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 {2018, title = {Stochastic homogenisation of free-discontinuity problems}, number = {SISSA;05/2018/MATE}, year = {2018}, abstract = {In this paper we study the stochastic homogenisation of free-discontinuity functionals. Assuming stationarity for the random volume and surface integrands, we prove the existence of a homogenised random free-discontinuity functional, which is deterministic in the ergodic case. Moreover, by establishing a connection between the deterministic convergence of the functionals at any fixed realisation and the pointwise Subadditive Ergodic Theorem by Akcoglou and Krengel, we characterise the limit volume and surface integrands in terms of asymptotic cell formulas.}, url = {http://preprints.sissa.it/handle/1963/35309}, author = {Filippo Cagnetti and Gianni Dal Maso and Lucia Scardia and Caterina Ida Zeppieri} } @article {2018, title = {Transmission conditions obtained by homogenisation}, number = {SISSA;11/2018/MATE}, year = {2018}, abstract = {We study the asymptotic behaviour of solutions to variational problems in perforated domains with Neumann boundary conditions. We consider perforations that in the limit concentrate on a smooth manifold. We characterise the limits of the solutions and show that they solve a variational problem with a transmission condition across the manifold. This is expressed through a measure on the manifold, vanishing on sets of capacity zero. Then, we prove that every such measure can be obtained by homogenising suitable perforations. Eventually, we provide an asymptotic formula for this measure by using some auxiliary minimum problems.}, url = {http://preprints.sissa.it/handle/1963/35310}, author = {Gianni Dal Maso and Giovanni Franzina and Davide Zucco} } @article {feltrin2017, title = {An application of coincidence degree theory to cyclic feedback type systems associated with nonlinear differential operators}, journal = {Topol. Methods Nonlinear Anal.}, volume = {50}, number = {2}, year = {2017}, pages = {683{\textendash}726}, publisher = {Nicolaus Copernicus University, Juliusz P. Schauder Centre for Nonlinear Studies}, doi = {10.12775/TMNA.2017.038}, url = {https://doi.org/10.12775/TMNA.2017.038}, author = {Guglielmo Feltrin and Fabio Zanolin} } @article {2017, title = {Gamma-Convergence of Free-discontinuity problems}, number = {SISSA;18/2017/MATE}, year = {2017}, institution = {SISSA}, abstract = {We study the Gamma-convergence of sequences of free-discontinuity functionals depending on vector-valued functions u which can be discontinuous across hypersurfaces whose shape and location are not known a priori. The main novelty of our result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Further, we consider the case of surface integrands which are not bounded from below by the amplitude of the jump of u. We obtain three main results: compactness with respect to Gamma-convergence, representation of the Gamma-limit in an integral form and identification of its integrands, and homogenisation formulas without periodicity assumptions. In particular, the classical case of periodic homogenisation follows as a by-product of our analysis. Moreover, our result covers also the case of stochastic homogenisation, as we will show in a forthcoming paper.}, url = {http://preprints.sissa.it/handle/1963/35276}, author = {Filippo Cagnetti and Gianni Dal Maso and Lucia Scardia and Caterina Ida Zeppieri} } @article {FELTRIN20174255, title = {Multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree}, journal = {Journal of Differential Equations}, volume = {262}, number = {8}, year = {2017}, pages = {4255 - 4291}, abstract = {We study the periodic boundary value problem associated with the second order nonlinear differential equationu"+cu'+(a+(t)-μa-(t))g(u)=0, where g(u) has superlinear growth at zero and at infinity, a(t) is a periodic sign-changing weight, c∈R and μ\>0 is a real parameter. Our model includes (for c=0) the so-called nonlinear Hill{\textquoteright}s equation. We prove the existence of 2m-1 positive solutions when a(t) has m positive humps separated by m negative ones (in a periodicity interval) and μ is sufficiently large, thus giving a complete solution to a problem raised by G.J. Butler in 1976. The proof is based on Mawhin{\textquoteright}s coincidence degree defined in open (possibly unbounded) sets and applies also to Neumann boundary conditions. Our method also provides a topological approach to detect subharmonic solutions.

}, keywords = {Coincidence degree, Multiplicity results, Neumann boundary value problems, Positive periodic solutions, subharmonic solutions, Superlinear indefinite problems}, issn = {0022-0396}, doi = {https://doi.org/10.1016/j.jde.2017.01.009}, url = {http://www.sciencedirect.com/science/article/pii/S0022039617300219}, author = {Guglielmo Feltrin and Fabio Zanolin} } @article {2016, title = {Confinement of dislocations inside a crystal with a prescribed external strain}, year = {2016}, note = {Preprint SISSA 20/2016/MATE}, abstract = {We study screw dislocations in an isotropic crystal undergoing antiplane shear. In the framework of linear elasticity, by fixing a suitable boundary condition for the strain (prescribed non-vanishing boundary integral), we manage to confine the dislocations inside the material. More precisely, in the presence of an external strain with circulation equal to n times the lattice spacing, it is energetically convenient to have n distinct dislocations lying inside the crystal. The novelty of introducing a Dirichlet boundary condition for the tangential strain is crucial to the confinement: it is well known that, if Neumann boundary conditions are imposed, the dislocations tend to migrate to the boundary. The results are achieved using PDE techniques and Ƭ-convergence theory, in the framework of the so-called core radius approach.}, url = {http://urania.sissa.it/xmlui/handle/1963/35247}, author = {Ilaria Lucardesi and Marco Morandotti and Riccardo Scala and Davide Zucco} } @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 {2015, title = {A bridging mechanism in the homogenisation of brittle composites with soft inclusions}, number = {SISSA;02/2015/MATE}, year = {2015}, institution = {SISSA}, abstract = {We provide a homogenisation result for the energy-functional associated with a purely brittle composite whose microstructure is characterised by soft periodic inclusions embedded in a stiffer matrix. We show that the two constituents as above can be suitably arranged on a microscopic scale ε to obtain, in the limit as ε tends to zero, a homogeneous macroscopic energy-functional explicitly depending on the opening of the crack.}, url = {http://urania.sissa.it/xmlui/handle/1963/7492}, author = {Marco Barchiesi and Giuliano Lazzaroni and Caterina Ida Zeppieri} } @article {2015, title = {Convex combinations of low eigenvalues, Fraenkel asymmetries and attainable sets}, year = {2015}, institution = {SISSA}, abstract = {We consider the problem of minimizing convex combinations of the first two eigenvalues of the Dirichlet-Laplacian among open set of $R^N$ of fixed measure. We show that, by purely elementary arguments, based on the minimality condition, it is possible to obtain informations on the geometry of the minimizers of convex combinations: we study, in particular, when these minimizers are no longer convex, and the optimality of balls. As an application of our results we study the boundary of the attainable set for the Dirichlet spectrum. Our techniques involve symmetry results {\`a} la Serrin, explicit constants in quantitative inequalities, as well as a purely geometrical problem: the minimization of the Fraenkel 2-asymmetry among convex sets of fixed measure.}, url = {http://urania.sissa.it/xmlui/handle/1963/35140}, author = {Dario Mazzoleni and Davide Zucco} } @article {2015, title = {Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems}, journal = {Adv. Differential Equations 20 (2015), 937{\textendash}982.}, number = {Advances in Differential Equations;volume 20; issue 9/10; pages 937-982;}, year = {2015}, note = {AMS Subject Classification: 34B18, 34B15, 34C25, 47H11.}, publisher = {Khayyam Publishing}, abstract = {We prove the existence of positive periodic solutions for the second order nonlinear equation u{\textquoteright}{\textquoteright} + a(x) g(u) = 0, where g(u) has superlinear growth at zero and at infinity. The weight function a(x) is allowed to change its sign. Necessary and sufficient conditions for the existence of nontrivial solutions are obtained. The proof is based on Mawhin{\textquoteright}s coincidence degree and applies also to Neumann boundary conditions. Applications are given to the search of positive solutions for a nonlinear PDE in annular domains and for a periodic problem associated to a non-Hamiltonian equation.

