We prove local in time well-posedness for a large class of quasilinear Hamiltonian, or parity preserving, Schrödinger equations on the circle. After a paralinearization of the equation, we perform several paradifferential changes of coordinates in order to transform the system into a paradifferential one with symbols which, at the positive order, are constant and purely imaginary. This allows to obtain a priori energy estimates on the Sobolev norms of the solutions.

VL - 36 UR - http://www.sciencedirect.com/science/article/pii/S0294144918300428 ER - TY - JOUR T1 - A Localized Reduced-Order Modeling Approach for PDEs with Bifurcating Solutions JF - Computer Methods in Applied Mechanics and Engineering Y1 - 2019 A1 - Hess, Martin A1 - Alla, Alessandro A1 - Quaini, Annalisa A1 - Rozza, Gianluigi A1 - Gunzburger, Max AB -Reduced-order modeling (ROM) commonly refers to the construction, based on a few solutions (referred to as snapshots) of an expensive discretized partial differential equation (PDE), and the subsequent application of low-dimensional discretizations of partial differential equations (PDEs) that can be used to more efficiently treat problems in control and optimization, uncertainty quantification, and other settings that require multiple approximate PDE solutions. In this work, a ROM is developed and tested for the treatment of nonlinear PDEs whose solutions bifurcate as input parameter values change. In such cases, the parameter domain can be subdivided into subregions, each of which corresponds to a different branch of solutions. Popular ROM approaches such as proper orthogonal decomposition (POD), results in a global low-dimensional basis that does no respect not take advantage of the often large differences in the PDE solutions corresponding to different subregions. Instead, in the new method, the k-means algorithm is used to cluster snapshots so that within cluster snapshots are similar to each other and are dissimilar to those in other clusters. This is followed by the construction of local POD bases, one for each cluster. The method also can detect which cluster a new parameter point belongs to, after which the local basis corresponding to that cluster is used to determine a ROM approximation. Numerical experiments show the effectiveness of the method both for problems for which bifurcation cause continuous and discontinuous changes in the solution of the PDE.

VL - 351 UR - https://arxiv.org/abs/1807.08851 ER - TY - RPRT T1 - Local moduli of semisimple Frobenius coalescent structures Y1 - 2018 A1 - Giordano Cotti A1 - Boris Dubrovin A1 - Davide Guzzetti AB -There is a conjectural relation, formulated by the second author, between the enumerative geometry of a wide class of smooth projective varieties and their derived category of coherent sheaves. In particular, there is an increasing interest for an explicit description of certain local invariants, called monodromy data, of semisimple quantum cohomologies in terms of characteristic classes of exceptional collections in the derived categories. Being intentioned to address this problem, which, to our opinion, is still not well understood, we have realized that some issues in the theory of Frobenius manifolds need to be preliminarily clarified, and that an extension of the theory itself is necessary, in view of the fact that quantum cohomologies of certain classes of homogeneous spaces may show a coalescence phenomenon.

PB - SISSA UR - http://preprints.sissa.it/handle/1963/35304 U1 - 35610 U2 - Mathematics U4 - 1 U5 - MAT/03 ER - TY - RPRT T1 - Long time existence for fully nonlinear NLS with small Cauchy data on the circle Y1 - 2018 A1 - Feola Roberto A1 - Felice Iandoli ER - TY - JOUR T1 - Lp-Boundedness of Wave Operators for the Three-Dimensional Multi-Centre Point Interaction JF - Annales Henri Poincaré Y1 - 2018 A1 - Gianfausto Dell'Antonio A1 - Alessandro Michelangeli A1 - Raffaele Scandone A1 - Kenji Yajima AB -We prove that, for arbitrary centres and strengths, the wave operators for three-dimensional Schrödinger operators with multi-centre local point interactions are bounded in Lp(R3)for 1<p<3 and unbounded otherwise.

VL - 19 UR - https://doi.org/10.1007/s00023-017-0628-4 ER - TY - RPRT T1 - A Lagrangian approach for scalar multi-d conservation laws Y1 - 2017 A1 - Stefano Bianchini A1 - Paolo Bonicatto A1 - Elio Marconi UR - http://preprints.sissa.it/handle/1963/35290 U1 - 35596 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Lagrangian representations for linear and nonlinear transport JF - Contemporary Mathematics. Fundamental Directions Y1 - 2017 A1 - Stefano Bianchini A1 - Paolo Bonicatto A1 - Elio Marconi AB -In this note we present a unifying approach for two classes of first order partial differential equations: we introduce the notion of Lagrangian representation in the settings of continuity equation and scalar conservation laws. This yields, on the one hand, the uniqueness of weak solutions to transport equation driven by a two dimensional BV nearly incompressible vector field. On the other hand, it is proved that the entropy dissipation measure for scalar conservation laws in one space dimension is concentrated on countably many Lipschitz curves.

