@article {2023, title = {Benjamin-Feir Instability of Stokes Waves in Finite Depth}, journal = {ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS}, volume = {247}, year = {2023}, month = {OCT}, pages = {91}, type = {Article}, abstract = {

Whitham and Benjamin predicted in 1967 that small-amplitude periodic traveling Stokes waves of the 2d-gravity water waves equations are linearly unstable with respect to long-wave perturbations, if the depth h is larger than a critical threshold h(WB) approximate to 1.363. In this paper, we completely describe, for any finite value of h \> 0, the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent mu is turned on. We prove, in particular, the existence of a unique depth h(WB), which coincides with the one predicted by Whitham and Benjamin, such that, for any 0 \< h \< h(WB), the eigenvalues close to zero are purely imaginary and, for any h \> h(WB), a pair of non-purely imaginary eigenvalues depicts a closed figure {\textquoteleft}{\textquoteleft}8{\textquoteright}{\textquoteright}, parameterized by the Floquet exponent. As h -\> h(WB)(+) the {\textquoteleft}{\textquoteleft}8{\textquoteright}{\textquoteright} collapses to the origin of the complex plane. The complete bifurcation diagram of the spectrum is not deduced as in deep water, since the limits h -\> +infinity (deep water) and mu -\> 0 (long waves) do not commute. In finite depth, the four eigenvalues have all the same size O(mu), unlike in deep water, and the analysis of their splitting is much more delicate, requiring, as a new ingredient, a non-perturbative step of block-diagonalization. Along the whole proof, the explicit dependence of the matrix entries with respect to the depth h is carefully tracked.

}, issn = {0003-9527}, doi = {10.1007/s00205-023-01916-2}, author = {Berti, Massimiliano and Maspero, Alberto and Ventura, Paolo} } @article {2023, title = {Stokes waves at the critical depth are modulational unstable}, year = {2023}, author = {Berti, Massimiliano and Maspero, Alberto and Ventura, Paolo} } @article {2022, title = {On the analyticity of the Dirichlet-Neumann operator and Stokes waves}, journal = {RENDICONTI LINCEI-MATEMATICA E APPLICAZIONI}, volume = {33}, year = {2022}, pages = {611-650}, type = {Article}, abstract = {

We prove an analyticity result for the Dirichlet-Neumann operator under space periodic boundary conditions in any dimension in an unbounded domain with infinite depth. We derive an analytic bifurcation result of analytic Stokes waves-i.e., space periodic traveling solutions-of the water waves equations in deep water.

}, keywords = {Bifurcation theory, Dirichlet-Neumann operator, traveling waves, water waves}, issn = {1120-6330}, doi = {10.4171/RLM/983}, author = {Berti, Massimiliano and Maspero, Alberto and Ventura, Paolo} } @article {2022, title = {Benjamin-Feir instability of Stokes waves}, journal = {RENDICONTI LINCEI-MATEMATICA E APPLICAZIONI}, volume = {33}, year = {2022}, pages = {399-412}, type = {Article}, abstract = {

We present the recent results in Berti et al. [Invent. Math. (2022), to appear] regarding the Benjamin-Feir instability of small amplitude Stokes waves in deep water. We completely describe the behavior of the four eigenvalues close to zero of the linearized water waves equations at the Stokes solution, as the Floquet exponent is turned on, proving the conjecture that a pair of non-purely imaginary eigenvalues depicts a closed figure {\textquoteleft}{\textquoteleft}8{\textquoteright}{\textquoteright}, in full agreement with numerical simulations.

}, keywords = {Kato perturbation theory, modulational instability, traveling waves, water waves}, issn = {1120-6330}, doi = {10.4171/RLM/975}, author = {Berti, Massimiliano and Maspero, Alberto and Ventura, Paolo} } @article {2022, title = {Full description of Benjamin-Feir instability of stokes waves in deep water}, volume = {230}, year = {2022}, month = {2022/11/01}, pages = {651 - 711}, abstract = {

Small-amplitude, traveling, space periodic solutions {\textendash}called Stokes waves{\textendash} of the 2 dimensional gravity water waves equations in deep water are linearly unstable with respect to long-wave perturbations, as predicted by Benjamin and Feir in 1967. We completely describe the behavior of the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent is turned on. We prove in particular the conjecture that a pair of non-purely imaginary eigenvalues depicts a closed figure {\textquotedblleft}8{\textquotedblright}, parameterized by the Floquet exponent, in full agreement with numerical simulations. Our new spectral approach to the Benjamin-Feir instability phenomenon uses a symplectic version of Kato{\textquoteright}s theory of similarity transformation to reduce the problem to determine the eigenvalues of a $ 4 \times 4 $ complex Hamiltonian and reversible matrix. Applying a procedure inspired by KAM theory, we block-diagonalize such matrix into a pair of $2 \times 2 $ Hamiltonian and reversible matrices, thus obtaining the full description of its eigenvalues.

}, isbn = {1432-1297}, url = {https://doi.org/10.1007/s00222-022-01130-z}, author = {Berti, Massimiliano and Maspero, Alberto and Ventura, Paolo} } @conference {2021, title = {Discontinuous Galerkin Model Order Reduction of Geometrically Parametrized Stokes Equation}, booktitle = {Numerical Mathematics and Advanced Applications ENUMATH 2019}, year = {2021}, month = {2021//}, publisher = {Springer International Publishing}, organization = {Springer International Publishing}, address = {Cham}, abstract = {

The present work focuses on the geometric parametrization and the reduced order modeling of the Stokes equation. We discuss the concept of a parametrized geometry and its application within a reduced order modeling technique. The full order model is based on the discontinuous Galerkin method with an interior penalty formulation. We introduce the broken Sobolev spaces as well as the weak formulation required for an affine parameter dependency. The operators are transformed from a fixed domain to a parameter dependent domain using the affine parameter dependency. The proper orthogonal decomposition is used to obtain the basis of functions of the reduced order model. By using the Galerkin projection the linear system is projected onto the reduced space. During this process, the offline-online decomposition is used to separate parameter dependent operations from parameter independent operations. Finally this technique is applied to an obstacle test problem.The numerical outcomes presented include experimental error analysis, eigenvalue decay and measurement of online simulation time.

