Many-body states whose wave-function admits a representation in terms of a uniform binary-tree tensor decomposition are shown to obey to power-law two-body correlations functions. Any such state can be associated with the ground state of a translational invariant Hamiltonian which, depending on the dimension of the systems sites, involve at most couplings between third-neighboring sites. A detailed analysis of their spectra shows that they admit an exponentially large ground space.

}, doi = {10.1103/PhysRevA.81.062335}, url = {http://hdl.handle.net/1963/3909}, author = {Pietro Silvi and Vittorio Giovannetti and Simone Montangero and Matteo Rizzi and J. Ignacio Cirac and Rosario Fazio} } @article {2010, title = {Homogeneous multiscale entanglement renormalization ansatz tensor networks for quantum critical systems}, journal = {New J. Phys. 12 (2010) 075018}, year = {2010}, publisher = {IOP Publishing}, abstract = {In this paper, we review the properties of homogeneous multiscale entanglement renormalization ansatz (MERA) to describe quantum critical systems.We discuss in more detail our results for one-dimensional (1D) systems (the Ising and Heisenberg models) and present new data for the 2D Ising model. Together with the results for the critical exponents, we provide a detailed description of the numerical algorithm and a discussion of new optimization\\nstrategies. The relation between the critical properties of the system and the tensor structure of the MERA is expressed using the formalism of quantum channels, which we review and extend.

}, doi = {10.1088/1367-2630/12/7/075018}, url = {http://hdl.handle.net/1963/4067}, author = {Matteo Rizzi and Simone Montangero and Pietro Silvi and Vittorio Giovannetti and Rosario Fazio} } @article {2010, title = {Homogenization of fiber reinforced brittle material: the intermediate case}, journal = {Adv. Calc. Var. 3 (2010) 345-370}, number = {SISSA;14/2009/M}, year = {2010}, publisher = {Walter de Gruyter}, abstract = {We derive a cohesive fracture model by homogenizing a periodic composite material whose microstructure is characterized by the presence of brittle inclusions in a reticulated unbreakable elastic structure.}, doi = {10.1515/ACV.2010.011}, url = {http://hdl.handle.net/1963/3607}, author = {Gianni Dal Maso and Caterina Ida Zeppieri} } @inbook {2009, title = {Hamiltonian perturbations of hyperbolic PDEs: from classification results to the properties of solutions}, booktitle = {New Trends in Mathematical Physics : Selected contributions of the XVth International Congress on Mathematical Physics, Springer Netherlands, 2009, pp. 231-276.}, year = {2009}, publisher = {SISSA}, organization = {SISSA}, abstract = {We begin with presentation of classi cation results in the theory of Hamiltonian\\r\\nPDEs with one spatial dimension depending on a small parameter. Special\\r\\nattention is paid to the deformation theory of integrable hierarchies, including an\\r\\nimportant subclass of the so-called integrable hierarchies of the topological type\\r\\nassociated with semisimple Frobenius manifolds. Many well known equations of\\r\\nmathematical physics, such as KdV, NLS, Toda, Boussinesq etc., belong to this\\r\\nsubclass, but there are many new integrable PDEs, some of them being of interest\\r\\nfor applications. Connections with the theory of Gromov{Witten invariants\\r\\nand random matrices are outlined. We then address the problem of comparative\\r\\nstudy of singularities of solutions to the systems of first order quasilinear\\r\\nPDEs and their Hamiltonian perturbations containing higher derivatives. We\\r\\nformulate Universality Conjectures describing different types of critical behavior\\r\\nof perturbed solutions near the point of gradient catastrophe of the unperturbed\\r\\none.