Families of Monads and Instantons from a Noncommutative ADHM Construction - Simon Brain; Giovanni Landi (2009)
We give a \\\\theta-deformed version of the ADHM construction of SU(2) instantons with arbitrary topological charge on the sphere S^4. Classically the instanton gauge fields are constructed from suitable monad data; we show that in the deformed case the set of monads is itself a noncommutative space. We use these monads to construct noncommutative `families\\\' of SU(2) instantons on the deformed sphere S^4_\\\\theta. We also compute the topological charge of each of the families. Finally we discuss what it means for such families to be gauge equivalent.

First colonization of a spectral outpost in random matrix theory - Marco Bertola; Lee, S. Y. (Constr. Approx., 2009)

Foliations of small tubes in Riemannian manifolds by capillary minimal discs - Fall, Mouhamed Moustapha; Mercuri, Carlo (Nonlinear Analysis: Theory, Methods & Applications, 2009)
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font-size: 18px;"><span id="MathJax-Element-3-Frame" class="MathJax_SVG" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" data-mathml="Γ"><span class="MJX_Assistive_MathML" style="box-sizing: border-box; margin: 0px; padding: 1px 0px 0px !important; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); user-select: none; position: absolute !important; border: 0px !important; height: 1px !important; width: 1px !important; overflow: hidden !important; display: block !important; transition: none 0s ease 0s;">Γ<span style="color: #2e2e2e; font-family: NexusSerif, Georgia, 'Times New Roman', Times, STIXGeneral, 'Cambria Math', 'Lucida Sans Unicode', 'Microsoft Sans Serif', 'Segoe UI Symbol', 'Arial Unicode MS', serif; font-size: 18px;">inside the surface of revolution of <span class="math" style="box-sizing: border-box; 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font-family: NexusSerif, Georgia, 'Times New Roman', Times, STIXGeneral, 'Cambria Math', 'Lucida Sans Unicode', 'Microsoft Sans Serif', 'Segoe UI Symbol', 'Arial Unicode MS', serif; font-size: 18px;"> around <span class="math" style="box-sizing: border-box; margin: 0px; padding: 0px; color: #2e2e2e; font-family: NexusSerif, Georgia, 'Times New Roman', Times, STIXGeneral, 'Cambria Math', 'Lucida Sans Unicode', 'Microsoft Sans Serif', 'Segoe UI Symbol', 'Arial Unicode MS', serif; font-size: 18px;"><span id="MathJax-Element-5-Frame" class="MathJax_SVG" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" data-mathml="Γ"><span class="MJX_Assistive_MathML" style="box-sizing: border-box; margin: 0px; padding: 1px 0px 0px !important; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); user-select: none; position: absolute !important; border: 0px !important; height: 1px !important; width: 1px !important; overflow: hidden !important; display: block !important; transition: none 0s ease 0s;">Γ<span style="color: #2e2e2e; font-family: NexusSerif, Georgia, 'Times New Roman', Times, STIXGeneral, 'Cambria Math', 'Lucida Sans Unicode', 'Microsoft Sans Serif', 'Segoe UI Symbol', 'Arial Unicode MS', serif; font-size: 18px;">, having small radius, and intersecting it with constant angles. In particular we obtain that small tubular neighborhoods can be foliated by minimal discs.

Gauged Laplacians on quantum Hopf bundles - Giovanni Landi; Cesare Reina; Alessandro Zampini (Comm. Math. Phys. 287 (2009) 179-209, 2009)
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.

Hamiltonian perturbations of hyperbolic PDEs: from classification results to the properties of solutions - Boris Dubrovin (New Trends in Mathematical Physics : Selected contributions of the XVth International Congress on Mathematical Physics, Springer Netherlands, 2009, pp. 231-276., 2009)
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.

Hardy-Sobolev-Maz\\\'ja inequalities: symmetry and breaking symmetry of extremal functions - Marita Gazzini; Roberta Musina (Commun. Contemp. Math. 11 (2009) 993-1007, 2009)

A higher order model for image restoration: the one dimensional case - Gianni Dal Maso; Irene Fonseca; Giovanni Leoni; Massimiliano Morini (SIAM J. Math. Anal. 40 (2009) 2351-2391, 2009)
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.

Holomorphic equivariant cohomology of Atiyah algebroids and localization - Ugo Bruzzo; Vladimir Rubtsov (2009)

Homogenization of fiber reinforced brittle materials: the extremal cases - Marco Barchiesi; Gianni Dal Maso (SIAM J. Math. Anal. 41 (2009) 1874-1889, 2009)
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.

Initial value problem of the Whitham equations for the Camassa-Holm equation - Tamara Grava; Virgil U. Pierce; Fei-Ran Tian (Physica D 238 (2009) 55-66, 2009)
We study the Whitham equations for the Camassa-Holm equation. The equations are neither strictly hyperbolic nor genuinely nonlinear. We are interested in the initial value problem of the Whitham equations. When the initial values are given by a step function, the Whitham solution is self-similar. When the initial values are given by a smooth function, the Whitham solution exists within a cusp in the x-t plane. On the boundary of the cusp, the Whitham equation matches the Burgers solution, which exists outside the cusp.

The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups - Andrei A. Agrachev; Ugo Boscain; Jean-Paul Gauthier; Francesco Rossi (J. Funct. Anal. 256 (2009) 2621-2655, 2009)
We present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector, using the Popp\\\'s volume form introduced by Montgomery. This definition generalizes the one of the Laplace-Beltrami operator in Riemannian geometry. In the case of left-invariant problems on unimodular Lie groups we prove that it coincides with the usual sum of squares.\\nWe then extend a method (first used by Hulanicki on the Heisenberg group) to compute explicitly the kernel of the hypoelliptic heat equation on any unimodular Lie group of type I. The main tool is the noncommutative Fourier transform. We then study some relevant cases: SU(2), SO(3), SL(2) (with the metrics inherited by the Killing form), and the group SE(2) of rototranslations of the plane.\\nOur study is motivated by some recent results about the cut and conjugate loci on these sub-Riemannian manifolds. The perspective is to understand how singularities of the sub-Riemannian distance reflect on the kernel of the corresponding hypoelliptic heat equation.

Investigating the Conformational Stability of Prion Strains through a Kinetic Replication Model - Mattia Zampieri; Giuseppe Legname; Claudio Altafini (PLoS Comput Biol 2009;5(7): e1000420, 2009)
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.

Jacobi Equations and Comparison Theorems for Corank 1 Sub-Riemannian structures with symmetries - Li Chengbo; Igor Zelenko (2009)
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.

Low-Frequency Variations of Force Coefficients on Square Cylinders with Sharp and Rounded Corners - Andrea Mola; Giancarlo Bordonaro; Muhammad R. Hajj (Journal of Structural Engineering, 2009)

Mesoscopic colonization in a spectral band - Marco Bertola; Lee, S. Y.; Mo, M. Y. (J. Phys. A, 2009)

Minimal disc-type surfaces embedded in a perturbed cylinder - Fall, Mouhamed Moustapha; Mercuri, Carlo (Differential and Integral Equations, 2009)
In the present note we deal with small perturbations of an infinite cylinder in the 3D euclidian space. We find minimal disc-type surfaces embedded in the cylinder and intersecting its boundary perpendicularly. The existence and localization of those minimal discs is a consequence of a non-degeneracy condition for the critical points of a functional related to the oscillations of the cylinder from the flat configuration.

A model for the dynamics of rowing boats - L. Formaggia; Edie Miglio; Andrea Mola; Antonio Montano (International Journal for Numerical Methods in Fluids, 2009)

A model for the orbifold Chow ring of weighted projective spaces - Samuel Boissiere; Etienne Mann; Fabio Perroni (Comm. Algebra 37 (2009) 503-514, 2009)
We construct an isomorphism of graded Frobenius algebras between the orbifold Chow ring of weighted projective spaces and graded algebras of groups of roots of the unity.

Moment determinants as isomonodromic tau functions - Marco Bertola (Nonlinearity, 2009)

mRNA stability and the unfolding of gene expression in the long-period yeast metabolic cycle - Nicola Soranzo; Mattia Zampieri; Lorenzo Farina; Claudio Altafini (BMC Systems Biology (2009) 3:18, 2009)
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.

A nonlinear theory for shells with slowly varying thickness - Marta Lewicka; Maria Giovanna Mora; Mohammad Reza Pakzad (C. R. Math. 347 (2009) 211-216, 2009)
We study the Γ-limit of 3d nonlinear elasticity for shells of small, variable thickness, around an arbitrary smooth 2d surface.

A note on the paper \\\"Optimizing improved Hardy inequalities\\\" by S. Filippas and A. Tertikas - Roberta Musina (J. Funct. Anal. 256 (2009) 2741-2745, 2009)

Optimal transportation under nonholonomic constraints - Andrei A. Agrachev; Paul Lee (Trans. Amer. Math. Soc. 361 (2009) 6019-6047, 2009)
We study the Monge\\\'s optimal transportation problem where the cost is given by optimal control cost. We prove the existence and uniqueness of optimal map under certain regularity conditions on the Lagrangian, absolute continuity of the measures and most importantly the absent of sharp abnormal minimizers. In particular, this result is applicable in the case of subriemannian manifolds with a 2-generating distribution and cost given by d2, where d is the subriemannian distance. Also, we discuss some properties of the optimal plan when abnormal minimizers are present. Finally, we consider some examples of displacement interpolation in the case of Grushin plane.

The partition function of the two-matrix model as an isomonodromic τ function - Marco Bertola; Marchal, O. (J. Math. Phys., 2009)

Quasistatic evolution for Cam-Clay plasticity: examples of spatially homogeneous solutions - Gianni Dal Maso; Antonio DeSimone (Math. Models Methods Appl. Sci. 19 (2009) 1643-1711, 2009)
We study a quasistatic evolution problem for Cam-Clay plasticity under a special loading program which leads to spatially homogeneous solutions. Under some initial conditions, the solutions exhibit a softening behaviour and time discontinuities.\\nThe behavior of the solutions at the jump times is studied by a viscous approximation.

Quasistatic evolution problems for nonhomogeneous elastic plastic materials - Francesco Solombrino (J. Convex Anal., 2009)
The paper studies the quasistatic evolution for elastoplastic materials when the yield surface depends on the position in the reference configuration. The main results are obtained when the yield surface is continuous with respect to the space variable. The case of piecewise constant dependence is also considered. The evolution is studied in the framework of the variational formulation for rate independent problems developed by Mielke. The results are proved by adapting the arguments introduced for a constant yield surface, using some properties of convex valued semicontinuous multifunctions. A strong formulation of the problem is also obtained, which includes a pointwise version of the plastic flow rule. Some examples are considered, which show that strain concentration may occur as a consequence of a nonconstant yield surface.

Regularity of a vector potential problem and its spectral curve - Ferenc Balogh; Marco Bertola (J. Approx. Theory, 2009)

Relaxation dynamics of fluid membranes - Marino Arroyo; Antonio DeSimone (Phys. Rev. E 79 (2009) 031915, 2009)
We study the effect of membrane viscosity in the dynamics of liquid membranes-possibly with free or internal boundaries-driven by conservative forces (curvature elasticity and line tension) and dragged by the bulk dissipation of the ambient fluid and the friction occurring when the amphiphilic molecules move relative to each other. To this end, we formulate a continuum model which includes a form of the governing equations for a two-dimensional viscous fluid moving on a curved, time-evolving surface. The effect of membrane viscosity has received very limited attention in previous continuum studies of the dynamics of fluid membranes, although recent coarse-grained discrete simulations suggest its importance. By applying our model to the study of vesiculation and membrane fusion in a simplified geometry, we conclude that membrane viscosity plays a dominant role in the relaxation dynamics of fluid membranes of sizes comparable to those found in eukaryotic cells, and is not negligible in many large synthetic systems of current interest.

On a Sobolev type inequality related to the weighted p-Laplace operator - Marita Gazzini; Roberta Musina (J. Math. Anal. Appl. 352 (2009) 99-111, 2009)

Solutions of the Schrödinger–Poisson problem concentrating on spheres, part I: necessary conditions - Ianni, Isabella; Giusi Vaira (Mathematical Models and Methods in Applied Sciences, 2009)
In this paper we study a coupled nonlinear Schrödinger–Poisson problem with radial functions. This system has been introduced as a model describing standing waves for the nonlinear Schrödinger equations in the presence of the electrostatic field. We provide necessary conditions for concentration on sphere for the solutions of this kind of problem extending the results already known.