}, isbn = {978-3-030-55874-1}, author = {Nirav Shah and Martin W. Hess and Gianluigi Rozza}, editor = {Vermolen, Fred J. and Vuik, Cornelis} } @booklet {2021, title = {Monotonicity formulas for harmonic functions in RCD(0,N) spaces}, year = {2021}, abstract = {

We generalize to the\ RCD(0,N)\ setting a family of monotonicity formulas by Colding and Minicozzi for positive harmonic functions in Riemannian manifolds with non-negative Ricci curvature. Rigidity and almost rigidity statements are also proven, the second appearing to be new even in the smooth setting. Motivated by the recent work in [AFM] we also introduce the notion of electrostatic potential in\ RCD\ spaces, which also satisfies our monotonicity formulas. Our arguments are mainly based on new estimates for harmonic functions in\ RCD(K,N)\ spaces and on a new functional version of the {\textquoteleft}(almost) outer volume cone implies (almost) outer metric cone{\textquoteright} theorem.

}, author = {Nicola Gigli and Ivan Yuri Violo} } @conference {2021, title = {Reduced Order Methods for Parametrized Non-linear and Time Dependent Optimal Flow Control Problems, Towards Applications in Biomedical and Environmental Sciences}, booktitle = {Numerical Mathematics and Advanced Applications ENUMATH 2019}, year = {2021}, month = {2021//}, publisher = {Springer International Publishing}, organization = {Springer International Publishing}, address = {Cham}, abstract = {

We introduce reduced order methods as an efficient strategy to solve parametrized non-linear and time dependent optimal flow control problems governed by partial differential equations. Indeed, the optimal control problems require a huge computational effort in order to be solved, most of all in physical and/or geometrical parametrized settings. Reduced order methods are a reliable and suitable approach, increasingly gaining popularity, to achieve rapid and accurate optimal solutions in several fields, such as in biomedical and environmental sciences. In this work, we employ a POD-Galerkin reduction approach over a parametrized optimality system, derived from the Karush-Kuhn-Tucker conditions. The methodology presented is tested on two boundary control problems, governed respectively by (1) time dependent Stokes equations and (2) steady non-linear Navier-Stokes equations.

}, isbn = {978-3-030-55874-1}, doi = {https://doi.org/10.1007/978-3-030-55874-1_83}, url = {https://www.springerprofessional.de/en/reduced-order-methods-for-parametrized-non-linear-and-time-depen/19122676}, author = {Maria Strazzullo and Zakia Zainib and F. Ballarin and Gianluigi Rozza}, editor = {Fred J Vermolen and Cornelis Vuik} } @booklet {2021, title = {A remark on two notions of flatness for sets in the Euclidean space}, year = {2021}, abstract = {

In this note we compare two ways of measuring the\ n-dimensional "flatness" of a set\ SRd, where\ nN\ and\ d\>n. The first one is to consider the classical Reifenberg-flat numbers\ α(x,r)\ (xS,\ r\>0), which measure the minimal scaling-invariant Hausdorff distances in\ Br(x)\ between\ S\ and\ n-dimensional affine subspaces of\ Rd. The second is an {\textquoteleft}intrinsic{\textquoteright} approach in which we view the same set\ S\ as a metric space (endowed with the induced Euclidean distance). Then we consider numbers\ a(x,r){\textquoteright}s, that are the scaling-invariant Gromov-Hausdorff distances between balls centered at\ x\ of radius\ r\ in\ S\ and the\ n-dimensional Euclidean ball of the same radius. As main result of our analysis we make rigorous a phenomenon, first noted by David and Toro, for which the numbers\ a(x,r){\textquoteright}s behaves as the square of the numbers\ α(x,r){\textquoteright}s. Moreover we show how this result finds application in extending the Cheeger-Colding intrinsic-Reifenberg theorem to the biLipschitz case. As a by-product of our arguments, we deduce analogous results also for the Jones{\textquoteright} numbers\ β{\textquoteright}s (i.e. the one-sided version of the numbers\ α{\textquoteright}s).

}, author = {Ivan Yuri Violo} } @booklet {2021, title = {Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds}, year = {2021}, abstract = {

We prove that ifMis a closedn-dimensional Riemannian manifold,n>=3, withRic>=n-1and for which the optimal constant in the critical Sobolev inequality equals the one of then-dimensional sphereSn, thenMis isometric toSn. An almost-rigidity result is also established, saying that if equality is almost achieved, thenMis close in the measure Gromov-Hausdorff sense to a spherical suspension. These statements are obtained in theRCD-setting of (possibly non-smooth) metric measure spaces satisfying synthetic lower Ricci curvature bounds.
An independent result of our analysis is the characterization of the best constant in the Sobolev inequality on any compactCDspace, extending to the non-smooth setting a classical result by Aubin. Our arguments are based on a new concentration compactness result for mGH-converging sequences ofRCDspaces and on a Polya-Szego inequality of Euclidean-type inCDspaces.
As an application of the technical tools developed we prove both an existence result for the Yamabe equation and the continuity of the generalized Yamabe constant under measure Gromov-Hausdorff convergence, in theRCD-setting.
}, author = {Francesco Nobili and Ivan Yuri Violo} } @booklet {2020, title = {Minimality of the ball for a model of charged liquid droplets}, year = {2020}, author = {Ekaterina Mukoseeva and Giulia Vescovo} } @article {2020, title = {POD{\textendash}Galerkin reduced order methods for combined Navier{\textendash}Stokes transport equations based on a hybrid FV-FE solver}, journal = {Computers and Mathematics with Applications}, volume = {79}, year = {2020}, pages = {256-273}, abstract = {

The purpose of this work is to introduce a novel POD{\textendash}Galerkin strategy for the semi-implicit hybrid high order finite volume/finite element solver introduced in Berm{\'u}dez et al. (2014) and Busto et al. (2018). The interest is into the incompressible Navier{\textendash}Stokes equations coupled with an additional transport equation. The full order model employed in this article makes use of staggered meshes. This feature will be conveyed to the reduced order model leading to the definition of reduced basis spaces in both meshes. The reduced order model presented herein accounts for velocity, pressure, and a transport-related variable. The pressure term at both the full order and the reduced order level is reconstructed making use of a projection method. More precisely, a Poisson equation for pressure is considered within the reduced order model. Results are verified against three-dimensional manufactured test cases. Moreover a modified version of the classical cavity test benchmark including the transport of a species is analysed.