PB - Peoples' Friendship University of Russia VL - 63 UR - http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=cmfd&paperid=327&option_lang=eng ER - TY - JOUR T1 - Limit of viscous dynamic processes in delamination as the viscosity and inertia vanish JF - ESAIM: COCV Y1 - 2017 A1 - Riccardo Scala AB -We introduce a model of dynamic evolution of a delaminated visco-elastic body with viscous adhesive. We prove the existence of solutions of the corresponding system of PDEs and then study the behavior of such solutions when the data of the problem vary slowly. We prove that a rescaled version of the dynamic evolutions converge to a “local” quasistatic evolution, which is an evolution satisfying an energy inequality and a momentum balance at all times. In the one-dimensional case we give a more detailed description of the limit evolution and we show that it behaves in a very similar way to the limit of the solutions of the dynamic model in [T. Roubicek, SIAM J. Math. Anal. 45 (2013) 101–126], where no viscosity in the adhesive is taken into account.

VL - 23 UR - https://doi.org/10.1051/cocv/2016006 ER - TY - RPRT T1 - Linear hyperbolic systems in domains with growing cracks Y1 - 2017 A1 - Maicol Caponi AB - We consider the hyperbolic system $\ddot u-{\rm div}\,(\mathbb A\nabla u)=f$ in the time varying cracked domain $\Omega\setminus\Gamma_t$, where the set $\Omega\subset\mathbb R^d$ is open, bounded, and with Lipschitz boundary, the cracks $\Gamma_t$, $t\in[0,T]$, are closed subsets of $\overline\Omega$, increasing with respect to inclusion, and $u(t):\Omega\setminus\Gamma_t\to\mathbb R^d$ for every $t\in[0,T]$. We assume the existence of suitable regular changes of variables, which reduce our problem to the transformed system $\ddot v-{\rm div}\,(\mathbb B\nabla v)+\mathbf a\nabla v -2\nabla\dot vb=g$ on the fixed domain $\Omega\setminus\Gamma_0$. Under these assumptions, we obtain existence and uniqueness of weak solutions for these two problems. Moreover, we show an energy equality for the functions $v$, which allows us to prove a continuous dependence result for both systems. UR - http://urania.sissa.it/xmlui/handle/1963/35271 U1 - 35577 U2 - Mathematics U4 - 1 ER - TY - RPRT T1 - Linearisation of multiwell energies Y1 - 2017 A1 - Roberto Alicandro A1 - Gianni Dal Maso A1 - Giuliano Lazzaroni A1 - Mariapia Palombaro AB - Linear elasticity can be rigorously derived from finite elasticity under the assumption of small loadings in terms of Gamma-convergence. This was first done in the case of one-well energies with super-quadratic growth and later generalised to different settings, in particular to the case of multi-well energies where the distance between the wells is very small (comparable to the size of the load). In this paper we study the case when the distance between the wells is independent of the size of the load. In this context linear elasticity can be derived by adding to the multi-well energy a singular higher order term which penalises jumps from one well to another. The size of the singular term has to satisfy certain scaling assumptions whose optimality is shown in most of the cases. Finally, the derivation of linear elasticty from a two-well discrete model is provided, showing that the role of the singular perturbation term is played in this setting by interactions beyond nearest neighbours. UR - http://preprints.sissa.it/handle/1963/35288 U1 - 35594 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Lower semicontinuity of a class of integral functionals on the space of functions of bounded deformation JF - Advances in Calculus of Variations Y1 - 2017 A1 - Gianni Dal Maso A1 - Gianluca Orlando A1 - Rodica Toader AB -We study the lower semicontinuity of some free discontinuity functionals with linear growth defined on the space of functions with bounded deformation. The volume term is convex and depends only on the Euclidean norm of the symmetrized gradient. We introduce a suitable class of surface terms, which make the functional lower semicontinuous with respect to $L^1$ convergence.

PB - De Gruyter VL - 10 ER - TY - JOUR T1 - A lower semicontinuity result for a free discontinuity functional with a boundary term JF - Journal de Mathématiques Pures et Appliquées Y1 - 2017 A1 - Stefano Almi A1 - Gianni Dal Maso A1 - Rodica Toader AB -We study the lower semicontinuity in $GSBV^{p}(\Omega;\mathbb{R}^{m})$ of a free discontinuity functional $\mathcal{F}(u)$ that can be written as the sum of a crack term, depending only on the jump set $S_{u}$, and of a boundary term, depending on the trace of $u$ on $\partial\Omega$. We give sufficient conditions on the integrands for the lower semicontinuity of $\mathcal{F}$. Moreover, we prove a relaxation result, which shows that, if these conditions are not satisfied, the lower semicontinuous envelope of $\mathcal{F}$ can be represented by the sum of two integrals on $S_{u}$ and $\partial\Omega$, respectively.