}, isbn = {978-90-481-2810-5}, url = {http://hdl.handle.net/1963/6470}, author = {Boris Dubrovin} } @article {2009, title = {Hardy-Sobolev-Maz\\\'ja inequalities: symmetry and breaking symmetry of extremal functions}, journal = {Commun. Contemp. Math. 11 (2009) 993-1007}, number = {SISSA;06/2008/M}, year = {2009}, doi = {10.1142/S0219199709003636}, url = {http://hdl.handle.net/1963/2569}, author = {Marita Gazzini and Roberta Musina} } @article {2009, title = {A higher order model for image restoration: the one dimensional case}, journal = {SIAM J. Math. Anal. 40 (2009) 2351-2391}, number = {SISSA;66/2008/M}, year = {2009}, abstract = {The higher order total variation-based model for image restoration proposed by Chan, Marquina, and Mulet in [6] is analyzed in one dimension. A suitable functional framework in which the minimization problem is well posed is being proposed and it is proved analytically that the\\nhigher order regularizing term prevents the occurrence of the staircase effect. The generalized version of the model considered here includes, as particular cases, some curvature dependent functionals.}, doi = {10.1137/070697823}, url = {http://hdl.handle.net/1963/3174}, author = {Gianni Dal Maso and Irene Fonseca and Giovanni Leoni and Massimiliano Morini} } @article {2009, title = {Holomorphic equivariant cohomology of Atiyah algebroids and localization}, number = {SISSA;65/2009/FM}, year = {2009}, url = {http://hdl.handle.net/1963/3774}, author = {Ugo Bruzzo and Vladimir Rubtsov} } @article {2009, title = {Homogenization of fiber reinforced brittle materials: the extremal cases}, journal = {SIAM J. Math. Anal. 41 (2009) 1874-1889}, number = {SISSA;49/2008/M}, year = {2009}, publisher = {SIAM}, abstract = {We analyze the behavior of a fragile material reinforced by a reticulated elastic unbreakable structure in the case of antiplane shear. The microscopic geometry of this material is described by means of two small parameters: the period $\\\\varepsilon$ of the grid and the ratio $\\\\delta$ between the thickness of the fibers and the period $\\\\varepsilon$. We show that the asymptotic behavior as $\\\\varepsilon\\\\to0^+$ and $\\\\delta\\\\to0^+$ depends dramatically on the relative size of these parameters. Indeed, in the two cases considered, i.e., $\\\\varepsilon\\\\ll\\\\delta$ and $\\\\varepsilon\\\\gg\\\\delta$, we obtain two different limit models: a perfectly elastic model and an elastic model with macroscopic cracks, respectively.}, url = {http://hdl.handle.net/1963/2705}, author = {Marco Barchiesi and Gianni Dal Maso} } @article {2008, title = {Hamiltonian partial differential equations and Frobenius manifolds}, journal = {Russian Mathematical Surveys. Volume 63, Issue 6, 2008, Pages 999-1010}, year = {2008}, publisher = {SISSA}, abstract = {In the first part of this paper the theory of Frobenius manifolds\\r\\nis applied to the problem of classification of Hamiltonian systems of partial\\r\\ndifferential equations depending on a small parameter. Also developed is\\r\\na deformation theory of integrable hierarchies including the subclass of\\r\\nintegrable hierarchies of topological type. Many well-known examples\\r\\nof integrable hierarchies, such as the Korteweg{\textendash}de Vries, non-linear\\r\\nSchr{\textasciidieresis}odinger, Toda, Boussinesq equations, and so on, belong to this\\r\\nsubclass that also contains new integrable hierarchies. Some of these new\\r\\nintegrable hierarchies may be important for applications. Properties of the\\r\\nsolutions to these equations are studied in the second part. Consideration\\r\\nis given to the comparative study of the local properties of perturbed and\\r\\nunperturbed solutions near a point of gradient catastrophe. A Universality\\r\\nConjecture is formulated describing the various types of critical behaviour\\r\\nof solutions to perturbed Hamiltonian systems near the point of gradient\\r\\ncatastrophe of the unperturbed solution.}, doi = {10.1070/RM2008v063n06ABEH004575}, url = {http://hdl.handle.net/1963/6471}, author = {Boris Dubrovin} } @article {MR2435752, title = {Harish-Chandra integrals as nilpotent integrals}, journal = {Int. Math. Res. Not. IMRN}, number = {16}, year = {2008}, pages = {Art. ID rnn062, 15}, issn = {1073-7928}, author = {Marco Bertola and Ferrer, Aleix Prats} } @article {2007, title = {High-order angles in almost-Riemannian geometry}, number = {SISSA;59/2007/M}, year = {2007}, abstract = {Let X and Y be two smooth vector fields on a two-dimensional manifold M. If X and Y are everywhere linearly independent, then they define a Riemannian metric on M (the metric for which