}, doi = {10.1016/j.camwa.2019.06.026}, url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85068068567\&doi=10.1016\%2fj.camwa.2019.06.026\&partnerID=40\&md5=a8dcce1b53b8ee872d174bbc4c20caa3}, author = {S. Busto and Giovanni Stabile and Gianluigi Rozza and M.E. V{\'a}zquez-Cend{\'o}n} } @conference {2019, title = {Efficient reduction in shape parameter space dimension for ship propeller blade design}, booktitle = {8th International Conference on Computational Methods in Marine Engineering, MARINE 2019}, 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.

}, url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85075395143\&partnerID=40\&md5=b6aa0fcedc2f88e78c295d0f437824d0}, author = {Andrea Mola and Marco Tezzele and Mahmoud Gadalla and Valdenazzi, Federica and Grassi, Davide and Padovan, Roberta and Gianluigi Rozza} } @article {2019, title = {A neutrally stable shell in a Stokes flow: a rotational Taylor{\textquoteright}s sheet}, journal = {Proceedings of the Royal Society A: Mathematical, Physical and Engineering SciencesProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences}, volume = {475}, year = {2019}, month = {2019/07/26}, pages = {20190178}, url = {https://doi.org/10.1098/rspa.2019.0178}, author = {Giovanni Corsi and Antonio DeSimone and C. Maurini and S. Vidoli} } @booklet {2019, title = {Regularity of minimizers for a model of charged droplets}, year = {2019}, author = {Guido De Philippis and Jonas Hirsch and Giulia Vescovo} } @article {2019, title = {Weighted Reduced Order Methods for Parametrized Partial Differential Equations with Random Inputs}, journal = {PoliTO Springer Series}, year = {2019}, pages = {27-40}, abstract = {

In this manuscript we discuss weighted reduced order methods for stochastic partial differential equations. Random inputs (such as forcing terms, equation coefficients, boundary conditions) are considered as parameters of the equations. We take advantage of the resulting parametrized formulation to propose an efficient reduced order model; we also profit by the underlying stochastic assumption in the definition of suitable weights to drive to reduction process. Two viable strategies are discussed, namely the weighted reduced basis method and the weighted proper orthogonal decomposition method. A numerical example on a parametrized elasticity problem is shown.

}, doi = {10.1007/978-3-030-04870-9_2}, url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85084009379\&doi=10.1007\%2f978-3-030-04870-9_2\&partnerID=40\&md5=446bcc1f331167bbba67bc00fb170150}, author = {L. Venturi and D. Torlo and F. Ballarin and Gianluigi Rozza} } @inbook {AuricchioContiLefieuxMorgantiRealiRozzaVeneziani2018, title = {Computational methods in cardiovascular mechanics}, booktitle = {Cardiovascular Mechanics}, year = {2018}, pages = {54}, publisher = {CRC Press}, organization = {CRC Press}, chapter = {Computational methods in cardiovascular mechanics}, abstract = {

The introduction of computational models in cardiovascular sciences has been progressively bringing new and unique tools for the investigation of the physiopathology. Together with the dramatic improvement of imaging and measuring devices on one side, and of computational architectures on the other one, mathematical and numerical models have provided a new, clearly noninvasive, approach for understanding not only basic mechanisms but also patient-specific conditions, and for supporting the design and the development of new therapeutic options. The terminology in silico is, nowadays, commonly accepted for indicating this new source of knowledge added to traditional in vitro and in vivo investigations. The advantages of in silico methodologies are basically the low cost in terms of infrastructures and facilities, the reduced invasiveness and, in general, the intrinsic predictive capabilities based on the use of mathematical models. The disadvantages are generally identified in the distance between the real cases and their virtual counterpart required by the conceptual modeling that can be detrimental for the reliability of numerical simulations.

}, url = {https://www.taylorfrancis.com/books/e/9781315280288/chapters/10.1201\%2Fb21917-5}, author = {Auricchio, Ferdinando and Conti, Michele and Lefieux, Adrian and Morganti, Simone and Alessandro Reali and Gianluigi Rozza and Veneziani, Alessandro}, editor = {Michel F. Labrosse} } @article {scala2016currents, title = {Currents and dislocations at the continuum scale}, journal = {Methods and Applications of Analysis}, volume = {23}, number = {1}, year = {2016}, pages = {1{\textendash}34}, publisher = {International Press of Boston}, abstract = {

A striking geometric property of elastic bodies with dislocations is that the deformation tensor cannot be written as the gradient of a one-to-one immersion, its curl being nonzero and equal to the density of the dislocations, a measure concentrated in the dislocation lines. In this work, we discuss the mathematical properties of such constrained deformations and study a variational problem in finite-strain elasticity, where Cartesian maps allow us to consider deformations in $L^p$ with $1\leq p\<2$, as required for dislocation-induced strain singularities. Firstly, we address the problem of mathematical modeling of dislocations. It is a key purpose of the paper to build a framework where dislocations are described in terms of integral 1-currents and to extract from this theoretical setting a series of notions having a mechanical meaning in the theory of dislocations. In particular, the paper aims at classifying integral 1-currents, with modeling purposes. In the second part of the paper, two variational problems are solved for two classes of dislocations, at the mesoscopic and at the continuum scale. By continuum it is here meant that a countable family of dislocations is considered, allowing for branching and cluster formation, with possible complex geometric patterns. Therefore, modeling assumptions of the defect part of the energy must also be provided, and discussed.

}, doi = {10.4310/MAA.2016.v23.n1.a1}, author = {Riccardo Scala and Nicolas Van Goethem} } @article {doi:10.1002/mma.3450, title = {A compatible-incompatible decomposition of symmetric tensors in Lp with application to elasticity}, journal = {Mathematical Methods in the Applied Sciences}, volume = {38}, number = {18}, year = {2015}, pages = {5217-5230}, abstract = {

In this paper, we prove the Saint-Venant compatibility conditions in $L^p$ for $p\in(1,$\infty$)$, in a simply connected domain of any space dimension. As a consequence, alternative, simple, and direct proofs of some classical Korn inequalities in Lp are provided. We also use the Helmholtz decomposition in $L^p$ to show that every symmetric tensor in a smooth domain can be decomposed in a compatible part, which is the symmetric part of a displacement gradient, and in an incompatible part, which is the incompatibility of a certain divergence-free tensor. Moreover, under a suitable Dirichlet boundary condition, this Beltrami-type decomposition is proved to be unique. This decomposition result has several applications, one of which being in dislocation models, where the incompatibility part is related to the dislocation density and where $1 \< p \< 2$. This justifies the need to generalize and prove these rather classical results in the Hilbertian case ($p = 2$), to the full range $p\in(1,$\infty$)$. Copyright {\textcopyright} 2015\ John Wiley \& Sons, Ltd.