VL - 108 UR - http://hdl.handle.net/20.500.11767/15979 IS - 6 U1 - 34731 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - RPRT T1 - Large KAM tori for perturbations of the dNLS equation Y1 - 2016 A1 - Massimiliano Berti A1 - Thomas Kappeler A1 - Riccardo Montalto AB - We prove that small, semi-linear Hamiltonian perturbations of the defocusing nonlinear Schr\"odinger (dNLS) equation on the circle have an abundance of invariant tori of any size and (finite) dimension which support quasi-periodic solutions. When compared with previous results the novelty consists in considering perturbations which do not satisfy any symmetry condition (they may depend on x in an arbitrary way) and need not be analytic. The main difficulty is posed by pairs of almost resonant dNLS frequencies. The proof is based on the integrability of the dNLS equation, in particular the fact that the nonlinear part of the Birkhoff coordinates is one smoothing. We implement a Newton-Nash-Moser iteration scheme to construct the invariant tori. The key point is the reduction of linearized operators, coming up in the iteration scheme, to 2×2 block diagonal ones with constant coefficients together with sharp asymptotic estimates of their eigenvalues. UR - http://preprints.sissa.it/handle/1963/35284 U1 - 35589 U2 - Mathematics ER - TY - JOUR T1 - LinearOperator – a generic, high-level expression syntax for linear algebra JF - COMPUTERS & MATHEMATICS WITH APPLICATIONS Y1 - 2016 A1 - Matthias Maier A1 - Mauro Bardelloni A1 - Luca Heltai VL - 72 ER - TY - JOUR T1 - Liquid crystal elastomer strips as soft crawlers JF - Journal of the Mechanics and Physics of Solids Y1 - 2015 A1 - Antonio DeSimone A1 - Paolo Gidoni A1 - Giovanni Noselli KW - Crawling motility KW - Directional surfaces KW - Frictional interactions KW - Liquid crystal elastomers KW - Soft biomimetic robots AB -In this paper, we speculate on a possible application of Liquid Crystal Elastomers to the field of soft robotics. In particular, we study a concept for limbless locomotion that is amenable to miniaturisation. For this purpose, we formulate and solve the evolution equations for a strip of nematic elastomer, subject to directional frictional interactions with a flat solid substrate, and cyclically actuated by a spatially uniform, time-periodic stimulus (e.g., temperature change). The presence of frictional forces that are sensitive to the direction of sliding transforms reciprocal, ‘breathing-like’ deformations into directed forward motion. We derive formulas quantifying this motion in the case of distributed friction, by solving a differential inclusion for the displacement field. The simpler case of concentrated frictional interactions at the two ends of the strip is also solved, in order to provide a benchmark to compare the continuously distributed case with a finite-dimensional benchmark. We also provide explicit formulas for the axial force along the crawler body.

VL - 84 UR - http://www.sciencedirect.com/science/article/pii/S0022509615300430 ER - TY - JOUR T1 - Laplace equation in a domain with a rectilinear crack: higher order derivatives of the energy with respect to the crack length Y1 - 2014 A1 - Gianni Dal Maso A1 - Gianluca Orlando A1 - Rodica Toader KW - cracked domains, energy release rate, higher order derivatives, asymptotic expansion of solutions AB -We consider the weak solution of the Laplace equation in a planar domain with a straight crack, prescribing a homogeneous Neumann condition on the crack and a nonhomogeneous Dirichlet condition on the rest of the boundary. For every k we express the k-th derivative of the energy with respect to the crack length in terms of a finite number of coefficients of the asymptotic expansion of the solution near the crack tip and of a finite number of other parameters, which only depend on the shape of the domain.