}, url = {http://projecteuclid.org/euclid.ade/1435064518}, author = {Guglielmo Feltrin and Fabio Zanolin} } @article {2015, title = {Extended affine Weyl groups of BCD type, Frobenius manifolds and their Landau-Ginzburg superpotentials}, year = {2015}, institution = {SISSA}, abstract = {For the root systems of type Bl, Cl and Dl, we generalize the result of [7] by showing the existence of Frobenius manifold structures on the orbit spaces of the extended affine Weyl groups that correspond to any vertex of the Dynkin diagram instead of a particular choice made in [7]. It also depends on certain additional data. We also construct LG superpotentials for these Frobenius manifold structures.}, url = {http://preprints.sissa.it/handle/1963/35316}, author = {Boris Dubrovin and Ian A.B. Strachan and Youjin Zhang and Dafeng Zuo} } @article {2015, title = {Multiple positive solutions for a superlinear problem: a topological approach}, journal = {J. Differential Equations 259 (2015), 925{\textendash}963.}, number = {Journal of Differential Equations;volume 259; issue 3; pages 925-963;}, year = {2015}, note = {Work presented at the "Special Session 21" of the "10th AIMS Conference on Dynamical Systems, Differential Equations and Applications" (Madrid, July 7-11, 2014).}, publisher = {Elsevier}, abstract = {We study the multiplicity of positive solutions for a two-point boundary value problem associated to the nonlinear second order equation u{\textquoteright}{\textquoteright}+f(x,u)=0. We allow x ↦ f(x,s) to change its sign in order to cover the case of scalar equations with indefinite weight. Roughly speaking, our main assumptions require that f(x,s)/s is below λ_1 as s{\textrightarrow}0^+ and above λ_1 as s{\textrightarrow}+$\infty$. In particular, we can deal with the situation in which f(x,s) has a superlinear growth at zero and at infinity. We propose a new approach based on the topological degree which provides the multiplicity of solutions. Applications are given for u{\textquoteright}{\textquoteright} + a(x) g(u) = 0, where we prove the existence of 2^n-1 positive solutions when a(x) has n positive humps and a^-(x) is sufficiently large.

}, doi = {10.1016/j.jde.2015.02.032}, url = {http://urania.sissa.it/xmlui/handle/1963/35147}, author = {Guglielmo Feltrin and Fabio Zanolin} } @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 {2014, title = {Dirac operators on noncommutative principal circle bundles}, number = {International Journal of Geometric Methods in Modern Physics;volume 11; issue 1; article number 1450012;}, year = {2014}, publisher = {World Scientific Publishing}, abstract = {We study spectral triples over noncommutative principal U(1)-bundles of arbitrary dimension and a compatibility condition between the connection and the Dirac operator on the total space and on the base space of the bundle. Examples of low-dimensional noncommutative tori are analyzed in more detail and all connections found that are compatible with an admissible Dirac operator. Conversely, a family of new Dirac operators on the noncommutative tori, which arise from the base-space Dirac operator and a suitable connection is exhibited. These examples are extended to the theta-deformed principal U(1)-bundle S 3 θ {\textrightarrow} S2.}, doi = {10.1142/S0219887814500121}, url = {http://urania.sissa.it/xmlui/handle/1963/35125}, author = {Andrzej Sitarz and Alessandro Zucca and Ludwik Dabrowski} } @article {2014, title = {Infinite-dimensional Frobenius manifolds underlying the Toda lattice hierarchy}, number = {Advances in Mathematics;volume 255; pages 487-524;}, year = {2014}, publisher = {Elsevier}, abstract = {Following the approach of Carlet et al. (2011) [9], we construct a class of infinite-dimensional Frobenius manifolds underlying the Toda lattice hierarchy, which are defined on the space of pairs of meromorphic functions with possibly higher-order poles at the origin and at infinity. We also show a connection between these infinite-dimensional Frobenius manifolds and the finite-dimensional Frobenius manifolds on the orbit space of extended affine Weyl groups of type A defined by Dubrovin and Zhang.}, doi = {10.1016/j.aim.2014.01.013}, url = {http://urania.sissa.it/xmlui/handle/1963/35026}, author = {Chaozhong Wu and Dafeng Zuo} } @article {2014, title = {Maximal generalized solution of eikonal equation}, number = {Journal of differential equations;volume 257; issue 1; pages 231-263;}, year = {2014}, publisher = {Elsevier}, abstract = {We study the Dirichlet problem for the eikonal equation: 1/2 |∇u(x)|^2-a(x)=0 in Ω u(x)=(x) on Ω, without continuity assumptions on the map a(.). We find a class of maps a(.) contained in the space L$\infty$(Ω) for which the problem admits a (maximal) generalized solution, providing a generalization of the notion of viscosity solution.}, doi = {10.1016/j.jde.2014.04.001}, url = {http://urania.sissa.it/xmlui/handle/1963/34642}, author = {Sandro Zagatti} } @article {2012, title = {New results on Gamma-limits of integral functionals}, number = {Annales de l{\textquoteright}Institut Henri Poincare. Analyse Non Lineaire}, year = {2014}, publisher = {Elsevier}, keywords = {Gamma-convergence}, doi = {10.1016/j.anihpc.2013.02.005}, url = {http://hdl.handle.net/1963/5880}, author = {Nadia Ansini and Gianni Dal Maso and Caterina Ida Zeppieri} } @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 {2014, title = {Where best to place a Dirichlet condition in an anisotropic membrane?}, number = {SISSA;61/2014/MATE}, year = {2014}, institution = {SISSA}, abstract = {We study a shape optimization problem for the first eigenvalue of an elliptic operator in divergence form, with non constant coefficients, over a fixed domain $\Omega$. Dirichlet conditions are imposed along $\partial\Omega$ and, in addition, along a set $\Sigma$ of prescribed length ($1$-dimensional Hausdorff measure). We look for the best shape and position for the supplementary Dirichlet region $\Sigma$ in order to maximize the first eigenvalue. We characterize the limit distribution of the optimal sets, as their prescribed length tends to infinity, via $\Gamma$-convergence.}, url = {http://urania.sissa.it/xmlui/handle/1963/7481}, author = {Paolo Tilli and Davide Zucco} } @article {2013, title = {Asymptotics of the first Laplace eigenvalue with Dirichlet regions of prescribed length}, number = {SIAM Journal on Mathematical Analysis;volume 45; issue 6; pp. 3266-3282}, year = {2013}, publisher = {Society for Industrial and Applied Mathematics}, abstract = {We consider the problem of maximizing the first eigenvalue of the $p$-Laplacian (possibly with nonconstant coefficients) over a fixed domain $\Omega$, with Dirichlet conditions along $\partial\Omega$ and along a supplementary set $\Sigma$, which is the unknown of the optimization problem. The set $\Sigma$, which plays the role of a supplementary stiffening rib for a membrane $\Omega$, is a compact connected set (e.g., a curve or a connected system of curves) that can be placed anywhere in $\overline{\Omega}$ and is subject to the constraint of an upper bound $L$ to its total length (one-dimensional Hausdorff measure). This upper bound prevents $\Sigma$ from spreading throughout $\Omega$ and makes the problem well-posed. We investigate the behavior of optimal sets $\Sigma_L$ as $L\to\infty$ via $\Gamma$-convergence, and we explicitly construct certain asymptotically optimal configurations. We also study the behavior as $p\to\infty$ with $L$ fixed, finding connections with maximum-distance problems related to the principal frequency of the $\infty$-Laplacian.}, doi = {10.1137/130916825}, url = {http://urania.sissa.it/xmlui/handle/1963/35141}, author = {Paolo Tilli and Davide Zucco} } @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 {1078-0947_2013_1_89, title = {Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions}, journal = {Discrete \& Continuous Dynamical Systems - A}, volume = {33}, number = {1078-0947_2013_1_8}, year = {2013}, pages = {89}, abstract = {We study the problem of existence and multiplicity of subharmonic solutions for a second order nonlinear ODE in presence of lower and upper solutions. We show how such additional information can be used to obtain more precise multiplicity results. Applications are given to pendulum type equations and to Ambrosetti-Prodi results for parameter dependent equations.