they are orthonormal) and they give to M the structure of metric space. If X and Y become linearly dependent somewhere on M, then the corresponding Riemannian metric has singularities, but under generic conditions the metric structure is still well defined. Metric structures that can be defined locally in this way are called almost-Riemannian structures. The main result of the paper is a generalization to almost-Riemannian structures of the Gauss-Bonnet formula for domains with piecewise-C2 boundary. The main feature of such formula is the presence of terms that play the role of high-order angles at the intersection points with the set of singularities.}, url = {http://hdl.handle.net/1963/1995}, author = {Ugo Boscain and Mario Sigalotti} } @article {2007, title = {The holomorphic anomaly for open string moduli}, journal = {JHEP 10 (2007) 060}, number = {SISSA;55/2007/EP}, year = {2007}, abstract = {We complete the holomorphic anomaly equations for topological strings with their dependence on open moduli. We obtain the complete system by standard path integral arguments generalizing the analysis of BCOV (Commun. Math. Phys. 165 (1994) 311) to strings with boundaries. We study both the anti-holomorphic dependence on open moduli and on closed moduli in presence of Wilson lines. By providing the compactification a\\\' la Deligne-Mumford of the moduli space of Riemann surfaces with boundaries, we show that the open holomorphic anomaly equations are structured on the (real codimension one) boundary components of this space.}, doi = {10.1088/1126-6708/2007/10/060}, url = {http://hdl.handle.net/1963/2113}, author = {Giulio Bonelli and Alessandro Tanzini} } @article {2006, title = {On Hamiltonian perturbations of hyperbolic systems of conservation laws, II: universality of critical behaviour}, number = {SISSA;89/2005/FM}, year = {2006}, abstract = {Hamiltonian perturbations of the simplest hyperbolic equation $u_t + a(u) u_x=0$ are studied. We argue that the behaviour of solutions to the perturbed equation near the point of gradient catastrophe of the unperturbed one should be essentially independent on the choice of generic perturbation neither on the choice of generic solution. Moreover, this behaviour is described by a special solution to an integrable fourth order ODE.}, doi = {10.1007/s00220-006-0021-5}, url = {http://hdl.handle.net/1963/1786}, author = {Boris Dubrovin} } @article {2006, title = {On Hamiltonian perturbations of hyperbolic systems of conservation laws I: quasitriviality of bihamiltonian perturbations}, journal = {Comm. Pure Appl. Math. 59 (2006) 559-615}, number = {arXiv.org;math/0410027}, year = {2006}, abstract = {We study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially one-dimensional systems of hyperbolic PDEs. Under certain genericity assumptions it is proved that any bihamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on the derivatives. The main tools is in constructing of the so-called quasi-Miura transformation of jet coordinates eliminating an arbitrary deformation of a semisimple bihamiltonian structure of hydrodynamic type (the quasitriviality theorem). We also describe, following \\\\cite{LZ1}, the invariants of such bihamiltonian structures with respect to the group of Miura-type transformations depending polynomially on the derivatives.}, doi = {10.1002/cpa.20111}, url = {http://hdl.handle.net/1963/2535}, author = {Boris Dubrovin and Liu Si-Qi and Zhang Youjin} } @article {2006, title = {Homogeneous polynomial forms for simultaneous stabilizability of families of linear control systems: a tensor product approach}, journal = {IEEE Trans. Automat. Control 51 (2006) 1566-1571}, year = {2006}, abstract = {The paper uses the formalism of tensor products in order to deal with the problem of simultaneous\\nstabilizability of a family of linear control systems by means of Lyapunov functions which are homogeneous polynomial forms. While the feedback synthesis seems to be nonconvex, the simultaneous stability by means of homogeneous polynomial forms of the uncontrollable modes yields (convex) necessary but not sufficient conditions for simultaneous stabilizability.