Some new entire solutions of semilinear elliptic equations on Rn - Andrea Malchiodi (Adv. Math. 221 (2009) 1843-1909, 2009)

Strain-order coupling in nematic elastomers: equilibrium configurations - Pierluigi Cesana; Antonio DeSimone (Math. Models Methods Appl. Sci. 19 (2009) 601-630, 2009)
We consider models that describe liquid crystal elastomers either in a biaxial or in a uniaxial phase and in the framework of Frank\\\'s director theory. We prove existence of static equilibrium solutions in the presence of frustrations due to electro-mechanical boundary conditions and to applied loads and fields. We find explicit solutions arising in connection with special boundary conditions and the corresponding phase diagrams, leading to significant implications on possible experimental observations.

Stratos: a code for 3D free surface flows with floating constraints - Antonio DeSimone; B. Bianchi; Luca Heltai (2009)
This report presents a brief discussion of the theoretical aspects and practical implementation of STRATOS . STRATOS is a 3D code for the simulation\\nof hydrodynamic flows for incompressible fluids, in the presence of a free surface, capable of simulating the interaction between the free surface and a\\nfloating object via Lagrange multipliers......

Tools for the Solution of PDEs Defined on Curved Manifolds with deal.II - Antonio DeSimone; Luca Heltai; Cataldo Manigrasso (2009)
The deal.II finite element library was originally designed to solve partial differential equations defined on one, two or three space dimensions, mostly\\nvia the Finite Element Method. In its versions prior to version 6.2, the user could not solve problems defined on curved manifolds embedded in two or\\nthree spacial dimensions. This infrastructure is needed if one wants to solve, for example, Boundary Integral Equations.

Topological branes, p-algebras and generalized Nahm equations - Giulio Bonelli; Alessandro Tanzini; Maxim Zabzine (Phys. Lett. B 672 (2009) 390-395, 2009)
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

Topological expansion for the Cauchy two-matrix model - Marco Bertola; Ferrer, A. Prats (J. Phys. A, 2009)

Twisted Covariance vs Weyl Quantisation - Gherardo Piacitelli (2009)
In this letter we wish to clarify in which sense the tensor nature of the commutation relations [x^mu,x^nu]=i theta ^{mu nu} underlying Minkowski spacetime quantisation cannot be suppressed even in the twisted approach to Lorentz covariance. We then address the vexata quaestio \\\"why theta\\\"?

On universality of critical behaviour in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the \\\\it tritronquée solution to the Painlevé-I equation - Boris Dubrovin; Tamara Grava; Christian Klein (J. Nonlinear Sci. 19 (2009) 57-94, 2009)
We argue that the critical behaviour near the point of ``gradient catastrophe\\\" of the solution to the Cauchy problem for the focusing nonlinear Schr\\\\\\\"odinger equation $ i\\\\epsilon \\\\psi_t +\\\\frac{\\\\epsilon^2}2\\\\psi_{xx}+ |\\\\psi|^2 \\\\psi =0$ with analytic initial data of the form $\\\\psi(x,0;\\\\epsilon) =A(x) e^{\\\\frac{i}{\\\\epsilon} S(x)}$ is approximately described by a particular solution to the Painlev\\\\\\\'e-I equation.

Universality of the break-up profile for the KdV equation in the small dispersion limit using the Riemann-Hilbert approach - Tamara Grava; Tom Claeys (Comm. Math. Phys. 286 (2009) 979-1009, 2009)
We obtain an asymptotic expansion for the solution of the Cauchy problem for the Korteweg-de Vries (KdV) equation in the small dispersion limit near the point of gradient catastrophe (x_c,t_c) for the solution of the dispersionless equation.\\nThe sub-leading term in this expansion is described by the smooth solution of a fourth order ODE, which is a higher order analogue to the Painleve I equation. This is in accordance with a conjecture of Dubrovin, suggesting that this is a universal phenomenon for any Hamiltonian perturbation of a hyperbolic equation. Using the Deift/Zhou steepest descent method applied on the Riemann-Hilbert problem for the KdV equation, we are able to prove the asymptotic expansion rigorously in a double scaling limit.

A variational model for quasistatic crack growth in nonlinear elasticity: some qualitative properties of the solutions - Gianni Dal Maso; Alessandro Giacomini; Marcello Ponsiglione (Boll. Unione Mat. Ital. (9) 2 (2009) 371-390, 2009)

On viscosity solutions of Hamilton-Jacobi equations - Sandro Zagatti (Trans. Amer. Math. Soc. 361 (2009) 41-59, 2009)
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.

Asymptotic evolution for the semiclassical nonlinear Schrödinger equation in presence of electric and magnetic fields - Alessandro Selvitella (Journal of Differential Equations, 2008)
In this paper we study the semiclassical limit for the solutions of a subcritical focusing NLS with electric and magnetic potentials. We consider in particular the Cauchy problem for initial data close to solitons and show that, when the Planck constant goes to zero, the motion shadows that of a classical particle. Several works were devoted to the case of standing waves: differently from these we show that, in the dynamic version, the Lorentz force appears crucially.

Cantor families of periodic solutions for completely resonant wave equations - Massimiliano Berti; Philippe Bolle (Frontiers of Mathematics in China, 2008)
We present recent existence results of Cantor families of small amplitude periodic solutions for completely resonant nonlinear wave equations. The proofs rely on the Nash-Moser implicit function theory and variational methods. © 2008 Higher Education Press.
cited By (since 1996)0

Cantor families of periodic solutions for wave equations via a variational principle - Massimiliano Berti; Philippe Bolle (Advances in Mathematics, 2008)
We prove existence of small amplitude periodic solutions of completely resonant wave equations with frequencies in a Cantor set of asymptotically full measure, via a variational principle. A Lyapunov-Schmidt decomposition reduces the problem to a finite dimensional bifurcation equation-variational in nature-defined on a Cantor set of non-resonant parameters. The Cantor gaps are due to "small divisors" phenomena. To solve the bifurcation equation we develop a suitable variational method. In particular, we do not require the typical "Arnold non-degeneracy condition" of the known theory on the nonlinear terms. As a consequence our existence results hold for new generic sets of nonlinearities. © 2007 Elsevier Inc. All rights reserved.
cited By (since 1996)6

Cantor families of periodic solutions of wave equations with C k nonlinearities - Massimiliano Berti; Philippe Bolle (Nonlinear Differential Equations and Applications, 2008)
We prove bifurcation of Cantor families of periodic solutions for wave equations with nonlinearities of class C k . It requires a modified Nash-Moser iteration scheme with interpolation estimates for the inverse of the linearized operators and for the composition operators. © 2008 Birkhaueser.
cited By (since 1996)10

Concentrating solutions of some singularly perturbed elliptic equations - Andrea Malchiodi (Front. Math. China 3 (2008) 239-252, 2008)
We study singularly perturbed elliptic equations arising from models in physics or biology, and investigate the asymptotic behavior of some special solutions. We also discuss some connections with problems arising in differential geometry.

On concentration of positive bound states for the Schrödinger-Poisson problem with potentials - Ianni, Isabella; Giusi Vaira (Advanced nonlinear studies, 2008)
We study the existence of semiclassical states for a nonlinear Schrödinger-Poisson system that concentrate near critical points of the external potential and of the density charge function. We use a perturbation scheme in a variational setting, extending the results in [1]. We also discuss necessary conditions for concentration.

Convergence of equilibria of three-dimensional thin elastic beams - Maria Giovanna Mora; Stefan Müller (Proc. Roy. Soc. Edinburgh Sect. A 138 (2008) 873-896, 2008)
A convergence result is proved for the equilibrium configurations of a three-dimensional thin elastic beam, as the diameter $h$ of the cross-section tends to zero. More precisely, we show that stationary points of the nonlinear elastic functional $E^h$, whose energies (per unit cross-section) are bounded by $Ch^2$, converge to stationary points of the $\\\\varGamma$-limit of $E^h/h^2$. This corresponds to a nonlinear one-dimensional model for inextensible rods, describing bending and torsion effects. The proof is based on the rigidity estimate for low-energy deformations by Friesecke, James and Müller and on a compensated compactness argument in a singular geometry. In addition, possible concentration effects of the strain are controlled by a careful truncation argument.

Decomposition results for functions with bounded variation - Gianni Dal Maso; Rodica Toader (Boll. Unione Mat. Ital. (9) 1 (2008) 497-505, 2008)

Discerning static and causal interactions in genome-wide reverse engineering problems - Mattia Zampieri; Nicola Soranzo; Claudio Altafini (Bioinformatics 24 (2008) 1510-1515, 2008)
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.

Entire solutions of autonomous equations on Rn with nontrivial asymptotics - Andrea Malchiodi (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19 (2008) 65-72, 2008)
We prove existence of a new type of solutions for the semilinear equation $- \\\\D u + u = u^p$ on $\\\\R^n$, with $1 < p < \\\\frac{n+2}{n-2}$. These solutions are positive, bounded, decay exponentially to zero away from three half-lines with a common origin, and at infinity are asymptotically periodic.

An entropy based Glimm-type functional - Laura Caravenna (J. Hyperbolic Differ. Equ. 5 (2008) 643-662, 2008)

Equivalent definitions of asymptotic 100% B.E.C. - Alessandro Michelangeli (Nuovo Cimento B 123 (2008) 181-192, 2008)
In the mathematical analysis Bose-Einstein condensates, in particular in the study of the quantum dynamics, some kind of factorisation property has been recently proposed as a convenient technical assumption of condensation. After having surveyed both the standard definition of complete Bose-Einstein condensation in the limit of infinitely many particles and some forms of asymptotic factorisation, we prove that these characterisations are equivalent.

Eulerian calculus for the displacement convexity in the Wasserstein distance - Sara Daneri; Giuseppe Savarè (SIAM J. Math. Anal. 40 (2008) 1104-1122, 2008)
In this paper we give a new proof of the (strong) displacement convexity of a class of integral functionals defined on a compact Riemannian manifold satisfying a lower Ricci curvature bound. Our approach does not rely on existence and regularity results for optimal transport maps on Riemannian manifolds, but it is based on the Eulerian point of view recently introduced by Otto and Westdickenberg [SIAM J. Math. Anal., 37 (2005), pp. 1227-1255] and on the metric characterization of the gradient flows generated by the functionals in the Wasserstein space.

Existence of conformal metrics with constant $Q$-curvature - Zindine Djadli; Andrea Malchiodi (Ann. of Math. 168 (2008) 813-858, 2008)
Given a compact four dimensional manifold, we prove existence of conformal metrics with constant $Q$-curvature under generic assumptions. The problem amounts to solving a fourth-order nonlinear elliptic equation with variational structure. Since the corresponding Euler functional is in general unbounded from above and from below, we employ topological methods and minimax schemes, jointly with a compactness result by the second author.

Fluid–structure interaction problems in free surface flows: Application to boat dynamics - L. Formaggia; Edie Miglio; Andrea Mola; N Parolini (International Journal for Numerical Methods in Fluids, 2008)

Flux reconstruction and solution post-processing in mimetic finite difference methods - Andrea Cangiani; Manzini, Gianmarco (Comput. Methods Appl. Mech. Engrg., 2008)

Forced Vibrations of a Nonhomogeneous String - P Baldi; Massimiliano Berti (SIAM J. Math. Anal. 40 (2008) 382-412, 2008)
We prove existence of vibrations of a nonhomogeneous string under a nonlinear time periodic forcing term in the case in which the forcing frequency avoids resonances with the vibration modes of the string (nonresonant case). The proof relies on a Lyapunov-Schmidt reduction and a Nash-Moser iteration scheme.

Frobenius Manifolds and Central Invariants for the Drinfeld - Sokolov Bihamiltonian Structures - Boris Dubrovin; Liu Si-Qi; Zhang Youjin (Adv. Math. 219 (2008) 780-837, 2008)
The Drinfeld - Sokolov construction associates a hierarchy of bihamiltonian integrable systems with every untwisted affine Lie algebra. We compute the complete set of invariants of the related bihamiltonian structures with respect to the group of Miura type transformations.