PB - SISSA UR - http://hdl.handle.net/1963/7271 U1 - 7316 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - CHAP T1 - Lecture notes on gradient flows and optimal transport Y1 - 2014 A1 - Sara Daneri A1 - Giuseppe Savarè AB - We present a short overview on the strongest variational formulation for gradient flows of geodesically λ-convex functionals in metric spaces, with applications to diffusion equations in Wasserstein spaces of probability measures. These notes are based on a series of lectures given by the second author for the Summer School "Optimal transportation: Theory and applications" in Grenoble during the week of June 22-26, 2009. PB - Cambridge University Press UR - http://urania.sissa.it/xmlui/handle/1963/35093 N1 - Book title: Optimal transportation U1 - 35348 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Legendre duality on hypersurfaces in Kähler manifolds Y1 - 2014 A1 - Vittorio Martino AB - We give a sufficient condition on real strictly Levi-convex hypersurfaces M, embedded in four-dimensional Kähler manifolds V , such that Legendre duality can be performed. We consider the contact form onM whose kernel is the restriction of the holomorphic tangent space of V and show that if there exists a Legendrian Killing vector field v, then the dual form β(̇) := d(v, ̇) is a contact form on M with the same orientation than theta. PB - Walter de Gruyter and Co. UR - http://urania.sissa.it/xmlui/handle/1963/34777 U1 - 34998 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Linearized plastic plate models as Γ-limits of 3D finite elastoplasticity JF - ESAIM: Control, Optimisation and Calculus of Variations Y1 - 2014 A1 - Elisa Davoli AB -The subject of this paper is the rigorous derivation of reduced models for a thin plate by means of $\Gamma$-convergence, in the framework of finite plasticity. Denoting by $\epsilon$ the thickness of the plate, we analyse the case where the scaling factor of the elasto-plastic energy per unit volume is of order $\epsilon^{2 \alpha -2}$, with $\alpha \geq 3$. According to the value of $\alpha$, partially or fully linearized models are deduced, which correspond, in the absence of plastic deformation, to the Von Kármán plate theory and the linearized plate theory.

PB - EDP Sciences VL - 20 ER - TY - JOUR T1 - Lipschitz continuous viscosity solutions for a class of fully nonlinear equations on lie groups Y1 - 2014 A1 - Vittorio Martino A1 - Annamaria Montanari AB - In this paper, we prove existence and uniqueness of Lipschitz continuous viscosity solutions for Dirichlet problems involving a class a fully non-linear operators on Lie groups. In particular, we consider the elementary symmetric functions of the eigenvalues of the Hessian built with left-invariant vector fields. PB - Springer UR - http://urania.sissa.it/xmlui/handle/1963/34699 U1 - 34910 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Local and global minimality results for a nonlocal isoperimetric problem on R^N JF - SIAM Journal on Mathematical Analysis Y1 - 2014 A1 - Marco Bonacini A1 - Riccardo Cristoferi KW - Nonlocal isoperimetric problem AB -We consider a nonlocal isoperimetric problem defined in the whole space R^N, whose nonlocal part is given by a Riesz potential with exponent $\alpha\in(0, N-1)$. We show that critical configurations with positive second variation are local minimizers and satisfy a quantitative inequality with respect to the L^1-norm. This criterion provides the existence of a (explicitly determined) critical threshold determining the interval of volumes for which the ball is a local minimizer, and allows to address several global minimality issues.

PB - SIAM Publications VL - 46 UR - http://hdl.handle.net/1963/6984 IS - 4 U1 - 6976 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - Local behavior of fractional p-minimizers Y1 - 2014 A1 - Agnese Di Castro A1 - Tuomo Kuusi A1 - Giampiero Palatucci KW - fractional Sobolev spaces AB -We extend the De Giorgi-Nash Moser theory to nonlocal, possibly degerate integro-differential operators

PB - SISSA U1 - 7301 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - On the Lp-differentiability of certain classes of functions Y1 - 2014 A1 - Giovanni Alberti A1 - Stefano Bianchini A1 - Gianluca Crippa AB - We prove the Lp-differentiability at almost every point for convolution products on ℝd of the form K*μ, where μ is bounded measure and K is a homogeneous kernel of degree 1-d. From this result we derive the Lp-differentiability for vector fields on R d whose curl and divergence are measures, and also for vector fields with bounded deformation. PB - European Mathematical Society UR - http://urania.sissa.it/xmlui/handle/1963/34695 U1 - 34909 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - JOUR T1 - Lipschitz Classification of Almost-Riemannian Distances on Compact Oriented Surfaces JF - Journal of Geometric Analysis Y1 - 2013 A1 - Ugo Boscain A1 - Grégoire Charlot A1 - Roberta Ghezzi A1 - Mario Sigalotti AB -Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. We consider the Carnot–Carathéodory distance canonically associated with an almost-Riemannian structure and study the problem of Lipschitz equivalence between two such distances on the same compact oriented surface. We analyze the generic case, allowing in particular for the presence of tangency points, i.e., points where two generators of the distribution and their Lie bracket are linearly dependent. The main result of the paper provides a characterization of the Lipschitz equivalence class of an almost-Riemannian distance in terms of a labeled graph associated with it.

VL - 23 UR - https://doi.org/10.1007/s12220-011-9262-4 ER - TY - JOUR T1 - Linear elasticity obtained from finite elasticity by Gamma-convergence under weak coerciveness conditions JF - Ann. Inst. H. Poincare Anal. Non Lineaire Y1 - 2012 A1 - Virginia Agostiniani A1 - Gianni Dal Maso A1 - Antonio DeSimone KW - Nonlinear elasticity AB -The energy functional of linear elasticity is obtained as G-limit of suitable rescalings of the energies of finite elasticity...

