We study the periodic boundary value problem associated with the second order nonlinear equation u''+(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.

PB - American Mathematical Society UR - http://urania.sissa.it/xmlui/handle/1963/35264 N1 - AMS Subject Classification: 34B15, 34B18, 34C25, 34C28, 47H11. U1 - 35568 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - RPRT T1 - Stochastic homogenisation of free-discontinuity problems Y1 - 2018 A1 - Filippo Cagnetti A1 - Gianni Dal Maso A1 - Lucia Scardia A1 - Caterina Ida Zeppieri AB - 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. UR - http://preprints.sissa.it/handle/1963/35309 U1 - 35617 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - RPRT T1 - Transmission conditions obtained by homogenisation Y1 - 2018 A1 - Gianni Dal Maso A1 - Giovanni Franzina A1 - Davide Zucco AB - 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. UR - http://preprints.sissa.it/handle/1963/35310 U1 - 35618 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - JOUR T1 - An application of coincidence degree theory to cyclic feedback type systems associated with nonlinear differential operators JF - Topol. Methods Nonlinear Anal. Y1 - 2017 A1 - Guglielmo Feltrin A1 - Fabio Zanolin PB - Nicolaus Copernicus University, Juliusz P. Schauder Centre for Nonlinear Studies VL - 50 UR - https://doi.org/10.12775/TMNA.2017.038 ER - TY - RPRT T1 - Gamma-Convergence of Free-discontinuity problems Y1 - 2017 A1 - Filippo Cagnetti A1 - Gianni Dal Maso A1 - Lucia Scardia A1 - Caterina Ida Zeppieri AB - 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. PB - SISSA UR - http://preprints.sissa.it/handle/1963/35276 U1 - 35583 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree JF - Journal of Differential Equations Y1 - 2017 A1 - Guglielmo Feltrin A1 - Fabio Zanolin KW - Coincidence degree KW - Multiplicity results KW - Neumann boundary value problems KW - Positive periodic solutions KW - subharmonic solutions KW - Superlinear indefinite problems AB -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'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'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.

VL - 262 UR - http://www.sciencedirect.com/science/article/pii/S0022039617300219 ER - TY - JOUR T1 - Confinement of dislocations inside a crystal with a prescribed external strain Y1 - 2016 A1 - Ilaria Lucardesi A1 - Marco Morandotti A1 - Riccardo Scala A1 - Davide Zucco AB - 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. UR - http://urania.sissa.it/xmlui/handle/1963/35247 N1 - Preprint SISSA 20/2016/MATE U1 - 35558 U2 - Mathematics ER - TY - JOUR T1 - Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super-sublinear case JF - Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), 449–474. Y1 - 2016 A1 - Alberto Boscaggin A1 - Guglielmo Feltrin A1 - Fabio Zanolin AB -We study the periodic and Neumann boundary value problems associated with the second order nonlinear differential equation u''+cu'+λa(t)g(u)=0, where g:[0,+∞[→[0,+∞[ 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's coincidence degree theory and index computations.

PB - Cambridge University Press UR - http://urania.sissa.it/xmlui/handle/1963/35262 N1 - AMS Subject Classification: Primary 34B18; 34C25; Secondary 34B15; 47H11; U1 - 35566 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - RPRT T1 - A bridging mechanism in the homogenisation of brittle composites with soft inclusions Y1 - 2015 A1 - Marco Barchiesi A1 - Giuliano Lazzaroni A1 - Caterina Ida Zeppieri AB - 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. PB - SISSA UR - http://urania.sissa.it/xmlui/handle/1963/7492 U1 - 7621 ER - TY - RPRT T1 - Convex combinations of low eigenvalues, Fraenkel asymmetries and attainable sets Y1 - 2015 A1 - Dario Mazzoleni A1 - Davide Zucco AB - 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 à 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. PB - SISSA UR - http://urania.sissa.it/xmlui/handle/1963/35140 U1 - 35378 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - JOUR T1 - Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems JF - Adv. Differential Equations 20 (2015), 937–982. Y1 - 2015 A1 - Guglielmo Feltrin A1 - Fabio Zanolin AB -We prove the existence of positive periodic solutions for the second order nonlinear equation u'' + 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'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.