Fulde-Ferrell-Larkin-Ovchinnikov pairing in one-dimensional optical lattices - Matteo Rizzi; Marco Polini; Miguel A. Cazalilla; M.R. Bakhtiari; Mario P. Tosi; Rosario Fazio (Phys. Rev. B 77 (2008) 245105, 2008)
Spin-polarized attractive Fermi gases in one-dimensional (1D) optical lattices are expected to be remarkably good candidates for the observation of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. We model these systems with an attractive Hubbard model with population imbalance. By means of the density-matrix renormalization-group method we compute the pairing correlations as well as the static spin and charge structure factors in the whole range from weak to strong coupling. We demonstrate that pairing correlations exhibit quasi-long range order and oscillations at the wave number expected from FFLO theory. However, we also show by numerically computing the mixed spin-charge static structure factor that charge and spin degrees of freedom appear to be coupled already for small imbalance. We discuss the consequences of this coupling for the observation of the FFLO phase, as well as for the stabilization of the quasi-long range order into long-range order by coupling many identical 1D systems, as in quasi-1D optical lattices.

A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds - Andrei A. Agrachev; Ugo Boscain; Mario Sigalotti (Discrete Contin. Dyn. Syst. 20 (2008) 801-822, 2008)
We consider a generalization of Riemannian geometry that naturally arises in the framework of control theory. 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 classical 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. They are special cases of rank-varying sub-Riemannian structures, which are naturally defined in terms of submodules of the space of smooth vector fields on $M$. Almost-Riemannian structures show interesting phenomena, in particular for what concerns the relation between curvature, presence of conjugate points, and topology of the manifold. The main result of the paper is a generalization to almost-Riemannian structures of the Gauss-Bonnet formula.

Globally stable quasistatic evolution in plasticity with softening - Gianni Dal Maso; Antonio DeSimone; Maria Giovanna Mora; Massimiliano Morini (Netw. Heterog. Media 3 (2008) 567-614, 2008)
We study a relaxed formulation of the quasistatic evolution problem in the context of small strain associative elastoplasticity with softening. The relaxation takes place in spaces of generalized Young measures. The notion of solution is characterized by the following properties: global stability at each time and energy balance on each\\ntime interval. An example developed in detail compares the solutions obtained by this method with the ones provided by a vanishing viscosity approximation, and shows that only the latter capture a decreasing branch in the stress-strain response.

Gradient bounds for minimizers of free discontinuity problems related to cohesive zone models in fracture mechanics - Gianni Dal Maso; Adriana Garroni (Calc. Var. Partial Differential Equations 31 (2008) 137-145, 2008)
In this note we consider a free discontinuity problem for a scalar function, whose energy depends also on the size of the jump. We prove that the gradient of every smooth local minimizer never exceeds a constant, determined only by the data of the problem.

Hamiltonian partial differential equations and Frobenius manifolds - Boris Dubrovin (Russian Mathematical Surveys. Volume 63, Issue 6, 2008, Pages 999-1010, 2008)
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.

Harish-Chandra integrals as nilpotent integrals - Marco Bertola; Ferrer, Aleix Prats (Int. Math. Res. Not. IMRN, 2008)

Instanton counting on Hirzebruch surfaces - Ugo Bruzzo; Rubik Poghossian; Alessandro Tanzini (2008)
We perform a study of the moduli space of framed torsion free sheaves on Hirzebruch surfaces by using localization techniques. After discussing general properties of this moduli space, we classify its fixed points under the appropriate toric action and compute its Poincare\\\' polynomial. From the physical viewpoint, our results provide the partition function of N=4 Vafa-Witten theory on Hirzebruch surfaces, which is relevant in black hole entropy counting problems according to a conjecture due to Ooguri, Strominger and Vafa.

Invariant Carnot-Caratheodory metrics on S3, SO(3), SL(2) and Lens Spaces - Ugo Boscain; Francesco Rossi (SIAM J. Control Optim. 47 (2008) 1851-1878, 2008)
In this paper we study the invariant Carnot-Caratheodory metrics on SU(2) \\\' S3,\\nSO(3) and SL(2) induced by their Cartan decomposition. Beside computing explicitly geodesics and conjugate loci, we compute the cut loci (globally) and we give the expression of the Carnot-Caratheodory distance as the inverse of an elementary function. We then prove that the metric\\ngiven on SU(2) projects on the so called Lens Spaces L(p; q). Also for Lens Spaces, we compute\\nthe cut loci (globally).

Invariant Manifolds for Viscous Profiles of a Class of Mixed Hyperbolic-Parabolic Systems - Stefano Bianchini; Laura Spinolo (2008)

The Isospectral Dirac Operator on the 4-dimensional Orthogonal Quantum Sphere - Francesco D'Andrea; Ludwik Dabrowski; Giovanni Landi (Comm. Math. Phys. 279 (2008) 77-116, 2008)
Equivariance under the action of Uq(so(5)) is used to compute the left regular and (chiral) spinorial representations of the algebra of the quantum Euclidean 4-sphere S^4_q. These representations are the constituents of a spectral triple on this sphere with a Dirac operator which is isospectral to the canonical one of the spin structure of the round undeformed four-sphere and which gives metric dimension four for the noncommutative geometry. Non-triviality of the geometry is proved by pairing the associated Fredholm module with an `instanton\\\' projection. A real structure which satisfies all required properties modulo a suitable ideal of `infinitesimals\\\' is also introduced.

Limit Time Optimal Syntheses for a control-affine system on S² - Paolo Mason; Rebecca Salmoni; Ugo Boscain; Yacine Chitour (SIAM J. Control Optim. 47 (2008) 111-143, 2008)
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 $U_\\\\alpha$.

On the Logarithmic Asymptotics of the Sixth Painleve\' Equation (Summer 2007) - Davide Guzzetti (J.Phys.A: Math.Theor. 41,(2008), 205201-205247, 2008)
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.
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

Minimization of non quasiconvex functionals by integro-extremization method - Sandro Zagatti (Discrete Contin. Dyn. Syst. 21 (2008) 625-641, 2008)

Minimizers of non convex scalar functionals and viscosity solutions of Hamilton-Jacobi equations - Sandro Zagatti (Calc. Var. Partial Differential Equations 31 (2008) 511-519, 2008)

Morse theory and a scalar field equation on compact surfaces - Andrea Malchiodi (Adv. Differential Equations 13 (2008) 1109-1129, 2008)

Multiple bound states for the Schroedinger-Poisson problem - Antonio Ambrosetti; David Ruiz (Commun. Contemp. Math. 10 (2008) 391-404, 2008)

Noncommutative families of instantons - Giovanni Landi; Chiara Pagani; Cesare Reina; Walter van Suijlekom (Int. Math. Res. Not. vol. 2008, Article ID rnn038, 2008)
We construct $\\\\theta$-deformations of the classical groups SL(2,H) and Sp(2). Coacting on the basic instanton on a noncommutative four-sphere $S^4_\\\\theta$, we construct a noncommutative family of instantons of charge 1. The family is parametrized by the quantum quotient of $SL_\\\\theta(2,H)$ by $Sp_\\\\theta(2)$.

The Noncommutative Geometry of the Quantum Projective Plane - Francesco D'Andrea; Ludwik Dabrowski; Giovanni Landi (Rev. Math. Phys. 20 (2008) 979-1006, 2008)
We study the spectral geometry of the quantum projective plane CP^2_q. In particular, we construct a Dirac operator which gives a 0^+ summable triple, equivariant under U_q(su(3)).

A note on the differentiability of Lipschitz functions and the chain rule in Sobolev spaces - Massimiliano Morini (2008)

Numerical study of a multiscale expansion of the Korteweg-de Vries equation and Painlevé-II equation - Tamara Grava; Christian Klein (Proc. R. Soc. A 464 (2008) 733-757, 2008)
The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order $\\\\e^2$, $\\\\e\\\\ll 1$, is characterized by the appearance of a zone of rapid modulated oscillations. These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wave-number and frequency are not constant but evolve according to the Whitham equations. Whereas the difference between the KdV and the asymptotic solution decreases as $\\\\epsilon$ in the interior of the Whitham oscillatory zone, it is known to be only of order $\\\\epsilon^{1/3}$ near the leading edge of this zone. To obtain a more accurate description near the leading edge of the oscillatory zone we present a multiscale expansion of the solution of KdV in terms of the Hastings-McLeod solution of the Painlev\\\\\\\'e-II equation. We show numerically that the resulting multiscale solution approximates the KdV solution, in the small dispersion limit, to the order $\\\\epsilon^{2/3}$.

Optimal Strokes for Low Reynolds Number Swimmers: An Example - François Alouges; Antonio DeSimone; Aline Lefebvre (J. Nonlinear Sci. 18 (2008) 277-302, 2008)
Swimming, i.e., being able to advance in the absence of external forces by performing cyclic shape changes, is particularly demanding at low Reynolds numbers. This is the regime of interest for micro-organisms and micro- or nano-robots. We focus in this paper on a simple yet representative example: the three-sphere swimmer of Najafi and Golestanian (Phys. Rev. E, 69, 062901-062904, 2004). For this system, we show how to cast the problem of swimming in the language of control theory, prove global controllability (which implies that the three-sphere swimmer can indeed swim), and propose a numerical algorithm to compute optimal strokes (which turn out to be suitably defined sub-Riemannian geodesics).

Origin of Co-Expression Patterns in E.coli and S.cerevisiae Emerging from Reverse Engineering Algorithms - Mattia Zampieri; Nicola Soranzo; Daniele Bianchini; Claudio Altafini (PLoS ONE 3 (2008) e2981, 2008)
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.

On periodic elliptic equations with gradient dependence - Massimiliano Berti; Matzeu, Michele; Valdinoci, Enrico (Commun. Pure Appl. Anal., 2008)

Positive solutions of nonlinear Schrödinger-Poisson systems with radial potentials vanishing at infinity - Mercuri, Carlo (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl, 2008)
We deal with a weighted nonlinear Schr¨odinger-Poisson system, allowing the potentials to vanish at infinity.

Relaxation of some transversally isotropic energies and applications to smectic A elastomers - James Adams; Sergio Conti; Antonio DeSimone; Georg Dolzmann (Math. Models Methods Appl. Sci. 18 (2008) 1-20, 2008)

A second order minimality condition for the Mumford-Shah functional - Filippo Cagnetti; Maria Giovanna Mora; Massimiliano Morini (Calc. Var. Partial Differential Equations 33 (2008) 37-74, 2008)
A new necessary minimality condition for the Mumford-Shah functional is derived by means of second order variations. It is expressed in terms of a sign condition for a nonlocal quadratic form on $H^1_0(\\\\Gamma)$, $\\\\Gamma$ being a submanifold of the regular part of the discontinuity set of the critical point. Two equivalent formulations are provided: one in terms of the first eigenvalue of a suitable compact operator, the other involving a sort of nonlocal capacity of $\\\\Gamma$. A sufficient condition for minimality is also deduced. Finally, an explicit example is discussed, where a complete characterization of the domains where the second variation is nonnegative can be given.

On semistable principal bundles over a complex projective manifold - Indranil Biswas; Ugo Bruzzo (Int. Math. Res. Not. vol. 2008, article ID rnn035, 2008)
Let G be a simple linear algebraic group defined over the complex numbers. Fix a proper parabolic subgroup P of G and a nontrivial antidominant character \\\\chi of P. We prove that a holomorphic principal G-bundle E over a connected complex projective manifold M is semistable and the second Chern class of its adjoint bundle vanishes in rational cohomology if and only if the line bundle over E/P defined by \\\\chi is numerically effective. Similar results remain valid for principal bundles with a reductive linear algebraic group as the structure group. These generalize an earlier work of Y. Miyaoka where he gave a characterization of semistable vector bundles over a smooth projective curve. Using these characterizations one can also produce similar criteria for the semistability of parabolic principal bundles over a compact Riemann surface.

Solitons of linearly coupled systems of semilinear non-autonomous equations on Rn - Antonio Ambrosetti; Giovanna Cerami; David Ruiz (J. Funct. Anal. 254 (2008) 2816-2845, 2008)
Using concentration compactness type arguments, we prove some results about the existence of positive ground and bound state of linearly coupled systems of nonlinear Schrödinger equations.