PB - Gauthier-Villars;Elsevier VL - 29 UR - http://hdl.handle.net/1963/4267 U1 - 3996 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - On localization in holomorphic equivariant cohomology JF - Central European Journal of Mathematics, Volume 10, Issue 4, August 2012, Pages 1442-1454 Y1 - 2012 A1 - Ugo Bruzzo A1 - Vladimir Rubtsov KW - Lie algebroids AB - We prove a localization formula for a "holomorphic equivariant cohomology" attached to the Atiyah algebroid of an equivariant holomorphic vector bundle. This generalizes Feng-Ma, Carrell-Liebermann, Baum-Bott and K. Liu's localization formulas. PB - Springer UR - http://hdl.handle.net/1963/6584 U1 - 6543 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - On the location of poles for the Ablowitz-Segur family of solutions to the second Painlevé equation JF - Nonlinearity Y1 - 2012 A1 - Marco Bertola VL - 25 UR - http://0-dx.doi.org.mercury.concordia.ca/10.1088/0951-7715/25/4/1179 ER - TY - JOUR T1 - Large Time Existence for Thin Vibrating Plates JF - Communication in Partial Differential Equations 36 (2011) 2062-2102 Y1 - 2011 A1 - Helmut Abels A1 - Maria Giovanna Mora A1 - Stefan Müller AB - We construct strong solutions for a nonlinear wave equation for a thin vibrating plate described by nonlinear elastodynamics. For sufficiently small thickness we obtain existence of strong solutions for large\\r\\ntimes under appropriate scaling of the initial values such that the limit system as h --> 0 is either the nonlinear von Karman plate equation or the linear fourth order Germain-Lagrange equation. In the case of the\\r\\nlinear Germain-Lagrange equation we even obtain a convergence rate of the three-dimensional solution to the solution of the two-dimensional linear plate equation. PB - Taylor & Francis UR - http://hdl.handle.net/1963/3755 U1 - 562 U2 - Mathematics U3 - Functional Analysis and Applications 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 - Lorentz Covariant k-Minkowski Spacetime JF - Phys. Rev. D 81 (2010) 125024 Y1 - 2010 A1 - Ludwik Dabrowski A1 - Michal Godlinski A1 - Gherardo Piacitelli AB - In recent years, different views on the interpretation of Lorentz covariance of non commuting coordinates were discussed. Here, by a general procedure, we construct the minimal canonical central covariantisation of the k-Minkowski spacetime. We then show that, though the usual k-Minkowski spacetime is covariant under deformed (or twisted) Lorentz action, the resulting framework is equivalent to taking a non covariant restriction of the covariantised model. We conclude with some general comments on the approach of deformed covariance. UR - http://hdl.handle.net/1963/3829 U1 - 498 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Low-Frequency Variations of Force Coefficients on Square Cylinders with Sharp and Rounded Corners JF - Journal of Structural Engineering Y1 - 2009 A1 - Andrea Mola A1 - Giancarlo Bordonaro A1 - Muhammad R. Hajj PB - American Society of Civil Engineers ({ASCE}) VL - 135 UR - https://doi.org/10.1061/(asce)st.1943-541x.0000034 ER - TY - JOUR T1 - Limit Time Optimal Syntheses for a control-affine system on S² JF - SIAM J. Control Optim. 