}, keywords = {35J58, 35Q74, compatibility conditions, elasticity, Korn inequality, strain decomposition, subclass74B05}, doi = {10.1002/mma.3450}, url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/mma.3450}, author = {Maggiani, Giovanni Battista and Riccardo Scala and Nicolas Van Goethem} } @article {2015, title = {Dynamics of screw dislocations: a generalised minimising-movements scheme approach}, number = {SISSA;38/2015/MATE}, year = {2015}, institution = {SISSA}, abstract = {The gradient flow structure of the model introduced in [CG99] for the dynamics of screw dislocations is investigated by means of a generalised minimising-movements scheme approach. The assumption of a finite number of available glide directions, together with the "maximal dissipation criterion" that governs the equations of motion, results into solving a differential inclusion rather than an ODE. This paper addresses how the model in [CG99] is connected to a time-discrete evolution scheme which explicitly confines dislocations to move each time step along a single glide direction. It is proved that the time-continuous model in [CG99] is the limit of these time-discrete minimising-movement schemes when the time step converges to 0. The study presented here is a first step towards a generalization of the setting in [AGS08, Chap. 2 and 3] that allows for dissipations which cannot be described by a metric.}, url = {http://urania.sissa.it/xmlui/handle/1963/34495}, author = {Giovanni A. Bonaschi and Patrick Van Meurs and Marco Morandotti} } @article {2014, title = {Adler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras}, number = {arXiv:1401.2082}, year = {2014}, note = {45 pages}, publisher = {SISSA}, abstract = {We put the Adler-Gelfand-Dickey approach to classical W-algebras in the framework of Poisson vertex algebras. We show how to recover the bi-Poisson structure of the KP hierarchy, together with its generalizations and reduction to the N-th KdV hierarchy, using the formal distribution calculus and the lambda-bracket formalism. We apply the Lenard-Magri scheme to prove integrability of the corresponding hierarchies. We also give a simple proof of a theorem of Kupershmidt and Wilson in this framework. Based on this approach, we generalize all these results to the matrix case. In particular, we find (non-local) bi-Poisson structures of the matrix KP and the matrix N-th KdV hierarchies, and we prove integrability of the N-th matrix KdV hierarchy.}, url = {http://hdl.handle.net/1963/7242}, author = {Alberto De Sole and Victor G. Kac and Daniele Valeri} } @article {2013, title = {Classical W-algebras and generalized Drinfeld-Sokolov hierarchies for minimal and short nilpotents}, journal = {Communications in Mathematical Physics 331, nr. 2 (2014) 623-676}, number = {arXiv:1306.1684;}, year = {2014}, note = {46 pages}, publisher = {SISSA}, abstract = {We derive explicit formulas for lambda-brackets of the affine classical W-algebras attached to the minimal and short nilpotent elements of any simple Lie algebra g. This is used to compute explicitly the first non-trivial PDE of the corresponding intgerable generalized Drinfeld-Sokolov hierarchies. It turns out that a reduction of the equation corresponding to a short nilpotent is Svinolupov{\textquoteright}s equation attached to a simple Jordan algebra, while a reduction of the equation corresponding to a minimal nilpotent is an integrable Hamiltonian equation on 2h-3 functions, where h is the dual Coxeter number of g. In the case when g is sl_2 both these equations coincide with the KdV equation. In the case when g is not of type C_n, we associate to the minimal nilpotent element of g yet another generalized Drinfeld-Sokolov hierarchy.}, doi = {10.1007/s00220-014-2049-2}, url = {http://hdl.handle.net/1963/6979}, author = {Alberto De Sole and Victor G. Kac and Daniele Valeri} } @article {2014, title = {A correction and an extension of Stampacchia{\textquoteright}s work on the geometric BVP}, number = {Advanced Nonlinear Studies;volume 14; issue 4; pages 813-837;}, year = {2014}, publisher = {Advanced Nonlinear Studies}, abstract = {G. Stampacchia introduced the geometric boundary value problem for ODEs in his doctoral thesis and published four papers related to it. Here we point out that the proof of his last theorem on the subject is incorrect and we provide a substitute for it as well as a generalizations of some of his earlier results.}, url = {http://urania.sissa.it/xmlui/handle/1963/35023}, author = {Giovanni Vidossich} } @article {2013, title = {Dirac reduction for Poisson vertex algebras}, journal = {Communications in Mathematical Physics 331, nr. 3 (2014) 1155-1190}, number = {arXiv:1306.6589;}, year = {2014}, note = {31 pages}, publisher = {SISSA}, abstract = {We construct an analogue of Dirac{\textquoteright}s reduction for an arbitrary local or non-local Poisson bracket in the general setup of non-local Poisson vertex algebras. This leads to Dirac{\textquoteright}s reduction of an arbitrary non-local Poisson structure. We apply this construction to an example of a generalized Drinfeld-Sokolov hierarchy.}, doi = {10.1007/s00220-014-2103-0}, url = {http://hdl.handle.net/1963/6980}, author = {Alberto De Sole and Victor G. Kac and Daniele Valeri} } @article {scala2014dislocations, title = {Dislocations at the continuum scale: functional setting and variational properties}, year = {2014}, url = {http://cvgmt.sns.it/paper/2294/}, author = {Riccardo Scala and Nicolas Van Goethem} } @article {2014, title = {Editorial}, number = {Rendiconti del circolo matematico di Palermo;volume 63; issue 3; pages 317-318;}, year = {2014}, publisher = {Springer}, doi = {10.1007/s12215-014-0172-8}, url = {http://urania.sissa.it/xmlui/handle/1963/34712}, author = {Ciro Ciliberto and Gianni Dal Maso and Pasquale Vetro} } @article {2014, title = {Integrability of Dirac reduced bi-Hamiltonian equations}, number = {arXiv:1401.6006;}, year = {2014}, note = {15 pages}, institution = {SISSA}, abstract = {First, we give a brief review of the theory of the Lenard-Magri scheme for a non-local bi-Poisson structure and of the theory of Dirac reduction. These theories are used in the remainder of the paper to prove integrability of three hierarchies of bi-Hamiltonian PDE{\textquoteright}s, obtained by Dirac reduction from some generalized Drinfeld-Sokolov hierarchies.}, url = {http://hdl.handle.net/1963/7247}, author = {Alberto De Sole and Victor G. Kac and Daniele Valeri} } @conference {IapichinoQuarteroniRozzaVolkwein2014, title = {Reduced basis method for the Stokes equations in decomposable domains using greedy optimization}, booktitle = {ECMI 2014 proceedings}, year = {2014}, pages = {1{\textendash}7}, author = {Laura Iapichino and Alfio Quarteroni and Gianluigi Rozza and Volkwein, Stefan} } @article {2014, title = {Six-dimensional supersymmetric gauge theories, quantum cohomology of instanton moduli spaces and gl(N) Quantum Intermediate Long Wave Hydrodynamics}, number = {Journal of high energy physics;volume 2014; issue 7; article number 141;}, year = {2014}, publisher = {Springer}, abstract = {We show that the exact partition function of U(N) six-dimensional gauge theory with eight supercharges on C^2 x S^2 provides the quantization of the integrable system of hydrodynamic type known as gl(N) periodic Intermediate Long Wave (ILW). We characterize this system as the hydrodynamic limit of elliptic Calogero-Moser integrable system. We compute the Bethe equations from the effective gauged linear sigma model on S^2 with target space the ADHM instanton moduli space, whose mirror computes the Yang-Yang function of gl(N) ILW. The quantum Hamiltonians are given by the local chiral ring observables of the six-dimensional gauge theory. As particular cases, these provide the gl(N) Benjamin-Ono and Korteweg-de Vries quantum Hamiltonians. In the four dimensional limit, we identify the local chiral ring observables with the conserved charges of Heisenberg plus W_N algebrae, thus providing a gauge theoretical proof of AGT correspondence.