Stability of planar switched systems: the nondiagonalizable case - Ugo Boscain; Moussa Balde (Commun. Pure Appl. Anal. 7 (2008) 1-21, 2008)

Symmetric obstruction theories and Hilbert schemes of points on threefolds - Kai Behrend; Barbara Fantechi (Algebra Number Theory 2 (2008) 313-345, 2008)
In an earlier paper by one of us (Behrend), Donaldson-Thomas type invariants were expressed as certain weighted Euler characteristics of the moduli space. The Euler characteristic is weighted by a certain canonical\\nZ-valued constructible function on the moduli space. This constructible function associates to\\nany point of the moduli space a certain invariant of the singularity of the space at the point. Here we evaluate this invariant for the case of a singularity that is an isolated point of a C∗-action and that admits a symmetric obstruction theory compatible with the C∗-action. The answer is (-1)d, where d\\nis the dimension of the Zariski tangent space. We use this result to prove that for any threefold, proper or not, the weighted Euler characteristic of the Hilbert scheme of n points on the threefold is, up to sign, equal to the usual Euler characteristic. For the case of a projective Calabi-Yau threefold, we deduce that the Donaldson-Thomas invariant of the Hilbert scheme of n points is, up to sign, equal to the Euler characteristic. This proves a conjecture of Maulik, Nekrasov, Okounkov and Pandharipande.

Symmetries of noncommutative spaces and equivariant cohomology - Lucio Cirio (SISSA, 2008)
As the title suggests, the main subject of this thesis is the study of symmetries of noncommutative spaces and related equivariant cohomologies. We focus on deformations of classical geometries coming from the action of some symmetry. A close relation between the deformation of the symmetry and the deformation of the space on which it acts is at the heart of our approach; we will use this idea to generate noncommutative geometries, and to de¯ne algebraic models for the equivariant cohomology of such actions.
Keywords: Noncommutative spaces

Topological Gauge Theories on Local Spaces and Black Hole Entropy Countings - Giulio Bonelli; Alessandro Tanzini (Adv. Theor. Math. Phys. 12 (2008) 1429-1446, 2008)
We study cohomological gauge theories on total spaces of holomorphic line bundles over complex manifolds and obtain their reduction to the base manifold by U(1) equivariant localization of the path integral. We exemplify this general mechanism by proving via exact path integral localization a reduction for local curves conjectured in hep-th/0411280, relevant to the calculation of black hole entropy/Gromov-Witten invariants. Agreement with the four-dimensional gauge theory is recovered by taking into account in the latter non-trivial contributions coming from one-loop fluctuations determinants at the boundary of the total space. We also study a class of abelian gauge theories on Calabi-Yau local surfaces, describing the quantum foam for the A-model, relevant to the calculation of Donaldson-Thomas invariants.

Topological methods for an elliptic equation with exponential nonlinearities - Andrea Malchiodi (Discrete Contin. Dyn. Syst. 21 (2008) 277-294, 2008)
We consider a class of variational equations with exponential nonlinearities on compact surfaces. From considerations involving the Moser-Trudinger inequality, we characterize some sublevels of the Euler-Lagrange functional in terms of the topology of the surface and of the data of the equation. This is used together with a min-max argument to obtain existence results.

Transition layer for the heterogeneous Allen-Cahn equation - Fethi Mahmoudi; Andrea Malchiodi; Juncheng Wei (Ann. Inst. H. Poincare Anal. Non Lineaire 25 (2008) 609-631, 2008)
We consider the equation $\\\\e^{2}\\\\Delta u=(u-a(x))(u^2-1)$ in $\\\\Omega$, $\\\\frac{\\\\partial u}{\\\\partial \\\\nu} =0$ on $\\\\partial \\\\Omega$, where $\\\\Omega$ is a smooth and bounded domain in $\\\\R^n$, $\\\\nu$ the outer unit normal to $\\\\pa\\\\Omega$, and $a$ a smooth function satisfying $-1<a(x)<1$ in $\\\\ov{\\\\Omega}$. We set $K$, $\\\\Omega_+$ and $\\\\Omega_-$ to be respectively the zero-level set of $a$, {a>0} and {a<0}. Assuming $\\\\nabla a \\\\neq 0$ on $K$ and $a\\\\ne 0$ on $\\\\partial \\\\Omega$, we show that there exists a sequence $\\\\e_j \\\\to 0$ such that the above equation has a solution $u_{\\\\e_j}$ which converges uniformly to $\\\\pm 1$ on the compact sets of $\\\\O_{\\\\pm}$ as $j \\\\to + \\\\infty$.

Transport Rays and Applications to Hamilton–Jacobi Equations - Stefano Bianchini; Matteo Gloyer (Nonlinear PDE’s and Applications : C.I.M.E. Summer School, Cetraro, Italy 2008 / Stefano Bianchini, Eric A. Carlen, Alexander Mielke, Cédric Villani. Eds. Luigi Ambrosio, Giuseppe Savaré. - Berlin : Springer, 2011. - (Lecture Notes in Mathematics ; 20, 2008)
The aim of these notes is to introduce the readers to the use of the Disintegration Theorem for measures as an effective tool for reducing problems in transport equations to simpler ones. The basic idea is to partition Rd into one dimensional sets, on which the problem under consideration becomes one space dimensional (and thus much easier, hopefully).
This volume collects the notes of the CIME course Nonlinear PDE’s and\\r\\napplications held in Cetraro (Italy) on June 23–28, 2008. The school consisted\\r\\nin 5 series of lectures, delivered by Stefano Bianchini (SISSA, Trieste), Eric A. Carlen (Rutgers University), Alexander Mielke (WIAS, Berlin), Felix Otto (Bonn University), Cedric Villani (Ecole Normale Superieure de Lyon).

A vanishing viscosity approach to quasistatic evolution in plasticity with softening - Gianni Dal Maso; Antonio DeSimone; Maria Giovanna Mora; Massimiliano Morini (Arch. Ration. Mech. Anal. 189 (2008) 469-544, 2008)
We deal with quasistatic evolution problems in plasticity with softening, in the framework of small strain associative elastoplasticity. The presence of a nonconvex term due to the softening phenomenon requires a nontrivial extension of the variational framework for rate-independent problems to the case of a nonconvex energy functional. We argue that, in this case, the use of global minimizers in the corresponding incremental problems is not justified from the mechanical point of view. Thus, we analize a different selection criterion for the solutions of the quasistatic evolution problem, based on a viscous approximation. This leads to a generalized formulation in terms of Young measures, developed in the first part of the paper. In the second part we apply our approach to some concrete examples.

Variational methods for Hamiltonian PDEs - Massimiliano Berti (NATO Science for Peace and Security Series B: Physics and Biophysics, 2008)
We present recent existence results of periodic solutions for completely resonant nonlinear wave equations in which both "small divisor" difficulties and infinite dimensional bifurcation phenomena occur. These results can be seen as generalizations of the classical finite-dimensional resonant center theorems of Weinstein-Moser and Fadell-Rabinowitz. The proofs are based on variational bifurcation theory: after a Lyapunov-Schmidt reduction, the small divisor problem in the range equation is overcome with a Nash-Moser implicit function theorem for a Cantor set of non-resonant parameters. Next, the infinite dimensional bifurcation equation, variational in nature, possesses minimax mountain-pass critical points. The big difficulty is to ensure that they are not in the "Cantor gaps". This is proved under weak non-degeneracy conditions. Finally, we also discuss the existence of forced vibrations with rational frequency. This problem requires variational methods of a completely different nature, such as constrained minimization and a priori estimates derivable from variational inequalities. © 2008 Springer Science + Business Media B.V.
cited By (since 1996)0

Asymptotic behaviour of smooth solutions for partially dissipative hyperbolic systems with a convex entropy - Stefano Bianchini; Bernard Hanouzet; Roberto Natalini (Comm. Pure Appl. Math. 60 (2007) 1559-1622, 2007)
We study the asymptotic time behavior of global smooth solutions to general entropy dissipative hyperbolic systems of balance law in m space dimensions, under the Shizuta-Kawashima condition.

The Asymptotic Behaviour of the Fourier Transforms of Orthogonal Polynomials II: L.I.F.S. Measures and Quantum Mechanics - Davide Guzzetti; Giorgio Mantica (Ann. Henri Poincar´e 8 (2007), 301–336, 2007)
We study measures generated by systems of linear iterated functions,\r\ntheir Fourier transforms, and those of their orthogonal polynomials. We\r\ncharacterize the asymptotic behaviours of their discrete and continuous averages.\r\nFurther related quantities are analyzed, and relevance of this analysis\r\nto quantum mechanics is briefly discussed

Asymptotic variational wave equations - Alberto Bressan; Zhang Ping; Zheng Yuxi (Arch. Ration. Mech. Anal. 183 (2007) 163-185, 2007)
We investigate the equation $(u_t + (f(u))_x)_x = f\\\'\\\'(u) (u_x)^2/2$ where $f(u)$ is a given smooth function. Typically $f(u)= u^2/2$ or $u^3/3$. This equation models unidirectional and weakly nonlinear waves for the variational wave equation $u_{tt} - c(u) (c(u)u_x)_x =0$ which models some liquid crystals with a natural sinusoidal $c$. The equation itself is also the Euler-Lagrange equation of a variational problem. Two natural classes of solutions can be associated with this equation. A conservative solution will preserve its energy in time, while a dissipative weak solution loses energy at the time when singularities appear. Conservative solutions are globally defined, forward and backward in time, and preserve interesting geometric features, such as the Hamiltonian structure. On the other hand, dissipative solutions appear to be more natural from the physical point of view.\\nWe establish the well-posedness of the Cauchy problem within the class of conservative solutions, for initial data having finite energy and assuming that the flux function $f$ has Lipschitz continuous second-order derivative. In the case where $f$ is convex, the Cauchy problem is well-posed also within the class of dissipative solutions. However, when $f$ is not convex, we show that the dissipative solutions do not depend continuously on the initial data.

Biorthogonal Laurent polynomials, Töplitz determinants, minimal Toda orbits and isomonodromic tau functions - Marco Bertola; Gekhtman, M. (Constr. Approx., 2007)

Biorthogonal polynomials for two-matrix models with semiclassical potentials - Marco Bertola (J. Approx. Theory, 2007)

Black Holes, Instanton Counting on Toric Singularities and q-Deformed Two-Dimensional Yang-Mills Theory - Luca Griguolo; Domenico Seminara; Richard J. Szabo; Alessandro Tanzini (Nucl. Phys. B 772 (2007) 1-24, 2007)
We study the relationship between instanton counting in N=4 Yang-Mills theory on a generic four-dimensional toric orbifold and the semi-classical expansion of q-deformed Yang-Mills theory on the blowups of the minimal resolution of the orbifold singularity, with an eye to clarifying the recent proposal of using two-dimensional gauge theories to count microstates of black holes in four dimensions. We describe explicitly the instanton contributions to the counting of D-brane bound states which are captured by the two-dimensional gauge theory. We derive an intimate relationship between the two-dimensional Yang-Mills theory and Chern-Simons theory on generic Lens spaces, and use it to show that the correct instanton counting is only reproduced when the Chern-Simons contributions are treated as non-dynamical boundary conditions in the D4-brane gauge theory. We also use this correspondence to discuss the counting of instantons on higher genus ruled Riemann surfaces.

Bose-Einstein condensation: analysis of problems and rigorous results - Alessandro Michelangeli (2007)

Boundary interface for the Allen-Cahn equation - Andrea Malchiodi; Juncheng Wei (J. Fixed Point Theory Appl. 1 (2007) 305-336, 2007)

Boundary-clustered interfaces for the Allen–Cahn equation - Andrea Malchiodi; Wei-Ming Ni; Juncheng Wei (Pacific Journal of Mathematics 229 (2007), No. 2, 447–468, 2007)

BV instability for the Lax-Friedrichs scheme - Paolo Baiti; Alberto Bressan; Helge Kristian Jenssen (2007)
It is proved that discrete shock profiles (DSPs) for the Lax-Friedrichs scheme for a system of conservation laws do not necessarily depend continuously in BV on their speed. We construct examples of $2 \\\\times 2$-systems for which there are sequences of DSPs with speeds converging to a rational number. Due to a resonance phenomenon, the difference between the limiting DSP and any DSP in the sequence will contain an order-one amount of variation.