}, keywords = {lower and upper solutions, parameter dependent equations, Periodic solutions, Poincar{\'e}-Birkhoff twist theorem, subharmonic solutions}, issn = {1078-0947}, doi = {10.3934/dcds.2013.33.89}, url = {http://aimsciences.org//article/id/3638a93e-4f3e-4146-a927-3e8a64e6863f}, author = {Alberto Boscaggin and Fabio Zanolin} } @article {2012, title = {Detection of transcriptional triggers in the dynamics of microbial growth: application to the respiratory-versatile bacterium Shewanella oneidensis}, journal = {Nucleic Acids Research, Volume 40, Issue 15, August 2012, Pages 7132-7149}, year = {2012}, publisher = {SISSA}, abstract = {The capacity of microorganisms to respond to variable external conditions requires a coordination of environment-sensing mechanisms and decisionmaking regulatory circuits. Here, we seek to understand the interplay between these two processes by combining high-throughput measurement of time-dependent mRNA profiles with a novel computational approach that searches for key genetic triggers of transcriptional changes. Our approach helped us understand the regulatory strategies of a respiratorily versatile bacterium with promising bioenergy and bioremediation applications, Shewanella oneidensis, in minimal and rich media. By comparing expression profiles across these two conditions, we unveiled components of the transcriptional program that depend mainly on the growth phase. Conversely, by integrating our time-dependent data with a previously available large compendium of static perturbation responses, we identified transcriptional changes that cannot be explained solely by internal network dynamics, but are rather triggered by specific genes acting as key mediators of an environment-dependent response. These transcriptional triggers include known and novel regulators that respond to carbon, nitrogen and oxygen limitation. Our analysis suggests a sequence of physiological responses, including a coupling between nitrogen depletion and glycogen storage, partially recapitulated through dynamic flux balance analysis, and experimentally confirmed by metabolite measurements. Our approach is broadly applicable to other systems}, doi = {10.1093/nar/gks467}, url = {http://hdl.handle.net/1963/6506}, author = {Q Beg and Mattia Zampieri and N Klitgord and S Collins and M Serres and Daniel Segr{\`e} and Claudio Altafini} } @article {2012, title = {Gamma-convergence and H-convergence of linear elliptic operators}, journal = {Journal de Math{\'e}matiques Pures et Appliqu{\'e}es, Available online 12 September 2012}, number = {SISSA;04/2012/M}, year = {2012}, publisher = {Elsevier}, keywords = {Linear elliptic operators}, doi = {10.1016/j.matpur.2012.09.004}, url = {http://hdl.handle.net/1963/5878}, author = {Nadia Ansini and Gianni Dal Maso and Caterina Ida Zeppieri} } @article {2012, title = {On the genus two free energies for semisimple Frobenius manifolds}, journal = {Russian Journal of Mathematical Physics. Volume 19, Issue 3, September 2012, Pages 273-298}, number = {arXiv:1205.5990;}, year = {2012}, note = {36 pages, 3 figures}, publisher = {SISSA}, abstract = {We represent the genus two free energy of an arbitrary semisimple Frobenius\\r\\nmanifold as a sum of contributions associated with dual graphs of certain\\r\\nstable algebraic curves of genus two plus the so-called \\\"genus two G-function\\\".\\r\\nConjecturally the genus two G-function vanishes for a series of important\\r\\nexamples of Frobenius manifolds associated with simple singularities as well as\\r\\nfor ${\\\\bf P}^1$-orbifolds with positive Euler characteristics. We explain the\\r\\nreasons for such Conjecture and prove it in certain particular cases.}, doi = {10.1134/S1061920812030028}, url = {http://hdl.handle.net/1963/6464}, author = {Boris Dubrovin and Si-Qi Liu and Youjin Zhang} } @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 {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 {2011, title = {Adaptation as a genome-wide autoregulatory principle in the stress response of yeast.}, journal = {IET systems biology. 2011 Jul; 5(4):269-79}, number = {PMID:21823758;}, year = {2011}, publisher = {The Institution of Engineering and Technology}, abstract = {The gene expression response of yeast to various types of stresses/perturbations shows a common functional and dynamical pattern for the vast majority of genes, characterised by a quick transient peak (affecting primarily short genes) followed by a return to the pre-stimulus level. Kinetically, this process of adaptation following the transient excursion can be modelled using a genome-wide autoregulatory mechanism by means of which yeast aims at maintaining a preferential concentration in its mRNA levels. The resulting feedback system explains well the different time constants observable in the transient response, while being in agreement with all the known experimental dynamical features. For example, it suggests that a very rapid transient can be induced also by a slowly varying concentration of the gene products.}, doi = {10.1049/iet-syb.2009.0050}, url = {http://hdl.handle.net/1963/5106}, author = {F Eduati and B Di Camillo and G Toffolo and Claudio Altafini and Giovanna De Palo and Mattia Zampieri} } @article {2011, title = {Compactness by maximality}, year = {2011}, abstract = {We derive a compactness property in the Sobolev space $W^{1,1}(\O)$ in order to study the Dirichlet problem for the eikonal equation \begin{displaymath} \begin{cases} \ha |\n u(x)|^2 - a(x) = 0 \& \ \textrm{in} \ \O\cr u(x)=\varphi(x) \& \ \textrm{on} \ \partial \O, \end{cases} \end{displaymath} without continuity assumptions on the map $a$.}, url = {http://preprints.sissa.it/handle/1963/35317}, author = {Sandro Zagatti} } @article {2011, title = {D-branes, surface operators, and ADHM quiver representations}, number = {SISSA;01/2011/FM}, year = {2011}, note = {45 pages, v2: minor corrections}, institution = {SISSA}, abstract = {A supersymmetric quantum mechanical model is constructed for BPS states bound to surface operators in five dimensional SU(r) gauge theories using D-brane engineering. This model represents the effective action of a certain D2-brane configuration, and is naturally obtained by dimensional reduction of a quiver $(0,2)$ gauged linear sigma model. In a special stability chamber, the resulting moduli space of quiver representations is shown to be smooth and isomorphic to a moduli space of framed quotients on the projective plane. A precise conjecture relating a K-theoretic partition function of this moduli space to refined open string invariants of toric lagrangian branes is formulated for conifold and local P^1 x P^1 geometries.}, url = {http://hdl.handle.net/1963/4133}, author = {Ugo Bruzzo and Duiliu-Emanuel Diaconescu and M. Yardim and G. Pan and Yi Zhang and Chuang Wu-yen} } @article {2011, title = {An Integro-Extremization Approach for Non Coercive and Evolution Hamilton-Jacobi Equations}, journal = {Journal of Convex Analysis 18 (2011) 1141-1170}, year = {2011}, publisher = {Heldermann Verlag}, abstract = {We devote the \\\\textit{integro-extremization} method to the study of the Dirichlet problem for homogeneous Hamilton-Jacobi equations \\\\begin{displaymath} \\\\begin{cases} F(Du)=0 \& \\\\quad \\\\textrm{in} \\\\quad\\\\O\\\\cr u(x)=\\\\varphi(x) \& \\\\quad \\\\textrm{for} \\\\quad x\\\\in \\\\partial \\\\O, \\\\end{cases} \\\\end{displaymath} with a particular interest for non coercive hamiltonians $F$, and to the Cauchy-Dirichlet problem for the corresponding homogeneous time-dependent equations \\\\begin{displaymath} \\\\begin{cases} \\\\frac{\\\\partial u}{\\\\partial t}+ F(\\\\nabla u)=0 \& \\\\quad \\\\textrm{in} \\\\quad ]0,T[\\\\times \\\\O\\\\cr u(0,x)=\\\\eta(x) \& \\\\quad \\\\textrm{for} \\\\quad x\\\\in\\\\O \\\\cr u(t,x)=\\\\psi(x) \& \\\\quad \\\\textrm{for} \\\\quad (t,x)\\\\in[0,T]\\\\times \\\\partial \\\\O. \\\\end{cases} \\\\end{displaymath} We prove existence and some qualitative results for viscosity and almost everywhere solutions, under suitably convexity conditions on the hamiltonian $F$, on the domain $\\\\O$ and on the boundary datum, without any growth assumptions on $F$.}, url = {http://hdl.handle.net/1963/5538}, author = {Sandro Zagatti} } @article {2011, title = {Linearly degenerate Hamiltonian PDEs and a new class of solutions to the WDVV associativity equations}, journal = {Functional Analysis and Its Applications. Volume 45, Issue 4, December 2011, Pages 278-290}, year = {2011}, publisher = {Springer}, abstract = {We define a new class of solutions to the WDVV associativity equations. This class is determined by the property that one of the commuting PDEs associated with such a WDVV solution is linearly degenerate. We reduce the problem of classifying such solutions of the WDVV equations to the particular case of the so-called algebraic Riccati equation and, in this way, arrive at a complete classification of irreducible solutions.