}, doi = {10.1109/TAC.2006.880811}, url = {http://hdl.handle.net/1963/2226}, author = {Claudio Altafini} } @article {2006, title = {A Hopf bundle over a quantum four-sphere from the symplectic group}, journal = {Commun. Math. Phys. 263 (2006) 65-88}, number = {arXiv.org;math/0407342v2}, year = {2006}, abstract = {We construct a quantum version of the SU(2) Hopf bundle $S^7 \\\\to S^4$. The quantum sphere $S^7_q$ arises from the symplectic group $Sp_q(2)$ and a quantum 4-sphere $S^4_q$ is obtained via a suitable self-adjoint idempotent $p$ whose entries generate the algebra $A(S^4_q)$ of polynomial functions over it. This projection determines a deformation of an (anti-)instanton bundle over the classical sphere $S^4$. We compute the fundamental $K$-homology class of $S^4_q$ and pair it with the class of $p$ in the $K$-theory getting the value -1 for the topological charge. There is a right coaction of $SU_q(2)$ on $S^7_q$ such that the algebra $A(S^7_q)$ is a non trivial quantum principal bundle over $A(S^4_q)$ with structure quantum group $A(SU_q(2))$.}, doi = {10.1007/s00220-005-1494-3}, url = {http://hdl.handle.net/1963/2179}, author = {Giovanni Landi and Chiara Pagani and Cesare Reina} } @article {2005, title = {Hybrid necessary principle}, journal = {SIAM J. Control Optim. 43 (2005) 1867-1887}, number = {SISSA;71/2002/M}, year = {2005}, note = {Proceedings of IFAC Conference on Analysis and Design of Hybrid Systems, Saint Malo, France}, publisher = {SIAM}, abstract = {We consider a hybrid control system and general optimal control problems for this system. We suppose that the switching strategy imposes restrictions on control sets and we provide necessary conditions for an optimal hybrid trajectory, stating a hybrid necessary principle (HNP). Our result generalizes various necessary principles available in the literature.}, doi = {10.1137/S0363012903416219}, url = {http://hdl.handle.net/1963/1641}, author = {Mauro Garavello and Benedetto Piccoli} } @article {2004, title = {H-bubbles in a perturbative setting: the finite-dimensional reduction\\\'s method}, journal = {Duke Math. J. 122 (2004), no. 3, 457--484}, number = {SISSA;36/2002/M}, year = {2004}, publisher = {SISSA Library}, abstract = {Given a regular function $H\\\\colon\\\\mathbb{R}^{3}\\\\to\\\\mathbb{R}$, we look for $H$-bubbles, that is, regular surfaces in $\\\\mathbb{R}^{3}$ parametrized on the sphere $\\\\mathbb{S}+^{2}$ with mean curvature $H$ at every point. Here we study the case of $H(u)=H_{0}+\\\\varepsilon H_{1}(u)=:H_{\\\\varepsilon}(u)$, where $H_{0}$ is a nonzero constant, $\\\\varepsilon$ is the smallness parameter, and $H_{1}$ is any $C^{2}$-function. We prove that if $\\\\bar p\\\\in\\\\mathbb{R}^{3}$ is a {\textquoteleft}{\textquoteleft}good\\\'\\\' stationary point for the Melnikov-type function $\\\\Gamma(p)=-\\\\int_{|q-p|<|H_{0}|^{-1}}H_{1}(q)\\\\,dq$, then for $|\\\\varepsilon|$ small there exists an $H_{\\\\varepsilon}$-bubble $\\\\omega^{\\\\varepsilon}$ that converges to a sphere of radius $|H_{0}|^{-1}$ centered at $\\\\bar p$, as $\\\\varepsilon\\\\to 0$.}, doi = {10.1215/S0012-7094-04-12232-8}, url = {http://hdl.handle.net/1963/1607}, author = {Paolo Caldiroli and Roberta Musina} } @article {2004, title = {Higher order quasiconvexity reduces to quasiconvexity}, journal = {Arch. Ration. Mech. Anal. 171 (2004) 55-81}, number = {SISSA;41/2003/M}, year = {2004}, publisher = {Springer}, abstract = {In this paper it is shown that higher order quasiconvex functions suitable in the variational treatment of problems involving second derivatives may be extended to the space of all matrices as classical quasiconvex functions. Precisely, it is proved that a smooth strictly 2-quasiconvex function with p-growth at infinity, p>1, is the restriction to symmetric matrices of a 1-quasiconvex function with the same growth. As a consequence, lower semicontinuity results for second-order variational problems are deduced as corollaries of well-known first order theorems.