically full measure sets of frequencies, extending the results of [7] to new types of nonlinearities.}, issn = {11206330}, author = {P Baldi and Massimiliano Berti} } @article {2005, title = {Periodic solutions of nonlinear wave equations with non-monotone forcing terms}, journal = {Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16 (2005), no. 2, 117-124}, year = {2005}, publisher = {Accademia Nazionale dei Lincei}, url = {http://hdl.handle.net/1963/4581}, author = {Massimiliano Berti and Luca Biasco} } @article {2005, title = {Principal fibrations from noncommutative spheres}, journal = {Comm. Math. Phys. 260 (2005) 203-225}, number = {SISSA;68/2004/FM}, year = {2005}, abstract = {We construct noncommutative principal fibrations S_\\\\theta^7 \\\\to S_\\\\theta^4 which are deformations of the classical SU(2) Hopf fibration over the four sphere. We realize the noncommutative vector bundles associated to the irreducible representations of SU(2) as modules of coequivariant maps and construct corresponding projections. The index of Dirac operators with coefficients in the associated bundles is computed with the Connes-Moscovici local index formula. The algebra inclusion $A(S_\\\\theta^4) \\\\into A(S_\\\\theta^7)$ is an example of a not trivial quantum principal bundle.}, doi = {10.1007/s00220-005-1377-7}, url = {http://hdl.handle.net/1963/2284}, author = {Giovanni Landi and Walter van Suijlekom} } @article {2004, title = {Periodic orbits close to elliptic tori and applications to the three-body problem}, journal = {Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004) 87-138}, number = {SISSA;28/2003/M}, year = {2004}, publisher = {Scuola Normale Superiore di Pisa}, abstract = {We prove, under suitable non-resonance and non-degeneracy {\textquoteleft}{\textquoteleft}twist\\\'\\\' conditions, a Birkhoff-Lewis type result showing the existence of infinitely many periodic solutions, with larger and larger minimal period, accumulating onto elliptic invariant tori (of Hamiltonian systems). We prove the applicability of this result to the spatial planetary three-body problem in the small eccentricity-inclination regime. Furthermore, we find other periodic orbits under some restrictions on the period and the masses of the {\textquoteleft}{\textquoteleft}planets\\\'\\\'. The proofs are based on averaging theory, KAM theory and variational methods. (Supported by M.U.R.S.T. Variational Methods and Nonlinear Differential Equations.)}, url = {http://hdl.handle.net/1963/2985}, author = {Massimiliano Berti and Luca Biasco and Enrico Valdinoci} } @article {2003, title = {Parameter differentiation and quantum state decomposition for time varying Schr{\"o}dinger equations}, journal = {Rep. Math. Phys. 52 (2003) 381-400}, number = {arXiv.org;quant-ph/0201034v2}, year = {2003}, publisher = {Elsevier}, abstract = {For the unitary operator, solution of the Schroedinger equation corresponding to a time-varying Hamiltonian, the relation between the Magnus and the product of exponentials expansions can be expressed in terms of a system of first order differential equations in the parameters of the two expansions. A method is proposed to compute such differential equations explicitly and in a closed form.}, doi = {10.1016/S0034-4877(03)80037-X}, url = {http://hdl.handle.net/1963/3017}, author = {Claudio Altafini} } @article {Bertola:PartitionJPA, title = {Partition functions for matrix models and isomonodromic tau functions}, journal = {J. Phys. A}, volume = {36}, number = {12}, year = {2003}, note = {Random matrix theory}, pages = {3067{\textendash}3083}, issn = {0305-4470}, author = {Marco Bertola and B. Eynard and Harnad, J.} } @article {2003, title = {Periodic solutions of nonlinear wave equations with general nonlinearities}, journal = {Comm.Math.Phys. 243 (2003) no.2, 315}, number = {SISSA;78/2002/M}, year = {2003}, publisher = {SISSA Library}, doi = {10.1007/s00220-003-0972-8}, url = {http://hdl.handle.net/1963/1648}, author = {Massimiliano Berti and Philippe Bolle} } @article {2003, title = {Poisson Pencils, Integrability, and Separation of Variables}, number = {SISSA;90/2003/FM}, year = {2003}, institution = {SISSA}, abstract = {In this paper we will review a recently introduced method for solving the Hamilton-Jacobi equations by the method of Separation of Variables. This method is based on the notion of pencil of Poisson brackets and on the bihamiltonian approach to integrable systems. We will discuss how separability conditions can be intrinsically characterized within such a geometrical set-up, the definition of the separation coordinates being encompassed in the \\\\bih structure itself. We finally discuss these constructions studying in details a particular example, based on a generalization of the classical Toda Lattice.}, url = {http://hdl.handle.net/1963/3026}, author = {Gregorio Falqui} } @article {2003, title = {Positive solutions to a class of quasilinear elliptic equations on R}, journal = {Discrete Contin.Dyn.Syst. 9 (2003), no.1, 55-68}, number = {SISSA;58/2002/M}, year = {2003}, publisher = {American Institute of Mathematical Sciences}, abstract = {We discuss the existence of positive solutions of perturbation to a class of quasilinear elliptic equations on R.