PB - Khayyam Publishing UR - http://projecteuclid.org/euclid.ade/1435064518 N1 - AMS Subject Classification: 34B18, 34B15, 34C25, 47H11. U1 - 35388 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - RPRT T1 - Extended affine Weyl groups of BCD type, Frobenius manifolds and their Landau-Ginzburg superpotentials Y1 - 2015 A1 - Boris Dubrovin A1 - Ian A.B. Strachan A1 - Youjin Zhang A1 - Dafeng Zuo AB - 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. PB - SISSA UR - http://preprints.sissa.it/handle/1963/35316 U1 - 35625 U2 - Mathematics U4 - 1 U5 - MAT/07 ER - TY - JOUR T1 - Multiple positive solutions for a superlinear problem: a topological approach JF - J. Differential Equations 259 (2015), 925–963. Y1 - 2015 A1 - Guglielmo Feltrin A1 - Fabio Zanolin AB -We study the multiplicity of positive solutions for a two-point boundary value problem associated to the nonlinear second order equation u''+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→0^+ and above λ_1 as s→+∞. 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'' + 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.

PB - Elsevier UR - http://urania.sissa.it/xmlui/handle/1963/35147 N1 - Work presented at the "Special Session 21" of the "10th AIMS Conference on Dynamical Systems, Differential Equations and Applications" (Madrid, July 7-11, 2014). U1 - 35387 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - JOUR T1 - The phototransduction machinery in the rod outer segment has a strong efficacy gradient Y1 - 2015 A1 - Monica Mazzolini A1 - Giuseppe Facchetti A1 - L. Andolfi A1 - R. Proietti Zaccaria A1 - S. Tuccio A1 - J. Treud A1 - Claudio Altafini A1 - Enzo M. Di Fabrizio A1 - Marco Lazzarino A1 - G. Rapp A1 - Vincent Torre PB - National Academy of Sciences UR - http://urania.sissa.it/xmlui/handle/1963/35157 N1 - Open Access article U1 - 35382 U2 - Neuroscience ER - TY - JOUR T1 - Dirac operators on noncommutative principal circle bundles Y1 - 2014 A1 - Andrzej Sitarz A1 - Alessandro Zucca A1 - Ludwik Dabrowski AB - 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 → S2. PB - World Scientific Publishing UR - http://urania.sissa.it/xmlui/handle/1963/35125 U1 - 35363 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Infinite-dimensional Frobenius manifolds underlying the Toda lattice hierarchy Y1 - 2014 A1 - Chaozhong Wu A1 - Dafeng Zuo AB - 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. PB - Elsevier UR - http://urania.sissa.it/xmlui/handle/1963/35026 U1 - 35264 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Maximal generalized solution of eikonal equation Y1 - 2014 A1 - Sandro Zagatti AB - 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∞(Ω) for which the problem admits a (maximal) generalized solution, providing a generalization of the notion of viscosity solution. PB - Elsevier UR - http://urania.sissa.it/xmlui/handle/1963/34642 U1 - 34846 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - New results on Gamma-limits of integral functionals Y1 - 2014 A1 - Nadia Ansini A1 - Gianni Dal Maso A1 - Caterina Ida Zeppieri KW - Gamma-convergence PB - Elsevier UR - http://hdl.handle.net/1963/5880 U1 - 5745 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Pseudo-automorphisms of positive entropy on the blowups of products of projective spaces Y1 - 2014 A1 - Fabio Perroni A1 - Deqi Zhang AB - 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. PB - Springer UR - http://urania.sissa.it/xmlui/handle/1963/34714 U1 - 34921 U2 - Mathematics U4 - 1 