Canonical structure and symmetries of the Schlesinger equations - Boris Dubrovin; Marta Mazzocco (Comm. Math. Phys. 271 (2007) 289-373, 2007)
The Schlesinger equations S (n,m) describe monodromy preserving deformations of order m Fuchsian systems with n+1 poles. They can be considered as a family of commuting time-dependent Hamiltonian systems on the direct product of n copies of m×m matrix algebras equipped with the standard linear Poisson bracket. In this paper we present a new canonical Hamiltonian formulation ofthe general Schlesinger equations S (n,m) for all n, m and we compute the action of the symmetries of the Schlesinger equations in these coordinates.

Chen-Ruan cohomology of ADE singularities - Fabio Perroni (International Journal of Mathematics. Volume 18, Issue 9, October 2007, Pages 1009-1059, 2007)
We study Ruan\'s \\textit{cohomological crepant resolution conjecture} for\r\norbifolds with transversal ADE singularities. In the $A_n$-case we compute both\r\nthe Chen-Ruan cohomology ring $H^*_{\\rm CR}([Y])$ and the quantum corrected\r\ncohomology ring $H^*(Z)(q_1,...,q_n)$. The former is achieved in general, the\r\nlater up to some additional, technical assumptions. We construct an explicit\r\nisomorphism between $H^*_{\\rm CR}([Y])$ and $H^*(Z)(-1)$ in the $A_1$-case,\r\nverifying Ruan\'s conjecture. In the $A_n$-case, the family\r\n$H^*(Z)(q_1,...,q_n)$ is not defined for $q_1=...=q_n=-1$. This implies that\r\nthe conjecture should be slightly modified. We propose a new conjecture in the\r\n$A_n$-case which we prove in the $A_2$-case by constructing an explicit\r\nisomorphism.
This is a short version of my Ph.D. Thesis math.AG/0510528. Version\r\n 2: chapters 2,3,4 and 5 has been rewritten using the language of groupoids; a\r\n link with the classical McKay correpondence is given. International Journal\r\n of Mathematics (to appear)
Keywords: Chen-Ruan cohomology, Ruan\'s conjecture, McKay correspondence

The cohomological crepant resolution conjecture for P(1,3,4,4) - Samuel Boissiere; Fabio Perroni; Etienne Mann (SISSA, 2007)
We prove the cohomological crepant resolution conjecture of Ruan for the\r\nweighted projective space P(1,3,4,4). To compute the quantum corrected\r\ncohomology ring we combine the results of Coates-Corti-Iritani-Tseng on\r\nP(1,1,1,3) and our previous results.
11 pages, 1 figure

Comparing association network algorithms for reverse engineering of large scale gene regulatory networks: synthetic vs real data - Nicola Soranzo; Ginestra Bianconi; Claudio Altafini (Bioinformatics 23 (2007) 1640-1647, 2007)
Motivation: Inferring a gene regulatory network exclusively from microarray expression profiles is a difficult but important task. The aim of this work is to compare the predictive power of some of the most popular algorithms in different conditions (like data taken at equilibrium or time courses) and on both synthetic and real microarray data. We are in particular interested in comparing similarity measures both of linear type (like correlations and partial correlations) and of nonlinear type (mutual information and conditional mutual information), and in investigating the underdetermined case (less samples than genes). Results: In our simulations we see that all network inference algorithms obtain better performances from data produced with \\\"structural\\\" perturbations, like gene knockouts at steady state, than with any dynamical perturbation. The predictive power of all algorithms is confirmed on a reverse engineering problem from E. coli gene profiling data: the edges of the \\\"physical\\\" network of transcription factor-binding sites are significantly overrepresented among the highest weighting edges of the graph that we infer directly from the data without any structure supervision. Comparing synthetic and in vivo data on the same network graph allows us to give an indication of how much more complex a real transcriptional regulation program is with respect to an artificial model. Availability: Software and supplementary material are freely available at the URL http://people.sissa.it/~altafini/papers/SoBiAl07/

The complete one-loop spin chain for N=2 Super-Yang-Mills - Paolo Di Vecchia; Alessandro Tanzini (2007)
We show that the complete planar one-loop mixing matrix of the N=2 Super Yang--Mills theory can be obtained from a reduction of that of the N=4 theory. For composite operators of scalar fields, this yields an anisotropic XXZ spin chain, whose spectrum of excitations displays a mass gap.
Contributed to NATO Advanced Study Institute and EC Summer School on String Theory: From Gauge Interactions to Cosmology, Cargese, France, 7-19 Jun 2004.

Computing Amplitudes in topological M-theory - Giulio Bonelli; Alessandro Tanzini; Maxim Zabzine (JHEP 03 (2007) 023, 2007)
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.

Concentration on minimal submanifolds for a singularly perturbed Neumann problem - Fethi Mahmoudi; Andrea Malchiodi (Adv. Math. 209 (2007) 460-525, 2007)
We consider the equation $- \\\\e^2 \\\\D u + u= u^p$ in $\\\\Omega \\\\subseteq \\\\R^N$, where $\\\\Omega$ is open, smooth and bounded, and we prove concentration of solutions along $k$-dimensional minimal submanifolds of $\\\\partial \\\\O$, for $N \\\\geq 3$ and for $k \\\\in \\\\{1, ..., N-2\\\\}$. We impose Neumann boundary conditions, assuming $1<p <\\\\frac{N-k+2}{N-k-2}$ and $\\\\e \\\\to 0^+$. This result settles in full generality a phenomenon previously considered only in the particular case $N = 3$ and $k = 1$.

Concentration phenomena for nonlinear Schrödinger equations: Recent results and new perspectives - Antonio Ambrosetti; Andrea Malchiodi (Perspectives in Nonlinear Partial Differential Equations: In Honor of Haïm Brezis / ed. by Henri Beresticky [et al.]. - Providence : American Mathematical Society, 2007. - (Contemporary mathematics ; 446). - p. 19-30, 2007)
We survey some results on (NLSepsilon), discussing also new perspectives and open problems.

Conformal Metrics with Constant Q-Curvature - Andrea Malchiodi (SIGMA 3 (2007) 120, 2007)
We consider the problem of varying conformally the metric of a four dimensional manifold in order to obtain constant $Q$-curvature. The problem is variational, and solutions are in general found as critical points of saddle type. We show how the problem leads naturally to consider the set of formal barycenters of the manifold.
This paper is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of\\nThomas P. Branson.

Convergence of equilibria of planar thin elastic beams - Maria Giovanna Mora; Stefan Müller; Maximilian G. Schultz (Indiana Univ. Math. J. 56 (2007) 2413-2438, 2007)
We consider a thin elastic strip of thickness h and we show that stationary points of the nonlinear elastic energy (per unit height) whose energy is of order h^2 converge to stationary points of the Euler-Bernoulli functional. The proof uses the rigidity estimate for low-energy deformations by Friesecke, James, and Mueller (Comm. Pure Appl. Math. 2002), and a compensated compactness argument in a singular geometry. In addition, possible concentration effects are ruled out by a careful truncation argument.

Critical voltages and blocking stresses in nematic gels - Antonio DeSimone; Antonio Di Carlo; Luciano Teresi (Eur. Phys. J. E 24 (2007) 303-310, 2007)
We use a recently proposed model to study the dynamics of director remodeling in nematic gels under combined electro-mechanical loading. Focusing on a model specimen, we describe the critical volt-ages that must be exceeded to achieve mesogen reorientation, and the blocking stresses that prevent alignment of the nematic mesogens with the electric field. The corresponding phase diagram shows that the dynamic thresholds defined above are different from those predicted on the sole basis of energetics. Multistep loading programs are used to explore the energy landscape of our model specimen, showing the existence of multiple local minima under the same voltage and applied stress. This leads us to conclude that hysteresis should be expected in the electro-mechanical response of nematic gels.

Derivation of a rod theory for phase-transforming materials - Maria Giovanna Mora; Stefan Müller (Calc. Var. Partial Differential Equations 28 (2007) 161-178, 2007)
A rigorous derivation is given of a rod theory for a multiphase material,starting from three-dimensional nonlinear elasticity. The stored energy density is supposed to be nonnegative and to vanish exactly on a set consisting of two copies of the group of rotations SO(3). The two potential wells correspond to the two crystalline configurations preferred by the material. We find the optimal scaling of the energy in terms of the diameter of the rod and we identify the limit, as the diameter goes to zero, in the sense of Gamma-convergence.

Dirac operators on all Podles quantum spheres - Francesco D'Andrea; Ludwik Dabrowski; Giovanni Landi; Elmar Wagner (J. Noncomm. Geom. 1 (2007) 213-239, 2007)
We construct spectral triples on all Podles quantum spheres. These noncommutative geometries are equivariant for a left action of $U_q(su(2))$ and are regular, even and of metric dimension 2. They are all isospectral to the undeformed round geometry of the 2-sphere. There is also an equivariant real structure for which both the commutant property and the first order condition for the Dirac operators are valid up to infinitesimals of arbitrary order.

Effective inverse spectral problem for rational Lax matrices and applications - Marco Bertola; Gekhtman, M. (Int. Math. Res. Not. IMRN, 2007)

Equilibrium configurations of epitaxially strained crystalline films: existence and regularity results - Irene Fonseca; Nicola Fusco; Giovanni Leoni; Massimiliano Morini (Arch. Ration. Mech. Anal. 186 (2007) 477-537, 2007)
Strained epitaxial films grown on a relatively thick substrate are considered in the context of plane linear elasticity. The total free energy of the system is assumed to be the sum of the energy of the free surface of the film and the strain energy. Because of the lattice mismatch between film and substrate, flat configurations are in general energetically unfavorable and a corrugated or islanded morphology is the preferred growth mode of the strained film. After specifying the functional setup in which the existence problem can be properly framed, a study of the qualitative properties of the solutions is undertaken. New regularity results for volume-constrained local minimizers of the total free energy are established, leading, as a byproduct, to a rigorous proof of the zero-contact-angle condition between islands and wetting layers.

On the Euler-Lagrange equation for a variational problem - Stefano Bianchini (Discrete Contin. Dynam. Systems A 17 (2007) 449-480, 2007)

On feedback classification of control-affine systems with one and two-dimensional inputs - Andrei A. Agrachev; Igor Zelenko (SIAM J. Control Optim. 46 (2007) 1431-1460, 2007)
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.

Feedback control of spin systems - Claudio Altafini (Quantum Inf. Process. 6 (2007) 9-36, 2007)
The feedback stabilization problem for ensembles of coupled spin 1/2 systems is discussed from a control theoretic perspective. The noninvasive nature of the bulk measurement allows for a fully unitary and deterministic closed loop. The Lyapunov-based feedback design presented does not require spins that are selectively addressable. With this method, it is possible to obtain control inputs also for difficult tasks, like suppressing undesired couplings in identical spin systems.
Proc. of the 44th IEEE Conf. on Decision and Control,. Seville, Spain, December 2005 (to appear).

Feedback stabilization of quantum ensembles: a global convergence analysis on complex flag manifolds - Claudio Altafini (45th IEEE Conference on Decision and Control (2007) 2471-2476, 2007)
In an N-level quantum mechanical system, the problem of unitary feedback stabilization of mixed density operators to periodic orbits admits a natural Lyapunov-based time-varying feedback design. A global description of the domain of attraction of the closed-loop system can be provided based on a \\\"root-space\\\"-like structure of the space of density operators. This convex set foliates as a complex flag manifold where each leaf is identified with the coadjoint orbit of the eigenvalues of the density operator. The converging conditions are time-independent but depend from the topology of the flag manifold: it is shown that the closed loop must have a number of equilibria at least equal to the Euler characteristic of the manifold, thus imposing obstructions of topological nature to global stabilizability.

On finite-dimensional projections of distributions for solutions of randomly forced PDE\\\'s - Andrei A. Agrachev; Sergei Kuksin; Andrey Sarychev; Armen Shirikyan (Ann. Inst. Henri Poincare-Prob. Stat. 43 (2007) 399-415, 2007)
The paper is devoted to studying the image of probability measures on a Hilbert space under finite-dimensional analytic maps. We establish sufficient conditions under which the image of a measure has a density with respect to the Lebesgue measure and continuously depends on the map. The results obtained are applied to the 2D Navier-Stokes equations perturbed by various random forces of low dimension.