}, doi = {10.3934/dcds.2003.9.55}, url = {http://hdl.handle.net/1963/1628}, author = {Antonio Ambrosetti and Wang Zhi-Qiang} } @article {2003, title = {Prescribing scalar and boundary mean curvature on the three dimensional half sphere}, journal = {J. Geom. Anal. 13 (2003) 255-289}, number = {SISSA;36/2001/M}, year = {2003}, publisher = {Springer}, abstract = {We consider the problem of prescribing the scalar curvature and the boundary mean curvature of the standard half three sphere, by deforming conformally its standard metric. Using blow up analysis techniques and minimax arguments, we prove some existence and compactness results.}, doi = {10.1007/BF02930697}, url = {http://hdl.handle.net/1963/3086}, author = {Zindine Djadli and Andrea Malchiodi and Mohameden Ould Ahmedou} } @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 {2002, title = {On a Poisson reduction for Gel\\\'fand-Zakharevich manifolds}, journal = {Rep.Math.Phys.50 (2002), no.3, 395}, number = {SISSA;31/2002/FM}, year = {2002}, publisher = {SISSA Library}, doi = {10.1016/S0034-4877(02)80068-4}, url = {http://hdl.handle.net/1963/1602}, author = {Gregorio Falqui and Marco Pedroni} } @article {2002, title = {Prescribing a fourth oder conformal invariant on the standard sphere - Part I: a perturbation result}, journal = {Commun. Contemp. Math., 2002, 4, 375}, number = {SISSA;82/00/M}, year = {2002}, publisher = {SISSA Library}, doi = {10.1142/S0219199702000695}, url = {http://hdl.handle.net/1963/1539}, author = {Zindine Djadli and Mohameden Ould Ahmedou and Andrea Malchiodi} } @article {2002, title = {Prescribing a fourth oder conformal invariant on the standard sphere - Part II: blow up analysis and applications}, journal = {Ann. Sc. Norm. Super. Pisa Cl. Sci., 2002, 1, 387}, number = {SISSA;83/00/M}, year = {2002}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1540}, author = {Zindine Djadli and Andrea Malchiodi and Mohameden Ould Ahmedou} } @article {2001, title = {Picard and Chazy solutions to the Painlev{\'e} VI equation}, journal = {Math. Ann. 321 (2001) 157-195}, number = {SISSA;89/98/FM}, year = {2001}, publisher = {Springer}, abstract = {I study the solutions of a particular family of Painlev{\'e} VI equations with the parameters $\beta=\gamma=0, \delta=1/2$ and $2\alpha=(2\mu-1)^2$, for $2\mu\in\mathbb{Z}$. I show that the case of half-integer $\mu$ is integrable and that the solutions are of two types: the so-called Picard solutions and the so-called Chazy solutions. I give explicit formulae for them and completely determine their asymptotic behaviour near the singular points $0,1,\infty$ and their nonlinear monodromy. I study the structure of analytic continuation of the solutions to the PVI$\mu$ equation for any $\mu$ such that $2\mu\in\mathbb{Z}$. As an application, I classify all the algebraic solutions. For $\mu$ half-integer, I show that they are in one to one correspondence with regular polygons or star-polygons in the plane. For $\mu$ integer, I show that all algebraic solutions belong to a one-parameter family of rational solutions.

}, doi = {10.1007/PL00004500}, url = {http://hdl.handle.net/1963/3118}, author = {Marta Mazzocco} } @inbook {2000, title = {Principal invariants of Jacobi curves}, booktitle = {Nonlinear control in the Year 2000 / Alberto Isidori, Francoise Lamnabhi-Lagarrigue, Witold Respondek (eds.) - Springer : Berlin, 2001. - (Lecture notes in control and information sciences ; 258). - ISBN 1-85233-363-4 (v. 1). - p. 9-22.}, year = {2000}, publisher = {Springer}, organization = {Springer}, abstract = {Jacobi curves are far going generalizations of the spaces of \\\"Jacobi fields\\\" along Riemannian geodesics. Actually, Jacobi curves are curves in the Lagrange Grassmannians. Differential geometry of these curves provides basic feedback or gauge invariants for a wide class of smooth control systems and geometric structures. In the present paper we mainly discuss two principal invariants: the generalized Ricci curvature, which is an invariant of the parametrized curve in the Lagrange Grassmanian providing the curve with a natural projective structure, and a fundamental form, which is a 4-oder differential on the curve.}, doi = {10.1007/BFb0110204}, url = {http://hdl.handle.net/1963/3825}, author = {Andrei A. Agrachev and Igor Zelenko} } @inbook {1999, title = {Painlev{\'e} transcendents in two-dimensional topological field theory}, booktitle = {The Painlev{\'e} property : one century later / Robert Conte ed. - New York : Springer-Verlag, 1999. - (CRM series in mathematical physics). - p. 287-412}, number = {SISSA;24/98/FM}, year = {1999}, publisher = {Springer}, organization = {Springer}, isbn = {0-387-98888-2}, url = {http://hdl.handle.net/1963/3238}, author = {Boris Dubrovin} } @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 = {Projection singularities of extremals for planar systems}, number = {SISSA;90/99/M}, year = {1999}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1304}, author = {Ugo Boscain and Benedetto Piccoli} } @article {1989, title = {A pointwise regularity theory for the two-obstacle problem}, journal = {Acta Math. 163 (1989), no. 1-2, 57-107}, number = {SISSA;47/88/M}, year = {1989}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/643}, author = {Gianni Dal Maso and Umberto Mosco and Maria Agostina Vivaldi} }