47 (2008) 111-143 Y1 - 2008 A1 - Paolo Mason A1 - Rebecca Salmoni A1 - Ugo Boscain A1 - Yacine Chitour AB - For $\\\\alpha \\\\in ]0,\\\\pi/2[$, let $(\\\\Sigma)_\\\\alpha$ be the control system $\\\\dot{x}=(F+uG)x$, where $x$ belongs to the two-dimensional unit sphere $S^2$, $u\\\\in [-1,1]$, and $F,G$ are $3\\\\times3$ skew-symmetric matrices generating rotations with perpendicular axes and of respective norms $\\\\cos(\\\\alpha)$ and $\\\\sin(\\\\alpha)$. In this paper, we study the time optimal synthesis (TOS) from the north pole $(0,0,1)^T$ associated to $(\\\\Sigma)_\\\\alpha$, as the parameter $\\\\alpha$ tends to zero; this problem is motivated by specific issues in the control of quantum systems. We first prove that the TOS is characterized by a \\\"two-snakes\\\" configuration on the whole $S^2$, except for a neighborhood $U_\\\\alpha$ of the south pole $(0,0,-1)^T$ of diameter at most ${\\\\cal O}(\\\\alpha)$. We next show that, inside $U_\\\\alpha$, the TOS depends on the relationship between $r(\\\\alpha):=\\\\pi/2\\\\alpha-[\\\\pi/2\\\\alpha]$ and $\\\\alpha$. More precisely, we characterize three main relationships by considering sequences $(\\\\alpha_k)_{k\\\\geq 0}$ satisfying (a) $r(\\\\alpha_k)=\\\\bar{r}$, (b) $r(\\\\alpha_k)=C\\\\alpha_k$, and (c) $r(\\\\alpha_k)=0$, where $\\\\bar{r}\\\\in (0,1)$ and $C>0$. In each case, we describe the TOS and provide, after a suitable rescaling, the limiting behavior, as $\\\\alpha$ tends to zero, of the corresponding TOS inside $U_\\\\alpha$. UR - http://hdl.handle.net/1963/1862 U1 - 2360 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - On the Logarithmic Asymptotics of the Sixth Painleve\' Equation (Summer 2007) JF - J.Phys.A: Math.Theor. 41,(2008), 205201-205247 Y1 - 2008 A1 - Davide Guzzetti AB - We study the solutions of the sixth Painlev\'e equation with a logarithmic\r\nasymptotic behavior at a critical point. We compute the monodromy group\r\nassociated to the solutions by the method of monodromy preserving deformations\r\nand we characterize the asymptotic behavior in terms of the monodromy itself. PB - SISSA UR - http://hdl.handle.net/1963/6521 N1 - This paper appeared as a preprint in August 2007. It is published in Journal of Physics A: Mathematical and Theoretical, Volume 41, Issue 20, 6 May 2008, p. 205201-205247. It was on the archive in January 2008 (arXiv:0801.1157). This version does not differ from the published one except for two facts: 1)the addition of subsection 8.2, which proves that tr(M0Mx) = −2 for solutions y(x) ∼ a (ln x)n , n = 1, 2, x → 0. 2). The title of the journal article is : The logarithmic asymptotics of the sixth Painlevé equation U1 - 6473 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - JOUR T1 - Luther-Emery Phase and Atomic-Density Waves in a Trapped Fermion Gas JF - Phys. Rev. Lett. 98 (2007) 030404 Y1 - 2007 A1 - Gao Xianlong A1 - Matteo Rizzi A1 - Marco Polini A1 - Rosario Fazio A1 - Mario P. Tosi A1 - Vivaldo L. Jr. Campo A1 - Klaus Capelle AB -The Luther-Emery liquid is a state of matter that is predicted to occur in one-dimensional systems of interacting fermions and is characterized by a gapless charge spectrum and a gapped spin spectrum. In this Letter we discuss a realization of the Luther-Emery phase in a trapped cold-atom gas. We study by means of the density-matrix renormalization-group technique a two-component atomic Fermi gas with attractive interactions subject to parabolic trapping inside an optical lattice. We demonstrate how this system exhibits compound phases characterized by the coexistence of spin pairing and atomic-density waves. A smooth crossover occurs with increasing magnitude of the atom-atom attraction to a state in which tightly bound spin-singlet dimers occupy the center of the trap. The existence of atomic-density waves could be detected in the elastic contribution to the light-scattering diffraction pattern.