ER - TY - RPRT T1 - Where best to place a Dirichlet condition in an anisotropic membrane? Y1 - 2014 A1 - Paolo Tilli A1 - Davide Zucco AB - 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. PB - SISSA UR - http://urania.sissa.it/xmlui/handle/1963/7481 U1 - 7592 ER - TY - JOUR T1 - Asymptotics of the first Laplace eigenvalue with Dirichlet regions of prescribed length Y1 - 2013 A1 - Paolo Tilli A1 - Davide Zucco AB - 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. PB - Society for Industrial and Applied Mathematics UR - http://urania.sissa.it/xmlui/handle/1963/35141 U1 - 35379 U2 - Physics U4 - 1 U5 - MAT/05 ER - TY - JOUR T1 - Pairs of nodal solutions for a class of nonlinear problems with one-sided growth conditions JF - Advanced Nonlinear Studies Y1 - 2013 A1 - Alberto Boscaggin A1 - Fabio Zanolin PB - Advanced Nonlinear Studies, Inc. VL - 13 ER - TY - JOUR T1 - Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions JF - Discrete & Continuous Dynamical Systems - A Y1 - 2013 A1 - Alberto Boscaggin A1 - Fabio Zanolin KW - lower and upper solutions KW - parameter dependent equations KW - Periodic solutions KW - Poincaré-Birkhoff twist theorem KW - subharmonic solutions AB -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.

VL - 33 UR - http://aimsciences.org//article/id/3638a93e-4f3e-4146-a927-3e8a64e6863f ER - TY - JOUR T1 - Detection of transcriptional triggers in the dynamics of microbial growth: application to the respiratory-versatile bacterium Shewanella oneidensis JF - Nucleic Acids Research, Volume 40, Issue 15, August 2012, Pages 7132-7149 Y1 - 2012 A1 - Q Beg A1 - Mattia Zampieri A1 - N Klitgord A1 - S Collins A1 - M Serres A1 - Daniel Segrè A1 - Claudio Altafini AB - 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 PB - SISSA UR - http://hdl.handle.net/1963/6506 U1 - 6452 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Gamma-convergence and H-convergence of linear elliptic operators JF - Journal de Mathématiques Pures et Appliquées, Available online 12 September 2012 Y1 - 2012 A1 - Nadia Ansini A1 - Gianni Dal Maso A1 - Caterina Ida Zeppieri KW - Linear elliptic operators PB - Elsevier UR - http://hdl.handle.net/1963/5878 U1 - 5746 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - On the genus two free energies for semisimple Frobenius manifolds JF - Russian Journal of Mathematical Physics. Volume 19, Issue 3, September 2012, Pages 273-298 Y1 - 2012 A1 - Boris Dubrovin A1 - Si-Qi Liu A1 - Youjin Zhang AB - 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. PB - SISSA UR - http://hdl.handle.net/1963/6464 N1 - 36 pages, 3 figures U1 - 6411 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - JOUR T1 - Pairs of positive periodic solutions of second order nonlinear equations with indefinite weight JF - Journal of Differential Equations Y1 - 2012 A1 - Alberto Boscaggin A1 - Fabio Zanolin KW - Critical points KW - Necessary conditions KW - Pairs of positive solutions KW - Periodic solutions AB -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.

VL - 252 UR - http://www.sciencedirect.com/science/article/pii/S0022039611003895 ER - TY - JOUR T1 - Positive periodic solutions of second order nonlinear equations with indefinite weight: Multiplicity results and complex dynamics JF - Journal of Differential Equations Y1 - 2012 A1 - Alberto Boscaggin A1 - Fabio Zanolin KW - Complex dynamics KW - Poincaré map KW - Positive periodic solutions KW - Subharmonics AB -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.

