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.

PB - American Physical Society UR - http://hdl.handle.net/1963/3909 U1 - 800 U2 - Physics U3 - Condensed Matter Theory ER - TY - JOUR T1 - Homogeneous multiscale entanglement renormalization ansatz tensor networks for quantum critical systems JF - New J. Phys. 12 (2010) 075018 Y1 - 2010 A1 - Matteo Rizzi A1 - Simone Montangero A1 - Pietro Silvi A1 - Vittorio Giovannetti A1 - Rosario Fazio AB -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.

PB - IOP Publishing UR - http://hdl.handle.net/1963/4067 U1 - 335 U2 - Physics U3 - Condensed Matter Theory 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 - CHAP T1 - Hamiltonian perturbations of hyperbolic PDEs: from classification results to the properties of solutions T2 - New Trends in Mathematical Physics : Selected contributions of the XVth International Congress on Mathematical Physics, Springer Netherlands, 2009, pp. 231-276. Y1 - 2009 A1 - Boris Dubrovin AB - 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. JF - New Trends in Mathematical Physics : Selected contributions of the XVth International Congress on Mathematical Physics, Springer Netherlands, 2009, pp. 231-276. PB - SISSA SN - 978-90-481-2810-5 UR - http://hdl.handle.net/1963/6470 U1 - 6415 U2 - Mathematics U4 - 1 U5 - MAT/03 GEOMETRIA ER - TY - JOUR T1 - Hardy-Sobolev-Maz\\\'ja inequalities: symmetry and breaking symmetry of extremal functions JF - Commun. Contemp. Math. 11 (2009) 993-1007 Y1 - 2009 A1 - Marita Gazzini A1 - Roberta Musina UR - http://hdl.handle.net/1963/2569 U1 - 1551 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A higher order model for image restoration: the one dimensional case JF - SIAM J. Math. Anal. 40 (2009) 2351-2391 Y1 - 2009 A1 - Gianni Dal Maso A1 - Irene Fonseca A1 - Giovanni Leoni A1 - Massimiliano Morini AB - 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. UR - http://hdl.handle.net/1963/3174 U1 - 1127 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - RPRT T1 - Holomorphic equivariant cohomology of Atiyah algebroids and localization Y1 - 2009 A1 - Ugo Bruzzo A1 - Vladimir Rubtsov UR - http://hdl.handle.net/1963/3774 U1 - 551 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Homogenization of fiber reinforced brittle materials: the extremal cases JF - SIAM J. Math. Anal. 41 (2009) 1874-1889 Y1 - 2009 A1 - Marco Barchiesi A1 - Gianni Dal Maso AB - 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. PB - SIAM UR - http://hdl.handle.net/1963/2705 U1 - 1396 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Hamiltonian partial differential equations and Frobenius manifolds JF - Russian Mathematical Surveys. Volume 63, Issue 6, 2008, Pages 999-1010 Y1 - 2008 A1 - Boris Dubrovin AB - 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–de Vries, non-linear\\r\\nSchr¨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. PB - SISSA UR - http://hdl.handle.net/1963/6471 U1 - 6416 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - JOUR T1 - Harish-Chandra integrals as nilpotent integrals JF - Int. Math. Res. Not. IMRN Y1 - 2008 A1 - Marco Bertola A1 - Ferrer, Aleix Prats ER - TY - RPRT T1 - High-order angles in almost-Riemannian geometry Y1 - 2007 A1 - Ugo Boscain A1 - Mario Sigalotti AB - 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. UR - http://hdl.handle.net/1963/1995 U1 - 2201 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - The holomorphic anomaly for open string moduli JF - JHEP 10 (2007) 060 Y1 - 2007 A1 - Giulio Bonelli A1 - Alessandro Tanzini AB - 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. UR - http://hdl.handle.net/1963/2113 U1 - 2576 U2 - Physics U3 - Elementary Particle Theory ER - TY - RPRT T1 - On Hamiltonian perturbations of hyperbolic systems of conservation laws, II: universality of critical behaviour Y1 - 2006 A1 - Boris Dubrovin AB - 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. JF - Comm. Math. Phys. 267 (2006) 117-139 UR - http://hdl.handle.net/1963/1786 U1 - 2758 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - On Hamiltonian perturbations of hyperbolic systems of conservation laws I: quasitriviality of bihamiltonian perturbations JF - Comm. Pure Appl. Math. 59 (2006) 559-615 Y1 - 2006 A1 - Boris Dubrovin A1 - Liu Si-Qi A1 - Zhang Youjin AB - 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. UR - http://hdl.handle.net/1963/2535 U1 - 1583 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Homogeneous polynomial forms for simultaneous stabilizability of families of linear control systems: a tensor product approach JF - IEEE Trans. Automat. Control 51 (2006) 1566-1571 Y1 - 2006 A1 - Claudio Altafini AB - 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. UR - http://hdl.handle.net/1963/2226 U1 - 2018 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A Hopf bundle over a quantum four-sphere from the symplectic group JF - Commun. Math. Phys. 263 (2006) 65-88 Y1 - 2006 A1 - Giovanni Landi A1 - Chiara Pagani A1 - Cesare Reina AB - 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 $A(SU_q(2))$. UR - http://hdl.handle.net/1963/2179 U1 - 2065 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Hybrid necessary principle JF - SIAM J. Control Optim. 43 (2005) 1867-1887 Y1 - 2005 A1 - Mauro Garavello A1 - Benedetto Piccoli AB - 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. PB - SIAM UR - http://hdl.handle.net/1963/1641 N1 - Proceedings of IFAC Conference on Analysis and Design of Hybrid Systems, Saint Malo, France U1 - 2477 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - H-bubbles in a perturbative setting: the finite-dimensional reduction\\\'s method JF - Duke Math. J. 122 (2004), no. 3, 457--484 Y1 - 2004 A1 - Paolo Caldiroli A1 - Roberta Musina AB - 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 ``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 $\\\\varepsilon\\\\to 0$. PB - SISSA Library UR - http://hdl.handle.net/1963/1607 U1 - 2511 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Higher order quasiconvexity reduces to quasiconvexity JF - Arch. Ration. Mech. Anal. 171 (2004) 55-81 Y1 - 2004 A1 - Gianni Dal Maso A1 - Irene Fonseca A1 - Giovanni Leoni A1 - Massimiliano Morini AB - 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. PB - Springer UR - http://hdl.handle.net/1963/2911 U1 - 1789 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Hybrid optimal control: case study of a car with gears JF - Int. J. Control 76 (2003) 1272-1284 Y1 - 2003 A1 - Ciro D'Apice A1 - Mauro Garavello A1 - Rosanna Manzo A1 - Benedetto Piccoli AB - 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. PB - Taylor and Francis UR - http://hdl.handle.net/1963/3022 U1 - 1311 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - High-order Averaging and Stability of Time-Varying Systems Y1 - 2000 A1 - Andrey Sarychev PB - SISSA Library UR - http://hdl.handle.net/1963/1465 U1 - 3075 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Hyperbolic Systems of Conservation Laws JF - Rev. Mat. Complut. 12 (1999) 135-200 Y1 - 1999 A1 - Alberto Bressan AB - 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. UR - http://hdl.handle.net/1963/1855 U1 - 77 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Homogeneous tangent vectors and high order necessary conditions for optimal controls JF - J. Dynam. Control Systems 3 (1997), no. 2, 205--240 Y1 - 1997 A1 - Fabio Ancona PB - SISSA Library UR - http://hdl.handle.net/1963/1015 U1 - 2841 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Hilbert schemes of points on some K3 surfaces and Gieseker stable boundles JF - MATH PROC CAMBRIDGE 120: 255-261 Part 2 Y1 - 1994 A1 - Ugo Bruzzo A1 - Antony Maciocia AB -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.