Gaussian estimates for hypoelliptic operators via optimal control - Ugo Boscain; Sergio Polidoro (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 18 (2007) 333-342, 2007)
We obtain Gaussian lower bounds for the fundamental solution of a class of hypoelliptic equations, by using repeatedly an invariant Harnack inequality. Our main result is given in terms of the value function of a suitable optimal control problem.

High-order angles in almost-Riemannian geometry - Ugo Boscain; Mario Sigalotti (2007)
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.

The holomorphic anomaly for open string moduli - Giulio Bonelli; Alessandro Tanzini (JHEP 10 (2007) 060, 2007)
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.

Luther-Emery Phase and Atomic-Density Waves in a Trapped Fermion Gas - Gao Xianlong; Matteo Rizzi; Marco Polini; Rosario Fazio; Mario P. Tosi; Vivaldo L. Jr. Campo; Klaus Capelle (Phys. Rev. Lett. 98 (2007) 030404, 2007)
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.

Massless scalar field in a two-dimensional de Sitter universe - Marco Bertola; Corbetta, Francesco; Moschella, Ugo (Rigorous quantum field theory, 2007)

On the Maz\\\'ya inequalities: existence and multiplicity results for an elliptic problem involving cylindrical weights - Marita Gazzini; Roberta Musina (2007)

Metrics on semistable and numerically effective Higgs bundles - Ugo Bruzzo; Beatriz Grana-Otero (J. Reine Angew. Math. 612 (2007) 59-79, 2007)
We consider fibre metrics on Higgs vector bundles on compact K\\\\\\\"ahler manifolds, providing notions of numerical effectiveness and numerical flatness in terms of such metrics. We prove several properties of bundles satisfying such conditions and in particular we show that numerically flat Higgs bundles have vanishing Chern classes, and that they admit filtrations whose quotients are stable flat Higgs bundles. We compare these definitions with those previously given in the case of projective varieties. Finally we study the relations between numerically effectiveness and semistability, providing semistability criteria for Higgs bundles on projective manifolds of any dimension.

Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations - Antonio Ambrosetti; Eduardo Colorado; David Ruiz (Calc. Var. Partial Differential Equations 30 (2007) 85-112, 2007)

Nearly time optimal stabilizing patchy feedbacks - Fabio Ancona; Alberto Bressan (Ann. Inst. H. Poincare Anal. Non Lineaire 24 (2007) 279-310, 2007)
We consider the time optimal stabilization problem for a nonlinear control system $\\\\dot x=f(x,u)$. Let $\\\\tau(y)$ be the minimum time needed to steer the system from the state $y\\\\in\\\\R^n$ to the origin, and call $\\\\A(T)$ the set of initial states that can be steered to the origin in time $\\\\tau(y)\\\\leq T$. Given any $\\\\ve>0$, in this paper we construct a patchy feedback $u=U(x)$ such that every solution of $\\\\dot x=f(x, U(x))$, $x(0)=y\\\\in \\\\A(T)$ reaches an $\\\\ve$-neighborhood of the origin within time $\\\\tau(y)+\\\\ve$.

Necessary and sufficient conditions for the chainrule in W1,1loc(RN;Rd) and BVloc(RN;Rd) - Giovanni Leoni; Massimiliano Morini (J. Eur. Math. Soc. (JEMS) 9 (2007) 219-252, 2007)
In this paper we prove necessary and sufficient conditions for the validity of the classical chain rule in Sobolev spaces and in the space of functions of bounded variation.

A new model for contact angle hysteresis - Antonio DeSimone; Natalie Gruenewald; Felix Otto (Netw. Heterog. Media 2 (2007) 211-225, 2007)
We present a model which explains several experimental observations relating contact angle hysteresis with surface roughness. The model is based on the balance between released energy and dissipation, and it describes the stick-slip behavior of drops on a rough surface using ideas similar to those employed in dry friction, elasto-plasticity and fracture mechanics. The main results of our analysis are formulas giving the interval of stable contact angles as a function of the surface roughness. These formulas show that the difference between advancing and receding angles is much larger for a drop in complete contact with the substrate (Wenzel drop) than for one whose cavities are filled with air (Cassie-Baxter drop). This fact is used as the key tool to interpret the experimental evidence.

Noncommutative geometry and quantum group symmetries - Francesco D'Andrea (SISSA, 2007)
It is a widespread belief that mathematics originates from the desire to understand (and eventually to formalize) some aspects of the real world. Quoting [Man07], «we are doing mathematics in order to understand, create, and handle things, and perhaps this understanding is mathematics» . Let me thus begin with a brief discussion of the physical ideas that motivated the development of Noncommutative Geometry and Quantum Group Theory - the areas of mathematics to which this dissertation belongs. Some physicists believe, and Einstein himself expressed this view in [Ein98a], that physics progresses in stages: there is no `final\\\' theory of Nature, but simply a sequence of theories which provide more and more accurate descriptions of the real world...
Keywords: Noncommutative geometry

Nonlinear oscillations of Hamiltonian PDEs - Massimiliano Berti (Progress in Nonlinear Differential Equations and their Applications, 2007)

Nonlinear vibrations of completely resonant wave equations - Massimiliano Berti (Fixed point theory and its applications, 2007)

On a notion of unilateral slope for the Mumford-Shah functional - Gianni Dal Maso; Rodica Toader (NoDEA 13 (2007) 713-734, 2007)
In this paper we introduce a notion of unilateral slope for the Mumford-Shah functional, and provide an explicit formula in the case of smooth cracks. We show that the slope is not lower semicontinuous and study the corresponding relaxed functional.

The number of eigenvalues of three-particle Schrödinger operators on lattices - Sergio Albeverio; Gianfausto Dell'Antonio; Saidakhmat N. Lakaev (J. Phys. A 40 (2007) 14819-14842, 2007)
We consider the Hamiltonian of a system of three quantum mechanical particles (two identical fermions and boson)on the three-dimensional lattice $\\\\Z^3$ and interacting by means of zero-range attractive potentials. We describe the location and structure of the essential spectrum of the three-particle discrete Schr\\\\\\\"{o}dinger operator $H_{\\\\gamma}(K),$ $K$ being the total quasi-momentum and $\\\\gamma>0$ the ratio of the mass of fermion and boson.\\nWe choose for $\\\\gamma>0$ the interaction $v(\\\\gamma)$ in such a way the system consisting of one fermion and one boson has a zero energy resonance.\\nWe prove for any $\\\\gamma> 0$ the existence infinitely many eigenvalues of the operator $H_{\\\\gamma}(0).$ We establish for the number $N(0,\\\\gamma; z;)$ of eigenvalues lying below $z<0$ the following asymptotics $$ \\\\lim_{z\\\\to 0-}\\\\frac{N(0,\\\\gamma;z)}{\\\\mid \\\\log \\\\mid z\\\\mid \\\\mid}={U} (\\\\gamma) .$$ Moreover, for all nonzero values of the quasi-momentum $K \\\\in T^3 $ we establish the finiteness of the number $ N(K,\\\\gamma;\\\\tau_{ess}(K))$ of eigenvalues of $H(K)$ below the bottom of the essential spectrum and we give an asymptotics for the number $N(K,\\\\gamma;0)$ of eigenvalues below zero.

Numerical solution of the small dispersion limit of Korteweg de Vries and Whitham equations - Tamara Grava; Christian Klein (Comm. Pure Appl. Math. 60 (2007) 1623-1664, 2007)
The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order $\\\\epsilon^2$, is characterized by the appearance of a zone of rapid modulated oscillations of wave-length of order $\\\\epsilon$. These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wave-number and frequency are not constant but evolve according to the Whitham equations. In this manuscript we give a quantitative analysis of the discrepancy between the numerical solution of the KdV equation in the small dispersion limit and the corresponding approximate solution for values of $\\\\epsilon$ between $10^{-1}$ and $10^{-3}$. The numerical results are compatible with a difference of order $\\\\epsilon$ within the `interior\\\' of the Whitham oscillatory zone, of order $\\\\epsilon^{1/3}$ at the left boundary outside the Whitham zone and of order $\\\\epsilon^{1/2}$ at the right boundary outside the Whitham zone.

Numerical study of a multiscale expansion of KdV and Camassa-Holm equation - Tamara Grava; Christian Klein (2007)
We study numerically solutions to the Korteweg-de Vries and Camassa-Holm equation close to the breakup of the corresponding solution to the dispersionless equation. The solutions are compared with the properly rescaled numerical solution to a fourth order ordinary differential equation, the second member of the Painlev\\\\\\\'e I hierarchy. It is shown that this solution gives a valid asymptotic description of the solutions close to breakup. We present a detailed analysis of the situation and compare the Korteweg-de Vries solution quantitatively with asymptotic solutions obtained via the solution of the Hopf and the Whitham equations. We give a qualitative analysis for the Camassa-Holm equation

Numerically flat Higgs vector bundles - Ugo Bruzzo; Beatriz Grana-Otero (Commun. Contemp. Math. 9 (2007) 437-446, 2007)
After providing a suitable definition of numerical effectiveness for Higgs bundles, and a related notion of numerical flatness, in this paper we prove, together with some side results, that all Chern classes of a Higgs-numerically flat Higgs bundle vanish, and that a Higgs bundle is Higgs-numerically flat if and only if it is has a filtration whose quotients are flat stable Higgs bundles. We also study the relation between these numerical properties of Higgs bundles and (semi)stability.

Parametrized curves in Lagrange Grassmannians - Igor Zelenko; Li Chengbo (C. R. Math. 345 (2007) 647-652, 2007)

Perturbation techniques applied to the real vanishing viscosity approximation of an initial boundary value problem - Stefano Bianchini (SISSA, 2007)

Quasistatic crack growth for a cohesive zone model with prescribed crack path - Gianni Dal Maso; Chiara Zanini (Proc. Roy. Soc. Edinburgh Sect. A 137 (2007) 253-279, 2007)
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.

Quasistatic evolution problems for pressure-sensitive plastic materials - Gianni Dal Maso; Alexey Demyanov; Antonio DeSimone (Milan J. Math. 75 (2007) 117-134, 2007)
We study quasistatic evolution problems for pressure-sensitive plastic materials in the context of small strain associative perfect plasticity.

Reciprocal transformations and flat metrics on Hurwitz spaces - Simonetta Abenda; Tamara Grava (J. Phys. A 40 (2007) 10769-10790, 2007)
We consider hydrodynamic systems which possess a local Hamiltonian structure of Dubrovin-Novikov type. To such a system there are also associated an infinite number of nonlocal Hamiltonian structures. We give necessary and sufficient conditions so that, after a nonlinear transformation of the independent variables, the reciprocal system still possesses a local Hamiltonian structure of Dubrovin-Novikov type. We show that, under our hypotheses, bi-hamiltonicity is preserved by the reciprocal transformation. Finally we apply such results to reciprocal systems of genus g Whitham-KdV modulation equations.

Reduced density matrices and Bose-Einstein condensation - Alessandro Michelangeli (2007)
Emergence and applications of the ubiquitous tool of reduced density matrices in the rigorous analysis of Bose Einstein condensation is reviewed, and new related results are added. The need and the nature of scaling limits of infinitely many particles is discussed, which imposes that a physically meaningful and mathematically well-posed definition of asymptotic condensation is placed at the level of marginals.\\nThe topic of correlations in the condensed state is addressed in order to show their influence at this level of marginals, both in the true condensed state and in the suitable trial functions one introduces to approximate the many-body structure and energy. Complete condensation is shown to be equivalently defined at any fixed k-body level, both for pure and mixed states. Further, it is proven to be equivalent to some other characterizations in terms of asymptotic factorization of the many-body state, which are currently present in the literature.

On the reductions and classical solutions of the Schlesinger equations - Boris Dubrovin; Marta Mazzocco (Differential equations and quantum groups, IRMA Lect. Math. Theor. Phys. 9 (2007) 157-187, 2007)
The Schlesinger equations S(n,m) describe monodromy preserving\\r\\ndeformations of order m Fuchsian systems with n+1 poles. They\\r\\ncan be considered as a family of commuting time-dependent Hamiltonian\\r\\nsystems on the direct product of n copies of m×m matrix algebras\\r\\nequipped with the standard linear Poisson bracket. In this paper we address\\r\\nthe problem of reduction of particular solutions of “more complicated”\\r\\nSchlesinger equations S(n,m) to “simpler” S(n′,m′) having n′ < n\\r\\nor m′ < m.