}, keywords = {Frobenius manifold}, doi = {10.1007/s10688-011-0030-9}, url = {http://hdl.handle.net/1963/6430}, author = {Boris Dubrovin and M.V. Pavlov and Sergei A. Zykov} } @article {2011, title = {The Liouville side of the vortex}, journal = {JHEP 09(2011)096}, number = {arXiv:1107.2787v1;}, year = {2011}, note = {25pages,11figures}, publisher = {SISSA}, abstract = {We analyze conformal blocks with multiple (semi-)degenerate field insertions in Liouville/Toda conformal field theories an show that their vector space is fully reproduced by the four-dimensional limit of open topological string amplitudes on the strip with generic boundary conditions associated to a suitable quiver gauge theory. As a byproduct we identify the non-abelian vortex partition function with a specific fusion channel of degenerate conformal blocks.}, doi = {10.1007/JHEP09(2011)096}, url = {http://hdl.handle.net/1963/4304}, author = {Giulio Bonelli and Alessandro Tanzini and Jian Zhao} } @article {2011, title = {A system-level approach for deciphering the transcriptional response to prion infection}, journal = {Bioinformatics (Oxford, England). 2011 Dec; 27(24):3407-14}, number = {PMID:22016408;}, year = {2011}, publisher = {Oxford University Press}, abstract = {MOTIVATION: Deciphering the response of a complex biological system to an insulting event, at the gene expression level, requires adopting theoretical models that are more sophisticated than a one-to-one comparison (i.e. t-test). Here, we investigate the ability of a novel reverse engineering approach (System Response Inference) to unveil non-obvious transcriptional signatures of the system response induced by prion infection.\\r\\nRESULTS: To this end, we analyze previously published gene expression data, from which we extrapolate a putative full-scale model of transcriptional gene-gene dependencies in the mouse central nervous system. Then, we use this nominal model to interpret the gene expression changes caused by prion replication, aiming at selecting the genes primarily influenced by this perturbation. Our method sheds light on the mode of action of prions by identifying key transcripts that are the most likely to be responsible for the overall transcriptional rearrangement from a nominal regulatory network. As a first result of our inference, we have been able to predict known targets of prions (i.e. PrP(C)) and to unveil the potential role of previously unsuspected genes.\\r\\nCONTACT: altafini@sissa.it\\r\\nSUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.}, doi = {10.1093/bioinformatics/btr580}, url = {http://hdl.handle.net/1963/5745}, author = {Mattia Zampieri and Giuseppe Legname and Daniel Segr{\`e} and Claudio Altafini} } @article {2010, title = {Homogenization of fiber reinforced brittle material: the intermediate case}, journal = {Adv. Calc. Var. 3 (2010) 345-370}, number = {SISSA;14/2009/M}, year = {2010}, publisher = {Walter de Gruyter}, abstract = {We derive a cohesive fracture model by homogenizing a periodic composite material whose microstructure is characterized by the presence of brittle inclusions in a reticulated unbreakable elastic structure.}, doi = {10.1515/ACV.2010.011}, url = {http://hdl.handle.net/1963/3607}, author = {Gianni Dal Maso and Caterina Ida Zeppieri} } @article {2009, title = {1D periodic potentials with gaps vanishing at k=0}, journal = {Mem. Differential Equations Math. Phys. 47 (2009) 133-158}, number = {SISSA;12/2006/FM}, year = {2009}, abstract = {Appearance of energy bands and gaps in the dispersion relations of a periodic potential is a standard feature of Quantum Mechanics. We investigate the class of one-dimensional periodic potentials for which all gaps vanish at the center of the Brillouin zone. We characterise themthrough a necessary and sufficient condition. Potentials of the form we focus on arise in different fields of Physics, from supersymmetric Quantum Mechanics, to Korteweg-de Vries equation theory and classical diffusion problems. The O.D.E. counterpart to this problem is the characterisation of periodic potentials for which coexistence occurs of linearly independent solutions of the corresponding Schr{\"o}dinger equation (Hill\\\'s equation). This result is placed in the perspective of the previous related results available in the literature.}, url = {http://hdl.handle.net/1963/1818}, author = {Alessandro Michelangeli and Osvaldo Zagordi} } @article {2009, title = {Differential geometry of curves in Lagrange Grassmannians with given Young diagram}, journal = {Differential Geom. Appl. 27 (2009) 723-742}, number = {arXiv.org;0708.1100v1}, year = {2009}, publisher = {Elsevier}, abstract = {Curves in Lagrange Grassmannians appear naturally in the intrinsic study of geometric structures on manifolds. By a smooth geometric structure on a manifold we mean any submanifold of its tangent bundle, transversal to the fibers. One can consider the time-optimal problem naturally associate with a geometric structure. The Pontryagin extremals of this optimal problem are integral curves of certain Hamiltonian system in the cotangent bundle. The dynamics of the fibers of the cotangent bundle w.r.t. this system along an extremal is described by certain curve in a Lagrange Grassmannian, called Jacobi curve of the extremal. Any symplectic invariant of the Jacobi curves produces the invariant of the original geometric structure. The basic characteristic of a curve in a Lagrange Grassmannian is its Young diagram. The number of boxes in its kth column is equal to the rank of the kth derivative of the curve (which is an appropriately defined linear mapping) at a generic point. We will describe the construction of the complete system of symplectic invariants for parameterized curves in a Lagrange Grassmannian with given Young diagram. It allows to develop in a unified way local differential geometry of very wide classes of geometric structures on manifolds, including both classical geometric structures such as Riemannian and Finslerian structures and less classical ones such as sub-Riemannian and sub-Finslerian structures, defined on nonholonomic distributions.}, doi = {10.1016/j.difgeo.2009.07.002}, url = {http://hdl.handle.net/1963/3819}, author = {Igor Zelenko and Li Chengbo} } @article {2009, title = {Discrete-to-continuum limits for strain-alignment-coupled systems: Magnetostrictive solids, ferroelectric crystals and nematic elastomers}, journal = {Netw. Heterog. Media 4 (2009) 667-708}, year = {2009}, publisher = {American Institute of Mathematical Sciences}, abstract = {In the framework of linear elasticity, we study the limit of a class of discrete free energies modeling strain-alignment-coupled systems by a rigorous coarse-graining procedure, as the number of molecules diverges. We focus on three paradigmatic examples: magnetostrictive solids, ferroelectric crystals and nematic elastomers, obtaining in the limit three continuum models consistent with those commonly employed in the current literature. We also derive the correspondent macroscopic energies in the presence of displacement boundary conditions and of various kinds of applied external fields.}, doi = {10.3934/nhm.2009.4.667}, url = {http://hdl.handle.net/1963/3788}, author = {Marco Cicalese and Antonio DeSimone and Caterina Ida Zeppieri} } @article {2009, title = {Gauged Laplacians on quantum Hopf bundles}, journal = {Comm. Math. Phys. 287 (2009) 179-209}, number = {arXiv.org;0801.3376v2}, year = {2009}, publisher = {Springer}, abstract = {We study gauged Laplacian operators on line bundles on a quantum 2-dimensional sphere. Symmetry under the (co)-action of a quantum group allows for their complete diagonalization. These operators describe {\textquoteleft}excitations moving on the quantum sphere\\\' in the field of a magnetic monopole. The energies are not invariant under the exchange monopole/antimonopole, that is under inverting the direction of the magnetic field. There are potential applications to models of quantum Hall effect.}, doi = {10.1007/s00220-008-0672-5}, url = {http://hdl.handle.net/1963/3540}, author = {Giovanni Landi and Cesare Reina and Alessandro Zampini} } @article {2009, title = {Investigating the Conformational Stability of Prion Strains through a Kinetic Replication Model}, journal = {PLoS Comput Biol 2009;5(7): e1000420}, year = {2009}, publisher = {PLoS}, abstract = {Prion proteins are known to misfold into a range of different aggregated forms, showing different phenotypic and pathological states. Understanding strain specificities is an important problem in the field of prion disease. Little is known about which PrPSc structural properties and molecular mechanisms determine prion replication, disease progression and strain phenotype. The aim of this work is to investigate, through a mathematical model, how the structural stability of different aggregated forms can influence the kinetics of prion replication. The model-based results suggest that prion strains with different conformational stability undergoing in vivo replication are characterizable in primis by means of different rates of breakage. A further role seems to be played by the aggregation rate (i.e. the rate at which a prion fibril grows). The kinetic variability introduced in the model by these two parameters allows us to reproduce the different characteristic features of the various strains (e.g., fibrils\\\' mean length) and is coherent with all experimental observations concerning strain-specific behavior.