}, doi = {10.1007/s00205-003-0278-1}, url = {http://hdl.handle.net/1963/2911}, author = {Gianni Dal Maso and Irene Fonseca and Giovanni Leoni and Massimiliano Morini} } @article {2003, title = {Hybrid optimal control: case study of a car with gears}, journal = {Int. J. Control 76 (2003) 1272-1284}, year = {2003}, publisher = {Taylor and Francis}, abstract = {The purpose of this paper is to show the use of some analytical tools for hybrid optimal control. We illustrate both the hybrid maximum principle and the hybrid necessary principle at work on a simple example of a car with gears. The model is sufficiently rich to generate non-trivial optimization problems and the obtained results match with intuition. Finally, computer simulations confirm the theoretical analysis.}, doi = {10.1080/0020717031000147520}, url = {http://hdl.handle.net/1963/3022}, author = {Ciro D{\textquoteright}Apice and Mauro Garavello and Rosanna Manzo and Benedetto Piccoli} } @article {2000, title = {High-order Averaging and Stability of Time-Varying Systems}, number = {SISSA;7/00/M}, year = {2000}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1465}, author = {Andrey Sarychev} } @article {1999, title = {Hyperbolic Systems of Conservation Laws}, journal = {Rev. Mat. Complut. 12 (1999) 135-200}, year = {1999}, abstract = {This is a survey paper, written in the occasion of an invited talk given by the author at the Universidad Complutense in Madrid, October 1998. Its purpose is to provide an account of some recent advances in the mathematical theory of hyperbolic systems of conservation laws in one space dimension. After a brief review of basic concepts, we describe in detail the method of wave-front tracking approximation and present some of the latest results on uniqueness and stability of entropy weak solutions.}, url = {http://hdl.handle.net/1963/1855}, author = {Alberto Bressan} } @article {1997, title = {Homogeneous tangent vectors and high order necessary conditions for optimal controls}, journal = {J. Dynam. Control Systems 3 (1997), no. 2, 205--240}, number = {SISSA;113/95/M}, year = {1997}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1015}, author = {Fabio Ancona} } @article {10572, title = {Hilbert schemes of points on some K3 surfaces and Gieseker stable boundles}, journal = {MATH PROC CAMBRIDGE 120: 255-261 Part 2}, number = {SISSA;34/95/GEO-FM}, year = {1994}, publisher = {SISSA Library}, abstract = {By using a Fourier-Mukai transform for sheaves on K3 surfaces we show that for a wide class of K3 surfaces $X$ the punctual Hilbert schemes $\\\\Hilb^n(X)$ can be identified, for all $n\\\\geq 1$, with moduli spaces of Gieseker stable vector bundles on $X$ of rank $1+2n$. We also introduce a new Fourier-Mukai type transform for such surfaces.

}, url = {http://hdl.handle.net/1963/937}, author = {Ugo Bruzzo and Antony Maciocia} } @article {1992, title = {Hamiltonian formalism of Whitham-type hierarchies and topological Landau - Ginsburg models}, journal = {Communications in Mathematical Physics. Volume 145, Issue 1, March 1992, Pages 195-207}, year = {1992}, publisher = {SISSA}, abstract = {We show that the bi-hamiltonian structure of the averaged Gelfand-Dikii\\r\\nhierarchy is involved in the Landau-Ginsburg topological models (for An-Series):\\r\\nthe Casimirs for the first P.B. give the correct coupling parameters for the perturbed\\r\\ntopological minimal model; the correspondence {coupling parameters} ~ {primary\\r\\nfields} is determined by the second P.B. The partition function (at the tree level) and\\r\\nthe chiral algebra for LG models are calculated for any genus g.}, doi = {10.1007/BF02099286}, url = {http://hdl.handle.net/1963/6476}, author = {Boris Dubrovin} } @article {1989, title = {Hyperbolic equations as ordinary differential equations in Banach spaces}, number = {SISSA;11/89/M}, year = {1989}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/773}, author = {Giovanni Vidossich} } @article {1988, title = {Holes and obstacles}, journal = {Ann. Inst. H. Poincare Anal. Non Lineaire 5 (1988), no. 4, 323-345}, number = {SISSA;20/87/M}, year = {1988}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/501}, author = {Roberta Musina and Giovanni Mancini} } @article {1988, title = {H-surfaces with obstacles. (Italian)}, journal = {Ann. Univ. Ferrara Sez. VII (N.S.) 34 (1988), 1-14}, number = {SISSA;9/87/M}, year = {1988}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/491}, author = {Roberta Musina} }