VL - 252 UR - http://www.sciencedirect.com/science/article/pii/S0022039611003883 ER - TY - JOUR T1 - Predicting and characterizing selective multiple drug treatments for metabolic diseases and cancer. JF - BMC Systems Biology. 29 August 2012, Page 115 Y1 - 2012 A1 - Giuseppe Facchetti A1 - Claudio Altafini A1 - Mattia Zampieri AB - 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. PB - BioMed Central UR - http://hdl.handle.net/1963/6515 U1 - 6450 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Adaptation as a genome-wide autoregulatory principle in the stress response of yeast. JF - IET systems biology. 2011 Jul; 5(4):269-79 Y1 - 2011 A1 - F Eduati A1 - B Di Camillo A1 - G Toffolo A1 - Claudio Altafini A1 - Giovanna De Palo A1 - Mattia Zampieri AB - 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. PB - The Institution of Engineering and Technology UR - http://hdl.handle.net/1963/5106 U1 - 4922 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - RPRT T1 - Compactness by maximality Y1 - 2011 A1 - Sandro Zagatti AB - 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 $a$. UR - http://preprints.sissa.it/handle/1963/35317 U1 - 35626 U2 - Mathematics U4 - 1 ER - TY - RPRT T1 - D-branes, surface operators, and ADHM quiver representations Y1 - 2011 A1 - Ugo Bruzzo A1 - Duiliu-Emanuel Diaconescu A1 - M. Yardim A1 - G. Pan A1 - Yi Zhang A1 - Chuang Wu-yen AB - 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. PB - SISSA UR - http://hdl.handle.net/1963/4133 N1 - 45 pages, v2: minor corrections U1 - 3873 U2 - Mathematics U3 - Mathematical Physics U4 - -1 ER - TY - JOUR T1 - An Integro-Extremization Approach for Non Coercive and Evolution Hamilton-Jacobi Equations JF - Journal of Convex Analysis 18 (2011) 1141-1170 Y1 - 2011 A1 - Sandro Zagatti AB - 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 $F$. PB - Heldermann Verlag UR - http://hdl.handle.net/1963/5538 U1 - 5375 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Linearly degenerate Hamiltonian PDEs and a new class of solutions to the WDVV associativity equations JF - Functional Analysis and Its Applications. Volume 45, Issue 4, December 2011, Pages 278-290 Y1 - 2011 A1 - Boris Dubrovin A1 - M.V. Pavlov A1 - Sergei A. Zykov KW - Frobenius manifold AB - 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. PB - Springer UR - http://hdl.handle.net/1963/6430 U1 - 6367 U2 - Mathematics U4 - -1 ER - TY - JOUR T1 - The Liouville side of the vortex JF - JHEP 09(2011)096 Y1 - 2011 A1 - Giulio Bonelli A1 - Alessandro Tanzini A1 - Jian Zhao AB - 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. PB - SISSA UR - http://hdl.handle.net/1963/4304 N1 - 25pages,11figures U1 - 4019 U2 - Physics U3 - Elementary Particle Theory U4 - -1 ER - TY - JOUR T1 - A system-level approach for deciphering the transcriptional response to prion infection JF - Bioinformatics (Oxford, England). 2011 Dec; 27(24):3407-14 Y1 - 2011 A1 - Mattia Zampieri A1 - Giuseppe Legname A1 - Daniel Segrè A1 - Claudio Altafini AB - 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. PB - Oxford University Press UR - http://hdl.handle.net/1963/5745 U1 - 5600 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Homogenization of fiber reinforced brittle material: the intermediate case JF - Adv. Calc. Var. 3 (2010) 345-370 Y1 - 2010 A1 - Gianni Dal Maso A1 - Caterina Ida Zeppieri AB - 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. PB - Walter de Gruyter UR - http://hdl.handle.net/1963/3607 U1 - 694 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - 1D periodic potentials with gaps vanishing at k=0 JF - Mem. Differential Equations Math. Phys. 47 (2009) 133-158 Y1 - 2009 A1 - Alessandro Michelangeli A1 - Osvaldo Zagordi AB - 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 Schrödinger equation (Hill\\\'s equation). This result is placed in the perspective of the previous related results available in the literature. UR - http://hdl.handle.net/1963/1818 U1 - 2396 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Differential geometry of curves in Lagrange Grassmannians with given Young diagram JF - Differential Geom. Appl. 27 (2009) 723-742 Y1 - 2009 A1 - Igor Zelenko