}, doi = {10.1371/journal.pcbi.1000420}, url = {http://hdl.handle.net/1963/3989}, author = {Mattia Zampieri and Giuseppe Legname and Claudio Altafini} } @article {2009, title = {Jacobi Equations and Comparison Theorems for Corank 1 Sub-Riemannian structures with symmetries}, number = {SISSA;53/2009/M}, year = {2009}, abstract = {The Jacobi curve of an extremal of optimal control problem is a curve in a Lagrangian Grassmannian defined up to a symplectic transformation and containing all information about the solutions of the Jacobi equations along this extremal. In our previous works we constructed the canonical\\nbundle of moving frames and the complete system of symplectic invariants, called curvature maps, for\\nparametrized curves in Lagrange Grassmannians satisfying very general assumptions. The structural\\nequation for a canonical moving frame of the Jacobi curve of an extremal can be interpreted as the\\nnormal form for the Jacobi equation along this extremal and the curvature maps can be seen as the\\n\\\"coefficients\\\"of this normal form. In the case of a Riemannian metric there is only one curvature map and it is naturally related to the Riemannian sectional curvature. In the present paper we study the curvature maps for a sub-Riemannian structure on a corank 1 distribution having an additional transversal infinitesimal symmetry. After the factorization by the integral foliation of this symmetry, such sub-Riemannian structure can be reduced to a Riemannian manifold equipped with a closed 2-form(a magnetic field). We obtain explicit expressions for the curvature maps of the original sub-Riemannian structure in terms of the curvature tensor of this Riemannian manifold and the magnetic field. We also estimate the number of conjugate points along the sub-Riemannian extremals in terms of the bounds for the curvature tensor of this Riemannian manifold and the magnetic field in the case of an uniform magnetic field. The language developed for the calculation of the curvature maps can be applied to more general sub-Riemannian structures with symmetries, including sub-Riemmannian structures appearing naturally in Yang-Mills fields.}, url = {http://hdl.handle.net/1963/3736}, author = {Li Chengbo and Igor Zelenko} } @article {2009, title = {mRNA stability and the unfolding of gene expression in the long-period yeast metabolic cycle}, journal = {BMC Systems Biology (2009) 3:18}, year = {2009}, publisher = {BioMed Central}, abstract = {Background: In yeast, genome-wide periodic patterns associated with energy-metabolic oscillations have been shown recently for both short (approx. 40 min) and long (approx. 300 min) periods.\\nResults: The dynamical regulation due to mRNA stability is found to be an important aspect of the genome-wide coordination of the long-period yeast metabolic cycle. It is shown that for periodic genes, arranged in classes according either to expression profile or to function, the pulses of mRNA abundance have phase and width which are directly proportional to the corresponding turnover rates.\\nConclusion: The cascade of events occurring during the yeast metabolic cycle (and their correlation with mRNA turnover) reflects to a large extent the gene expression program observable in other dynamical contexts such as the response to stresses/stimuli.}, doi = {10.1186/1752-0509-3-18}, url = {http://hdl.handle.net/1963/3630}, author = {Nicola Soranzo and Mattia Zampieri and Lorenzo Farina and Claudio Altafini} } @article {2009, title = {Topological branes, p-algebras and generalized Nahm equations}, journal = {Phys. Lett. B 672 (2009) 390-395}, number = {SISSA;47/2008/FM}, year = {2009}, abstract = {Inspired by the recent advances in multiple M2-brane theory, we consider the generalizations of Nahm equations for arbitrary p-algebras. We construct the topological p-algebra quantum mechanics associated to them and we show that this can be obtained as a truncation of the topological p-brane theory previously studied by the authors. The resulting topological p-algebra quantum mechanics is discussed in detail and the relation with the M2-M5 system is pointed out in the p=3 case, providing a geometrical argument for the emergence of the 3-algebra structure in the Bagger-Lambert-Gustavsson theory}, doi = {10.1016/j.physletb.2009.01.051}, url = {http://hdl.handle.net/1963/2702}, author = {Giulio Bonelli and Alessandro Tanzini and Maxim Zabzine} } @article {2009, title = {On viscosity solutions of Hamilton-Jacobi equations}, journal = {Trans. Amer. Math. Soc. 361 (2009) 41-59}, year = {2009}, publisher = {American Mathematical Society}, abstract = {We consider the Dirichlet problem for Hamilton-Jacobi equations and prove existence, uniqueness and continuous dependence on boundary data of Lipschitz continuous maximal viscosity solutions.}, doi = {10.1090/S0002-9947-08-04557-1}, url = {http://hdl.handle.net/1963/3420}, author = {Sandro Zagatti} } @article {2008, title = {Discerning static and causal interactions in genome-wide reverse engineering problems}, journal = {Bioinformatics 24 (2008) 1510-1515}, year = {2008}, abstract = {Background. In the past years devicing methods for discovering gene regulatory mechanisms at a genome-wide level has become a fundamental topic in the field of system biology. The aim is to infer gene-gene interactions in a more sophisticated and reliable way through the continuously improvement of reverse engineering algorithms exploiting microarray technologies. Motivation. This work is inspired by the several studies suggesting that co-expression is mostly related to \\\"static\\\" stable binding relationships, like belonging to the same protein complex, rather than other types of interactions more of a \\\"causal\\\" and transient nature (metabolic pathway or transcription factor-binding site interaction). Discerning static relationships from causal ones on the basis of their characteristic regulatory structures and in particular identifing \\\"dense modules\\\" with protein complex, and \\\"sparse modules\\\" with causal interactions such as those between transcription factor and corresponding binding site, the performances of different network inference algorithms in artificial and real networks (derived from E.coli and S.cerevisiae) can be tested and compared. Results. Our study shows that methods that try to prune indirect interactions from the inferred gene networks may fail to retrieve genes co-participating in a protein complex. On the other hand they are more robust in the identification of transcription factor-binding sites dependences when multiple transcription factors regulate the expression of the same gene. In the end we confirm the stronger co-expression regarding genes belonging to a protein complex than transcription factor-binding site, according, also, to the effect of multiple transcription factors and a low expression variance.}, doi = {10.1093/bioinformatics/btn220}, url = {http://hdl.handle.net/1963/2757}, author = {Mattia Zampieri and Nicola Soranzo and Claudio Altafini} } @article {2008, title = {Minimization of non quasiconvex functionals by integro-extremization method}, journal = {Discrete Contin. Dyn. Syst. 21 (2008) 625-641}, year = {2008}, doi = {10.3934/dcds.2008.21.625}, url = {http://hdl.handle.net/1963/2761}, author = {Sandro Zagatti} } @article {2008, title = {Minimizers of non convex scalar functionals and viscosity solutions of Hamilton-Jacobi equations}, journal = {Calc. Var. Partial Differential Equations 31 (2008) 511-519}, year = {2008}, doi = {10.1007/s00526-007-0124-7}, url = {http://hdl.handle.net/1963/2760}, author = {Sandro Zagatti} } @article {2008, title = {Origin of Co-Expression Patterns in E.coli and S.cerevisiae Emerging from Reverse Engineering Algorithms}, journal = {PLoS ONE 3 (2008) e2981}, year = {2008}, abstract = {Background: The concept of reverse engineering a gene network, i.e., of inferring a genome-wide graph of putative genegene interactions from compendia of high throughput microarray data has been extensively used in the last few years to deduce/integrate/validate various types of \\\"physical\\\" networks of interactions among genes or gene products. Results: This paper gives a comprehensive overview of which of these networks emerge significantly when reverse engineering large collections of gene expression data for two model organisms, E.coli and S.cerevisiae, without any prior information. For the first organism the pattern of co-expression is shown to reflect in fine detail both the operonal structure of the DNA and the regulatory effects exerted by the gene products when co-participating in a protein complex. For the second organism we find that direct transcriptional control (e.g., transcription factor-binding site interactions) has little statistical significance in comparison to the other regulatory mechanisms (such as co-sharing a protein complex, colocalization on a metabolic pathway or compartment), which are however resolved at a lower level of detail than in E.coli. Conclusion: The gene co-expression patterns deduced from compendia of profiling experiments tend to unveil functional categories that are mainly associated to stable bindings rather than transient interactions. The inference power of this systematic analysis is substantially reduced when passing from E.coli to S.cerevisiae. This extensive analysis provides a way to describe the different complexity between the two organisms and discusses the critical limitations affecting this type of methodologies.