}, doi = {10.1007/JHEP07(2014)141}, url = {http://urania.sissa.it/xmlui/handle/1963/34546}, author = {Giulio Bonelli and Antonio Sciarappa and Alessandro Tanzini and Petr Vasko} } @article {2014, title = {Some remarks on the seismic behaviour of embedded cantilevered retaining walls}, number = {Geotechnique;volume 64; issue 1; pages 40-50;}, year = {2014}, publisher = {Thomas Telford}, abstract = {This paper is a numerical investigation of the physical phenomena that control the dynamic behaviour of embedded cantilevered retaining walls. Recent experimental observations obtained from centrifuge tests have shown that embedded cantilevered retaining walls experience permanent displacements even before the acceleration reaches its critical value, corresponding to full mobilisation of the soil strength. The motivation for this work stems from the need to incorporate these observations in simplified design procedures. A parametric study was carried out on a pair of embedded cantilevered walls in dry sand, subjected to real earthquakes scaled at different values of the maximum acceleration. The results of these analyses indicate that, for the geotechnical design of the wall, the equivalent acceleration to be used in pseudo-static calculations can be related to the maximum displacement that the structure can sustain, and can be larger than the maximum acceleration expected at the site. For the structural design of the wall, it is suggested that the maximum bending moments of the wall can be computed using a realistic distribution of contact stress and a conservative value of the pseudo-static acceleration, taking into account two-dimensional amplification effects near the walls.}, doi = {10.1680/geot.13.P.031}, url = {http://urania.sissa.it/xmlui/handle/1963/35073}, author = {Riccardo Conti and F. Burali D{\textquoteright}Arezzo and Giulia M.B. Viggiani} } @article {2014, title = {The stringy instanton partition function}, number = {Journal of high energy physics;volume 2014; issue 1; article number 038;}, year = {2014}, publisher = {Springer}, abstract = {We perform an exact computation of the gauged linear sigma model associated to a D1-D5 brane system on a resolved A_1 singularity. This is accomplished via supersymmetric localization on the blown-up two-sphere. We show that in the blow-down limit C^2/Z_2 the partition function reduces to the Nekrasov partition function evaluating the equivariant volume of the instanton moduli space. For finite radius we obtain a tower of world-sheet instanton corrections, that we identify with the equivariant Gromov-Witten invariants of the ADHM moduli space. We show that these corrections can be encoded in a deformation of the Seiberg-Witten prepotential. From the mathematical viewpoint, the D1-D5 system under study displays a twofold nature: the D1-branes viewpoint captures the equivariant quantum cohomology of the ADHM instanton moduli space in the Givental formalism, and the D5-branes viewpoint is related to higher rank equivariant Donaldson-Thomas invariants of P^1 x C^2.}, doi = {10.1007/JHEP01(2014)038}, url = {http://urania.sissa.it/xmlui/handle/1963/34589}, author = {Giulio Bonelli and Antonio Sciarappa and Alessandro Tanzini and Petr Vasko} } @article {2014, title = {Structure of classical (finite and affine) W-algebras}, number = {arXiv:1404.0715;}, year = {2014}, note = {40 pages}, institution = {SISSA}, abstract = {First, we derive an explicit formula for the Poisson bracket of the classical finite W-algebra W^{fin}(g,f), the algebra of polynomial functions on the Slodowy slice associated to a simple Lie algebra g and its nilpotent element f. On the other hand, we produce an explicit set of generators and we derive an explicit formula for the Poisson vertex algebra structure of the classical affine W-algebra W(g,f). As an immediate consequence, we obtain a Poisson algebra isomorphism between W^{fin}(g,f) and the Zhu algebra of W(g,f). We also study the generalized Miura map for classical W-algebras.}, url = {http://hdl.handle.net/1963/7314}, author = {Alberto De Sole and Victor G. Kac and Daniele Valeri} } @article {2014, title = {Vortex Partition Functions, Wall Crossing and Equivariant Gromov{\textendash}Witten Invariants}, number = {Communications in mathematical physics;volume 333; issue 2; pages 717-760;}, year = {2014}, publisher = {Springer}, abstract = {In this paper we identify the problem of equivariant vortex counting in a (2,2) supersymmetric two dimensional quiver gauged linear sigma model with that of computing the equivariant Gromov{\textendash}Witten invariants of the GIT quotient target space determined by the quiver. We provide new contour integral formulae for the I and J-functions encoding the equivariant quantum cohomology of the target space. Its chamber structure is shown to be encoded in the analytical properties of the integrand. This is explained both via general arguments and by checking several key cases. We show how several results in equivariant Gromov{\textendash}Witten theory follow just by deforming the integration contour. In particular, we apply our formalism to compute Gromov{\textendash}Witten invariants of the C3/Zn orbifold, of the Uhlembeck (partial) compactification of the moduli space of instantons on C2, and of An and Dn singularities both in the orbifold and resolved phases. Moreover, we analyse dualities of quantum cohomology rings of holomorphic vector bundles over Grassmannians, which are relevant to BPS Wilson loop algebrae.}, doi = {10.1007/s00220-014-2193-8}, url = {http://urania.sissa.it/xmlui/handle/1963/34652}, author = {Giulio Bonelli and Antonio Sciarappa and Alessandro Tanzini and Petr Vasko} } @article {2012, title = {Classical W-algebras and generalized Drinfeld-Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras}, journal = {Communications in Mathematical Physics 323, nr. 2 (2013) 663-711}, number = {arXiv:1207.6286;}, year = {2013}, note = {43 pages. Second version with minor editing and corrections}, publisher = {Springer}, abstract = {We provide a description of the Drinfeld-Sokolov Hamiltonian reduction for the construction of classical W-algebras within the framework of Poisson vertex algebras. In this context, the gauge group action on the phase space is translated in terms of (the exponential of) a Lie conformal algebra action on the space of functions. Following the ideas of Drinfeld and Sokolov, we then establish under certain sufficient conditions the applicability of the Lenard-Magri scheme of integrability and the existence of the corresponding integrable hierarchy of bi-Hamiltonian equations.}, doi = {10.1007/s00220-013-1785-z}, url = {http://hdl.handle.net/1963/6978}, author = {Alberto De Sole and Victor G. Kac and Daniele Valeri} } @article {2013, title = {Dislocation dynamics in crystals: a macroscopic theory in a fractional Laplace setting}, number = {SISSA preprint;43/2013/MATE}, year = {2013}, institution = {SISSA}, abstract = {We consider an evolution equation arising in the Peierls-Nabarro model for crystal dislocation. We study the evolution of such dislocation function and show that, at a macroscopic scale, the dislocations have the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium. These dislocation points evolve according to the external stress and an interior repulsive potential.}, keywords = {nonlocal Allen-Cahn equation}, url = {http://hdl.handle.net/1963/7124}, author = {Serena Dipierro and Giampiero Palatucci and Enrico Valdinoci} } @article {2013, title = {Existence and symmetry results for a Schrodinger type problem involving the fractional Laplacian}, journal = {Le Matematiche (Catania), Vol. 68 (2013), no. 1: 201-216}, number = {arXiv:1202.0576v1;}, year = {2013}, publisher = {University of Catania}, abstract = {