A1 - Li Chengbo AB - 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. PB - Elsevier UR - http://hdl.handle.net/1963/3819 U1 - 508 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Discrete-to-continuum limits for strain-alignment-coupled systems: Magnetostrictive solids, ferroelectric crystals and nematic elastomers JF - Netw. Heterog. Media 4 (2009) 667-708 Y1 - 2009 A1 - Marco Cicalese A1 - Antonio DeSimone A1 - Caterina Ida Zeppieri AB - 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. PB - American Institute of Mathematical Sciences UR - http://hdl.handle.net/1963/3788 U1 - 538 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Gauged Laplacians on quantum Hopf bundles JF - Comm. Math. Phys. 287 (2009) 179-209 Y1 - 2009 A1 - Giovanni Landi A1 - Cesare Reina A1 - Alessandro Zampini AB - 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 `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. PB - Springer UR - http://hdl.handle.net/1963/3540 U1 - 1161 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Investigating the Conformational Stability of Prion Strains through a Kinetic Replication Model JF - PLoS Comput Biol 2009;5(7): e1000420 Y1 - 2009 A1 - Mattia Zampieri A1 - Giuseppe Legname A1 - Claudio Altafini AB - 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. PB - PLoS UR - http://hdl.handle.net/1963/3989 U1 - 413 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - RPRT T1 - Jacobi Equations and Comparison Theorems for Corank 1 Sub-Riemannian structures with symmetries Y1 - 2009 A1 - Li Chengbo A1 - Igor Zelenko AB - 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. UR - http://hdl.handle.net/1963/3736 U1 - 581 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - mRNA stability and the unfolding of gene expression in the long-period yeast metabolic cycle JF - BMC Systems Biology (2009) 3:18 Y1 - 2009 A1 - Nicola Soranzo A1 - Mattia Zampieri A1 - Lorenzo Farina A1 - Claudio Altafini AB - 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. PB - BioMed Central UR - http://hdl.handle.net/1963/3630 U1 - 674 U2 - Physics U3 - Statistical and Biological Physics ER - TY - JOUR T1 - Topological branes, p-algebras and generalized Nahm equations JF - Phys. Lett. B 672 (2009) 390-395 Y1 - 2009 A1 - Giulio Bonelli A1 - Alessandro Tanzini A1 - Maxim Zabzine AB - 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 UR - http://hdl.handle.net/1963/2702 U1 - 1398 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - On viscosity solutions of Hamilton-Jacobi equations JF - Trans. Amer. Math. Soc. 361 (2009) 41-59 Y1 - 2009 A1 - Sandro Zagatti AB - 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. PB - American Mathematical Society UR - http://hdl.handle.net/1963/3420 U1 - 915 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Discerning static and causal interactions in genome-wide reverse engineering problems JF - Bioinformatics 24 (2008) 1510-1515 Y1 - 2008 A1 - Mattia Zampieri A1 - Nicola Soranzo A1 - Claudio Altafini AB - 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. UR - http://hdl.handle.net/1963/2757 U1 - 1943 U2 - Physics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Minimization of non quasiconvex functionals by integro-extremization method JF - Discrete Contin. Dyn. Syst. 21 (2008) 625-641 Y1 - 2008 A1 - Sandro Zagatti UR - http://hdl.handle.net/1963/2761 U1 - 1939 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Minimizers of non convex scalar functionals and viscosity solutions of Hamilton-Jacobi equations JF - Calc. Var. Partial Differential Equations 31 (2008) 511-519 Y1 - 2008 A1 - Sandro Zagatti UR - http://hdl.handle.net/1963/2760 U1 - 1940 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Origin of Co-Expression Patterns in E.coli and S.cerevisiae Emerging from Reverse Engineering Algorithms JF - PLoS ONE 3 (2008) e2981 Y1 - 2008 A1 - Mattia Zampieri A1 - Nicola Soranzo A1 - Daniele Bianchini A1 - Claudio Altafini AB - 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. UR - http://hdl.handle.net/1963/2722 U1 - 1379 U2 - Physics U3 - Functional Analysis and Applications ER - TY - RPRT T1 - Computing Amplitudes in topological M-theory Y1 - 2007 A1 - Giulio Bonelli A1 - Alessandro Tanzini A1 - Maxim Zabzine AB - 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. JF - JHEP 03 (2007) 023 UR - http://hdl.handle.net/1963/1901 U1 - 2335 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - On feedback classification of control-affine systems with one and two-dimensional inputs JF - SIAM J. Control Optim. 