}, doi = {10.1371/journal.pone.0002981}, url = {http://hdl.handle.net/1963/2722}, author = {Mattia Zampieri and Nicola Soranzo and Daniele Bianchini and Claudio Altafini} } @article {2007, title = {Computing Amplitudes in topological M-theory}, number = {SISSA;74/2006/FM}, year = {2007}, abstract = {We define a topological quantum membrane theory on a seven dimensional manifold of $G_2$ holonomy. We describe in detail the path integral evaluation for membrane geometries given by circle bundles over Riemann surfaces. We show that when the target space is $CY_3\\\\times S^1$ quantum amplitudes of non-local observables of membranes wrapping the circle reduce to the A-model amplitudes. \\nIn particular for genus zero we show that our model computes the Gopakumar-Vafa invariants. Moreover, for membranes wrapping calibrated homology spheres in the $CY_3$, we find that the amplitudes of our model are related to Joyce invariants.}, doi = {10.1088/1126-6708/2007/03/023}, url = {http://hdl.handle.net/1963/1901}, author = {Giulio Bonelli and Alessandro Tanzini and Maxim Zabzine} } @article {2007, title = {On feedback classification of control-affine systems with one and two-dimensional inputs}, journal = {SIAM J. Control Optim. 46 (2007) 1431-1460}, number = {arXiv.org;math/050231v2}, year = {2007}, abstract = {The paper is devoted to the local classification of generic control-affine systems on an n-dimensional manifold with scalar input for any n>3 or with two inputs for n=4 and n=5, up to state-feedback transformations, preserving the affine structure. First using the Poincare series of moduli numbers we introduce the intrinsic numbers of functional moduli of each prescribed number of variables on which a classification problem depends. In order to classify affine systems with scalar input we associate with such a system the canonical frame by normalizing some structural functions in a commutative relation of the vector fields, which define our control system. Then, using this canonical frame, we introduce the canonical coordinates and find a complete system of state-feedback invariants of the system. It also gives automatically the micro-local (i.e. local in state-input space) classification of the generic non-affine n-dimensional control system with scalar input for n>2. Further we show how the problem of feedback-equivalence of affine systems with two-dimensional input in state space of dimensions 4 and 5 can be reduced to the same problem for affine systems with scalar input. In order to make this reduction we distinguish the subsystem of our control system, consisting of the directions of all extremals in dimension 4 and all abnormal extremals in dimension 5 of the time optimal problem, defined by the original control system. In each classification problem under consideration we find the intrinsic numbers of functional moduli of each prescribed number of variables according to its Poincare series.}, doi = {10.1137/050623711}, url = {http://hdl.handle.net/1963/2186}, author = {Andrei A. Agrachev and Igor Zelenko} } @article {2007, title = {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 = {Quasistatic crack growth for a cohesive zone model with prescribed crack path}, number = {SISSA;44/2005/M}, year = {2007}, abstract = {In this paper we study the quasistatic crack growth for a cohesive zone model. We assume that the crack path is prescribed and we study the time evolution of the crack in the framework of the variational theory of rate-independent processes.}, doi = {10.1017/S030821050500079X}, url = {http://hdl.handle.net/1963/1686}, author = {Gianni Dal Maso and Chiara Zanini} } @article {2007, title = {Solutions of vectorial Hamilton-Jacobi equations and minimizers of nonquasiconvex functionals}, journal = {J. Math. Anal. Appl. 335 (2007) 1143-1160}, year = {2007}, abstract = {We provide a unified approach to prove existence results for the Dirichlet problem for Hamilton-Jacobi equations and for the minimum problem for nonquasiconvex functionals of the Calculus of Variations with affine boundary data. The idea relies on the definition of integro-extremal solutions introduced in the study of nonconvex scalar variational problem.}, doi = {10.1016/j.jmaa.2007.02.034}, url = {http://hdl.handle.net/1963/2763}, author = {Sandro Zagatti} } @article {2007, title = {Uniqueness and continuous dependence on boundary data for integro-extremal minimizers of the functional of the gradient}, journal = {J. Convex Anal. 14 (2007) 705-727}, year = {2007}, abstract = {We study some qualitative properties of the integro-extremal minimizers of the functional of the gradient defined on Sobolev spaces with Dirichlet boundary conditions. We discuss their use in the non-convex case via viscosity methods and give conditions under which they are unique and depend continuously on boundary data.}, url = {http://hdl.handle.net/1963/2762}, author = {Sandro Zagatti} } @article {2006, title = {An artificial viscosity approach to quasistatic crack growth}, number = {SISSA;43/2006/M}, year = {2006}, abstract = {We introduce a new model of irreversible quasistatic crack growth in which the evolution of cracks is the limit of a suitably modified $\\\\epsilon$-gradient flow of the energy functional, as the \\\"viscosity\\\" parameter $\\\\epsilon$ tends to zero.}, url = {http://hdl.handle.net/1963/1850}, author = {Rodica Toader and Chiara Zanini} } @article {2006, title = {A Canonical Frame for Nonholonomic Rank Two Distributions of Maximal Class}, number = {SISSA;25/2005/M}, year = {2006}, abstract = {In 1910 E. Cartan constructed the canonical frame and found the most symmetric case for maximally nonholonomic rank 2 distributions in R5. We solve the analogous problems for rank 2 distributions in Rn for arbitrary n > 5. Our method is a kind of symplectification of the problem and it is completely different from the Cartan method of equivalence.}, doi = {10.1016/j.crma.2006.02.010}, url = {http://hdl.handle.net/1963/1712}, author = {Boris Doubrov and Igor Zelenko} } @article {2006, title = {Fundamental form and Cartan tensor of (2,5)-distributions coincide}, journal = {J. Dyn. Control Syst. 12 (2006) 247-276}, number = {arXiv.org;math/0402195v2}, year = {2006}, abstract = {In our previous paper for generic rank 2 vector distributions on n-dimensional manifold (n greater or equal to 5) we constructed a special differential invariant, the fundamental form. In the case n=5 this differential invariant has the same algebraic nature, as the covariant binary biquadratic form, constructed by E.Cartan in 1910, using his {\textquoteleft}{\textquoteleft}reduction- prolongation\\\'\\\' procedure (we call this form Cartan\\\'s tensor). In the present paper we prove that our fundamental form coincides (up to constant factor -35) with Cartan\\\'s tensor. This result explains geometric reason for existence of Cartan\\\'s tensor (originally this tensor was obtained by very sophisticated algebraic manipulations) and gives the true analogs of this tensor in Riemannian geometry. In addition, as a part of the proof, we obtain a new useful formula for Cartan\\\'s tensor in terms of structural functions of any frame naturally adapted to the distribution.}, doi = {10.1007/s10450-006-0383-1}, url = {http://hdl.handle.net/1963/2187}, author = {Igor Zelenko} } @article {2006, title = {On geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on distributions of corank 1}, journal = {J. Math. Sci. 135 (2006) 3168-3194}, number = {arXiv.org;math/0406111v1}, year = {2006}, abstract = {The present paper is devoted to the problem of (local) geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on generic corank 1 distributions. Using Pontryagin Maximum Principle, we treat Riemannian and sub-Riemannian cases in an unified way and obtain some algebraic necessary conditions for the geodesic equivalence of (sub-)Riemannian metrics. In this way first we obtain a new elementary proof of classical Levi-Civita\\\'s Theorem about the classification of all Riemannian geodesically equivalent metrics in a neighborhood of so-called regular (stable) point w.r.t. these metrics. Secondly we prove that sub-Riemannian metrics on contact distributions are geodesically equivalent iff they are constantly proportional. Then we describe all geodesically equivalent sub-Riemannian metrics on quasi-contact distributions. Finally we make the classification of all pairs of geodesically equivalent Riemannian metrics on a surface, which proportional in an isolated point. This is the simplest case, which was not covered by Levi-Civita\\\'s Theorem.