This paper deals with the following class of nonlocal Schr\"odinger equations $$ \displaystyle (-\Delta)^s u + u = |u|^{p-1}u \ \ \text{in} \ \mathbb{R}^N, \quad \text{for} \ s\in (0,1). $$ We prove existence and symmetry results for the solutions $u$ in the fractional Sobolev space $H^s(\mathbb{R}^N)$. Our results are in clear accordance with those for the classical local counterpart, that is when $s=1$.

}, author = {Serena Dipierro and Giampiero Palatucci and Enrico Valdinoci} } @article {2013, title = {Fields of bounded deformation for mesoscopic dislocations}, year = {2013}, publisher = {SISSA}, abstract = {In this paper we discuss the consequences of the distributional approach to dislocations in terms of the mathematical properties\\r\\nof the auxiliary model fields such as displacement and displacement gradient which are obtained directly from \\r\\nthe main model field here considered as the linear strain. We show that these fields cannot be introduced rigourously without \\r\\nthe introduction of gauge fields, or equivalently, without cuts in the Riemann foliation associated to the dislocated crystal.\\r\\nIn a second step we show that the space of bounded deformations follows from the distributional approach in a natural way and \\r\\ndiscuss the reasons why it is adequate to model dislocations. The case of dislocation clusters is also addressed, as it represents an important issue in industrial crystal growth while from a mathematical point of view, peculiar phenomena might appear at the set of accumulation points. \\r\\nThe elastic-plastic decomposition of the strain within this approach is also given a precise meaning.}, url = {http://hdl.handle.net/1963/6378}, author = {Nicolas Van Goethem} } @article {2013, title = {Minimal partitions and image classification using a gradient-free perimeter approximation}, year = {2013}, institution = {SISSA}, abstract = {In this paper a new mathematically-founded method for the optimal partitioning of domains, with applications to the classification of greyscale and color images, is proposed. Since optimal partition problems are in general ill-posed, some regularization strategy is required. Here we regularize by a non-standard approximation of the total interface length, which does not involve the gradient of approximate characteristic functions, in contrast to the classical Modica-Mortola approximation. Instead, it involves a system of uncoupled linear partial differential equations and nevertheless shows $\Gamma$-convergence properties in appropriate function spaces. This approach leads to an alternating algorithm that ensures a decrease of the objective function at each iteration, and which always provides a partition, even during the iterations. The efficiency of this algorithm is illustrated by various numerical examples. Among them we consider binary and multilabel minimal partition problems including supervised or automatic image classification, inpainting, texture pattern identification and deblurring.}, keywords = {Image classification, deblurring, optimal partitions, perimeter approximation}, url = {http://hdl.handle.net/1963/6976}, author = {Samuel Amstutz and Nicolas Van Goethem and Antonio Andr{\'e} Novotny} } @article {2012, title = {Asymptotics of the s-perimeter as s {\textrightarrow}0 }, journal = {Discrete Contin. Dyn. Syst. 33, nr.7 (2012): 2777-2790}, number = {arXiv:1204.0750v2;}, year = {2012}, publisher = {American Institute of Mathematical Sciences}, abstract = {

We deal with the asymptotic behavior of the $s$-perimeter of a set $E$ inside a domain $\Omega$ as $s\searrow0$. We prove necessary and sufficient conditions for the existence of such limit, by also providing an explicit formulation in terms of the Lebesgue measure of $E$ and $\Omega$. Moreover, we construct examples of sets for which the limit does not exist.