PB - SISSA Library UR - http://hdl.handle.net/1963/937 U1 - 3517 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Hamiltonian formalism of Whitham-type hierarchies and topological Landau - Ginsburg models JF - Communications in Mathematical Physics. Volume 145, Issue 1, March 1992, Pages 195-207 Y1 - 1992 A1 - Boris Dubrovin AB - 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. PB - SISSA UR - http://hdl.handle.net/1963/6476 U1 - 6434 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - JOUR T1 - Hyperbolic equations as ordinary differential equations in Banach spaces Y1 - 1989 A1 - Giovanni Vidossich PB - SISSA Library UR - http://hdl.handle.net/1963/773 U1 - 3018 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Holes and obstacles JF - Ann. Inst. H. Poincare Anal. Non Lineaire 5 (1988), no. 4, 323-345 Y1 - 1988 A1 - Roberta Musina A1 - Giovanni Mancini PB - SISSA Library UR - http://hdl.handle.net/1963/501 U1 - 3403 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - H-surfaces with obstacles. (Italian) JF - Ann. Univ. Ferrara Sez. VII (N.S.) 34 (1988), 1-14 Y1 - 1988 A1 - Roberta Musina PB - SISSA Library UR - http://hdl.handle.net/1963/491 U1 - 3413 U2 - Mathematics U3 - Functional Analysis and Applications ER -