}, doi = {10.1007/s10958-006-0151-5}, url = {http://hdl.handle.net/1963/2205}, author = {Igor Zelenko} } @article {2006, title = {On topological M-theory}, number = {SISSA;64/2005/FM}, year = {2006}, abstract = {We construct a gauge fixed action for topological membranes on G2-manifolds such that its bosonic part is the standard membrane theory in a particular gauge. We prove that quantum mechanically the path-integral in this gauge localizes on associative submanifolds.}, url = {http://hdl.handle.net/1963/1765}, author = {Giulio Bonelli and Alessandro Tanzini and Maxim Zabzine} } @article {2006, title = {On variational approach to differential invariants of rank two distributions}, journal = {Differential Geom. Appl. 24 (2006) 235-259}, number = {arXiv.org;math/0402171v2}, year = {2006}, abstract = {n the present paper we construct differential invariants for generic rank 2 vector distributions on n-dimensional manifold. In the case n=5 (the first case containing functional parameters) E. Cartan found in 1910 the covariant fourth-order tensor invariant for such distributions, using his \\\"reduction-prolongation\\\" procedure. After Cartan\\\'s work the following questions remained open: first the geometric reason for existence of Cartan\\\'s tensor was not clear; secondly it was not clear how to generalize this tensor to other classes of distributions; finally there were no explicit formulas for computation of Cartan\\\'s tensor. Our paper is the first in the series of papers, where we develop an alternative approach, which gives the answers to the questions mentioned above. It is based on the investigation of dynamics of the field of so-called abnormal extremals (singular curves) of rank 2 distribution and on the general theory of unparametrized curves in the Lagrange Grassmannian, developed in our previous works with A. Agrachev . In this way we construct the fundamental form and the projective Ricci curvature of rank 2 vector distributions for arbitrary n greater than 4.\\nFor n=5 we give an explicit method for computation of these invariants and demonstrate it on several examples. In our next paper we show that in the case n=5 our fundamental form coincides with Cartan\\\'s tensor.}, doi = {10.1016/j.difgeo.2005.09.004}, url = {http://hdl.handle.net/1963/2188}, author = {Igor Zelenko} } @inbook {2005, title = {Complete systems of invariants for rank 1 curves in Lagrange Grassmannians}, booktitle = {Differential geometry and its applications, 367-382, Matfyzpress, Prague, 2005}, number = {SISSA;82/2004/M}, year = {2005}, note = {Proceedings of 9th Conference on Differential Geometry and its Applications, Prague 2004}, abstract = {Curves in Lagrange Grassmannians naturally appear when one studies intrinsically \\\"the Jacobi equations for extremals\\\", associated with control systems and geometric structures. In this way one reduces the problem of construction of the curvature-type invariants for these objects to the much more concrete problem of finding of invariants of curves in Lagrange Grassmannians w.r.t. the action of the linear Symplectic group. In the present paper we develop a new approach to differential geometry of so-called rank 1 curves in Lagrange Grassmannian, i.e., the curves with velocities being rank one linear mappings (under the standard identification of the tangent space to a point of the Lagrange Grassmannian with an appropriate space of linear mappings). The curves of this class are associated with \\\"the Jacobi equations for extremals\\\", corresponding to control systems with scalar control and to rank 2 vector distributions. In particular, we construct the tuple of m principal invariants, where m is equal to half of dimension of the ambient linear symplectic space, such that for a given tuple of arbitrary m smooth functions there exists the unique, up to a symplectic transformation, rank 1 curve having this tuple, as the tuple of the principal invariants. This approach extends and essentially simplifies some results of our previous paper (J. Dynamical and Control Systems, 8, 2002, No. 1, 93-140), where only the uniqueness part was proved and in rather cumbersome way. It is based on the construction of the new canonical moving frame with the most simple structural equation.}, url = {http://hdl.handle.net/1963/2310}, author = {Igor Zelenko} } @article {2005, title = {On curvatures and focal points of distributions of dynamical Lagrangian distributions and their reductions by first integrals}, journal = {J. Dyn. Control Syst. 11 (2005) 297-327}, number = {SISSA;58/2004/M}, year = {2005}, abstract = {Pairs (Hamiltonian system, Lagrangian distribution), called dynamical Lagrangian distributions, appear naturally in Differential Geometry, Calculus of Variations and Rational Mechanics. The basic differential invariants of a dynamical Lagrangian distribution w.r.t. the action of the group of symplectomorphisms of the ambient symplectic manifold are the curvature operator and the curvature form. These invariants can be seen as generalizations of the classical curvature tensor in Riemannian Geometry. In particular, in terms of these invariants one can localize the focal points along extremals of the corresponding variational problems. In the present paper we study the behavior of the curvature operator, the curvature form and the focal points of a dynamical Lagrangian distribution after its reduction by arbitrary first integrals in involution. The interesting phenomenon is that the curvature form of so-called monotone increasing Lagrangian dynamical distributions, which appear naturally in mechanical systems, does not decrease after reduction. It also turns out that the set of focal points to the given point w.r.t. the monotone increasing dynamical Lagrangian distribution and the corresponding set of focal points w.r.t. its reduction by one integral are alternating sets on the corresponding integral curve of the Hamiltonian system of the considered dynamical distributions. Moreover, the first focal point corresponding to the reduced Lagrangian distribution comes before any focal point related to the original dynamical distribution. We illustrate our results on the classical $N$-body problem.}, doi = {10.1007/s10883-005-6581-4}, url = {http://hdl.handle.net/1963/2254}, author = {Andrei A. Agrachev and Natalia N. Chtcherbakova and Igor Zelenko} } @article {2005, title = {On the Minimum Problem for Nonconvex Scalar Functionals}, journal = {SIAM J. Math. Anal. 37 (2005) 982-995}, year = {2005}, abstract = {We study the minimum problem for scalar nonconvex functionals defined on Sobolev maps satisfying a Dirichlet boundary condition and refine well-known existence results under standard regularity assumptions.}, doi = {10.1137/040612506}, url = {http://hdl.handle.net/1963/2764}, author = {Sandro Zagatti} } @article {2005, title = {Nonlinear Schr{\"o}dinger Equations with vanishing and decaying potentials}, number = {SISSA;52/2005/M}, year = {2005}, url = {http://hdl.handle.net/1963/1760}, author = {Antonio Ambrosetti and Wang Zhi-Qiang} } @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 {2002, title = {Geometry of Jacobi Curves I}, journal = {J. Dynam. Control Systems 8 (2002) 93-140}, number = {SISSA;75/2001/M}, year = {2002}, publisher = {Springer}, abstract = {Jacobi curves are deep generalizations of the spaces of \\\"Jacobi fields\\\" along Riemannian geodesics. Actually, Jacobi curves are curves in the Lagrange Grassmannians. In our paper we develop differential geometry of these curves which provides basic feedback or gauge invariants for a wide class of smooth control systems and geometric structures. Two principal invariants are the generalized Ricci curvature, which is an invariant of the parametrized curve in the Lagrange Grassmannian endowing the curve with a natural projective structure, and a fundamental form, which is a fourth-order differential on the curve. The so-called rank 1 curves are studied in more detail. Jacobi curves of this class are associated with systems with scalar controls and with rank 2 vector distributions.\\nIn the forthcoming second part of the paper we will present the comparison theorems (i.e., the estimates for the conjugate points in terms of our invariants( for rank 1 curves an introduce an important class of \\\"flat curves\\\".