}, doi = {10.3934/dcds.2013.33.2777}, author = {Serena Dipierro and Alessio Figalli and Giampiero Palatucci and Enrico Valdinoci} } @article {2012, title = {On the behaviour of flexible retaining walls under seismic actions}, journal = {Geotechnique, Volume 62, Issue 12, December 2012, Pages 1081-1094}, year = {2012}, publisher = {ICE Publishing}, abstract = {This paper describes an experimental investigation of the behaviour of embedded retaining walls under seismic actions. Nine centrifuge tests were carried out on reduced-scale models of pairs of retaining walls in dry sand, either cantilevered or with one level of props near the top. The experimental data indicate that, for maximum accelerations that are smaller than the critical limit equilibrium value, the retaining walls experience significant permanent displacements under increasing structural loads, whereas for larger accelerations the walls rotate under constant internal forces. The critical acceleration at which the walls start to rotate increases with increasing maximum acceleration. No significant displacements are measured if the current earthquake is less severe than earthquakes previously experienced by the wall. The increase of critical acceleration is explained in terms of redistribution of earth pressures and progressive mobilisation of the passive strength in front of the wall. The experimental data for cantilevered retaining walls indicate that the permanent displacements of the wall can be reasonably predicted adopting a Newmark-type calculation with a critical acceleration that is a fraction of the limit equilibrium value.}, keywords = {Centrifuge modelling}, doi = {10.1680/geot.11.P.029}, url = {http://hdl.handle.net/1963/6933}, author = {Riccardo Conti and G.S.P. Madabhushi and Giulia M.B. Viggiani} } @article {2012, title = {Numerical modelling of installation effects for diaphragm walls in sand}, journal = {Acta Geotechnica, Volume 7, Issue 3, September 2012, Pages 219-237}, year = {2012}, publisher = {Springer}, abstract = {The scopes of this work are to study the mechanisms of load transfer and the deformations of the ground during slurry trenching and concreting in dry sand and to evaluate their effects on service structural loads, wall deflections and ground displacements behind the wall caused by subsequent excavation. A series of three-dimensional finite element analyses was carried out modelling the installation of diaphragm walls consisting of panels of different length. The soil was modelled as either linearly elastic-perfectly plastic or incrementally non-linear (hypoplastic) with elastic strain range. Plane strain analyses of diaphragm walls of identical cross section were also carried out in which wall installation was either modelled or the wall was wished in place (WIP). The analyses predict ground movements consistent with the experimental observations both in magnitude and trend. The results also show that the maximum horizontal wall deflections and structural loads reduce with increasing panel aspect ratio towards a minimum which is about twice the value computed for WIP analyses. Panel aspect ratios should be larger than about three to take advantage of the three-dimensional effects. The pattern and magnitude of surface vertical displacements obtained from linearly elastic-perfectly plastic analyses, no matter whether three- or two-dimensional, are unrealistic.}, keywords = {Constitutive relations}, doi = {10.1007/s11440-011-0157-0}, url = {http://hdl.handle.net/1963/6934}, author = {Riccardo Conti and Luca de Sanctis and Giulia M.B. Viggiani} } @article {2012, title = {Topological sensitivity analysis for high order elliptic operators}, number = {SISSA;36/2012/M}, year = {2012}, institution = {SISSA}, abstract = {The topological derivative is defined as the first term of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of a singular domain perturbation. It has applications in many different fields such as shape and topology optimization, inverse problems, image processing and mechanical modeling including synthesis and/or optimal design of microstructures, fracture mechanics sensitivity analysis and damage evolution modeling. The topological derivative has been fully developed for a wide range of second order differential operators. In this paper we deal with the topological asymptotic expansion of a class of shape functionals associated with elliptic differential operators of order 2m, m>=1. The general structure of the polarization tensor is derived and the concept of degenerate polarization tensor is introduced. We provide full mathematical justifications for the derived formulas, including precise estimates of remainders.}, keywords = {Topological derivative, Elliptic operators, Polarization tensor}, url = {http://hdl.handle.net/1963/6343}, author = {Samuel Amstutz and Antonio Andr{\'e} Novotny and Nicolas Van Goethem} } @article {ruiz2011, title = {Cluster solutions for the Schr{\"o}dinger-Poisson-Slater problem around a local minimum of the potential}, journal = {Rev. Mat. Iberoamericana}, volume = {27}, number = {1}, year = {2011}, month = {01}, pages = {253{\textendash}271}, publisher = {Real Sociedad Matem{\'a}tica Espa{\~n}ola}, url = {https://projecteuclid.org:443/euclid.rmi/1296828834}, author = {David Ruiz and Giusi Vaira} } @article {DAVENIA20115705, title = {Infinitely many positive solutions for a Schr{\"o}dinger{\textendash}Poisson system}, journal = {Nonlinear Analysis: Theory, Methods \& Applications}, volume = {74}, number = {16}, year = {2011}, pages = {5705 - 5721}, abstract = {

We are interested in the existence of infinitely many positive solutions of the Schr{\"o}dinger{\textendash}Poisson system -Δu+u+V(|x|)ϕu=|u|p-1u,x∈R3,-Δϕ=V(|x|)u2,x∈R3, where V(|x|) is a positive bounded function, 1\<p\<5 and V(r

}, keywords = {Non-autonomous Schr{\"o}dinger{\textendash}Poisson system, Perturbation method}, issn = {0362-546X}, doi = {https://doi.org/10.1016/j.na.2011.05.057}, url = {http://www.sciencedirect.com/science/article/pii/S0362546X11003518}, author = {Pietro d{\textquoteright}Avenia and Alessio Pomponio and Giusi Vaira} } @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 {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 {doi:10.1142/S0218202509003589, title = {Solutions of the Schr{\"o}dinger{\textendash}Poisson problem concentrating on spheres, part I: necessary conditions}, journal = {Mathematical Models and Methods in Applied Sciences}, volume = {19}, number = {05}, year = {2009}, pages = {707-720}, abstract = {

In this paper we study a coupled nonlinear Schr{\"o}dinger{\textendash}Poisson problem with radial functions. This system has been introduced as a model describing standing waves for the nonlinear Schr{\"o}dinger equations in the presence of the electrostatic field. We provide necessary conditions for concentration on sphere for the solutions of this kind of problem extending the results already known.

}, doi = {10.1142/S0218202509003589}, url = {https://doi.org/10.1142/S0218202509003589}, author = {Ianni, Isabella and Giusi Vaira} } @article {ianni2008concentration, title = {On concentration of positive bound states for the Schr{\"o}dinger-Poisson problem with potentials}, journal = {Advanced nonlinear studies}, volume = {8}, number = {3}, year = {2008}, pages = {573{\textendash}595}, publisher = {Advanced Nonlinear Studies, Inc.}, abstract = {

We study the existence of semiclassical states for a nonlinear Schr{\"o}dinger-Poisson system that concentrate near critical points of the external potential and of the density charge function. We use a perturbation scheme in a variational setting, extending the results in [1]. We also discuss necessary conditions for concentration.