46 (2007) 1431-1460 Y1 - 2007 A1 - Andrei A. Agrachev A1 - Igor Zelenko AB - 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. UR - http://hdl.handle.net/1963/2186 U1 - 2058 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Parametrized curves in Lagrange Grassmannians JF - C. R. Math. 345 (2007) 647-652 Y1 - 2007 A1 - Igor Zelenko A1 - Li Chengbo UR - http://hdl.handle.net/1963/2560 U1 - 1559 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - RPRT T1 - Quasistatic crack growth for a cohesive zone model with prescribed crack path Y1 - 2007 A1 - Gianni Dal Maso A1 - Chiara Zanini AB - 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. JF - Proc. Roy. Soc. Edinburgh Sect. A 137 (2007) 253-279 UR - http://hdl.handle.net/1963/1686 U1 - 2447 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Solutions of vectorial Hamilton-Jacobi equations and minimizers of nonquasiconvex functionals JF - J. Math. Anal. Appl. 335 (2007) 1143-1160 Y1 - 2007 A1 - Sandro Zagatti AB - 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. UR - http://hdl.handle.net/1963/2763 U1 - 1937 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Uniqueness and continuous dependence on boundary data for integro-extremal minimizers of the functional of the gradient JF - J. Convex Anal. 14 (2007) 705-727 Y1 - 2007 A1 - Sandro Zagatti AB - 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. UR - http://hdl.handle.net/1963/2762 U1 - 1938 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - RPRT T1 - An artificial viscosity approach to quasistatic crack growth Y1 - 2006 A1 - Rodica Toader A1 - Chiara Zanini AB - 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. UR - http://hdl.handle.net/1963/1850 U1 - 2367 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - RPRT T1 - A Canonical Frame for Nonholonomic Rank Two Distributions of Maximal Class Y1 - 2006 A1 - Boris Doubrov A1 - Igor Zelenko AB - 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. JF - C. R. Math. Acad. Sci. Paris 342 (2006) 589-594 UR - http://hdl.handle.net/1963/1712 U1 - 2439 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Fundamental form and Cartan tensor of (2,5)-distributions coincide JF - J. Dyn. Control Syst. 12 (2006) 247-276 Y1 - 2006 A1 - Igor Zelenko AB - 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 ``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. UR - http://hdl.handle.net/1963/2187 U1 - 2057 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - On geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on distributions of corank 1 JF - J. Math. Sci. 135 (2006) 3168-3194 Y1 - 2006 A1 - Igor Zelenko AB - 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. UR - http://hdl.handle.net/1963/2205 U1 - 2039 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - RPRT T1 - On topological M-theory Y1 - 2006 A1 - Giulio Bonelli A1 - Alessandro Tanzini A1 - Maxim Zabzine AB - 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. JF - Adv. Theor. Math. Phys. 10 (2006) 239-260 UR - http://hdl.handle.net/1963/1765 U1 - 2779 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - On variational approach to differential invariants of rank two distributions JF - Differential Geom. Appl. 24 (2006) 235-259 Y1 - 2006 A1 - Igor Zelenko AB - 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. UR - http://hdl.handle.net/1963/2188 U1 - 2056 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - CHAP T1 - Complete systems of invariants for rank 1 curves in Lagrange Grassmannians T2 - Differential geometry and its applications, 367-382, Matfyzpress, Prague, 2005 Y1 - 2005 A1 - Igor Zelenko AB - 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. JF - Differential geometry and its applications, 367-382, Matfyzpress, Prague, 2005 UR - http://hdl.handle.net/1963/2310 N1 - Proceedings of 