UR - http://hdl.handle.net/1963/2056 U1 - 2140 U2 - Physics U3 - Condensed Matter Theory ER - TY - RPRT T1 - Large Parameter Behavior of Equilibrium Measures Y1 - 2006 A1 - Tamara Grava A1 - Fei-Ran Tian AB - We study the equilibrium measure for a logarithmic potential in the presence of an external field V*(x) + tp(x), where t is a parameter, V*(x) is a smooth function and p(x) a monic polynomial. When p(x) is of an odd degree, the equilibrium measure is shown to be supported on a single interval as |t| is sufficiently large. When p(x) is of an even degree, the equilibrium measure is supported on two disjoint intervals as t is negatively large; it is supported on a single interval for convex p(x) as t is positively large and is likely to be supported on multiple disjoint intervals for non-convex p(x). JF - Commun. Math. Sci. 4 (2006) 551-573 UR - http://hdl.handle.net/1963/1789 U1 - 2755 U2 - Mathematics U3 - Mathematical Physics ER - TY - RPRT T1 - Local Index Formula on the Equatorial Podles Sphere Y1 - 2006 A1 - Francesco D'Andrea A1 - Ludwik Dabrowski AB - We discuss spectral properties of the equatorial Podles sphere. As a preparation we also study the `degenerate\\\' (i.e. $q=0$) case (related to the quantum disk). We consider two different spectral triples: one related to the Fock representation of the Toeplitz algebra and the isopectral one.... JF - Lett. Math. Phys. 75 (2006) 235-254 UR - http://hdl.handle.net/1963/1782 U1 - 2762 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - The local index formula for SUq(2) JF - K-Theory 35 (2005) 375-394 Y1 - 2005 A1 - Walter van Suijlekom A1 - Ludwik Dabrowski A1 - Giovanni Landi A1 - Andrzej Sitarz A1 - Joseph C. Varilly AB - We discuss the local index formula of Connes-Moscovici for the isospectral noncommutative geometry that we have recently constructed on quantum SU(2). We work out the cosphere bundle and the dimension spectrum as well as the local cyclic cocycles yielding the index formula. UR - http://hdl.handle.net/1963/1713 U1 - 2438 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - A lemma and a conjecture on the cost of rearrangements JF - Rend. Sem. Mat. Univ. Padova 110 (2003) 97-102 Y1 - 2003 A1 - Alberto Bressan AB - Consider a stack of books, containing both white and black books. Suppose that we want to sort them out, putting the white books on the right, and the black books on the left (fig.~1). This will be done by a finite sequence of elementary transpositions. In other words, if we have a stack of all black books of length $a$ followed by a stack of all white books of length $b$, we are allowed to reverse their order at the cost of $a+b$. We are interested in a lower bound on the total cost of the rearrangement. PB - Università di Padova UR - http://hdl.handle.net/1963/2914 U1 - 1786 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - On the local structure of optimal trajectories in R3 JF - SIAM J. Control Optim. 42 (2003) 513-531 Y1 - 2003 A1 - Andrei A. Agrachev A1 - Mario Sigalotti AB - We analyze the structure of a control function u(t) corresponding to an optimal trajectory for the system $\\\\dot q =f(q)+u\\\\, g(q)$ in a three-dimensional manifold, near a point where some nondegeneracy conditions are satisfied. The kind of optimality which is studied includes time-optimality. The control turns out to be the concatenation of some bang and some singular arcs. Studying the index of the second variation of the switching times, the number of such arcs is bounded by four. PB - SISSA Library UR - http://hdl.handle.net/1963/1612 U1 - 2506 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Linearized elasticity as gamma-limit of finite elasticity JF - Set-Valued Anal. 10 (2002), p.165-183 Y1 - 2002 A1 - Gianni Dal Maso A1 - Matteo Negri A1 - Danilo Percivale PB - Springer UR - http://hdl.handle.net/1963/3052 U1 - 1281 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Local calibrations for minimizers of the Mumford-Shah functional with a triple junction JF - Commun. Contemp. Math. 4 (2002) 297-326 Y1 - 2002 A1 - Maria Giovanna Mora AB - We prove that, if u is a function satisfying all Euler conditions for the Mumford-Shah functional and the discontinuity set of u is given by three line segments meeting at the origin with equal angles, then there exists a neighbourhood U of the origin such that u is a minimizer of the Mumford-Shah functional on U with respect to its own boundary conditions on the boundary of U. The proof is obtained by using the calibration method. PB - World Scientific UR - http://hdl.handle.net/1963/3050 U1 - 1283 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - On a Lyapunov functional relating shortening curves and viscous conservation laws JF - Nonlinear Anal. 51 (2002) 649-662 Y1 - 2002 A1 - Stefano Bianchini A1 - Alberto Bressan AB - We study a nonlinear functional which controls the area swept by a curve moving in the plane in the direction of curvature. In turn, this yields a priori estimates on solutions to a class of parabolic equations and of scalar viscous conservation laws. A further application provides an estimate on the \\\"change of shape\\\" of a BV solution to a scalar conservation law. PB - Elsevier UR - http://hdl.handle.net/1963/1337 U1 - 3118 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Lax representation and Poisson geometry of the Kowalevski top JF - J. Phys. A 34 (2001) 2077-2085 Y1 - 2001 A1 - Gregorio Falqui AB - We discuss the Poisson structure underlying the two-field Kowalevski gyrostat and the Kowalevski top. We start from their Lax structure and construct a suitable pencil of Poisson brackets which