}, doi = {10.1515/ans-2008-0305.}, author = {Ianni, Isabella and Giusi Vaira} } @article {2008, title = {Noncommutative families of instantons}, journal = {Int. Math. Res. Not. vol. 2008, Article ID rnn038}, number = {arXiv.org;0710.0721v2}, year = {2008}, publisher = {Oxford University Press}, abstract = {We construct $\\\\theta$-deformations of the classical groups SL(2,H) and Sp(2). Coacting on the basic instanton on a noncommutative four-sphere $S^4_\\\\theta$, we construct a noncommutative family of instantons of charge 1. The family is parametrized by the quantum quotient of $SL_\\\\theta(2,H)$ by $Sp_\\\\theta(2)$.}, doi = {10.1093/imrn/rnn038}, url = {http://hdl.handle.net/1963/3417}, author = {Giovanni Landi and Chiara Pagani and Cesare Reina and Walter van Suijlekom} } @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 {2005, title = {The Dirac operator on SU_q(2)}, journal = {Commun. Math. Phys. 259 (2005) 729-759}, number = {arXiv:math/0411609;}, year = {2005}, note = {v2: minor changes}, publisher = {Springer}, abstract = {We construct a 3^+ summable spectral triple (A(SU_q(2)),H,D) over the quantum group SU_q(2) which is equivariant with respect to a left and a right action of U_q(su(2)). The geometry is isospectral to the classical case since the spectrum of the operator D is the same as that of the usual Dirac operator on the 3-dimensional round sphere. The presence of an equivariant real structure J demands a modification in the axiomatic framework of spectral geometry, whereby the commutant and first-order properties need be satisfied only modulo infinitesimals of arbitrary high order.}, doi = {10.1007/s00220-005-1383-9}, url = {http://hdl.handle.net/1963/4425}, author = {Ludwik Dabrowski and Giovanni Landi and Andrzej Sitarz and Walter van Suijlekom and Joseph C. Varilly} } @article {2005, title = {The local index formula for SUq(2)}, journal = {K-Theory 35 (2005) 375-394}, number = {SISSA;01/2005/FM}, year = {2005}, abstract = {We discuss the local index formula of Connes-Moscovici for the isospectral noncommutative geometry that we have recently constructed on quantum SU(2). We work out the cosphere bundle and the dimension spectrum as well as the local cyclic cocycles yielding the index formula.}, doi = {10.1007/s10977-005-3116-4}, url = {http://hdl.handle.net/1963/1713}, author = {Walter van Suijlekom and Ludwik Dabrowski and Giovanni Landi and Andrzej Sitarz and Joseph C. Varilly} } @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 {1996, title = {Solving Honig generic problem about Volterra integral equations}, journal = {Bull. Polish Acad. Sci. Math. 44 (1996), no. 4, 495--498}, number = {SISSA;38/95/M}, year = {1996}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/941}, author = {Giovanni Vidossich} } @article {1990, title = {A general existence theorem for boundary value problems for ordinary differential equations}, journal = {Nonlinear Anal. 15 (1990), no. 10, 897--914}, number = {SISSA;36/88/M}, year = {1990}, publisher = {SISSA Library}, doi = {10.1016/0362-546X(90)90073-P}, url = {http://hdl.handle.net/1963/632}, author = {Giovanni Vidossich} } @article {1989, title = {On the continuous dependence of solutions of boundary value problems for ordinary differential equations (Revised version)}, journal = {J. Differential Equations 82 (1989), no. 1, 1-14}, number = {SISSA;70/88/M}, year = {1989}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/666}, author = {Giovanni Vidossich} } @article {1989, title = {On the continuous dependence of solutions of boundary value problems for ordinary differential equations}, journal = {J. Differential Equations 82 (1989), no. 1, 1--14}, number = {SISSA;37/88/M}, year = {1989}, publisher = {SISSA Library}, doi = {10.1016/0022-0396(89)90164-2}, url = {http://hdl.handle.net/1963/633}, author = {Giovanni Vidossich} } @article {1989, title = {Hyperbolic equations as ordinary differential equations in Banach spaces}, number = {SISSA;11/89/M}, year = {1989}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/773}, author = {Giovanni Vidossich} } @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} } @article {1989, title = {On the solvability of boundary value problems for higher order ordinary differential equations (Revised version)}, journal = {Nonlinear Anal. 13 (1989), no. 10, 1171-1179}, number = {SISSA;66/88/M}, year = {1989}, publisher = {SISSA Library}, doi = {10.1016/0362-546X(89)90005-9}, url = {http://hdl.handle.net/1963/662}, author = {Giovanni Vidossich} } @article {1989, title = {On the solvability of boundary value problems for higher order ordinary differential equations}, journal = {Nonlinear Anal. 13 (1989), no. 10, 1171-179}, number = {SISSA;35/88/M}, year = {1989}, publisher = {SISSA Library}, doi = {10.1016/0362-546X(89)90005-9}, url = {http://hdl.handle.net/1963/631}, author = {Giovanni Vidossich} } @article {1985, title = {The two-point boundary value problem from the Cauchy problem}, journal = {J. Differential Equations 60 (1985), no. 1, 1--20}, number = {SISSA;29/84/M}, year = {1985}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/332}, author = {Giovanni Vidossich} } @article {1983, title = {On the asymptotic behaviour of solutions to Pazy\\\'s class of evolution equations}, number = {SISSA;37/83/M}, year = {1983}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/276}, author = {Giovanni Vidossich} } @article {1983, title = {Towards a theory for periodic solutions to first order ordinary differential equations.}, number = {SISSA;59/83/M}, year = {1983}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/295}, author = {Giovanni Vidossich} } @article {1982, title = {A criterion for he existence of maximal solutions of strongly nonlinear elliptic problems}, number = {SISSA;10/82/M}, year = {1982}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/161}, author = {Giovanni Vidossich} } @article {1982, title = {On the obstacle problem for strongly nonlinear elliptic equations}, number = {SISSA;11/82/M}, year = {1982}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/162}, author = {Giovanni Vidossich} } @article {1982, title = {Three uniqueness theorems for strongly non-linear elliptic problems}, number = {SISSA;16/82/M}, year = {1982}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/167}, author = {Giovanni Vidossich} } @article {9979, title = {Uniqueness and multiplicity of periodic solutions to first order ordinary differential equations}, journal = {Not Found}, number = {SISSA;18/84/M}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/321}, author = {Giovanni Vidossich} }