}, doi = {10.1023/A:1013904801414}, url = {http://hdl.handle.net/1963/3110}, author = {Andrei A. Agrachev and Igor Zelenko} } @article {2002, title = {Geometry of Jacobi curves II}, journal = {J. Dynam. Control Systems 8 (2002), no. 2, 167--215}, number = {SISSA;18/2002/M}, year = {2002}, publisher = {SISSA Library}, doi = {10.1023/A:1015317426164}, url = {http://hdl.handle.net/1963/1589}, author = {Andrei A. Agrachev and Igor Zelenko} } @article {2001, title = {A note on the super Krichever map}, journal = {J. Geom. Phys. 37 (2001), no. 1-2, 169-181}, number = {SISSA;36/00/FM}, year = {2001}, publisher = {SISSA Library}, abstract = {We consider the geometrical aspects of the Krichever map in the context of Jacobian Super KP hierarchy. We use the representation of the hierarchy based\\non the Fa{\textquoteleft}a di Bruno recursion relations, considered as the cocycle condition for the natural double complex associated with the deformations of super Krichever data. Our approach is based on the construction of the universal super divisor (of degree g), and a local universal family of geometric data which give the map into the Super Grassmannian.}, doi = {10.1016/S0393-0440(00)00037-1}, url = {http://hdl.handle.net/1963/1494}, author = {Gregorio Falqui and Cesare Reina and Alessandro Zampa} } @article {2000, title = {3D superconformal theories from Sasakian seven-manifolds: new nontrivial evidences for AdS_4/CFT_3}, journal = {Nucl.Phys. B577 (2000) 547-608}, number = {SISSA;113/99/FM}, year = {2000}, publisher = {SISSA Library}, doi = {10.1016/S0550-3213(00)00098-5}, url = {http://hdl.handle.net/1963/1327}, author = {Davide Fabbri and Pietro Fr{\'e} and Leonardo Gualtieri and Cesare Reina and Alessandro Tomasiello and Alberto Zaffaroni and Alessandro Zampa} } @article {2000, title = {A(SLq(2)) at roots of unity is a free module over A(SL(2))}, journal = {Lett. Math. Phys., 2000, 52, 339}, number = {SISSA;42/00/FM}, year = {2000}, publisher = {SISSA Library}, doi = {10.1023/A:1007601131002}, url = {http://hdl.handle.net/1963/1500}, author = {Ludwik Dabrowski and Cesare Reina and Alessandro Zampa} } @article {2000, title = {A bi-Hamiltonian theory for stationary KDV flows and their separability}, journal = {Regul. Chaotic Dyn. 5 (2000), no. 1, 33-52}, number = {SISSA;137/99/FM}, year = {2000}, publisher = {SISSA Library}, doi = {10.1070/rd2000v005n01ABEH000122}, url = {http://hdl.handle.net/1963/1352}, author = {Gregorio Falqui and Franco Magri and Marco Pedroni and Jorge P. Zubelli} } @article {2000, title = {An elementary approach to the polynomial $\\\\tau$-functions of the KP Hierarchy}, journal = {Theor. Math. Phys. 122 (2000) 17-28}, number = {SISSA;117/98/FM}, year = {2000}, publisher = {Springer}, abstract = {We give an elementary construction of the solutions of the KP hierarchy associated with polynomial τ-functions starting with a geometric approach to soliton equations based on the concept of a bi-Hamiltonian system. As a consequence, we establish a Wronskian formula for the polynomial τ-functions of the KP hierarchy. This formula, known in the literature, is obtained very directly.}, doi = {10.1007/BF02551166}, url = {http://hdl.handle.net/1963/3223}, author = {Gregorio Falqui and Franco Magri and Marco Pedroni and Jorge P. Zubelli} } @article {2000, title = {Minimization of functionals of the gradient by Baire{\textquoteright}s theorem}, journal = {SIAM J. Control Optim. 38 (2000) 384-399}, number = {SISSA;71/97/M}, year = {2000}, publisher = {SIAM}, abstract = {We give sufficient conditions for the existence of solutions of the minimum problem $$ {\mathcal{P}}_{u_0}: \qquad \hbox{Minimize}\quad \int_\Omega g(Du(x))dx, \quad u\in u_0 + W_0^{1,p}(\Omega,{\mathbb{R}}), $$ based on the structure of the epigraph of the lower convex envelope of g, which is assumed be lower semicontinuous and to grow at infinity faster than the power p with p larger than the dimension of the space. No convexity conditions are required on g, and no assumptions are made on the boundary datum $u_0\in W_0^{1,p}(\Omega,\mathbb{R})$.

}, doi = {10.1137/S0363012998335206}, url = {http://hdl.handle.net/1963/3511}, author = {Sandro Zagatti} } @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} } @article {2000, title = {Super KP equations and Darboux transformations: another perspective on the Jacobian super KP hierarchy}, journal = {J. Geom. Phys. 35 (2000), no. 2-3, 239-272}, number = {SISSA;152/99/FM}, year = {2000}, publisher = {SISSA Library}, doi = {10.1016/S0393-0440(00)00007-3}, url = {http://hdl.handle.net/1963/1367}, author = {Gregorio Falqui and Cesare Reina and Alessandro Zampa} } @article {1999, title = {Enhanced gauge symmetries on elliptic K3}, journal = {Phys.Lett. B452 (1999) 244-250}, number = {SISSA;74/98/EP-FM}, year = {1999}, publisher = {Elsevier}, abstract = {We show that the geometry of K3 surfaces with singularities of type A-D-E contains enough information to reconstruct a copy of the Lie algebra associated to the given Dynkin diagram. We apply this construction to explain the enhancement of symmetry in F and IIA theories compactified on singular K3\\\'s.}, doi = {10.1016/S0370-2693(99)00295-6}, url = {http://hdl.handle.net/1963/3366}, author = {Loriano Bonora and Cesare Reina and Alessandro Zampa} } @article {1998, title = {On the Dirichlet problem for vectorial Hamilton-Jacobi equations}, journal = {SIAM J. Math. Anal. 29 (1998) 1481-1491}, number = {SISSA;182/96/M}, year = {1998}, publisher = {SIAM}, abstract = {We give sufficient conditions for the existence of solutions to the Hamilton--Jacobi equations with Dirichlet boundary condition: $$ \\\\cases{ g(x,{\\\\hbox{\\\\rm det}}Du(x))=0, \\\\ \& for a.e. $x\\\\in\\\\Omega,$\\\\cr u(x)=\\\\varphi(x), \& for $x\\\\in\\\\partial\\\\Omega,$} $$ obtaining, in addition, an application to the theory of existence of minimizers for a class of nonconvex variational problems.}, doi = {10.1137/S0036141097321279}, url = {http://hdl.handle.net/1963/3512}, author = {Sandro Zagatti} } @article {1998, title = {Extended affine Weyl groups and Frobenius manifolds}, journal = {Compositio Mathematica. Volume 111, Issue 2, 1998, Pages 167-219}, year = {1998}, publisher = {Kluwer}, doi = {10.1023/A:1000258122329}, url = {http://hdl.handle.net/1963/6486}, author = {Boris Dubrovin and Youjin Zhang} } @article {1997, title = {Krichever maps, Fa{\`a} di Bruno polynomials, and cohomology in KP theory}, journal = {Lett. Math. Phys. 42 (1997) 349-361}, number = {SISSA;37/97/FM}, year = {1997}, publisher = {Springer}, abstract = {We study the geometrical meaning of the Faa\\\' di Bruno polynomials in the context of KP theory. They provide a basis in a subspace W of the universal Grassmannian associated to the KP hierarchy. When W comes from geometrical data via the Krichever map, the Faa\\\' di Bruno recursion relation turns out to be the cocycle condition for (the Welters hypercohomology group describing) the deformations of the dynamical line bundle on the spectral curve together with the meromorphic sections which give rise to the Krichever map. Starting from this, one sees that the whole KP hierarchy has a similar cohomological meaning.}, doi = {10.1023/A:1007323118991}, url = {http://hdl.handle.net/1963/3539}, author = {Gregorio Falqui and Cesare Reina and Alessandro Zampa} } @article {1995, title = {An existence result in a problem of the vectorial case of the calculus of variations}, year = {1995}, publisher = {SIAM}, abstract = {SIAM J. Control Optim. 33 (1995) 960-970}, url = {http://hdl.handle.net/1963/3513}, author = {Arrigo Cellina and Sandro Zagatti} } @inbook {1995, title = {Quantum homogeneous spaces at roots of unity}, booktitle = {Quantization, Coherent States and Poisson Structures, Proc. XIVth Workshop on Geometric Methods in Physics, Bialowieza, Poland, 9-15 July 1995, eds. A. Strasburger,\\nS.T. Ali, J.-P. Antoine, J.-P. Gazeau , A. Odzijewicz, Polish Scientific Publisher PWN 1}, number = {SISSA;120/95/FM}, year = {1995}, publisher = {SISSA Library}, organization = {SISSA Library}, url = {http://hdl.handle.net/1963/1022}, author = {Cesare Reina and Alessandro Zampa} } @article {1994, title = {A version of Olech\\\'s lemma in a problem of the calculus of variations}, journal = {SIAM J. Control Optim. 32 (1994) 1114-1127}, year = {1994}, publisher = {SIAM}, abstract = {This paper studies the solutions of the minimum problem for a functional of the gradient under linear boundary conditions. A necessary and sufficient condition, based on the facial structure of the epigraph of the integrand, is provided for the continuous dependence of the solutions on boundary data.}, doi = {10.1137/S0363012992234669}, url = {http://hdl.handle.net/1963/3514}, author = {Arrigo Cellina and Sandro Zagatti} } @mastersthesis {1992, title = {Some Problems in the Calculus of the Variations}, year = {1992}, school = {SISSA}, keywords = {Calculus of variations}, url = {http://hdl.handle.net/1963/5428}, author = {Sandro Zagatti} }