On the regularity of weak solutions to H-systems - Roberta Musina (Atti .Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 18 (2007) 209-219, 2007)
Abstract. In this paper we prove that every weak solution to the H-surface equation is locally bounded, provided the prescibed mean curvatore H is asymptotic to a constant at infinity (with a suitable decay rate). No smoothness ssumptions are required on H. We consider also the Dirichlet problem....

The residual-free-bubble finite element method on anisotropic partitions - Andrea Cangiani; Süli, Endre (SIAM J. Numer. Anal., 2007)

Role of scaling limits in the rigorous analysis of Bose-Einstein condensation - Alessandro Michelangeli (J. Math. Phys. 48 (2007) 102102, 2007)
In the context of the rigorous analysis of Bose-Einstein condensation, recent achievements have been obtained in the form of asymptotic results when some appropriate scaling is performed in the Hamiltonian, and the limit of infinite number of particles is taken. In particular, two modified thermodynamic limits of infinite dilution turned out to provide an insight in this analysis, the so-\\ncalled Gross-Pitaevskii limit and the related Tomas-Fermi limit. Here such scalings are discussed with respect to their physical and mathematical motivations, and to the currently known results obtained within this framework.

Semistable principal Higgs bundles - Ugo Bruzzo; Beatriz Grana-Otero (2007)

Smooth toric DM stacks - Barbara Fantechi; Etienne Mann; Fabio Nironi (2007)
We give a new definition of smooth toric DM stacks in the same spirit of toric varieties. We show that our definition is equivalent to the one of Borisov, Chen and Smith in terms of stacky fans. In particular, we give a geometric interpretation of the combinatorial data contained in a stacky fan. We also give a bottom up classification in terms of simplicial toric varieties and fiber products of root stacks.

Soft elasticity and microstructure in smectic C elastomers - Antonio DeSimone; James Adams; Sergio Conti (Contin. Mech. Thermodyn. 18 (2007) 319-334, 2007)
Smectic C elastomers are layered materials exhibiting a solid-like elastic response along the layer normal and a rubbery one in the plane. The set of strains minimizing the elastic energy contains a one-parameter family of simple stretches associated with an internal degree of freedom, coming from the in-plane component of the director. We investigate soft elasticity and the corresponding microstructure by determining the quasiconvex hull of the set , and use this to propose experimental tests that should make the predicted soft response observable.

Solutions of vectorial Hamilton-Jacobi equations and minimizers of nonquasiconvex functionals - Sandro Zagatti (J. Math. Anal. Appl. 335 (2007) 1143-1160, 2007)
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.

Solutions to the nonlinear Schroedinger equation carrying momentum along a curve. Part II: proof of the existence result - Fethi Mahmoudi; Andrea Malchiodi (2007)
We prove existence of a special class of solutions to the (elliptic) Nonlinear Schroedinger Equation $- \\\\epsilon^2 \\\\Delta \\\\psi + V(x) \\\\psi = |\\\\psi|^{p-1} \\\\psi$ on a manifold or in the Euclidean space. Here V represents the potential, p is an exponent greater than 1 and $\\\\epsilon$ a small parameter corresponding to the Planck constant. As $\\\\epsilon$ tends to zero (namely in the semiclassical limit) we prove existence of complex-valued solutions which concentrate along closed curves, and whose phase in highly oscillatory. Physically, these solutions carry quantum-mechanical momentum along the limit curves. In the first part of this work we identified the limit set and constructed approximate solutions, while here we give the complete proof of our main existence result.

Solutions to the nonlinear Schroedinger equation carrying momentum along a curve. Part I: study of the limit set and approximate solutions - Fethi Mahmoudi; Andrea Malchiodi; Marcelo Montenegro (2007)
We prove existence of a special class of solutions to the (elliptic) Nonlinear Schroeodinger Equation $- \\\\epsilon^2 \\\\Delta \\\\psi + V(x) \\\\psi = |\\\\psi|^{p-1} \\\\psi$, on a manifold or in the Euclidean space. Here V represents the potential, p an exponent greater than 1 and $\\\\epsilon$ a small parameter corresponding to the Planck constant. As $\\\\epsilon$ tends to zero (namely in the semiclassical limit) we prove existence of complex-valued solutions which concentrate along closed curves, and whose phase is highly oscillatory. Physically, these solutions carry quantum-mechanical momentum along the limit curves. In this first part we provide the characterization of the limit set, with natural stationarity and non-degeneracy conditions. We then construct an approximate solution up to order $\\\\epsilon^2$, showing that these conditions appear naturally in a Taylor expansion of the equation in powers of $\\\\epsilon$. Based on these, an existence result will be proved in the second part.

Some existence results for the Toda system on closed surfaces - Andrea Malchiodi; Cheikh Birahim Ndiaye (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 18 (2007) 391-412, 2007)
Given a compact closed surface $\\\\Sig$, we consider the {\\\\em generalized Toda} system of equations on $\\\\Sig$: $- \\\\D u_i = \\\\sum_{j=1}^2 \\\\rho_j a_{ij} \\\\left( \\\\frac{h_j e^{u_j}}{\\\\int_\\\\Sig h_j e^{u_j} dV_g} - 1 \\\\right)$ for $i = 1, 2$, where $\\\\rho_1, \\\\rho_2$ are real parameters and $h_1, h_2$ are smooth positive functions. Exploiting the variational structure of the problem and using a new minimax scheme we prove existence of solutions for generic values of $\\\\rho_1$ and for $\\\\rho_2 < 4 \\\\pi$.

Stability of front tracking solutions to the initial and boundary value problem for systems of conservation laws - Andrea Marson; Carlotta Donadello (NoDEA Nonlinear Differential Equations Appl. 14 (2007) 569-592, 2007)

Standing waves of some coupled Nonlinear Schrödinger Equations - Antonio Ambrosetti; Eduardo Colorado (J. Lond. Math. Soc. 75 (2007) 67-82, 2007)
We deal with a class of systems of NLS equations, proving the existence of bound and ground states provided the coupling parameter is small, respectively, large.

Strengthened convergence of marginals to the cubic nonlinear Schroedinger equation - Alessandro Michelangeli (2007)
We rewrite a recent derivation of the cubic non-linear Schroedinger equation by Adami, Golse, and Teta in the more natural formof the asymptotic factorisation of marginals at any fixed time and in the trace norm. This is the standard form in which the emergence of the\\nnon-linear effective dynamics of a large system of interacting bosons is\\nproved in the literature.

Surfactants in Foam Stability: A Phase-Field Model - Irene Fonseca; Massimiliano Morini; Valeriy Slastikov (Arch. Rational Mech. Anal. 183 (2007) 411-456, 2007)
The role of surfactants in stabilizing the formation of bubbles in foams is studied using a phase-field model. The analysis is centered on a van der Walls-Cahn-Hilliard-type energy with an added term accounting for the interplay between the presence of a surfactant density and the creation of interfaces. In particular, it is concluded that the surfactant segregates to the interfaces, and that the prescriptionof the distribution of surfactant will dictate the locus of interfaces, what is in agreement with experimentation.

Time optimal swing-up of the planar pendulum - Mireille E. Broucke; Paolo Mason; Benedetto Piccoli (46th IEEE Conference on Decision and Control (2007) 5389 - 5394, 2007)
This paper presents qualitative and numerical results on the global structure of the time optimal trajectories of the planar pendulum on a cart.

Time-dependent systems of generalized Young measures - Gianni Dal Maso; Antonio DeSimone; Maria Giovanna Mora; Massimiliano Morini (Netw. Heterog. Media 2 (2007) 1-36, 2007)
In this paper some new tools for the study of evolution problems in the framework of Young measures are introduced. A suitable notion of time-dependent system of generalized Young measures is defined, which allows to extend the classical notions of total variation and absolute continuity with respect to time, as well as the notion of time derivative. The main results are a Helly type theorem for sequences of systems of generalized Young measures and a theorem about the existence of the time derivative for systems with bounded variation with respect to time.

Twisted noncommutative equivariant - Lucio Cirio (2007)
We propose Weil and Cartan models for the equivariant cohomology of covariant actions on toric deformation manifolds. The construction is based on the noncommutative Weil algebra of Alekseev and Meinrenken; we show that one can implement a Drinfeld twist of their models in order to take into account the noncommutativity of the spaces we are acting on.

Uniqueness and continuous dependence on boundary data for integro-extremal minimizers of the functional of the gradient - Sandro Zagatti (J. Convex Anal. 14 (2007) 705-727, 2007)
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.

Viscosity solutions of Hamilton-Jacobi equations with discontinuous coefficients - Giuseppe Maria Coclite; Nils Henrik Risebro (J. Hyperbolic Differ. Equ. 4 (2007) 771-795, 2007)
We consider Hamilton--Jacobi equations, where the Hamiltonian depends discontinuously on both the spatial and temporal location. Our main results are the existence and well--posedness of a viscosity solution to the Cauchy problem. We define a viscosity solution by treating the discontinuities in the coefficients analogously to \\\"internal boundaries\\\". By defining an appropriate penalization function, we prove that viscosity solutions are unique. The existence of viscosity solutions is established by showing that a sequence of front tracking approximations is compact in $L^\\\\infty$, and that the limits are viscosity solutions.

2-d stability of the Néel wall - Antonio DeSimone; Hans Knuepfer; Felix Otto (Calc. Var. Partial Differential Equations 27 (2006) 233-253, 2006)
We are interested in thin-film samples in micromagnetism, where the magnetization m is a 2-d unit-length vector field. More precisely we are interested in transition layers which connect two opposite magnetizations, so called Néel walls.

4e-condensation in a fully frustrated Josephson junction diamond chain - Matteo Rizzi; Vittorio Cataudella; Rosario Fazio (Phys. Rev. B 73 (2006) 100502(R), 2006)
Fully frustrated one-dimensional diamond Josephson chains have been shown [B. Dou\\\\c{c}ot and J. Vidal, Phys. Rev. Lett. {\\\\bf 88}, 227005 (2002)] to posses a remarkable property: The superfluid phase occurs through the condensation of pairs of Cooper pairs. By means of Monte Carlo simulations we analyze quantitatively the Insulator to $4e$-Superfluid transition. We determine the location of the critical point and discuss the behaviour of the phase-phase correlators. For comparison we also present the case of a diamond chain at zero and 1/3 frustration where the standard $2e$-condensation is observed.

Almost Global Stochastic Feedback Stabilization of Conditional Quantum Dynamics - Claudio Altafini; Francesco Ticozzi (2006)
We propose several parametrization-free solutions to the problem of quantum state reduction control by means of continuous measurement and smooth quantum feedback. In particular, we design a feedback law for which almost global stochastic feedback stabilization can be proved analytically by means of Lyapunov techinques. This synthesis arises very naturally from the physics of the problem, as it relies on the variance associated with the quantum filtering process.

An artificial viscosity approach to quasistatic crack growth - Rodica Toader; Chiara Zanini (2006)
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.

A Birkhoff-Lewis-Type Theorem for Some Hamiltonian PDEs - Dario Bambusi; Massimiliano Berti (SIAM J. Math. Anal. 37 (2006) 83-102, 2006)
In this paper we give an extension of the Birkhoff--Lewis theorem to some semilinear PDEs. Accordingly we prove existence of infinitely many periodic orbits with large period accumulating at the origin. Such periodic orbits bifurcate from resonant finite dimensional invariant tori of the fourth order normal form of the system. Besides standard nonresonance and nondegeneracy assumptions, our main result is obtained assuming a regularizing property of the nonlinearity. We apply our main theorem to a semilinear beam equation and to a nonlinear Schr\\\\\\\"odinger equation with smoothing nonlinearity.