9th Conference on Differential Geometry and its Applications, Prague 2004 U1 - 1706 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - On curvatures and focal points of distributions of dynamical Lagrangian distributions and their reductions by first integrals JF - J. Dyn. Control Syst. 11 (2005) 297-327 Y1 - 2005 A1 - Andrei A. Agrachev A1 - Natalia N. Chtcherbakova A1 - Igor Zelenko AB - 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. UR - http://hdl.handle.net/1963/2254 U1 - 1993 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - On the Minimum Problem for Nonconvex Scalar Functionals JF - SIAM J. Math. Anal. 37 (2005) 982-995 Y1 - 2005 A1 - Sandro Zagatti AB - 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. UR - http://hdl.handle.net/1963/2764 U1 - 1936 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - RPRT T1 - Nonlinear Schrödinger Equations with vanishing and decaying potentials Y1 - 2005 A1 - Antonio Ambrosetti A1 - Wang Zhi-Qiang JF - Differential Integral Equations 18 (2005), no. 12, 1321-1332 UR - http://hdl.handle.net/1963/1760 U1 - 2784 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Positive solutions to a class of quasilinear elliptic equations on R JF - Discrete Contin.Dyn.Syst. 9 (2003), no.1, 55-68 Y1 - 2003 A1 - Antonio Ambrosetti A1 - Wang Zhi-Qiang AB - We discuss the existence of positive solutions of perturbation to a class of quasilinear elliptic equations on R. PB - American Institute of Mathematical Sciences UR - http://hdl.handle.net/1963/1628 U1 - 2490 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Geometry of Jacobi Curves I JF - J. Dynam. Control Systems 8 (2002) 93-140 Y1 - 2002 A1 - Andrei A. Agrachev A1 - Igor Zelenko AB - 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\\\". PB - Springer UR - http://hdl.handle.net/1963/3110 U1 - 1223 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Geometry of Jacobi curves II JF - J. Dynam. Control Systems 8 (2002), no. 2, 167--215 Y1 - 2002 A1 - Andrei A. Agrachev A1 - Igor Zelenko PB - SISSA Library UR - http://hdl.handle.net/1963/1589 U1 - 2529 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A note on the super Krichever map JF - J. Geom. Phys. 37 (2001), no. 1-2, 169-181 Y1 - 2001 A1 - Gregorio Falqui A1 - Cesare Reina A1 - Alessandro Zampa AB - 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`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. PB - SISSA Library UR - http://hdl.handle.net/1963/1494 U1 - 2669 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - 3D superconformal theories from Sasakian seven-manifolds: new nontrivial evidences for AdS_4/CFT_3 JF - Nucl.Phys. B577 (2000) 547-608 Y1 - 2000 A1 - Davide Fabbri A1 - Pietro Fré A1 - Leonardo Gualtieri A1 - Cesare Reina A1 - Alessandro Tomasiello A1 - Alberto Zaffaroni A1 - Alessandro Zampa PB - SISSA Library UR - http://hdl.handle.net/1963/1327 U1 - 3128 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - A(SLq(2)) at roots of unity is a free module over A(SL(2)) JF - Lett. Math. Phys., 2000, 52, 339 Y1 - 2000 A1 - Ludwik Dabrowski A1 - Cesare Reina A1 - Alessandro Zampa PB - SISSA Library UR - http://hdl.handle.net/1963/1500 U1 - 2663 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - A bi-Hamiltonian theory for stationary KDV flows and their separability JF - Regul. Chaotic Dyn. 5 (2000), no. 1, 33-52 Y1 - 2000 A1 - Gregorio Falqui A1 - Franco Magri A1 - Marco Pedroni A1 - Jorge P. Zubelli PB - SISSA Library UR - http://hdl.handle.net/1963/1352 U1 - 3103 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - An elementary approach to the polynomial $\\\\tau$-functions of the KP Hierarchy JF - Theor. Math. Phys. 122 (2000) 17-28 Y1 - 2000 A1 - Gregorio Falqui A1 - Franco Magri A1 - Marco Pedroni A1 - Jorge P. Zubelli AB - 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. PB - Springer UR - http://hdl.handle.net/1963/3223 U1 - 1078 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Minimization of functionals of the gradient by Baire's theorem JF - SIAM J. Control Optim. 38 (2000) 384-399 Y1 - 2000 A1 - Sandro Zagatti AB -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 $u_0\in W_0^{1,p}(\Omega,\mathbb{R})$.