endows these systems with the structure of bi-Hamiltonian completely integrable systems. We study the Casimir functions of such pencils, and show how it is possible to frame the Kowalevski systems within the so-called Gel\\\'fand-Zakharevich bi-Hamiltonian setting for integrable systems. PB - IOP Publishing UR - http://hdl.handle.net/1963/3244 U1 - 1457 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Lie triple systems and warped products JF - Rend. Mat. Appl. (7) Y1 - 2001 A1 - Marco Bertola A1 - Gouthier, D. VL - 21 ER - TY - JOUR T1 - Local calibrations for minimizers of the Mumford-Shah functional with a regular discontinuity set JF - Ann. I. H. Poincare - An., 2001, 18, 403 Y1 - 2001 A1 - Maria Giovanna Mora A1 - Massimiliano Morini PB - SISSA Library UR - http://hdl.handle.net/1963/1479 U1 - 2684 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Local calibrations for minimizers of the Mumford-Shah functional with rectilinear discontinuity sets JF - J. Math. Pures Appl. 79, 2 (2000) 141-162 Y1 - 2000 A1 - Gianni Dal Maso A1 - Maria Giovanna Mora A1 - Massimiliano Morini PB - SISSA Library UR - http://hdl.handle.net/1963/1261 U1 - 3194 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - L-1 stability estimates for n x n conservation laws JF - Arch. Ration. Mech. Anal. 149 (1999), no. 1, 1--22 Y1 - 1999 A1 - Alberto Bressan A1 - Tai-Ping Liu A1 - Tong Yang AB - Let $u_t+f(u)_x=0$ be a strictly hyperbolic $n\\\\times n$ system of conservation laws, each characteristic field being linearly degenerate or genuinely nonlinear. In this paper we explicitly define a functional $\\\\Phi=\\\\Phi(u,v)$, equivalent to the $L^1$ distance, which is `almost decreasing\\\', i.e., $\\\\Phi(u(t),v(t))-\\\\Phi(u(s),v(s))\\\\leq\\\\break O (\\\\epsilon)·(t-s)$ for all $t>s\\\\geq 0$, for every pair of $\\\\epsilon$-approximate solutions $u,v$ with small total variation, generated by a wave-front-tracking algorithm. The small parameter $\\\\epsilon$ here controls the errors in the wave speeds, the maximum size of rarefaction fronts and the total strength of all non-physical waves in $u$ and in $v$. From the above estimate, it follows that front-tracking approximations converge to a unique limit solution, depending Lipschitz continuously on the initial data, in the $L^1$ norm. This provides a new proof of the existence of the standard Riemann semigroup generated by an $n\\\\times n$ system of conservation laws.\\\'\\\' PB - Springer UR - http://hdl.handle.net/1963/3373 U1 - 957 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A Lipschitz selection from the set of minimizers of a nonconvex functional of the gradient JF - Nonlinear Analysis, Theory, Methods and Applications. Volume 37, Issue 6, September 1999, Pages 707-717 Y1 - 1999 A1 - Gianni Dal Maso A1 - Vladimir V. Goncharov A1 - Antonio Ornelas AB - A constructive and improved version of the proof that there exist a continuous map that solves the convexified problem is presented. A Lipschitz continuous map is analyzed such that a map vector minimizes the functional at each vector satisfying Cellina\\\'s condition of existence of minimum. This map is explicitly given by a direct constructive algorithm. PB - SISSA UR - http://hdl.handle.net/1963/6439 U1 - 6379 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - Limits of variational problems for Dirichlet forms in varying domains JF - Journal des Mathematiques Pures et Appliquees. Volume 77, Issue 1, January 1998, Pages 89-116 Y1 - 1998 A1 - Gianni Dal Maso A1 - Virginia De Cicco A1 - Lino Notarantonio A1 - Nicoletta A. Tchou PB - SISSA UR - http://hdl.handle.net/1963/6440 U1 - 6377 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - Limits of Dirichlet problems in perforated domains: a new formulation JF - Rend. Istit. Mat. Univ. Trieste 26 (1994) 339-360 Y1 - 1994 A1 - Gianni Dal Maso A1 - Rodica Toader PB - Università degli Studi di Trieste, Dipartimento di Scienze Matematiche UR - http://hdl.handle.net/1963/3649 U1 - 656 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Limits of obstacle problems for the area functional. JF - Partial differential equations and the calculus of variations : essays in honor of Ennio De Giorgi. - Boston : Birkhauser, 1989. - p. 285-309 Y1 - 1989 A1 - Gianni Dal Maso A1 - Michele Carriero A1 - Antonio Leaci A1 - Eduardo Pascali PB - SISSA Library UR - http://hdl.handle.net/1963/577 U1 - 3327 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Limits of nonlinear Dirichlet problems in varying domains. JF - Manuscripta Math. 61 (1988), no. 3, 251-278. Y1 - 1988 A1 - Gianni Dal Maso A1 - Anneliese Defranceschi AB - We study the general form of the limit, in the sense of gamma-convergence, of a sequence of nonlinear variational problems in varying domains with Dirichlet boudary conditions. The asymptotic problem is characterized in terms of the limit of suitable nonlinear capacities associated to the domains. PB - SISSA Library UR - http://hdl.handle.net/1963/536 U1 - 3368 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Lie algebra extensions and abelian monopoles. JF - Phys. Lett. B 195 (1987), no. 3, 429-434 Y1 - 1987 A1 - Giovanni Landi A1 - Giuseppe Marmo PB - SISSA Library UR - http://hdl.handle.net/1963/506 U1 - 3398 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Limits of nonlinear Dirichlet problems in varying domains. (Italian) JF - Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 81, 1987, no. 2, 111-118 Y1 - 1987 A1 - Gianni Dal Maso A1 - Anneliese Defranceschi PB - SISSA Library UR - http://hdl.handle.net/1963/486 U1 - 3418 U2 - Mathematics U3 - Functional Analysis and Applications ER -