Born approximation in the problem of the rigorous derivation of the Gross-Pitaevskii equation - Alessandro Michelangeli (2006)
\\\"It has a flavour of Mathematical Physics...\\\"With these words, just few years ago, prof. Di Giacomo\\nused to introduce the topic of the Born approximation within a nonrelativistic potential theory, in his `oversize\\\' course of Theoretical Physics in Pisa. Something maybe too fictitious inside the formal theory of the scattering he was teaching us at that point of the course. Now that I\\\'m (studying to become) a Mathematical Physicist indeed, dealing with such an `exotic tasting\\\' topic, those words come back to the mind, into a new perspective. Here the very recent problem of the rigorous derivation of\\nthe cubic nonlinear Schrödinger equation (the Gross-Pitaevskiî equation) is reviewed and discussed, with respect to the role of the Born approximation that one ends up with in an appropriate scaling limit

Bound and ground states of coupled nonlinear Schrödinger equations - Antonio Ambrosetti; Eduardo Colorado (C. R. Acad. Sci. Paris, Ser. I 342 (2006) 453-458, 2006)
We prove existence of bound and ground states of some systems of coupled nonlinear Schrodinger equations.

Bound states of Nonlinear Schroedinger Equations with Potentials Vanishing at Infinity - Antonio Ambrosetti; Andrea Malchiodi; David Ruiz (J. Anal. Math. 98 (2006) 317-348, 2006)

On Bressan\\\'s conjecture on mixing properties of vector fields - Stefano Bianchini (Self-Similar Solutions of Nonlinear PDE / Ed. Piotr Biler and Grzegorz Karch. - Banach Center Publ. 74 (2006) 13-31, 2006)

On a Camassa-Holm type equation with two dependent variables - Gregorio Falqui (J. Phys. A 39 (2006) 327-342, 2006)
We consider a generalization of the Camassa Holm (CH) equation with two dependent variables, called CH2, introduced in [16]. We briefly provide an alternative derivation of it based on the theory of Hamiltonian structures\\non (the dual of) a Lie Algebra. The Lie Algebra here involved is the same algebra underlying the NLS hierarchy. We study the structural properties of the CH2 hierarchy within the bihamiltonian theory of integrable PDEs, and\\nprovide its Lax representation. Then we explicitly discuss how to construct classes of solutions, both of peakon and of algebro-geometrical type. We finally sketch the construction of a class of singular solutions, defined by setting to zero one of the two dependent variables.

A Canonical Frame for Nonholonomic Rank Two Distributions of Maximal Class - Boris Doubrov; Igor Zelenko (C. R. Math. Acad. Sci. Paris 342 (2006) 589-594, 2006)
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.

Cantor families of periodic solutions for completely resonant nonlinear wave equations - Massimiliano Berti; Philippe Bolle (Duke Math. J. 134 (2006) 359-419, 2006)
We prove the existence of small amplitude, $2\\\\pi \\\\slash \\\\om$-periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions, for any frequency $ \\\\om $ belonging to a Cantor-like set of positive measure and for a new set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser Implicit Function Theorem. In spite of the complete resonance of the equation we show that we can still reduce the problem to a {\\\\it finite} dimensional bifurcation equation. Moreover, a new simple approach for the inversion of the linearized operators required by the Nash-Moser scheme is developed. It allows to deal also with nonlinearities which are not odd and with finite spatial regularity.

Classification of stable time-optimal controls on 2-manifolds - Ugo Boscain; Igor Nikolaev; Benedetto Piccoli (J. Math. Sci. 135 (2006) 3109-3124, 2006)

Common Polynomial Lyapunov Functions for Linear Switched Systems - Paolo Mason; Ugo Boscain; Yacine Chitour (SIAM J. Control Optim. 45 (2006) 226-245, 2006)
In this paper, we consider linear switched systems $\\\\dot x(t)=A_{u(t)} x(t)$, $x\\\\in\\\\R^n$, $u\\\\in U$, and the problem of asymptotic stability for arbitrary switching functions, uniform with respect to switching ({\\\\bf UAS} for short). We first prove that, given a {\\\\bf UAS} system, it is always possible to build a common polynomial Lyapunov function. Then our main result is that the degree of that common polynomial Lyapunov function is not uniformly bounded over all the {\\\\bf UAS} systems. This result answers a question raised by Dayawansa and Martin. A generalization to a class of piecewise-polynomial Lyapunov functions is given.

Compactness of solutions to some geometric fourth-order equations - Andrea Malchiodi (J. Reine Angew. Math. 594 (2006) 137-174, 2006)
We prove compactness of solutions to some fourth order equations with exponential nonlinearities on four manifolds. The proof is based on a refined bubbling analysis, for which the main estimates are given in integral form. Our result is used in a subsequent paper to find critical points (via minimax arguments) of some geometric functional, which give rise to conformal metrics of constant $Q$-curvature. As a byproduct of our method, we also obtain compactness of such metrics.

Concentration at manifolds of arbitrary dimension for a singularly perturbed Neumann problem - Fethi Mahmoudi; Andrea Malchiodi (Atti Accad. Naz Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 17 (2006) 279-290, 2006)
We consider the equation $- \\\\e^2 \\\\D u + u = u^p$ in $\\\\O \\\\subseteq \\\\R^N$, where $\\\\O$ is open, smooth and bounded, and we prove concentration of solutions along $k$-dimensional minimal submanifolds of $\\\\pa \\\\O$, for $N \\\\geq 3$ and for $k \\\\in \\\\{1, \\\\dots, N-2\\\\}$. We impose Neumann boundary conditions, assuming $1<\\\\frac{N-k+2}{N-k-2}$ and $\\\\e \\\\to 0^+$. This result settles in full generality a phenomenon previously considered only in the particular case $N = 3$ and $k = 1$.

Conservative Solutions to a Nonlinear Variational Wave Equation - Alberto Bressan; Zheng Yuxi (Comm. Math. Phys. 266 (2006) 471-497, 2006)
We establish the existence of a conservative weak solution to the Cauchy problem for the nonlinear variational wave equation $u_{tt} - c(u)(c(u)u_x)_x=0$, for initial data of finite energy. Here $c(\\\\cdot)$ is any smooth function with uniformly positive bounded values.

A cyclic integral on k-Minkowski noncommutative space-time - Alessandra Agostini; Giovanni Amelino-Camelia; Michele Arzano; Francesco D'Andrea (Int. J. Mod. Phys. A 21 (2006) 3133-3150, 2006)
We examine some alternative possibilities for an action functional for $\\\\kappa$-Minkowski noncommutative spacetime, with an approach which should be applicable to other spacetimes with coordinate-dependent commutators of the spacetime coordinates ($[x_\\\\mu,x_\\\\nu]=f_{\\\\mu,\\\\nu}(x)$). Early works on $\\\\kappa$-Minkowski focused on $\\\\kappa$-Poincar\\\\\\\'e covariance and the dependence of the action functional on the choice of Weyl map, renouncing to invariance under cyclic permutations of the factors composing the argument of the action functional. A recent paper (hep-th/0307149), by Dimitrijevic, Jonke, Moller, Tsouchnika, Wess and Wohlgenannt, focused on a specific choice of Weyl map and, setting aside the issue of $\\\\kappa$-Poincar\\\\\\\'e covariance of the action functional, introduced in implicit form a cyclicity-inducing measure. We provide an explicit formula for (and derivation of) a choice of measure which indeed ensures cyclicity of the action functional, and we show that the same choice of measure is applicable to all the most used choices of Weyl map. We find that this ``cyclicity-inducing measure\\\'\\\' is not covariant under $\\\\kappa$-Poincar\\\\\\\'e transformations. We also notice that the cyclicity-inducing measure can be straightforwardly derived using a map which connects the $\\\\kappa$-Minkowski spacetime coordinates and the spacetime coordinates of a ``canonical\\\'\\\' noncommutative spacetime, with coordinate-independent commutators.

The Dirichlet problem for H-systems with small boundary data: blowup phenomena and nonexistence results - Paolo Caldiroli; Roberta Musina (Arch. Ration. Mech. Anal. 181 (2006) 1-42, 2006)

An estimation of the controllability time for single-input systems on compact Lie Groups - Andrei A. Agrachev; Thomas Chambrion (ESAIM Control Optim. Calc. Var. 12 (2006) 409-441, 2006)
Geometric control theory and Riemannian techniques are used to describe the reachable set at time t of left invariant single-input control systems on semi-simple compact Lie groups and to estimate the minimal time needed to reach any point from identity. This method provides an effective way to give an upper and a lower bound for the minimal time needed to transfer a controlled quantum system with a drift from a given initial position to a given final position. The bounds include diameters of the flag manifolds; the latter are also explicitly computed in the paper.

Experimental and modeling studies of desensitization of P2X3 receptors. - Elena Sokolova; Andrei Skorinkin; Igor Moiseev; Andrei A. Agrachev; Andrea Nistri; Rashid Giniatullin (Molecular pharmacology. 2006 Jul; 70(1):373-82, 2006)
The function of ATP-activated P2X3 receptors involved in pain sensation is modulated by desensitization, a phenomenon poorly understood. The present study used patch-clamp recording from cultured rat or mouse sensory neurons and kinetic modeling to clarify the properties of P2X3 receptor desensitization. Two types of desensitization were observed, a fast process (t1/2 = 50 ms; 10 microM ATP) following the inward current evoked by micromolar agonist concentrations, and a slow process (t1/2 = 35 s; 10 nM ATP) that inhibited receptors without activating them. We termed the latter high-affinity desensitization (HAD). Recovery from fast desensitization or HAD was slow and agonist-dependent. When comparing several agonists, there was analogous ranking order for agonist potency, rate of desensitization and HAD effectiveness, with 2-methylthioadenosine triphosphate the strongest and beta,gamma-methylene-ATP the weakest. HAD was less developed with recombinant (ATP IC50 = 390 nM) than native P2X3 receptors (IC50 = 2.3 nM). HAD could also be induced by nanomolar ATP when receptors seemed to be nondesensitized, indicating that resting receptors could express high-affinity binding sites. Desensitization properties were well accounted for by a cyclic model in which receptors could be desensitized from either open or closed states. Recovery was assumed to be a multistate process with distinct kinetics dependent on the agonist-dependent dissociation rate from desensitized receptors. Thus, the combination of agonist-specific mechanisms such as desensitization onset, HAD, and resensitization could shape responsiveness of sensory neurons to P2X3 receptor agonists. By using subthreshold concentrations of an HAD-potent agonist, it might be possible to generate sustained inhibition of P2X3 receptors for controlling chronic pain.

Extended affine Weyl groups and Frobenius manifolds -- II - Boris Dubrovin; Zhang Youjin; Zuo Dafeng (2006)
For the root system of type $B_l$ and $C_l$, we generalize the result of \\\\cite{DZ1998} by showing the existence of a Frobenius manifold structure on the orbit space of the extended affine Weyl group that corresponds to any vertex of the Dynkin diagram instead of a particular choice of \\\\cite{DZ1998}.

Forced vibrations of wave equations with non-monotone nonlinearities - Massimiliano Berti; Luca Biasco (Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006) 439-474, 2006)
We prove existence and regularity of periodic in time solutions of completely resonant nonlinear forced wave equations with Dirichlet boundary conditions for a large class of non-monotone forcing terms. Our approach is based on a variational Lyapunov-Schmidt reduction. It turns out that the infinite dimensional bifurcation equation exhibits an intrinsic lack of compactness. We solve it via a minimization argument and a-priori estimate methods inspired to regularity theory of Rabinowitz.

Fundamental form and Cartan tensor of (2,5)-distributions coincide - Igor Zelenko (J. Dyn. Control Syst. 12 (2006) 247-276, 2006)
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.

On geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on distributions of corank 1 - Igor Zelenko (J. Math. Sci. 135 (2006) 3168-3194, 2006)
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.

Glimm interaction functional for BGK schemes - Stefano Bianchini (2006)

On Hamiltonian perturbations of hyperbolic systems of conservation laws, II: universality of critical behaviour - Boris Dubrovin (Comm. Math. Phys. 267 (2006) 117-139, 2006)
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.

On Hamiltonian perturbations of hyperbolic systems of conservation laws I: quasitriviality of bihamiltonian perturbations - Boris Dubrovin; Liu Si-Qi; Zhang Youjin (Comm. Pure Appl. Math. 59 (2006) 559-615, 2006)
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.

Homogeneous polynomial forms for simultaneous stabilizability of families of linear control systems: a tensor product approach - Claudio Altafini (IEEE Trans. Automat. Control 51 (2006) 1566-1571, 2006)
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.