PB - SIAM UR - http://hdl.handle.net/1963/3511 U1 - 753 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - CHAP T1 - Principal invariants of Jacobi curves T2 - 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. Y1 - 2000 A1 - Andrei A. Agrachev A1 - Igor Zelenko AB - 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. JF - 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. PB - Springer UR - http://hdl.handle.net/1963/3825 U1 - 502 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Super KP equations and Darboux transformations: another perspective on the Jacobian super KP hierarchy JF - J. Geom. Phys. 35 (2000), no. 2-3, 239-272 Y1 - 2000 A1 - Gregorio Falqui A1 - Cesare Reina A1 - Alessandro Zampa PB - SISSA Library UR - http://hdl.handle.net/1963/1367 U1 - 3088 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Enhanced gauge symmetries on elliptic K3 JF - Phys.Lett. B452 (1999) 244-250 Y1 - 1999 A1 - Loriano Bonora A1 - Cesare Reina A1 - Alessandro Zampa AB - 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. PB - Elsevier UR - http://hdl.handle.net/1963/3366 U1 - 964 U2 - Physics U3 - Elementary Particle Theory ER - TY - JOUR T1 - On the Dirichlet problem for vectorial Hamilton-Jacobi equations JF - SIAM J. Math. Anal. 29 (1998) 1481-1491 Y1 - 1998 A1 - Sandro Zagatti AB - 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. PB - SIAM UR - http://hdl.handle.net/1963/3512 U1 - 752 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Extended affine Weyl groups and Frobenius manifolds JF - Compositio Mathematica. Volume 111, Issue 2, 1998, Pages 167-219 Y1 - 1998 A1 - Boris Dubrovin A1 - Youjin Zhang PB - Kluwer UR - http://hdl.handle.net/1963/6486 U1 - 6424 U2 - Mathematics U4 - 1 U5 - MAT/03 GEOMETRIA ER - TY - JOUR T1 - Krichever maps, Faà di Bruno polynomials, and cohomology in KP theory JF - Lett. Math. Phys. 42 (1997) 349-361 Y1 - 1997 A1 - Gregorio Falqui A1 - Cesare Reina A1 - Alessandro Zampa AB - 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. PB - Springer UR - http://hdl.handle.net/1963/3539 U1 - 1162 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - An existence result in a problem of the vectorial case of the calculus of variations Y1 - 1995 A1 - Arrigo Cellina A1 - Sandro Zagatti AB - SIAM J. Control Optim. 33 (1995) 960-970 PB - SIAM UR - http://hdl.handle.net/1963/3513 U1 - 751 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - CHAP T1 - Quantum homogeneous spaces at roots of unity T2 - 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 Y1 - 1995 A1 - Cesare Reina A1 - Alessandro Zampa JF - 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 PB - SISSA Library UR - http://hdl.handle.net/1963/1022 U1 - 2834 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - A version of Olech\\\'s lemma in a problem of the calculus of variations JF - SIAM J. Control Optim. 32 (1994) 1114-1127 Y1 - 1994 A1 - Arrigo Cellina A1 - Sandro Zagatti AB - 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. PB - SIAM UR - http://hdl.handle.net/1963/3514 U1 - 750 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - THES T1 - Some Problems in the Calculus of the Variations Y1 - 1992 A1 - Sandro Zagatti KW - Calculus of variations PB - SISSA UR - http://hdl.handle.net/1963/5428 U1 - 5260 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER -