Vertices, vortices & interacting surface operators - Giulio Bonelli; Alessandro Tanzini; Zhao Jian (JHEP 06(2012)178, 2012)
We show that the vortex moduli space in non-abelian supersymmetric N=(2,2) gauge theories on the two dimensional plane with adjoint and anti-fundamental matter can be described as an holomorphic submanifold of the instanton moduli space in four dimensions. The vortex partition functions for these theories are computed via equivariant localization. We show that these coincide with the field theory limit of the topological vertex on the strip with boundary conditions corresponding to column diagrams. Moreover, we resum the field theory limit of the vertex partition functions in terms of generalized hypergeometric functions formulating their AGT dual description as interacting surface operators of simple type. Analogously we resum the topological open string amplitudes in terms of q-deformed generalized hypergeometric functions proving that they satisfy appropriate finite difference equations.
22 pages, 4 figures

A Viscosity-driven crack evolution - Simone Racca (Advances in Calculus of Variations 5 (2012) 433-483, 2012)
We present a model of crack growth in brittle materials which couples dissipative effects on the crack tip and viscous effects. We consider the 2 -dimensional antiplane case with pre-assigned crack path, and firstly prove an existence result for a rate-dependent evolution problem by means of time-discretization. The next goal is to describe the rate-independent evolution as limit of the rate-dependent ones when the dissipative and viscous effects vanish. The rate-independent evolution satisfies a Griffith’s criterion for the crack growth, but, in general, it does not fulfil a global minimality condition; its fracture set may exhibit jump discontinuities with respect to time. Under suitable regularity assumptions, the quasi-static crack growth is described by solving a finite-dimensional problem.

Weighted barycentric sets and singular Liouville equations on compact surfaces - Alessandro Carlotto; Andrea Malchiodi (Journal of Functional Analysis 262 (2012) 409-450, 2012)
Given a closed two dimensional manifold, we prove a general existence result\\r\\nfor a class of elliptic PDEs with exponential nonlinearities and negative Dirac\\r\\ndeltas on the right-hand side, extending a theory recently obtained for the\\r\\nregular case. This is done by global methods: since the associated Euler\\r\\nfunctional is in general unbounded from below, we need to define a new model\\r\\nspace, generalizing the so-called space of formal barycenters and\\r\\ncharacterizing (up to homotopy equivalence) its very low sublevels. As a\\r\\nresult, the analytic problem is reduced to a topological one concerning the\\r\\ncontractibility of this model space. To this aim, we prove a new functional\\r\\ninequality in the spirit of [16] and then we employ a min-max scheme based on a cone-style construction, jointly with the blow-up analysis given in [5] (after\\r\\n[6] and [8]). This study is motivated by abelian Chern- Simons theory in\\r\\nself-dual regime, or from the problem of prescribing the Gaussian curvature in\\r\\npresence of conical singularities (hence generalizing a problem raised by\\r\\nKazdan and Warner in [26]).

Wild quiver gauge theories - Giulio Bonelli; Kazunobu Maruyoshi; Alessandro Tanzini (JHEP 02(2012)031, 2012)
We study $N=2$ supersymmetric $SU(2)$ gauge theories coupled to non-Lagrangian superconformal field theories induced by compactifying the six dimensional $A_1 (2,0)$ theory on Riemann surfaces with irregular punctures. These are naturally associated to Hitchin systems with wild ramification whose spectral curves provide the relevant Seiberg-Witten geometries. We propose that the prepotential of these gauge theories on the Omega-background can be obtained from the corresponding irregular conformal blocks on the Riemann surfaces via a generalization of the coherent state construction to the case of higher order singularities.
34 pages

Adaptation as a genome-wide autoregulatory principle in the stress response of yeast. - F Eduati; B Di Camillo; G Toffolo; Claudio Altafini; Giovanna De Palo; Mattia Zampieri (IET systems biology. 2011 Jul; 5(4):269-79, 2011)
The gene expression response of yeast to various types of stresses/perturbations shows a common functional and dynamical pattern for the vast majority of genes, characterised by a quick transient peak (affecting primarily short genes) followed by a return to the pre-stimulus level. Kinetically, this process of adaptation following the transient excursion can be modelled using a genome-wide autoregulatory mechanism by means of which yeast aims at maintaining a preferential concentration in its mRNA levels. The resulting feedback system explains well the different time constants observable in the transient response, while being in agreement with all the known experimental dynamical features. For example, it suggests that a very rapid transient can be induced also by a slowly varying concentration of the gene products.

An asymptotic reduction of a Painlevé VI equation to a Painlevé III - Davide Guzzetti (J.Phys.A: Math.Theor. 44 (2011) 215203, 2011)
When the independent variable is close to a critical point, it is shown that\\r\\nPVI can be asymptotically reduced to PIII. In this way, it is possible to\\r\\ncompute the leading term of the critical behaviors of PVI transcendents\\r\\nstarting from the behaviors of PIII transcendents.

Axial symmetry of some steady state solutions to nonlinear Schrödinger equations - Changfeng Gui; Andrea Malchiodi; Haoyuan Xu; Paul Yang (Proc. Amer. Math. Soc. 139 (2011), 1023-1032, 2011)
In this note, we show the axial symmetry of steady state solutions of nonlinear Schrodinger equations when the exponent of the nonlinearity is between the critical Sobolev exponent of n dimensional space and n - 1 dimensional space.
Keywords: Nonlinear Schrödinger equation

Bishop and Laplacian Comparison Theorems on Three Dimensional Contact Subriemannian Manifolds with Symmetry - Andrei A. Agrachev; Paul Lee (SISSA, 2011)
We prove a Bishop volume comparison theorem and a Laplacian comparison\r\ntheorem for three dimensional contact subriemannian manifolds with symmetry.
25 pages

Boutroux curves with external field: equilibrium measures without a variational problem - Marco Bertola (Anal. Math. Phys., 2011)

Branching of Cantor Manifolds of Elliptic Tori and Applications to PDEs - Massimiliano Berti; Luca Biasco (Communications in Mathematical Physics, 2011)
We consider infinite dimensional Hamiltonian systems. We prove the existence of "Cantor manifolds" of elliptic tori-of any finite higher dimension-accumulating on a given elliptic KAM torus. Then, close to an elliptic equilibrium, we show the existence of Cantor manifolds of elliptic tori which are "branching" points of other Cantor manifolds of higher dimensional tori. We also answer to a conjecture of Bourgain, proving the existence of invariant elliptic tori with tangential frequency along a pre-assigned direction. The proofs are based on an improved KAM theorem. Its main advantages are an explicit characterization of the Cantor set of parameters and weaker smallness conditions on the perturbation. We apply these results to the nonlinear wave equation. © 2011 Springer-Verlag.
cited By (since 1996)8

A class of existence results for the singular Liouville equation - Alessandro Carlotto; Andrea Malchiodi (Comptes Rendus Mathematique 349 (2011) 161-166, 2011)
We consider a class of elliptic PDEs on closed surfaces with exponential nonlinearities and Dirac deltas on the right-hand side. The study arises from abelian Chern–Simons theory in self-dual regime, or from the problem of prescribing the Gaussian curvature in presence of conical singularities. A general existence result is proved using global variational methods: the analytic problem is reduced to a topological problem concerning the contractibility of a model space, the so-called space of formal barycenters, characterizing the very low sublevels of a suitable functional.

Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of the potential - David Ruiz; Giusi Vaira (Rev. Mat. Iberoamericana, 2011)

Compactness by maximality - Sandro Zagatti (2011)
We derive a compactness property in the Sobolev space $W^{1,1}(Ø)$ in order to study the Dirichlet problem for the eikonal equation \begin{displaymath} \begin{cases} \ha |\n u(x)|^2 - a(x) = 0 & \ in \ Ø\cr u(x)=φ(x) & \ on \ \partial Ø, \end{cases} \end{displaymath} without continuity assumptions on the map $a$.

Computing global structural balance in large-scale signed social networks. - Giuseppe Facchetti; Giovanni Iacono; Claudio Altafini (Proceedings of the National Academy of Sciences of the United States of America. Volume 108, Issue 52, 27 December 2011, Pages 20953-20958, 2011)
Structural balance theory affirms that signed social networks (i.e., graphs whose signed edges represent friendly/hostile interactions among individuals) tend to be organized so as to avoid conflictual situations, corresponding to cycles of negative parity. Using an algorithm for ground-state calculation in large-scale Ising spin glasses, in this paper we compute the global level of balance of very large online social networks and verify that currently available networks are indeed extremely balanced. This property is explainable in terms of the high degree of skewness of the sign distributions on the nodes of the graph. In particular, individuals linked by a large majority of negative edges create mostly \\\"apparent disorder,\\\" rather than true \\\"frustration.\\\"
Free fulltext article in Pubmed Central
Keywords: Combinatorial optimization

Concentration of solutions for a singularly perturbed Neumann problem in non-smooth domains - Serena Dipierro (Annales de l'I.H.P. Analyse non linéaire, 2011)

Cones of divisors of blow-ups of projective spaces - Alessio Lo Giudice; Salvatore Cacciola; M. Donten-Bury; O. Dumitrescu; J. Park (Le Matematiche (Catania), volume 66, Issue no.2, (2011), pages : 153-187, 2011)
We investigate Mori dream spaces obtained by blowing-up the n-dimensional complex projective space at n+1, n+2 or n+3 points in very general position. Using toric techniques we study the movable cone of the blow-up of Pn at n+1 points, its decomposition into nef chambers and the action of theWeyl group on the set of chambers. Moreover, using different methods, we explicitly write down the equations of the movable cone also for Pn blown-up at n+2 points.
Keywords: Mori dream space

Convergence of the mimetic finite difference method for eigenvalue problems in mixed form - Andrea Cangiani; Gardini, Francesca; Manzini, Gianmarco (Comput. Methods Appl. Mech. Engrg., 2011)

Covered by lines and Conic connected varieties - Simone Marchesi; Alex Massarenti; Saeed Tafazolian (Le Matematiche 66 (2011) 137-151, 2011)
We study some properties of an embedded variety covered by lines and give a\\r\\nnumerical criterion ensuring the existence of a singular conic through two of\\r\\nits general points. We show that our criterion is sharp. Conic-connected,\\r\\ncovered by lines, QEL, LQEL, prime Fano, defective, and dual defective\\r\\nvarieties are closely related. We study some relations between the above\\r\\nmentioned classes of objects using celebrated results by Ein and Zak.

Crack growth with non-interpenetration : a simplified proof for the pure Neumann problem - Gianni Dal Maso; Giuliano Lazzaroni (Discrete and Continuous Dynamical Systems - Series A 31 (2011) 1219-1231, 2011)
We present a recent existence result concerning the quasi-static evolution of cracks in hyperelastic brittle materials, in the frame-work of finite elasticity with non-interpenetration. In particular, here we consider the problem where no Dirichlet conditions are imposed, the boundary is traction-free, and the body is subject only to time-dependent volume forces. This allows us to present the main ideas of the proof in a simpler way, avoiding some of the technicalities needed in the general case, studied in.

Crepant resolutions of weighted projective spaces and quantum deformations - Samuel Boissiere; Etienne Mann; Fabio Perroni (This article will be published in 2011 in the \"Nagoya Mathematical Journal\" Volume 201, March 2011, Pages 1-22, under the title \"Computing certain Gromov-Witten invariants of the crepant resolution of Pdouble-strock(1, 3, 4, 4), 2011)
We compare the Chen-Ruan cohomology ring of the weighted projective spaces\r\n$\\IP(1,3,4,4)$ and $\\IP(1,...,1,n)$ with the cohomology ring of their crepant\r\nresolutions. In both cases, we prove that the Chen-Ruan cohomology ring is\r\nisomorphic to the quantum corrected cohomology ring of the crepant resolution\r\nafter suitable evaluation of the quantum parameters. For this, we prove a\r\nformula for the Gromov-Witten invariants of the resolution of a transversal\r\n${\\rm A}_3$ singularity.
Exposition improved, new title, typos corrected. The section\r\n concerning the model for the orbifold Chow ring has been removed (appears now\r\n in our new preprint 0709.4559)

Critical points of the Moser-Trudinger functional - Francesca De Marchis; Andrea Malchiodi; Luca Martinazzi (SISSA, 2011)
Keywords: Moser-Trudinger inequality

Cytoskeletal actin networks in motile cells are critically self-organized systems synchronized by mechanical interactions - Luca Cardamone; Alessandro Laio; Rajesh Shahapure; Antonio DeSimone (PNAS 108 (2011) 13978, 2011)

D-branes, surface operators, and ADHM quiver representations - Ugo Bruzzo; Duiliu-Emanuel Diaconescu; M. Yardim; G. Pan; Yi Zhang; Chuang Wu-yen (SISSA, 2011)
A supersymmetric quantum mechanical model is constructed for BPS states bound to surface operators in five dimensional SU(r) gauge theories using D-brane engineering. This model represents the effective action of a certain D2-brane configuration, and is naturally obtained by dimensional reduction of a quiver $(0,2)$ gauged linear sigma model. In a special stability chamber, the resulting moduli space of quiver representations is shown to be smooth and isomorphic to a moduli space of framed quotients on the projective plane. A precise conjecture relating a K-theoretic partition function of this moduli space to refined open string invariants of toric lagrangian branes is formulated for conifold and local P^1 x P^1 geometries.
45 pages, v2: minor corrections

A Decomposition Theorem for BV functions - Stefano Bianchini; Daniela Tonon (Communications on Pure and Applied Analysis, 2011)

Degenerate KAM theory for partial differential equations - Dario Bambusi; Massimiliano Berti; Elena Magistrelli (Journal of Differential Equations, 2011)
This paper deals with degenerate KAM theory for lower dimensional elliptic tori of infinite dimensional Hamiltonian systems, depending on one parameter only. We assume that the linear frequencies are analytic functions of the parameter, satisfy a weak non-degeneracy condition of Rüssmann type and an asymptotic behavior. An application to nonlinear wave equations is given. © 2010 Elsevier Inc.
cited By (since 1996)3

Dimensional Reduction and Approximation of Measures and Weakly Differentiable Homeomorphisms - Sara Daneri (SISSA, 2011)
This thesis is devoted to the study of two different problems: the properties of the disintegration of the Lebesgue measure on the faces of a convex function and the existence of smooth approximations of bi-Lipschitz orientation-preserving homeomorphisms in the plane.

Double resonance with Landesman–Lazer conditions for planar systems of ordinary differential equations - Alessandro Fonda; Maurizio Garrione (Journal of Differential Equations, 2011)
We prove the existence of periodic solutions for first order planar systems at resonance. The nonlinearity is indeed allowed to interact with two positively homogeneous Hamiltonians, both at resonance, and some kind of Landesman–Lazer conditions are assumed at both sides. We are thus able to obtain, as particular cases, the existence results proposed in the pioneering papers by Lazer and Leach (1969) [27], and by Frederickson and Lazer (1969) [18]. Our theorem also applies in the case of asymptotically piecewise linear systems, and in particular generalizes Fabry's results in Fabry (1995) [10], for scalar equations with double resonance with respect to the Dancer–Fučik spectrum.
Keywords: Double resonance; Landesman–Lazer conditions; Nonlinear planar systems

Embedding theorems and existence results for nonlinear Schrödinger–Poisson systems with unbounded and vanishing potentials - Bonheure, Denis; Mercuri, Carlo (Journal of Differential Equations, 2011)
<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;">Motivated by existence results for positive solutions of non-autonomous nonlinear Schrödinger–Poisson systems with potentials possibly unbounded or vanishing at infinity, we prove embedding theorems for weighted Sobolev spaces. We both consider a general framework and spaces of radially symmetric functions when assuming radial symmetry of the potentials.

Energy release rate and stress intensity factor in antiplane elasticity - Giuliano Lazzaroni; Rodica Toader (Journal de Mathematiques Pures et Appliquees 95 (2011) 565-584, 2011)
In the setting of antiplane linearized elasticity, we show the existence of the stress intensity factor and its relation with the energy release rate when the crack path is a C1,1 curve. Finally, we show that the energy release rate is continuous with respect to the Hausdorff convergence in a class of admissible cracks.

Ennio De Giorgi and Γ-convergence - Gianni Dal Maso (Discrete and Continuous Dynamical Systems - Series A 31 (2011) 1017-1021, 2011)
Γ-convergence was introduced by Ennio De Giorgi in a series of papers published between 1975 and 1983. In the same years he developed many applications of this tool to a great variety of asymptotic problems in the calculus of variations and in the theory of partial differential equations.

An Estimate on the Flow Generated by Monotone Operators - Stefano Bianchini; Matteo Gloyer (Communications in Partial Differential Equations 36 (2011) 777-796, 2011)

An Existence and Uniqueness Result for the Motion of Self-Propelled Microswimmers - Gianni Dal Maso; Antonio DeSimone; Marco Morandotti (SIAM J. Math. Anal., 2011)
We present an analytical framework to study the motion of micro-swimmers in a viscous fluid. Our main result is that, under very mild regularity assumptions, the change of shape determines uniquely the motion of the swimmer. We assume that the Reynolds number is very small, so that the velocity field of the surrounding, infinite fluid is governed by the Stokes system and all inertial effects can be neglected. Moreover, we enforce the self propulsion constraint (no external forces and torques). Therefore, Newton\\\'s equations of motion reduce to the vanishing of the viscous drag force and torque acting on the body. By exploiting an integral representation of viscous force and torque, the equations of motion can be reduced to a system of six ordinary differential equations. Variational techniques are used to prove the boundedness and measurability of its coefficients, so that classical results on ordinary differential equations can be invoked to prove existence and uniqueness of the solution.

Existence for wave equations on domains with arbitrary growing cracks - Gianni Dal Maso; Cristopher J. Larsen (Rend. Lincei Mat. Appl. 22 (2011) 387-408, 2011)
In this paper we formulate and study scalar wave equations on domains with arbitrary growing cracks. This includes a zero Neumann condition on the crack sets, and the only assumptions on these sets are that they have bounded surface measure and are growing in the sense of set inclusion. In particular, they may be dense, so the weak formulations must fall outside of the usual weak formulations using Sobolev spaces. We study both damped and undamped equations, showing existence and, for the damped equation, uniqueness and energy conservation.
Keywords: Wave equation

Fracture and plastic models as Gamma-limits of damage models under different regimes - Flaviana Iurlano (Advances in Calculus of Variations., to appear., 2011)
We consider a variational model for damaged elastic materials. This model depends on three small parameters, which are related to the cost of the damage, to the width of the damaged regions, and to the minimum elasticity constant attained in the damaged regions. As these parameters tend to zero, our models Gamma-converge to a model for brittle fracture, for fracture with a cohesive zone, or for perfect plasticity, depending on the asymptotic ratios of the three parameters.

Gamma-convergence of energies for nematic elastomers in the small strain limit - Virginia Agostiniani; Antonio DeSimone (Continuum. Mech. Therm., 2011)
We study two variational models recently proposed in the literature to describe the mechanical behaviour of nematic elastomers either in the fully nonlinear regime or in the framework of a geometrically linear theory. We show that, in the small strain limit, the energy functional of the first one I\\\"-converges to the relaxation of the second one, a functional for which an explicit representation formula is available.
Keywords: Liquid crystals

Generalised functions of bounded deformation - Gianni Dal Maso (J. Eur. Math. Soc. (JEMS), to appear, 2011)
We introduce the space GBD of generalized functions of bounded deformation and study the structure properties of these functions: the rectifiability and the slicing properties of their jump sets, and the existence of their approximate symmetric gradients. We conclude by proving a compactness results for GBD, which leads to a compactness result for the space GSBD of generalized special functions of bounded deformation. The latter is connected to the existence of solutions to a weak formulation of some variational problems arising in fracture mechanics in the framework of linearized elasticity.
Keywords: free discontinuity problems, special functions of bounded deformation, jump set, rec- tifiability, slicing, approximate differentiability

Generalized matrix models and AGT correspondence at all genera - Giulio Bonelli; Kazunobu Maruyoshi; Alessandro Tanzini; Futoshi Yagib (JHEP, Volume 2011, Issue 7, 2011, Article number055, 2011)
We study generalized matrix models corresponding to n-point Virasoro\r\nconformal blocks on Riemann surfaces with arbitrary genus g. Upon AGT\r\ncorrespondence, these describe four dimensional N=2 SU(2)^{n+3g-3} gauge\r\ntheories with generalized quiver diagrams. We obtain the generalized matrix\r\nmodels from the perturbative evaluation of the Liouville correlation functions\r\nand verify the consistency of the description with respect to degenerations of\r\nthe Riemann surface. Moreover, we derive the Seiberg-Witten curve for the N=2\r\ngauge theory as the spectral curve of the generalized matrix model, thus\r\nproviding a check of AGT correspondence at all genera.
This version is published in : Journal of High Energy Physics, Volume 2011, Issue 7, 2011, Article number055

Generalized Ricci Curvature Bounds for Three Dimensional Contact Subriemannian manifolds - Andrei A. Agrachev; Paul Lee (SISSA, 2011)
This is a revised extended version that contains new results.

The geometry of Maximum Principle - Andrei A. Agrachev; Revaz Gamkrelidze (Proceedings of the Steklov Institute of mathematics. vol. 273 (2011), page: 5-27 ; ISSN: 0081-5438, 2011)
An invariant formulation of the maximum principle in optimal control is presented, and some second-order invariants are discussed.

Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle - Indranil Biswas; Ugo Bruzzo (Differential Geometry and its Applications 29 (2011) 147-153, 2011)

Homology invariants of quadratic maps - Antonio Lerario (SISSA, 2011)
Given a real projective algebraic set X we could hope that the equations describing it can give some information on its topology, e.g. on the number of its connected components. Unfortunately in the general case this hope is too vague and there is no direct way to extract such information from the algebraic description of X: Even the problem to decide whether X is empty or not is far from an easy visualization and requires some complicated algebraic machinery. A fi rst step observation is that as long as we are interested only in the topology of X, we can replace, using some Veronese embedding, the original ambient space with a much bigger RPn and assume that X is cut by quadratic equations. The price for this is the increase of the number of equations de ning our set; the advantage is that quadratic polynomials are easier to handle and our hope becomes more concrete...

Infinite-dimensional Frobenius manifolds for 2 + 1 integrable systems - Guido Carlet; Boris Dubrovin; Luca Philippe Mertens (Matematische Annalen 349 (2011) 75-115, 2011)
We introduce a structure of an infinite-dimensional Frobenius manifold on a subspace in the space of pairs of functions analytic inside/outside the unit circle with simple poles at 0/infinity respectively. The dispersionless 2D Toda equations are embedded into a bigger integrable hierarchy associated with this Frobenius manifold.

Infinitely many positive solutions for a Schrödinger–Poisson system - Pietro d’Avenia; Alessio Pomponio; Giusi Vaira (Nonlinear Analysis: Theory, Methods & Applications, 2011)
We are interested in the existence of infinitely many positive solutions of the Schrödinger–Poisson system −Δu+u+V(|x|)ϕu=|u|p−1u,x∈R3,−Δϕ=V(|x|)u2,x∈R3, where V(|x|) is a positive bounded function, 1<p<5 and V(r
Keywords: Non-autonomous Schrödinger–Poisson system; Perturbation method

Instantons on ALE spaces and Super Liouville Conformal Field Theories - Giulio Bonelli; Kazunobu Maruyoshi; Alessandro Tanzini (SISSA, 2011)
We provide evidence that the conformal blocks of N=1 super Liouville\\r\\nconformal field theory are described in terms of the SU(2) Nekrasov partition\\r\\nfunction on the ALE space O_{P^1}(-2).
10 pages

An Integro-Extremization Approach for Non Coercive and Evolution Hamilton-Jacobi Equations - Sandro Zagatti (Journal of Convex Analysis 18 (2011) 1141-1170, 2011)
We devote the \\\\textit{integro-extremization} method to the study of the Dirichlet problem for homogeneous Hamilton-Jacobi equations \\\\begin{displaymath} \\\\begin{cases} F(Du)=0 & \\\\quad \\\\textrm{in} \\\\quad\\\\O\\\\cr u(x)=\\\\varphi(x) & \\\\quad \\\\textrm{for} \\\\quad x\\\\in \\\\partial \\\\O, \\\\end{cases} \\\\end{displaymath} with a particular interest for non coercive hamiltonians $F$, and to the Cauchy-Dirichlet problem for the corresponding homogeneous time-dependent equations \\\\begin{displaymath} \\\\begin{cases} \\\\frac{\\\\partial u}{\\\\partial t}+ F(\\\\nabla u)=0 & \\\\quad \\\\textrm{in} \\\\quad ]0,T[\\\\times \\\\O\\\\cr u(0,x)=\\\\eta(x) & \\\\quad \\\\textrm{for} \\\\quad x\\\\in\\\\O \\\\cr u(t,x)=\\\\psi(x) & \\\\quad \\\\textrm{for} \\\\quad (t,x)\\\\in[0,T]\\\\times \\\\partial \\\\O. \\\\end{cases} \\\\end{displaymath} We prove existence and some qualitative results for viscosity and almost everywhere solutions, under suitably convexity conditions on the hamiltonian $F$, on the domain $\\\\O$ and on the boundary datum, without any growth assumptions on $F$.

Invariant manifolds for a singular ordinary differential equation - Stefano Bianchini; Laura Spinolo (Journal of Differential Equations 250 (2011) 1788-1827, 2011)

Invariants, volumes and heat kernels in sub-Riemannian geometry - Davide Barilari (SISSA, 2011)
Sub-Riemannian geometry can be seen as a generalization of Riemannian geometry under non-holonomic constraints. From the theoretical point of view, sub-Riemannian geometry is the geometry underlying the theory of hypoelliptic operators (see [32, 57, 70, 92] and references therein) and many problems of geometric measure theory (see for instance [18, 79]). In applications it appears in the study of many mechanical problems (robotics, cars with trailers, etc.) and recently in modern elds of research such as mathematical models of human behaviour, quantum control or motion of self-propulsed micro-organism (see for instance [15, 29, 34])\\r\\nVery recently, it appeared in the eld of cognitive neuroscience to model the\\r\\nfunctional architecture of the area V1 of the primary visual cortex, as proposed by Petitot in [87, 86], and then by Citti and Sarti in [51]. In this context, the sub-Riemannian heat equation has been used as basis to new applications in image reconstruction (see [35]).
Keywords: Sub-Riemannian geometry

Large Time Existence for Thin Vibrating Plates - Helmut Abels; Maria Giovanna Mora; Stefan Müller (Communication in Partial Differential Equations 36 (2011) 2062-2102, 2011)
We construct strong solutions for a nonlinear wave equation for a thin vibrating plate described by nonlinear elastodynamics. For sufficiently small thickness we obtain existence of strong solutions for large\\r\\ntimes under appropriate scaling of the initial values such that the limit system as h --> 0 is either the nonlinear von Karman plate equation or the linear fourth order Germain-Lagrange equation. In the case of the\\r\\nlinear Germain-Lagrange equation we even obtain a convergence rate of the three-dimensional solution to the solution of the two-dimensional linear plate equation.

Linearly degenerate Hamiltonian PDEs and a new class of solutions to the WDVV associativity equations - Boris Dubrovin; M.V. Pavlov; Sergei A. Zykov (Functional Analysis and Its Applications. Volume 45, Issue 4, December 2011, Pages 278-290, 2011)
We define a new class of solutions to the WDVV associativity equations. This class is determined by the property that one of the commuting PDEs associated with such a WDVV solution is linearly degenerate. We reduce the problem of classifying such solutions of the WDVV equations to the particular case of the so-called algebraic Riccati equation and, in this way, arrive at a complete classification of irreducible solutions.
Keywords: Frobenius manifold

The Liouville side of the vortex - Giulio Bonelli; Alessandro Tanzini; Jian Zhao (JHEP 09(2011)096, 2011)
We analyze conformal blocks with multiple (semi-)degenerate field insertions in Liouville/Toda conformal field theories an show that their vector space is fully reproduced by the four-dimensional limit of open topological string amplitudes on the strip with generic boundary conditions associated to a suitable quiver gauge theory. As a byproduct we identify the non-abelian vortex partition function with a specific fusion channel of degenerate conformal blocks.
25pages,11figures

The matching property of infinitesimal isometries on elliptic surfaces and elasticity on thin shells - Marta Lewicka; Maria Giovanna Mora; Mohammad Reza Pakzad (Archive for Rational Mechanics and Analysis 200 (2011) 1023-1050, 2011)
Using the notion of Γ-convergence, we discuss the limiting behavior of the three-dimensional nonlinear elastic energy for thin elliptic shells, as their thickness h converges to zero, under the assumption that the elastic energy of deformations scales like h β with 2 < β < 4. We establish that, for the given scaling regime, the limiting theory reduces to linear pure bending. Two major ingredients of the proofs are the density of smooth infinitesimal isometries in the space of W 2,2 first order infinitesimal isometries, and a result on matching smooth infinitesimal isometries with exact isometric immersions on smooth elliptic surfaces.

Metastable equilibria of capillary drops on solid surfaces: a phase field approach - Livio Fedeli; Turco, Alessandro; Antonio DeSimone (Continuum Mechanics and Thermodynamics, 2011)
We discuss a phase field model for the numerical simulation of metastable equilibria of capillary drops resting on rough solid surfaces and for the description of contact angle hysteresis phenomena in wetting. The model is able to reproduce observed transitions of drops on micropillars from Cassie–Baxter to Wenzel states. When supplemented with a dissipation potential which describes energy losses due to frictional forces resisting the motion of the contact line, the model can describe metastable states such as drops in equilibrium on vertical glass plates. The reliability of the model is assessed by a detailed comparison of its predictions with experimental data on the maximal size of water drops that can stick on vertical glass plates which have undergone different surface treatments.

A MODEL FOR CRACK PROPAGATION BASED ON VISCOUS APPROXIMATION - Giuliano Lazzaroni; Rodica Toader ({MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES}, 2011)
{In the setting of antiplane linearized elasticity, we show the existence of quasistatic evolutions of cracks in brittle materials by using a vanishing viscosity approach, thus taking into account local minimization. The main feature of our model is that the path followed by the crack need not be prescribed a priori: indeed, it is found as the limit (in the sense of Hausdorff convergence) of curves obtained by an incremental procedure. The result is based on a continuity property for the energy release rate in a suitable class of admissible cracks.}
Keywords: Brittle fracture; Crack propagation; energy derivative; energy release rate; free-discontinuity problems; Griffith's criterion; local minimizers; stress intensity factor}; vanishing viscosity; {Variational models

Moduli of framed sheaves on projective surfaces - Ugo Bruzzo; Dimitri Markushevich (Doc. Math. 16 (2011) 399-410, 2011)
We show that there exists a fine moduli space for torsion-free sheaves on a\\r\\nprojective surface, which have a \\\"good framing\\\" on a big and nef divisor. This\\r\\nmoduli space is a quasi-projective scheme. This is accomplished by showing that such framed sheaves may be considered as stable pairs in the sense of\\r\\nHuybrechts and Lehn. We characterize the obstruction to the smoothness of the moduli space, and discuss some examples on rational surfaces.

The Monge Problem in Geodesic Spaces - Stefano Bianchini; Fabio Cavalletti (Nonlinear Conservation Laws and Applications, 2011)
We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish non branching geodesic space. We show that we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce an assumption on the transport problem π which implies that the conditional probabilities of the first marginal on each geodesic are continuous. It is known that this regularity is sufficient for the construction of an optimal transport map.

Multi-physics modelling and sensitivity analysis of olympic rowing boat dynamics - Andrea Mola; Mehdi Ghommem; Muhammad R. Hajj (Sports Engineering, 2011)

Multiplicity of solutions for a mean field equation on compact surfaces - Francesca De Marchis (Boll. Unione Mat. Ital.(9), 2011)
We consider a scalar field equation on compact surfaces which has variational structure. When the surface is a torus and a physical parameter ρ belongs to $(8π, 4π^2 )$ we show under some extra assumptions that, as conjectured in [9], the functional admits at least three saddle points other than a local minimum.

New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces - Andrea Malchiodi; David Ruiz (Geometric and Functional Analysis 21 (2011) 1196-1217, 2011)
We consider a singular Liouville equation on a compact surface, arising from the study of Chern-Simons vortices in a self dual regime. Using new improved versions of the Moser-Trudinger inequalities (whose main feature is to be scaling invariant) and a variational scheme, we prove new existence results.

Nonlinear resonance: a comparison between Landesman-Lazer and Ahmad-Lazer-Paul conditions - Alessandro Fonda; Maurizio Garrione (Advanced Nonlinear Studies, 2011)
We show that the Ahmad-Lazer-Paul condition for resonant problems is more general than the Landesman-Lazer one, discussing some relations with other existence conditions, as well. As a consequence, such a relation holds, for example, when considering resonant boundary value problems associated with linear elliptic operators, the p-Laplacian and, in the scalar case, with an asymmetric oscillator.

Nonlinear thin-walled beams with a rectangular cross-section - Part II - Lorenzo Freddi; Maria Giovanna Mora; Roberto Paroni (SISSA, 2011)
In this paper we report the second part of our results concerning the rigorous derivation of a hierarchy of one-dimensional models for thin-walled beams with rectangular cross-section..
Keywords: Thin-walled cross-section beams

Nonlinear wave and Schrödinger equations on compact Lie groups and homogeneous spaces - Massimiliano Berti; Michela Procesi (Duke Mathematical Journal, 2011)
We develop linear and nonlinear harmonic analysis on compact Lie groups and homogeneous spaces relevant for the theory of evolutionary Hamiltonian PDEs. A basic tool is the theory of the highest weight for irreducible representations of compact Lie groups. This theory provides an accurate description of the eigenvalues of the Laplace-Beltrami operator as well as the multiplication rules of its eigenfunctions. As an application, we prove the existence of Cantor families of small amplitude time-periodic solutions for wave and Schr¨odinger equations with differentiable nonlinearities. We apply an abstract Nash-Moser implicit function theorem to overcome the small divisors problem produced by the degenerate eigenvalues of the Laplace operator. We provide a new algebraic framework to prove the key tame estimates for the inverse linearized operators on Banach scales of Sobolev functions.

A note on a superlinear indefinite Neumann problem with multiple positive solutions - Alberto Boscaggin (Journal of Mathematical Analysis and Applications, 2011)
We prove the existence of three positive solutions for the Neumann problem associated to u″+a(t)uγ+1=0, assuming that a(t) has two positive humps and ∫0Ta−(t)dt is large enough. Actually, the result holds true for a more general class of superlinear nonlinearities.
Keywords: Indefinite weight; Nonlinear boundary value problems; positive solutions; Shooting method

On the number of eigenvalues of a model operator related to a system of three particles on lattices - Gianfausto Dell'Antonio; Zahriddin I. Muminov; Y.M. Shermatova (J. Phys. A 44 (2011) 315302, 2011)

Numerical Strategies for Stroke Optimization of Axisymmetric Microswimmers - François Alouges; Antonio DeSimone; Luca Heltai (Mathematical Models and Methods in Applied Sciences 21 (2011) 361-387, 2011)
We propose a computational method to solve optimal swimming problems, based on the boundary integral formulation of the hydrodynamic interaction between swimmer and surrounding fluid and direct constrained minimization of the energy consumed by the swimmer. We apply our method to axisymmetric model examples. We consider a classical model swimmer (the three-sphere swimmer of Golestanian et al.) as well as a novel axisymmetric swimmer inspired by the observation of biological micro-organisms.
Keywords: Optimal swimming

Numerical Study of breakup in generalized Korteweg-de Vries and Kawahara equations - Boris Dubrovin; Tamara Grava; Christian Klein (SIAM J. Appl. Math. 71 (2011) 983-1008, 2011)
This article is concerned with a conjecture in [B. Dubrovin, Comm. Math. Phys., 267 (2006), pp. 117–139] on the formation of dispersive shocks in a class of Hamiltonian dispersive regularizations of the quasi-linear transport equation. The regularizations are characterized by two arbitrary functions of one variable, where the condition of integrability implies that one of these functions must not vanish. It is shown numerically for a large class of equations that the local behavior of their solution near the point of gradient catastrophe for the transport equation is described by a special solution of a Painlevé-type equation. This local description holds also for solutions to equations where blowup can occur in finite time. Furthermore, it is shown that a solution of the dispersive equations away from the point of gradient catastrophe is approximated by a solution of the transport equation with the same initial data, modulo terms of order $\\\\epsilon^2$, where $\\\\epsilon^2$ is the small dispersion parameter. Corrections up to order $\\\\epsilon^4$ are obtained and tested numerically.

Osservazioni sui teoremi di inversione globale - Antonio Ambrosetti (Rendiconti Lincei - Matematica e Applicazioni 22 (2011) 3-15, 2011)
Some global inversion theorems with applications to semilinear elliptic equation are discussed.

A planar bi-Lipschitz extension Theorem - Sara Daneri; Aldo Pratelli (2011)

Planar loops with prescribed curvature: existence, multiplicity and uniqueness results - Roberta Musina (Proceedings of the American Mathematical Society 139 (2011) 4445-4459, 2011)
Keywords: Plane curves

Poincaré covariance and κ-Minkowski spacetime - Ludwik Dabrowski; Gherardo Piacitelli (Physics Letters A 375 (2011) 3496-3498, 2011)
A fully Poincaré covariant model is constructed out of the k-Minkowski spacetime. Covariance is implemented by a unitary representation of the Poincaré group, and thus complies with the original Wigner approach to quantum symmetries. This provides yet another example (besides the DFR model), where Poincaré covariance is realised á la Wigner in the presence of two characteristic dimensionful parameters: the light speed and the Planck length. In other words, a Doubly Special Relativity (DSR) framework may well be realised without deforming the meaning of \\\"Poincaré covariance\\\".

Poincaré polynomial of moduli spaces of framed sheaves on (stacky) Hirzebruch surfaces - Ugo Bruzzo; Rubik Poghossian; Alessandro Tanzini (Communications in Mathematical Physics 304 (2011) 395-409, 2011)
We perform a study of the moduli space of framed torsion-free sheaves on Hirzebruch surfaces by using localization techniques. We discuss some general properties of this moduli space by studying it in the framework of Huybrechts-Lehn theory of framed modules. We classify the fixed points under a toric action on the moduli space, and use this to compute the Poincare polynomial of the latter. This will imply that the moduli spaces we are considering are irreducible. We also consider fractional first Chern classes, which means that we are extending our computation to a stacky deformation of a Hirzebruch surface. From the physical viewpoint, our results provide the partition function of N=4 Vafa-Witten theory on total spaces of line bundles on P1, which is relevant in black hole entropy counting problems according to a conjecture due to Ooguri, Strominger and Vafa.

Product of real spectral triples - Ludwik Dabrowski; Giacomo Dossena (International Journal of Geometric Methods in Modern Physics 8 (2011) 1833-1848, 2011)
We construct the product of real spectral triples of arbitrary finite dimension (and arbitrary parity) taking into account the fact that in the even case there are two possible real structures, in the odd case there are two inequivalent representations of the gamma matrices (Clifford algebra), and in the even-even case there are two natural candidates for the Dirac operator of the product triple.
Based on the talk given at the conference \\\"Noncommutative Geometry and Quantum Physics, Vietri sul Mare, Aug 31 - Sept 5, 2009\\\"

A proof of Sudakov theorem with strictly convex norms - Laura Caravenna (Mathematische Zeitschrift 268 (2011) 371-407, 2011)
We establish a solution to the Monge problem in Rn, with an asymmetric, strictly convex norm cost function, when the initial measure is absolutely continuous. We focus on the strategy, based on disintegration of measures, initially proposed by Sudakov. As known, there is a gap to fill. The missing step is completed when the unit ball is strictly convex, but not necessarily differentiable nor uniformly convex. The key disintegration is achieved following a similar proof for a variational problem.

Q-factorial Laurent rings - Ugo Bruzzo; Antonella Grassi (SISSA, 2011)
Dolgachev proved that, for any field k, the ring naturally associated to a\\r\\ngeneric Laurent polynomial in d variables, $d \\\\geq 4$, is factorial. We prove a\\r\\nsufficient condition for the ring associated to a very general complex Laurent\\r\\npolynomial in d=3 variables to be Q-factorial.
5 pages

Quantum Geometry on Quantum Spacetime: Distance, Area and Volume Operators - Dorothea Bahns; Sergio Doplicher; Klaus Fredenhagen; Gherardo Piacitelli (Commun. Math. Phys. 308 (2011) 567-589, 2011)
We develop the first steps towards an analysis of geometry on the quantum\\r\\nspacetime proposed in Doplicher et al. (Commun Math Phys 172:187–220, 1995). The homogeneous elements of the universal differential algebra are naturally identified with operators living in tensor powers of Quantum Spacetime; this allows us to compute their spectra. In particular, we consider operators that can be interpreted as distances, areas, 3- and 4-volumes. The Minkowski distance operator between two independent events is shown to have pure Lebesgue spectrum with infinite multiplicity. The Euclidean distance operator is shown to have spectrum bounded below by a constant of the order of the Planck length. The corresponding statement is proved also for both the space-space and space-time area operators, as well as for the Euclidean length of the vector representing the 3-volume operators. However, the space 3-volume operator (the time component of that vector) is shown to have spectrum equal to the whole complex plane. All these operators are normal, while the distance operators are also selfadjoint. The Lorentz invariant spacetime volume operator, representing the 4- volume spanned by five\\r\\nindependent events, is shown to be normal. Its spectrum is pure point with a\\r\\nfinite distance (of the order of the fourth power of the Planck length) away\\r\\nfrom the origin. The mathematical formalism apt to these problems is developed and its relation to a general formulation of Gauge Theories on Quantum Spaces is outlined. As a byproduct, a Hodge Duality between the absolute differential and the Hochschild boundary is pointed out.

Quantum Hitchin Systems via beta-deformed Matrix Models - Giulio Bonelli; Kazunobu Maruyoshi; Alessandro Tanzini (SISSA, 2011)
We study the quantization of Hitchin systems in terms of beta-deformations of generalized matrix models related to conformal blocks of Liouville theory on punctured Riemann surfaces. We show that in a suitable limit, corresponding to the Nekrasov-Shatashvili one, the loop equations of the matrix model reproduce the Hamiltonians of the quantum Hitchin system on the sphere and the torus with marked points. The eigenvalues of these Hamiltonians are shown to be the epsilon1-deformation of the chiral observables of the corresponding N=2 four ndimensional gauge theory. Moreover, we find the exact wave-functions in terms of the matrix model representation of the conformal blocks with degenerate field insertions.
29 pages; v2. refs. added and typos corrected

Quantum Isometries of the finite noncommutative geometry of the Standard Model - Jyotishman Bhowmick; Francesco D'Andrea; Ludwik Dabrowski (Commun. Math. Phys. 307:101-131, 2011, 2011)
We compute the quantum isometry group of the finite noncommutative geometry F describing the internal degrees of freedom in the Standard Model of particle physics. We show that this provides genuine quantum symmetries of the spectral triple corresponding to M x F where M is a compact spin manifold. We also prove that the bosonic and fermionic part of the spectral action are preserved by these symmetries.

Quasiconvex envelopes of energies for nematic elastomers in the small strain regime and applications - Pierluigi Cesana; Antonio DeSimone (Journal of the Mechanics and Physics of Solids 59 (2011) 787-803, 2011)
We provide some explicit formulas for the quasiconvex envelope of energy densities for nematic elastomers in the small strain regime and plane strain conditions. We then demonstrate their use as a powerful tool for the interpretation of mechanical experiments.

Quasi-periodic solutions of nonlinear Schrödinger equations on $\Bbb T^d$ - Massimiliano Berti; Bolle, Philippe (Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 2011)

Quasistatic crack evolution for a cohesive zone model with different response to loading and unloading: a Young measures approach - Filippo Cagnetti; Rodica Toader (ESAIM: COCV 17 (2011) 1-27, 2011)
A new approach to irreversible quasistatic fracture growth is given, by means of Young measures. The study concerns a cohesive zone model with prescribed crack path, when the material gives different responses to loading and unloading phases. In the particular situation of constant unloading response, the result contained in [6] is recovered. In this case, the convergence of the discrete time approximations is improved.

Quasistatic crack growth in finite elasticity with Lipschitz data - Giuliano Lazzaroni (ANNALI DI MATEMATICA PURA ED APPLICATA, 2011)
{We extend the recent existence result of Dal Maso and Lazzaroni (Ann Inst H Poincare Anal Non Lineaire 27:257-290, 2010) for quasistatic evolutions of cracks in finite elasticity, allowing for boundary conditions and external forces with discontinuous first derivatives.}
Keywords: Brittle fracture; Crack propagation; Energy minimization; Finite elasticity; free-discontinuity problems; Griffith's criterion; Non-interpenetration}; Polyconvexity; Quasistatic evolution; Rate-independent processes; {Variational models

Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling - Gianni Dal Maso; Antonio DeSimone; Francesco Solombrino (Calculus of Variations and Partial Differential Equations 40 (2011) 125-181, 2011)
Cam-Clay nonassociative plasticity exhibits both hardening and softening behaviour, depending on the loading. For many initial data the classical formulation of the quasistatic evolution problem has no smooth solution. We propose here a notion of generalized solution, based on a viscoplastic approximation. To study the limit of the viscoplastic evolutions we rescale time, in such a way that the plastic strain is uniformly Lipschitz with respect to the rescaled time. The limit of these rescaled solutions, as the viscosity parameter tends to zero, is characterized through an energy-dissipation balance, that can be written in a natural way using the rescaled time. As shown in [4] and [6], the proposed solution may be discontinuous with respect to the original time. Our formulation allows to compute the amount of viscous dissipation occurring instantaneously at each discontinuity time.
Keywords: Cam-Clay plasticity

Quasistatic evolution of sessile drops and contact angle hysteresis - Giovanni Alberti; Antonio DeSimone (Arch. Rational Mech. Anal. 202 (2011) 295-348, 2011)
We consider the classical model of capillarity coupled with a rate-independent dissipation mechanism due to frictional forces acting on the contact line, and prove the existence of quasistatic evolutions with prescribed initial configuration. We also discuss in detail some explicit solutions to show that the model does account for contact angle hysteresis, and to compare its predictions with experimental observations.

Resonance and Landesman-Lazer conditions for first order systems in R^2 - Maurizio Garrione (Le Matematiche, 2011)
The first part of the paper surveys the concept of resonance for $T$-periodic nonlinear problems. In the second part, some new results about existence conditions for nonlinear planar systems are presented. In particular, the Landesman-Lazer conditions are generalized to systems in $\mathbbR^2$ where the nonlinearity interacts with two resonant Hamiltonians. Such results apply to second order equations, generalizing previous theorems by Fabry [4] (for the undamped case), and Frederickson-Lazer [9] (for the case with friction). The results have been obtained with A. Fonda, and have been published in [8].

Resonance and rotation numbers for planar Hamiltonian systems: Multiplicity results via the Poincaré–Birkhoff theorem - Alberto Boscaggin; Maurizio Garrione (Nonlinear Analysis: Theory, Methods & Applications, 2011)
In the general setting of a planar first order system (0.1)u′=G(t,u),u∈R2, with G:[0,T]×R2→R2, we study the relationships between some classical nonresonance conditions (including the Landesman–Lazer one) — at infinity and, in the unforced case, i.e. G(t,0)≡0, at zero — and the rotation numbers of “large” and “small” solutions of (0.1), respectively. Such estimates are then used to establish, via the Poincaré–Birkhoff fixed point theorem, new multiplicity results for T-periodic solutions of unforced planar Hamiltonian systems Ju′=∇uH(t,u) and unforced undamped scalar second order equations x″+g(t,x)=0. In particular, by means of the Landesman–Lazer condition, we obtain sharp conclusions when the system is resonant at infinity.
Keywords: Multiple periodic solutions; Poincaré–Birkhoff theorem; Resonance; Rotation number

SBV regularity for Hamilton-Jacobi equations in R^n - Stefano Bianchini; Camillo De Lellis; Roger Robyr (Arch. Rational Mech. Anal. 200 (2011) 1003-1021, 2011)
In this paper we study the regularity of viscosity solutions to the following Hamilton-Jacobi equations $$ \partial_t u + H(D_{x} u)=0 \qquad in\quad Ω\subset \mathbb{R}× \mathbb{R}^{n} . $$ In particular, under the assumption that the Hamiltonian $H\in C^2(\mathbb{R}^n)$ is uniformly convex, we prove that $D_{x}u$ and $\partial_t u$ belong to the class $SBV_{loc}(Ω)$.

Semistable and numerically effective principal (Higgs) bundles - Ugo Bruzzo; Beatriz Grana-Otero (Advances in Mathematics 226 (2011) 3655-3676, 2011)
We study Miyaoka-type semistability criteria for principal Higgs G-bundles E on complex projective manifolds of any dimension. We prove that E has the property of being semistable after pullback to any projective curve if and only if certain line bundles, obtained from some characters of the parabolic subgroups of G, are numerically effective. One also proves that these conditions are met for semistable principal Higgs bundles whose adjoint bundle has vanishing second Chern class.\\r\\n\\r\\nIn a second part of the paper, we introduce notions of numerical effectiveness and numerical flatness for principal (Higgs) bundles, discussing their main properties. For (non-Higgs) principal bundles, we show that a numerically flat principal bundle admits a reduction to a Levi factor which has a flat Hermitian–Yang–Mills connection, and, as a consequence, that the cohomology ring of a numerically flat principal bundle with coefficients in R is trivial. To our knowledge this notion of numerical effectiveness is new even in the case of (non-Higgs) principal bundles.

Singular perturbation models in phase transitions for second order materials - Milena Chermisi; Gianni Dal Maso; Irene Fonseca; Giovanni Leoni (Indiana Univ. Math. J. 60 (2011) 367-409, 2011)
A variational model proposed in the physics literature to describe the onset of pattern formation in two-component bilayer membranes and amphiphilic monolayers leads to the analysis of a Ginzburg-Landau type energy with a negative term depending on the first derivative of the phase function. Scaling arguments motivate the study of the family of second order singular perturbed energies Fe having a negative term depending on the first derivative of the phase function. Here, the asymptotic behavior of Fe is studied using G-convergence techniques. In particular, compactness results and an integral representation of the limit energy are obtained.

Solving PVI by Isomonodromy Deformations - Davide Guzzetti (Painlevé equations and related topics : proceedings of the international conference, Saint Petersburg, Russia, June 17-23, 2011 / Aleksandr Dmitrievich Briuno; Alexander B Batkhin. - Berlin : De Gruyter, [2012]. - p. 101-105, 2011)
The critical and asymptotic behaviors of solutions of the sixth Painlev\\\'e\r\nequation, an their parametrization in terms of monodromy data, are\r\nsynthetically reviewed. The explicit formulas are given. This paper has been\r\nwithdrawn by the author himself, because some improvements are necessary.\r\nThis is a proceedings of the international conference \"Painlevé Equations and Related Topics\" which was taking place at the Euler International Mathematical Institute, a branch of the Saint Petersburg Department of the SteklovInstitute of Mathematicsof theRussian Academy of Sciences, in Saint Petersburg on June 17 to 23, 2011.
12 pages, 1 figurethis paper has been
Keywords: Painlevé Equations

On the Space of Symmetric Operators with Multiple Ground States - Andrei A. Agrachev (Functional Analysis and its Applications, Volume 45, Issue 4, December 2011, Pages 241-251, 2011)
We study homological structure of the filtrations of the spaces of self-adjoint operators by the multiplicity of the ground state. We consider only operators acting in a finite dimensional complex or real Hilbert space but infinite dimensional generalizations are easily guessed.
Keywords: Multiple eigenvalue

The sphere and the cut locus at a tangency point in two-dimensional almost-Riemannian geometry - Bernard Bonnard; Grégoire Charlot; Roberta Ghezzi; Gabriel Janin (Journal of Dynamical and Control Systems, 2011)
We study the tangential case in 2-dimensional almost-Riemannian geometry. We\\r\\nanalyse the connection with the Martinet case in sub-Riemannian geometry. We\\r\\ncompute estimations of the exponential map which allow us to describe the\\r\\nconjugate locus and the cut locus at a tangency point. We prove that this last\\r\\none generically accumulates at the tangency point as an asymmetric cusp whose branches are separated by the singular set.

Structure of level sets and Sard-type properties of Lipschitz maps - Giovanni Alberti; Stefano Bianchini; Gianluca Crippa (SISSA, 2011)

Subharmonic solutions of planar Hamiltonian systems: a rotation number approach - Alberto Boscaggin (Advanced Nonlinear Studies, 2011)

Subharmonic solutions of planar Hamiltonian systems via the Poincaré́-Birkhoff theorem - Alberto Boscaggin (Le Matematiche, 2011)
We revisit some recent results obtained in [1] about the existence of subharmonic solutions for a class of (nonautonomous) planar Hamiltonian systems, and we compare them with the existing literature. New applications to undamped second order equations are discussed, as well.

Supercritical conformal metrics on surfaces with conical singularities - Mauro Bardelloni; Francesca De Marchis; Andrea Malchiodi (Int Math Res Notices (2011) 2011 (24): 5625-5643, 2011)
We study the problem of prescribing the Gaussian curvature on surfaces with conical singularities in supercritical regimes. Using a Morse-theoretical approach we prove a general existence theorem on surfaces with positive genus, with a generic multiplicity result.

A system-level approach for deciphering the transcriptional response to prion infection - Mattia Zampieri; Giuseppe Legname; Daniel Segrè; Claudio Altafini (Bioinformatics (Oxford, England). 2011 Dec; 27(24):3407-14, 2011)
MOTIVATION: Deciphering the response of a complex biological system to an insulting event, at the gene expression level, requires adopting theoretical models that are more sophisticated than a one-to-one comparison (i.e. t-test). Here, we investigate the ability of a novel reverse engineering approach (System Response Inference) to unveil non-obvious transcriptional signatures of the system response induced by prion infection.\\r\\nRESULTS: To this end, we analyze previously published gene expression data, from which we extrapolate a putative full-scale model of transcriptional gene-gene dependencies in the mouse central nervous system. Then, we use this nominal model to interpret the gene expression changes caused by prion replication, aiming at selecting the genes primarily influenced by this perturbation. Our method sheds light on the mode of action of prions by identifying key transcripts that are the most likely to be responsible for the overall transcriptional rearrangement from a nominal regulatory network. As a first result of our inference, we have been able to predict known targets of prions (i.e. PrP(C)) and to unveil the potential role of previously unsuspected genes.\\r\\nCONTACT: altafini@sissa.it\\r\\nSUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Thin-walled beams with a cross-section of arbitrary geometry: derivation of linear theories starting from 3D nonlinear elasticity - Elisa Davoli (2011)
The subject of this paper is the rigorous derivation of lower dimensional models for a nonlinearly elastic thin-walled beam whose cross-section is given by a thin tubular neighbourhood of a smooth curve. Denoting by h and δ_h, respectively, the diameter and the thickness of the cross-section, we analyse the case where the scaling factor of the elastic energy is of order ε_h^2, with ε_h/δ_h^2 \rightarrow l \in [0, +\infty). Different linearized models are deduced according to the relative order of magnitude of δ_h with respect to h.

The time-dependent von Kármán plate equation as a limit of 3d nonlinear elasticity - Helmut Abels; Maria Giovanna Mora; Stefan Müller (Calculus of Variations and Partial Differential Equations 41 (2011) 241-259, 2011)
The asymptotic behaviour of the solutions of three-dimensional nonlinear elastodynamics in a thin plate is studied, as the thickness $h$ of the plate tends to zero. Under appropriate scalings of the applied force and of the initial values in terms of $h$, it is shown that three-dimensional solutions of the nonlinear elastodynamic equation converge to solutions of the time-dependent von K\\\\\\\'arm\\\\\\\'an plate equation.

The Transition between the Gap Probabilities from the Pearcey to the Airy Process–a Riemann-Hilbert Approach - Marco Bertola; Mattia Cafasso (International Mathematics Research Notices, 2011)

Uniqueness and nondegeneracy of the ground state for a quasilinear Schrödinger equation with a small parameter - Alessandro Selvitella (Nonlinear Analysis: Theory, Methods & Applications, 2011)
We study least energy solutions of a quasilinear Schrödinger equation with a small parameter. We prove that the ground state is nondegenerate and unique up to translations and phase shifts using bifurcation theory.
Keywords: Bifurcation theory; Nonlinear Schrödinger equations; Stationary solutions

A uniqueness result for the continuity equation in two dimensions - Giovanni Alberti; Stefano Bianchini; Gianluca Crippa (SISSA, 2011)

The well-posedness issue for the density-dependent Euler equations in endpoint Besov spaces - Raphaël Danchin; Francesco Fanelli (Journal de Mathématiques Pures et Appliquées, 2011)
This work is the continuation of the recent paper (Danchin, 2010) [9] devoted to the density-dependent incompressible Euler equations. Here we concentrate on the well-posedness issue in Besov spaces of type B∞,rs embedded in the set of Lipschitz continuous functions, a functional framework which contains the particular case of Hölder spaces C1,α and of the endpoint Besov space B∞,11. For such data and under the non-vacuum assumption, we establish the local well-posedness and a continuation criterion in the spirit of that of Beale, Kato and Majda (1984) [2]. In the last part of the paper, we give lower bounds for the lifespan of a solution. In dimension two, we point out that the lifespan tends to infinity when the initial density tends to be a constant. This is, to our knowledge, the first result of this kind for the density-dependent incompressible Euler equations. Résumé Ce travail complète lʼarticle récent (Danchin, 2010) [9] consacré au système dʼEuler incompressible à densité variable. Lorsque lʼétat initial ne comporte pas de vide, on montre ici que le système est bien posé dans tous les espaces de Besov B∞,rs inclus dans lʼensemble des fonctions lipschitziennes. Ce cadre fonctionnel contient en particulier les espaces de Hölder C1,α et lʼespace de Besov limite B∞,11. On établit également un critère de prolongement dans lʼesprit de celui de Beale, Kato et Majda (1984) [2] pour le cas homogène. Dans la dernière partie de lʼarticle, on donne des minorations pour le temps de vie des solutions du système. En dimension deux, on montre que ce temps de vie tend vers lʼinfini lorsque la densité tend à être homogène. À notre connaissance, il sʼagit du premier résultat de ce type pour le système dʼEuler incompressible à densité variable.
Keywords: Blow-up criterion; Critical regularity; Incompressible Euler equations; Lifespan; Nonhomogeneous inviscid fluids

An abstract Nash-Moser theorem with parameters and applications to PDEs - Massimiliano Berti; Philippe Bolle; Michela Procesi (Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis, 2010)
We prove an abstract Nash-Moser implicit function theorem with parameters which covers the applications to the existence of finite dimensional, differentiable, invariant tori of Hamiltonian PDEs with merely differentiable nonlinearities. The main new feature of the abstract iterative scheme is that the linearized operators, in a neighborhood of the expected solution, are invertible, and satisfy the "tame" estimates, only for proper subsets of the parameters. As an application we show the existence of periodic solutions of nonlinear wave equations on Riemannian Zoll manifolds. A point of interest is that, in presence of possibly very large "clusters of small divisors", due to resonance phenomena, it is more natural to expect solutions with only Sobolev regularity. © 2009 Elsevier Masson SAS. All rights reserved.
cited By (since 1996)9
Keywords: Abstracting; Aircraft engines; Finite dimensional; Hamiltonian PDEs; Implicit function theorem; Invariant tori; Iterative schemes; Linearized operators; Mathematical operators; Moser theorem; Non-Linearity; Nonlinear equations; Nonlinear wave equation; Periodic solution; Point of interest; Resonance phenomena; Small divisors; Sobolev; Wave equations

Almost-Riemannian Geometry from a Control Theoretical Viewpoint - Roberta Ghezzi (SISSA, 2010)

Aspects of Quantum Field Theory on Quantum Spacetime - Gherardo Piacitelli (PoS CNCFG2010:027,2010, 2010)
We provide a minimal, self-contained introduction to the covariant DFR flat\\r\\nquantum spacetime, and to some partial results for the corresponding quantum field theory. Explicit equations are given in the Dirac notation.
25 pages, active hyperlinks. Corfu Summer Institute on Elementary\\r\\n Particles and Physics - Workshop on Non Commutative Field Theory and Gravity,\\r\\n September 8-12, 2010, Corfu Greece

Canonical k-Minkowski Spacetime - Gherardo Piacitelli; Ludwik Dabrowski (2010)
A complete classification of the regular representations of the relations [T,X_j] = (i/k)X_j, j=1,...,d, is given. The quantisation of RxR^d canonically (in the sense of Weyl) associated with the universal representation of the above relations is intrinsically \\\"radial\\\", this meaning that it only involves the time variable and the distance from the origin; angle variables remain classical. The time axis through the origin is a spectral singularity of the model: in the large scale limit it is topologically disjoint from the rest. The symbolic calculus is developed; in particular there is a trace functional on symbols. For suitable choices of states localised very close to the origin, the uncertainties of all spacetime coordinates can be made simultaneously small at wish. On the contrary, uncertainty relations become important at \\\"large\\\" distances: Planck scale effects should be visible at LHC energies, if processes are spread in a region of size 1mm (order of peak nominal beam size) around the origin of spacetime.

Cauchy biorthogonal polynomials - Marco Bertola; Gekhtman, M.; Szmigielski, J. (J. Approx. Theory, 2010)

Chern-Simons theory on L(p,q) lens spaces and Gopakumar-Vafa duality - Andrea Brini; Luca Griguolo; Domenico Seminara; Alessandro Tanzini (J. Geom. Phys. 60 (2010) 417-429, 2010)
We consider aspects of Chern-Simons theory on L(p,q) lens spaces and its relation with matrix models and topological string theory on Calabi-Yau threefolds, searching for possible new large N dualities via geometric transition for non-SU(2) cyclic quotients of the conifold. To this aim we find, on one hand, some novel matrix integral representations of the SU(N) CS partition function in a generic flat background for the whole L(p,q) family and provide a solution for its large N dynamics; on the other, we perform in full detail the construction of a family of would-be dual closed string backgrounds via conifold geometric transition from T^*L(p,q). We can then explicitly prove that Gopakumar-Vafa duality in a fixed vacuum fails in the case q>1, and briefly discuss how it could be restored in a non-perturbative setting.

Cohomology of Skew-holomorphic lie algebroids - Ugo Bruzzo; Vladimir Rubtsov (2010)
We introduce the notion of skew-holomorphic Lie algebroid on a complex manifold, and explore some cohomologies theories that one can associate to it. Examples are given in terms of holomorphic Poisson structures of various sorts.

Concentration of solutions for some singularly perturbed mixed problems: Asymptotics of minimal energy solutions - Jesus Garcia Azorero; Andrea Malchiodi; Luigi Montoro; Ireneo Peral (Ann. Inst. H. Poincare Anal. Non Lineaire 27 (2010) 37-56, 2010)
In this paper we carry on the study of asymptotic behavior of some solutions to a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions, started in the first paper. Here we are mainly interested in the analysis of the location and shape of least energy solutions when the singular perturbation parameter tends to zero. We show that in many cases they coincide with the new solutions produced in.

Concentration of solutions for some singularly perturbed mixed problems. Part I: existence results - Jesus Garcia Azorero; Andrea Malchiodi; Luigi Montoro; Ireneo Peral (Arch. Ration. Mech. Anal. 196 (2010) 907-950, 2010)
In this paper we study the asymptotic behavior of some solutions to a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions. We prove that, under suitable geometric conditions on the boundary of the domain, there exist solutions which approach the intersection of the Neumann and the Dirichlet parts as the singular perturbation parameter tends to zero.

Continuity of optimal control costs and its application to weak KAM theory - Andrei A. Agrachev; Paul Lee (Calculus of Variations and Partial Differential Equations. Volume 39, Issue 1, 2010, Pages 213-232, 2010)
We prove continuity of certain cost functions arising from optimal control of\\r\\naffine control systems. We give sharp sufficient conditions for this\\r\\ncontinuity. As an application, we prove a version of weak KAM theorem and\\r\\nconsider the Aubry-Mather problems corresponding to these systems.
23 pages, 1 figures

Convergence of equilibria of thin elastic rods under physical growth conditions for the energy density - Elisa Davoli; Maria Giovanna Mora (2010)
The subject of this paper is the study of the asymptotic behaviour of the equilibrium configurations of a nonlinearly elastic thin rod, as the diameter of the cross-section tends to zero. Convergence results are established assuming physical growth conditions for the elastic energy density and suitable scalings of the applied loads, that correspond at the limit to different rod models: the constrained linear theory, the analogous of von Kármán plate theory for rods, and the linear theory.

The dependence on the monodromy data of the isomonodromic tau function - Marco Bertola (Comm. Math. Phys., 2010)

Dirac Operators on Quantum Projective Spaces - Francesco D'Andrea; Ludwik Dabrowski (Comm. Math. Phys. 295 (2010) 731-790, 2010)
We construct a family of self-adjoint operators D_N which have compact resolvent and bounded commutators with the coordinate algebra of the quantum projective space CP_q(l), for any l>1 and 0<q<1. They provide 0^+ dimensional equivariant even spectral triples. If l is odd and N=(l+1)/2, the spectral triple is real with KO-dimension 2l mod 8.
Quantum Algebra

The disintegration of the Lebesgue measure on the faces of a convex function - Laura Caravenna; Sara Daneri (J. Funct. Anal. 258 (2010) 3604-3661, 2010)
We consider the disintegration of the Lebesgue measure on the graph of a convex function f:\\\\Rn-> \\\\R w.r.t. the partition into its faces, which are convex sets and therefore have a well defined linear dimension, and we prove that each conditional measure is equivalent to the k-dimensional Hausdorff measure of the k-dimensional face on which it is concentrated. The remarkable fact is that a priori the directions of the faces are just Borel and no Lipschitz regularity is known. Notwithstanding that, we also prove that a Green-Gauss formula for these directions holds on special sets.

Dynamics control by a time-varying feedback - Andrei A. Agrachev; Marco Caponigro (Journal of Dynamical and Control Systems. Volume 16, Issue 2, April 2010, Pages :149-162, 2010)
We consider a smooth bracket generating control-affine system in R^d and show that any orientation preserving diffeomorphism of R^d can be approximated, in the very strong sense, by a diffeomorphism included in the flow generated by a time-varying feedback control which is polynomial with respect to the state variables and trigonometric-polynomial with respect to the time variable.
Keywords: Discrete-time dynamics

Effective Schroedinger dynamics on $ ε$-thin Dirichlet waveguides via Quantum Graphs I: star-shaped graphs - Gianfausto Dell'Antonio; Emanuele Costa (J. Phys. A 43 (2010) 474014, 2010)
We describe the boundary conditions at the vertex that one must choose to obtain a dynamical system that best describes the low-energy part of the evolution of a quantum system confined to a very small neighbourhood of a star-shaped metric graph.

Estimates on path functionals over Wasserstein Spaces - Stefano Bianchini; Alessio Brancolini (SIAM J. Math. Anal. 42 (2010) 1179-1217, 2010)
In this paper we consider the class a functionals (introduced in [Brancolini, Buttazzo, and Santambrogio, J. Eur. Math. Soc. (JEMS), 8 (2006), pp. 415-434] $\\\\mathcal{G}_{r,p}$ defined on Lipschitz curves $\\\\gamma$ valued in the $p$-Wasserstein space. The problem considered is the following: given a measure $\\\\mu$, give conditions in order to assure the existence of a curve $\\\\gamma$ such that $\\\\gamma(0)=\\\\mu$, $\\\\gamma(1)=\\\\delta_{x_0}$, and $\\\\mathcal{G}_{r,p}(\\\\gamma)<+\\\\infty$. To this end, new estimates on $\\\\mathcal{G}_{r,p}(\\\\mu)$ are given, and a notion of dimension of a measure (called path dimension) is introduced: the path dimension specifies the values of the parameters $(r,p)$ for which the answer to the previous reachability problem is positive. Finally, we compare the path dimension with other known dimensions.

On the Euler-Lagrange equation for a variational problem : the general case II - Stefano Bianchini; Matteo Gloyer (Math. Z. 265 (2010) 889-923, 2010)

Exact reconstruction of damaged color images using a total variation model - Irene Fonseca; Giovanni Leoni; Francesco Maggi; Massimiliano Morini (Ann. Inst. H. Poincare Anal. Non Lineaire 27 (2010) 1291-1331, 2010)
In this paper the reconstruction of damaged piecewice constant color images is studied using a RGB total variation based model for colorization/inpainting. In particular, it is shown that when color is known in a uniformly distributed region, then reconstruction is possible with maximal fidelity.

Existence of planar curves minimizing length and curvature - Ugo Boscain; Grégoire Charlot; Francesco Rossi (Proc. Steklov Inst. Math. 270 (2010) 43-56, 2010)
In this paper we consider the problem of reconstructing a curve that is partially hidden or corrupted by minimizing the functional $\\\\int \\\\sqrt{1+K_\\\\gamma^2} ds$, depending both on length and curvature $K$. We fix starting and ending points as well as initial and final directions.\\nFor this functional we discuss the problem of existence of minimizers on various functional spaces. We find non-existence of minimizers in cases in which initial and final directions are considered with orientation. In this case, minimizing sequences of trajectories can converge to curves with angles.\\nWe instead prove existence of minimizers for the \\\"time-reparameterized\\\" functional $$\\\\int \\\\| \\\\dot\\\\gamma(t) \\\\|\\\\sqrt{1+K_\\\\ga^2} dt$$ for all boundary conditions if initial and final directions are considered regardless to orientation. In this case, minimizers can present cusps (at most two) but not angles.

Feedback schemes for radiation damping suppression in NMR: a control-theoretical perspective - Claudio Altafini; Paola Cappellaro; David Cory (Systems and Control Letters, 59 (12):782-786, 2010, 2010)
In NMR spectroscopy, the collective measurement is weakly invasive and its back-action is called radiation damping. The aim of this paper is to provide a control-theoretical analysis of the problem of suppressing this radiation damping. It is shown that the two feedback schemes commonly used in the NMR practice correspond one to a high gain oputput feedback for the simple case of maintaining the spin 1/2 in its inverted state, and the second to a 2-degree of freedom control design with a prefeedback that exactly cancels the radiation damping field. A general high gain feedback stabilization design not requiring the knowledge of the radiation damping time constant is also investigated.

First colonization of a hard-edge in random matrix theory - Marco Bertola; Lee, S. Y. (Constr. Approx., 2010)

Gauge theory: from physics to geometry - Ugo Bruzzo (Rend. Istit. Mat. Univ. Trieste 42 (2010) 103-128, 2010)
Maxwell theory may be regarded as a prototype of gauge theory and generalized to nonabelian gauge theory. We briey sketch the history of the gauge theories, from Maxwell to Yang-Mills theory, and the identification of gauge fields with connections on fibre bundles. We introduce the notion of instanton and consider the moduli spaces of such objects. Finally, we discuss some modern techniques for studying the topology of these moduli spaces.

Gene expression analysis of the emergence of epileptiform activity after focal injection of kainic acid into mouse hippocampus. - Dario Motti; Caroline Le Duigou; Nicole Chemaly; Lucia Wittner; Dejan Lazarevic; Helena Krmac; Troels Torben Marstrand; Eivind Valen; Remo Sanges; Elia Stupka; Albin Sandelin; Enrico Cherubini; et al. (The European journal of neuroscience. 2010 Oct; 32(8):1364-79, 2010)
We report gene profiling data on genomic processes underlying the progression towards recurrent seizures after injection of kainic acid (KA) into the mouse hippocampus. Focal injection enabled us to separate the effects of proepileptic stimuli initiated by KA injection. Both the injected and contralateral hippocampus participated in the status epilepticus. However, neuronal death induced by KA treatment was restricted to the injected hippocampus, although there was some contralateral axonal degeneration. We profiled gene expression changes in dorsal and ventral regions of both the injected and contralateral hippocampus. Changes were detected in the expression of 1526 transcripts in samples from three time-points: (i) during the KA-induced status epilepticus, (ii) at 2 weeks, before recurrent seizures emerged, and (iii) at 6 months after seizures emerged. Grouping genes with similar spatio-temporal changes revealed an early transcriptional response, strong immune, cell death and growth responses at 2 weeks and an activation of immune and extracellular matrix genes persisting at 6 months. Immunostaining for proteins coded by genes identified from array studies provided evidence for gliogenesis and suggested that the proteoglycan biglycan is synthesized by astrocytes and contributes to a glial scar. Gene changes at 6 months after KA injection were largely restricted to tissue from the injection site. This suggests that either recurrent seizures might depend on maintained processes including immune responses and changes in extracellular matrix proteins near the injection site or alternatively might result from processes, such as growth, distant from the injection site and terminated while seizures are maintained.

Generic multiplicity for a scalar field equation on compact surfaces - Francesca De Marchis (Journal of Functional Analysis, 2010)
We prove generic multiplicity of solutions for a scalar field equation on compact surfaces via Morse inequalities. In particular our result improves significantly the multiplicity estimate which can be deduced from the degree-counting formula in Chen and Lin (2003) [12]. Related results are derived for the prescribed Q-curvature equation.
Keywords: Generic multiplicity; Geometric PDE's; Morse inequalities; Scalar field equations

On the geometric origin of the bi-Hamiltonian structure of the Calogero-Moser system - Claudio Bartocci; Gregorio Falqui; Igor Mencattini; Giovanni Ortenzi; Marco Pedroni (Int. Math. Res. Not. (2010) 2010:279-296, 2010)
We show that the bi-Hamiltonian structure of the rational n-particle (attractive) Calogero-Moser system can be obtained by means of a double projection from a very simple Poisson pair on the cotangent bundle of gl(n,R). The relation with the Lax formalism is also discussed.

The geometry emerging from the symmetries of a quantum system - Giuseppe De Nittis; Gianluca Panati (2010)
We investigate the relation between the symmetries of a quantum system and its topological quantum numbers, in a general C*-algebraic framework. We prove that, under suitable assumptions on the symmetry algebra, there exists a generalization of the Bloch-Floquet transform which induces a direct-integral decomposition of the algebra of observables. Such generalized transform selects uniquely the set of \\\"continuous sections\\\" in the direct integral, thus yielding a Hilbert bundle. The emerging geometric structure provides some topological invariants of the quantum system. Two running examples provide an Ariadne\\\'s thread through the paper. For the sake of completeness, we review two related theorems by von Neumann and Maurin and compare them with our result.

A global compactness result for the p-Laplacian involving critical nonlinearities - Mercuri, Carlo; Willem, Michel (Discrete & Continuous Dynamical Systems-A, 2010)
We prove a representation theorem for Palais-Smale sequences involving the p-Laplacian and critical nonlinearities. Applications are given to a critical problem.

Hamiltonian PDEs: deformations, integrability, solutions - Boris Dubrovin (Journal of Physics A: Mathematical and Theoretical. Volume 43, Issue 43, 29 October 2010, Article number 434002, 2010)
We review recent classification results on the theory of systems of nonlinear\\r\\nHamiltonian partial differential equations with one spatial dimension, including\\r\\na perturbative approach to the integrability theory of such systems, and discuss\\r\\nuniversality conjectures describing critical behaviour of solutions to such\\r\\nsystems.

Hitchin systems, N=2 gauge theories and W-gravity - Giulio Bonelli; Alessandro Tanzini (Phys. Lett. B 691 (2010) 111-115, 2010)
We propose some arguments supporting an M-theory derivation of the duality recently discovered by Alday, Gaiotto and Tachikawa between two-dimensional conformal field theories and N=2 superconformal gauge theories in four dimensions. We find that A_{N-1} Toda field theory is the simplest two-dimensional conformal field theory quantizing the moduli of N M5-branes wrapped on a Riemann surface. This leads us to identify chiral operators of the N=2 gauge theories with W-algebra currents. As a check of this correspondence we study some relevant OPE\\\'s obtaining that Nekrasov\\\'s partition function satisfies W-geometry constraints.

Homogeneous binary trees as ground states of quantum critical Hamiltonians - Pietro Silvi; Vittorio Giovannetti; Simone Montangero; Matteo Rizzi; J. Ignacio Cirac; Rosario Fazio (Phys. Rev. A 81 (2010) 062335, 2010)
Many-body states whose wave-function admits a representation in terms of a uniform binary-tree tensor decomposition are shown to obey to power-law two-body correlations functions. Any such state can be associated with the ground state of a translational invariant Hamiltonian which, depending on the dimension of the systems sites, involve at most couplings between third-neighboring sites. A detailed analysis of their spectra shows that they admit an exponentially large ground space.

Homogeneous multiscale entanglement renormalization ansatz tensor networks for quantum critical systems - Matteo Rizzi; Simone Montangero; Pietro Silvi; Vittorio Giovannetti; Rosario Fazio (New J. Phys. 12 (2010) 075018, 2010)
In this paper, we review the properties of homogeneous multiscale entanglement renormalization ansatz (MERA) to describe quantum critical systems.We discuss in more detail our results for one-dimensional (1D) systems (the Ising and Heisenberg models) and present new data for the 2D Ising model. Together with the results for the critical exponents, we provide a detailed description of the numerical algorithm and a discussion of new optimization\\nstrategies. The relation between the critical properties of the system and the tensor structure of the MERA is expressed using the formalism of quantum channels, which we review and extend.

Homogenization of fiber reinforced brittle material: the intermediate case - Gianni Dal Maso; Caterina Ida Zeppieri (Adv. Calc. Var. 3 (2010) 345-370, 2010)
We derive a cohesive fracture model by homogenizing a periodic composite material whose microstructure is characterized by the presence of brittle inclusions in a reticulated unbreakable elastic structure.

Invariant Lagrange submanifolds of dissipative systems - Andrei A. Agrachev (Russian Mathematical Surveys. Volume 65, Issue 5, 2010, Pages: 977-978, 2010)
We study solutions of modified Hamilton-Jacobi equations H(du/dq,q) + cu(q) =\\r\\n0, q \\\\in M, on a compact manifold M .

A kinetic mechanism inducing oscillations in simple chemical reactions networks - Julien Coatleven; Claudio Altafini (Mathematical Biosciences and Engineering 7(2):301-312, 2010, 2010)
It is known that a kinetic reaction network in which one or more secondary substrates are acting as cofactors may exhibit an oscillatory behavior. The aim of this work is to provide a description of the functional form of such a cofactor action guaranteeing the\\r\\nonset of oscillations in sufficiently simple reaction networks.

Lorentz Covariant k-Minkowski Spacetime - Ludwik Dabrowski; Michal Godlinski; Gherardo Piacitelli (Phys. Rev. D 81 (2010) 125024, 2010)
In recent years, different views on the interpretation of Lorentz covariance of non commuting coordinates were discussed. Here, by a general procedure, we construct the minimal canonical central covariantisation of the k-Minkowski spacetime. We then show that, though the usual k-Minkowski spacetime is covariant under deformed (or twisted) Lorentz action, the resulting framework is equivalent to taking a non covariant restriction of the covariantised model. We conclude with some general comments on the approach of deformed covariance.

Monotonicity, frustration, and ordered response: an analysis of the energy landscape of perturbed large-scale biological networks - Giovanni Iacono; Claudio Altafini (BMC Systems Biology 2010, 4:83, 2010)
Background. \\nFor large-scale biological networks represented as signed graphs, the index of frustration measures how far a network is from a monotone system, i.e., how incoherently the system responds to perturbations.\\nResults. \\nIn this paper we find that the frustration is systematically lower in transcriptional networks (modeled at functional level) than in signaling and metabolic networks (modeled at stoichiometric level). A possible interpretation of this result is in terms of energetic cost of an interaction: an erroneous or contradictory transcriptional action costs much more than a signaling/metabolic error, and therefore must be avoided as much as possible. Averaging over all possible perturbations, however, we also find that unlike for transcriptional networks, in the signaling/metabolic networks the probability of finding the system in its least frustrated configuration tends to be high also in correspondence of a moderate energetic regime, meaning that, in spite of the higher frustration, these networks can achieve a globally ordered response to perturbations even for moderate values of the strength of the interactions. Furthermore, an analysis of the energy landscape shows that signaling and metabolic networks lack energetic barriers around their global optima, a property also favouring global order.\\nConclusion. \\nIn conclusion, transcriptional and signaling/metabolic networks appear to have systematic differences in both the index of frustration and the transition to global order. These differences are interpretable in terms of the different functions of the various classes of networks.

Moore-Read Fractional Quantum Hall wavefunctions and SU(2) quiver gauge theories - Raoul Santachiara; Alessandro Tanzini (2010)
We identify Moore-Read wavefunctions, describing non-abelian statistics in fractional quantum Hall systems, with the instanton partition of N=2 superconformal quiver gauge theories at suitable values of masses and \\\\Omega-background parameters. This is obtained by extending to rational conformal field theories the SU(2) gauge quiver/Liouville field theory duality recently found by Alday-Gaiotto-Tachikawa. A direct link between the Moore-Read Hall $n$-body wavefunctions and Z_n-equivariant Donaldson polynomials is pointed out.

New approximation results for free discontinuity problems - Flaviana Iurlano (Università degli Studi di Trieste and SISSA, 2010)

Nonlocal character of the reduced theory of thin films with higher order perturbations - Gianni Dal Maso; Irene Fonseca; Giovanni Leoni (Adv. Calc. Var. 3 (2010) 287-319, 2010)

A normal form for generic 2-dimensional almost-Riemannian structures at a tangency point - Ugo Boscain; Grégoire Charlot; Roberta Ghezzi (arXiv preprint arXiv:1008.5036, 2010)

On the number of positive solutions of some semilinear elliptic problems - Antonio Ambrosetti (2010)

Numerical Solution of the Small Dispersion Limit of the Camassa-Holm and Whitham Equations and Multiscale Expansions - Simonetta Abenda; Tamara Grava; Christian Klein (2010)
The small dispersion limit of solutions to the Camassa-Holm (CH) equation is characterized by the appearance of a zone of rapid modulated oscillations. An asymptotic description of these oscillations is given, for short times, by the one-phase solution to the CH equation, where the branch points of the corresponding elliptic curve depend on the physical coordinates via the Whitham equations. We present a conjecture for the phase of the asymptotic solution. A numerical study of this limit for smooth hump-like initial data provides strong evidence for the validity of this conjecture....

On optimality of c-cyclically monotone transference plans - Stefano Bianchini; Laura Caravenna (Comptes Rendus Mathematique 348 (2010) 613-618, 2010)
Abstract. This note deals with the equivalence between the optimality of a transport plan for the Monge-Kantorovich problem and the condition of c-cyclical monotonicity, as an outcome of the construction in [7]. We emphasize the measurability assumption on the hidden structure of linear preorder, applied also to extremality and uniqueness problems. Resume. Dans la presente note nous decrivons brievement la construction introduite dans [7] a propos de l\\\'equivalence entre l\\\'optimalite d\\\'un plan de transport pour le probleme de Monge-Kantorovich et la condition de monotonie c-cyclique ainsi que d\\\'autres sujets que cela nous amene a aborder. Nous souhaitons mettre en evidence l\\\'hypothese de mesurabilite sur la structure sous-jacente de pre-ordre lineaire.

Optimally swimming Stokesian Robots - François Alouges; Antonio DeSimone; Luca Heltai; Aline Lefebvre; Benoit Merlet (2010)
We study self propelled stokesian robots composed of assemblies of balls, in dimen-\\nsions 2 and 3, and prove that they are able to control their position and orientation. This is a result of controllability, and its proof relies on applying Chow\\\'s theorem in an analytic framework, similarly to what has been done in [3] for an axisymmetric system swimming along the axis of symmetry. However, we simplify drastically\\nthe analyticity result given in [3] and apply it to a situation where more complex swimmers move either in a plane or in three-dimensional space, hence experiencing also rotations. We then focus our attention on energetically optimal strokes, which we are able to compute numerically. Some examples of computed optimal strokes are discussed in detail.

Painlevé II asymptotics near the leading edge of the oscillatory zone for the Korteweg-de Vries equation in the small-dispersion limit - Tom Claeys; Tamara Grava (Comm. Pure Appl. Math. 63 (2010) 203-232, 2010)
In the small dispersion limit, solutions to the Korteweg-de Vries equation develop an interval of fast oscillations after a certain time. We obtain a universal asymptotic expansion for the Korteweg-de Vries solution near the leading edge of the oscillatory zone up to second order corrections. This expansion involves the Hastings-McLeod solution of the Painlev\\\\\\\'e II equation. We prove our results using the Riemann-Hilbert approach.

A Phase Field Approach to Wetting and Contact Angle Hysteresis Phenomena - Antonio DeSimone; Livio Fedeli; Turco, Alessandro (IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, 2010)
We discuss a phase field model for the numerical simulation of contact angle hysteresis phenomena in wetting. The performance of the model is assessed by comparing its predictions with experimental data on the critical size of drops that can stick on a vertical glass plate.

Picard group of hypersurfaces in toric varieties - Ugo Bruzzo; Antonella Grassi (2010)
We show that the usual sufficient criterion for a generic hypersurface in a smooth projective manifold to have the same Picard number as the ambient variety can be generalized to hypersurfaces in complete simplicial toric varieties. This sufficient condition is always satisfied by generic K3 surfaces embedded in Fano toric 3-folds.

Poles of Integrale Tritronquee and Anharmonic Oscillators. Asymptotic localization from WKB analysis - Davide Masoero (Nonlinearity. vol. 23, (2010), page 2501-2507, 2010)
Poles of integrale tritronquee are in bijection with cubic oscillators that admit the simultaneous solutions of two quantization conditions. We show that the poles lie near the solutions of a pair of Bohr-Sommerfeld quantization conditions (the Bohr-Sommerfeld-Boutroux system): the distance between a pole and the corresponding solution of the Bohr-Sommerfeld-Boutroux system vanishes asymptotically.

Positive solutions for some non-autonomous Schrödinger–Poisson systems - Giovanna Cerami; Giusi Vaira (Journal of Differential Equations, 2010)

Projective Reeds-Shepp car on $S^2$ with quadratic cost - Ugo Boscain; Francesco Rossi (ESAIM COCV 16 (2010) 275-297, 2010)
Fix two points $x,\\\\bar{x}\\\\in S^2$ and two directions (without orientation) $\\\\eta,\\\\bar\\\\eta$ of the velocities in these points. In this paper we are interested to the problem of minimizing the cost $$ J[\\\\gamma]=\\\\int_0^T g_{\\\\gamma(t)}(\\\\dot\\\\gamma(t),\\\\dot\\\\gamma(t))+\\nK^2_{\\\\gamma(t)}g_{\\\\gamma(t)}(\\\\dot\\\\gamma(t),\\\\dot\\\\gamma(t)) ~dt$$ along all smooth curves starting from $x$ with direction $\\\\eta$ and ending in $\\\\bar{x}$ with direction $\\\\bar\\\\eta$. Here $g$ is the standard Riemannian metric on $S^2$ and $K_\\\\gamma$ is the corresponding geodesic curvature.\\nThe interest of this problem comes from mechanics and geometry of vision. It can be formulated as a sub-Riemannian problem on the lens space L(4,1).\\nWe compute the global solution for this problem: an interesting feature is that some optimal geodesics present cusps. The cut locus is a stratification with non trivial topology.

Quantum Spacetime: a Disambiguation - Gherardo Piacitelli (2010)
We review an approach to non-commutative geometry, where models are constructed by quantisation of the coordinates. In particular we focus on the full DFR model and its irreducible components; the (arbitrary) restriction to a particular irreducible component is often referred to as the \\\"canonical quantum spacetime\\\". The aim is to distinguish and compare the approaches under various points of view, including motivations, prescriptions for quantisation, the choice of mathematical objects and concepts, approaches to dynamics and to covariance. Some incorrect statements as \\\"universality of Planck scale conflicts with Lorentz-Fitzgerald contraction and requires a modification of covariance\\\", or \\\"stability of the geometric background requires an absolute lower bound of (\\\\Delta x^\\\\mu)\\\", or \\\"violations of unitarity are due to time/space non-commutativity\\\" are put in context, and discussed.

Quasistatic crack growth in elasto-plastic materials: the two-dimensional case - Gianni Dal Maso; Rodica Toader (Arch. Ration. Mech. Anal. 196 (2010) 867-906, 2010)
We study a variational model for the quasistatic evolution of elasto-plastic materials with cracks in the case of planar small strain associative elasto-plasticity.

Quasistatic crack growth in finite elasticity with non-interpenetration - Gianni Dal Maso; Giuliano Lazzaroni (Ann. Inst. H. Poincare Anal. Non Lineaire 27 (2010) 257-290, 2010)
We present a variational model to study the quasistatic growth of brittle cracks in hyperelastic materials, in the framework of finite elasticity, taking\\ninto account the non-interpenetration condition.

Quasistatic evolution for Cam-Clay plasticity: the spatially homogeneous case - Gianni Dal Maso; Francesco Solombrino (Netw. Heterog. Media 5 (2010) 97-132, 2010)
We study the spatially uniform case of the problem of quasistatic evolution in small strain nonassociative elastoplasticity (Cam-Clay model). Through the introdution of a viscous approximation, the problem reduces to determine the limit behavior of the solutions of a singularly perturbed system of ODE\\\'s in a finite dimensional Banach space. Depending on the sign of two explicit scalar indicators, we see that the limit dynamics presents, under quite generic assumptions, the alternation of three possible regimes: the elastic regime, when the limit equation is just the equation of linearized elasticity, the slow dynamics, when the strain evolves smoothly on the yield surface and plastic flow is produced, and the fast dynamics, which may happen only in the softening regime, where\\nviscous solutions exhibit a jump across a heteroclinic orbit of an auxiliary system.
Keywords: Cam-Clay plasticity

Quasistatic evolution for plasticity with softening: The spatially homogeneous case - Francesco Solombrino (Discrete & Continuous Dynamical Systems - A, 2010)
The spatially uniform case of the problem of quasistatic evolution in small strain associative elastoplasticity with softening is studied. Through the introdution of a viscous approximation, the problem reduces to determine the limit behaviour of the solutions of a singularly perturbed system of ODE's in a finite dimensional Banach space. We see that the limit dynamics presents, for a generic choice of the initial data, the alternation of three possible regimes (elastic regime, slow dynamics, fast dynamics), which is determined by the sign of two scalar indicators, whose explicit expression is given.
Keywords: plasticity with softening; rate independent processes

The reductions of the dispersionless 2D Toda hierarchy and their Hamiltonian structures - Guido Carlet; Paolo Lorenzoni; Andrea Raimondo (J. Phys. A 43 (2010) 045201, 2010)
We study finite-dimensional reductions of the dispersionless 2D Toda hierarchy showing that the consistency conditions for such reductions are given by a system of radial Loewner equations. We then construct their Hamiltonian structures, following an approach proposed by Ferapontov.

Riemann-Roch theorems and elliptic genus for virtually smooth schemes - Barbara Fantechi; Lothar Göttsche (Geom. Topol. 14 (2010) 83-115, 2010)
For a proper scheme X with a fixed 1-perfect obstruction theory, we define virtual versions of holomorphic Euler characteristic, chi y-genus, and elliptic genus; they are deformation invariant, and extend the usual definition in the smooth case. We prove virtual versions of the Grothendieck-Riemann-Roch and Hirzebruch-Riemann-Roch theorems. We show that the virtual chi y-genus is a polynomial, and use this to define a virtual topological Euler characteristic. We prove that the virtual elliptic genus satisfies a Jacobi modularity property; we state and prove a localization theorem in the toric equivariant case. We show how some of our results apply to moduli spaces of stable sheaves.

The role of membrane viscosity in the dynamics of fluid membranes - Marino Arroyo; Antonio DeSimone; Luca Heltai (2010)
Fluid membranes made out of lipid bilayers are the fundamental separation structure in eukaryotic cells. Many physiological processes rely on dramatic shape and topological changes (e.g. fusion, fission) of fluid membrane systems. Fluidity is key to the versatility and constant reorganization of lipid bilayers. Here, we study the role of the membrane intrinsic viscosity, arising from the friction of the lipid molecules as they rearrange to accommodate shape changes, in the dynamics of morphological changes of fluid vesicles. In particular, we analyze the competition between the membrane viscosity and the viscosity of the bulk fluid surrounding the vesicle as the dominant dissipative mechanism. We consider the relaxation dynamics of fluid vesicles put in an out-of-equilibrium state, but conclusions can be drawn regarding the kinetics or power consumption in regulated shape changes in the cell. On the basis of numerical calculations, we find that the dynamics arising from the membrane viscosity are qualitatively different from the dynamics arising from the bulk viscosity. When these two dissipation mechanisms are put in competition, we find that for small vesicles the membrane dissipation dominates, with a relaxation time that scales as the size of the vesicle to the power 2. For large vesicles, the bulk dissipation dominates, and the exponent in the relaxation time vs. size relation is 3.

Semiclassical evolution of two rotating solitons for the Nonlinear Schrödinger Equation with electric potential - Alessandro Selvitella (Adv. Differential Equations, 2010)

On semistable principal bundles over complex projective manifolds, II - Indranil Biswas; Ugo Bruzzo (Geom. Dedicata 146 (2010) 27-41, 2010)
Let (X, \\\\omega) be a compact connected Kaehler manifold of complex dimension d and E_G a holomorphic principal G-bundle on X, where G is a connected reductive linear algebraic group defined over C. Let Z (G) denote the center of G. We prove that the following three statements are equivalent: (1) There is a parabolic subgroup P of G and a holomorphic reduction of the structure group of E_G to P (say, E_P) such that the bundle obtained by extending the structure group of E_P to L(P)/Z(G) (where L(P) is the Levi quotient of P) admits a flat connection; (2) The adjoint vector bundle ad(E_G) is numerically flat; (3) The principal G-bundle E_G is pseudostable, and the degree of the charateristic class c_2(ad(E_G) is zero.

Sharp nonexistence results for a linear elliptic inequality involving Hardy and Leray potentials - Mouhamed Moustapha Fall; Roberta Musina (2010)
In this paper we deal with nonnegative distributional supersolutions for a class of linear\\nelliptic equations involving inverse-square potentials and logarithmic weights. We prove sharp nonexistence results.

Shell theories arising as low energy Gamma-limit of 3d nonlinear elasticity - Marta Lewicka; Maria Giovanna Mora; Mohammad Reza Pakzad (Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. IX (2010) 253-295, 2010)
We discuss the limiting behavior (using the notion of gamma-limit) of the 3d nonlinear elasticity for thin shells around an arbitrary smooth 2d surface. In particular, under the assumption that the elastic energy of deformations scales like h4, h being the thickness of a shell, we derive a limiting theory which is a generalization of the von Karman theory for plates.

Sobolev periodic solutions of nonlinear wave equations in higher spatial dimensions - Massimiliano Berti; Philippe Bolle (Archive for Rational Mechanics and Analysis, 2010)
We prove the existence of Cantor families of periodic solutions for nonlinear wave equations in higher spatial dimensions with periodic boundary conditions. We study both forced and autonomous PDEs. In the latter case our theorems generalize previous results of Bourgain to more general nonlinearities of class C k and assuming weaker non-resonance conditions. Our solutions have Sobolev regularity both in time and space. The proofs are based on a differentiable Nash-Moser iteration scheme, where it is sufficient to get estimates of interpolation-type for the inverse linearized operators. Our approach works also in presence of very large "clusters of small divisors". © Springer-Verlag (2009).
cited By (since 1996)6

Solitonic asymptotics for the Korteweg-de Vries equation in the small dispersion limit - Tamara Grava; Tom Claeys (SIAM J. Math. Anal. 42 (2010) 2132-2154, 2010)
We study the small dispersion limit for the Korteweg-de Vries (KdV) equation $u_t+6uu_x+\\\\epsilon^{2}u_{xxx}=0$ in a critical scaling regime where $x$ approaches the trailing edge of the region where the KdV solution shows oscillatory behavior. Using the Riemann-Hilbert approach, we obtain an asymptotic expansion for the KdV solution in a double scaling limit, which shows that the oscillations degenerate to sharp pulses near the trailing edge. Locally those pulses resemble soliton solutions of the KdV equation.

A spatial model of cellular molecular trafficking including active transport along microtubules - Andrea Cangiani; Natalini, R. (J. Theoret. Biol., 2010)

Stable determination of an immersed body in a stationary Stokes fluid - Andrea Ballerini (Inverse Problems, 2010)
We consider the inverse problem of the detection of a single body, immersed in a bounded container filled with a fluid which obeys the Stokes equations, from a single measurement of force and velocity on a portion of the boundary. We obtain an estimate of the stability of log–log type.

Taming open/closed string duality with a Losev trick - Giulio Bonelli; Andrea Prudenziati; Alessandro Tanzini (JHEP 06(2010)063, 2010)
A target space string field theory formulation for open and closed B-model is provided by giving a Batalin-Vilkovisky quantization of the holomorphic Chern-Simons theory with off-shell gravity background. The target space expression for the coefficients of the holomorphic anomaly equation for open strings are obtained. Furthermore, open/closed string duality is proved from a judicious integration over the open string fields. In particular, by restriction to the case of independence on continuous open moduli, the shift formulas of [7] are reproduced and shown therefore to encode the data of a closed string dual.

A three-dimensional model for the dynamics and hydrodynamics of rowing boats - L. Formaggia; Andrea Mola; N Parolini; M Pischiutta (Proceedings of the Institution of Mechanical Engineers, Part P: Journal of Sports Engineering and Technology, 2010)
This paper proposes a new model describing the dynamics of a rowing boat for general three-dimensional motions. The complex interaction between the different components of the rowers–-oars–-boat system is analysed and reduced to a set of ordinary differential equations governing the rigid motion along the six degrees of freedom. To treat the unstable nature of the physical problem, a rather simple (but effective) control model is included, which mimics the main active control techniques adopted by the rowers during their action.

A time-dependent perturbative analysis for a quantum particle in a cloud chamber - Gianfausto Dell'Antonio; Rodolfo Figari; Alessandro Teta (Annales Henri Poincare 11 (2010) 539-564, 2010)
We consider a simple model of a cloud chamber consisting of a test particle (the alpha-particle) interacting with two other particles (the atoms of the vapour) subject to attractive potentials centered in $a_1, a_2 \\\\in \\\\mathbb{R}^3$. At time zero the alpha-particle is described by an outgoing spherical wave centered in the origin and the atoms are in their ground state. We show that, under suitable assumptions on the physical parameters of the system and up to second order in perturbation theory, the probability that both atoms are ionized is negligible unless $a_2$ lies on the line joining the origin with $a_1$. The work is a fully time-dependent version of the original analysis proposed by Mott in 1929.

Twisted Covariance as a Non Invariant Restriction of the Fully Covariant DFR Model - Gherardo Piacitelli (Comm. Math. Phys. 295 (2010) 701-729, 2010)
We discuss twisted covariance over the noncommutative spacetime algebra generated by the relations [q_theta^mu,q_theta^nu]=i theta^{mu nu}, where the matrix theta is treated as fixed (not a tensor), and we refrain from using the asymptotic Moyal expansion of the twists. We show that the tensor nature of theta is only hidden in the formalism: in particular if theta fulfils the DFR conditions, the twisted Lorentz covariant model of the flat quantum spacetime may be equivalently described in terms of the DFR model, if we agree to discard a huge non invariant set of localisation states; it is only this last step which, if taken as a basic assumption, severely breaks the relativity principle. We also will show that the above mentioned, relativity breaking, ad hoc rejection of localisation states is an independent, unnecessary assumption, as far as some popular approaches to quantum field theory on the quantum Minkowski spacetime are concerned. The above should raise some concerns about speculations on possible observable consequences of arbitrary choices of theta in arbitrarily selected privileged frames.

Two-dimensional almost-Riemannian structures with tangency points - Andrei A. Agrachev; Ugo Boscain; Grégoire Charlot; Roberta Ghezzi; Mario Sigalotti (Ann. Inst. H. Poincare Anal. Non Lineaire, 2010)
Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. We study the relation between the topological invariants of an almost-Riemannian structure on a compact oriented surface and the rank-two vector bundle over the surface which defines the structure. We analyse the generic case including the presence of tangency points, i.e. points where two generators of the distribution and their Lie bracket are linearly dependent. The main result of the paper provides a classification of oriented almost-Riemannian structures on compact oriented surfaces in terms of the Euler number of the vector bundle corresponding to the structure. Moreover, we present a Gauss-Bonnet formula for almost-Riemannian structures with tangency points.

Uhlenbeck-Donaldson compactification for framed sheaves on projective surfaces - Ugo Bruzzo; Dimitri Markushevich; Alexander Tikhomirov (2010)
We construct a compactification $M^{\\\\mu ss}$ of the Uhlenbeck-Donaldson type for the moduli space of slope stable framed bundles. This is a kind of a moduli space of slope semistable framed sheaves. We show that there exists a projective morphism $\\\\gamma \\\\colon M^s \\\\to M^{\\\\mu ss}$, where $M^s$ is the moduli space of S-equivalence classes of Gieseker-semistable framed sheaves. The space $M^{\\\\mu ss}$ has a natural set-theoretic stratification which allows one, via a Hitchin-Kobayashi correspondence, to compare it with the moduli spaces of framed ideal instantons.

Universality in the profile of the semiclassical limit solutions to the focusing nonlinear Schrödinger equation at the first breaking curve - Marco Bertola; Alexander Tovbis (Int. Math. Res. Not. IMRN, 2010)

Well-posed infinite horizon variational problems on a compact manifold - Andrei A. Agrachev (Proceedings of the Steklov Institute of Mathematics. Volume 268, Issue 1, 2010, Pages 17-31, 2010)
We give an effective sufficient condition for a variational problem with infinite horizon on a compact Riemannian manifold M to admit a smooth optimal synthesis, i. e., a smooth dynamical system on M whose positive semi-trajectories are solutions to the problem. To realize the synthesis, we construct an invariant Lagrangian submanifold (well-projected to M) of the flow of extremals in the cotangent bundle T*M. The construction uses the curvature of the flow in the cotangent bundle and some ideas of hyperbolic dynamics

1D periodic potentials with gaps vanishing at k=0 - Alessandro Michelangeli; Osvaldo Zagordi (Mem. Differential Equations Math. Phys. 47 (2009) 133-158, 2009)
Appearance of energy bands and gaps in the dispersion relations of a periodic potential is a standard feature of Quantum Mechanics. We investigate the class of one-dimensional periodic potentials for which all gaps vanish at the center of the Brillouin zone. We characterise themthrough a necessary and sufficient condition. Potentials of the form we focus on arise in different fields of Physics, from supersymmetric Quantum Mechanics, to Korteweg-de Vries equation theory and classical diffusion problems. The O.D.E. counterpart to this problem is the characterisation of periodic potentials for which coexistence occurs of linearly independent solutions of the corresponding Schrödinger equation (Hill\\\'s equation). This result is placed in the perspective of the previous related results available in the literature.

Biological Fluid Dynamics, Non-linear Partial Differential Equations - Antonio DeSimone; François Alouges; Aline Lefebvre (Encyclopedia of Complexity and Systems Science / Robert A. Meyers (ed.). - Springer, 2009, 548-554, 2009)

The boundary Riemann solver coming from the real vanishing viscosity approximation - Stefano Bianchini; Laura Spinolo (Arch. Ration. Mech. Anal. 191 (2009) 1-96, 2009)
We study the limit of the hyperbolic-parabolic approximation $$ \\\\begin{array}{lll} v_t + \\\\tilde{A} ( v, \\\\, \\\\varepsilon v_x ) v_x = \\\\varepsilon \\\\tilde{B}(v ) v_{xx} \\\\qquad v \\\\in R^N\\\\\\\\ \\\\tilde \\\\beta (v (t, \\\\, 0)) = \\\\bar g \\\\\\\\ v (0, \\\\, x) = \\\\bar v_0. \\\\\\\\ \\\\end{array} \\\\right. $$\\nThe function $\\\\tilde \\\\beta$ is defined in such a way to guarantee that the initial boundary value problem is well posed even if $\\\\tilde \\\\beta$ is not invertible.\\nThe data $\\\\bar g$ and $\\\\bar v_0$ are constant. When $\\\\tilde B$ is invertible, the previous problem takes the simpler form $$ \\\\left\\\\{ \\\\begin{array}{lll} v_t + \\\\tilde{A} \\\\big( v, \\\\, \\\\varepsilon v_x \\\\big) v_x = \\\\varepsilon \\\\tilde{B}(v ) v_{xx} \\\\qquad v \\\\in \\\\mathbb{R}^N\\\\\\\\ v (t, \\\\, 0) \\\\equiv \\\\bar v_b \\\\\\\\ v (0, \\\\, x) \\\\equiv \\\\bar{v}_0. \\\\\\\\ \\\\end{array} \\\\right. $$\\nAgain, the data $\\\\bar v_b$ and $\\\\bar v_0$ are constant. The conservative case is included in the previous formulations. It is assumed convergence of the v, smallness of the total variation and other technical hypotheses and it is provided a complete characterization of the limit. The most interesting points are the following two. First, the boundary characteristic case is considered, i.e. one eigenvalue of $\\\\tilde A$ can be 0.\\n Second, as pointed out before we take into account the possibility that $\\\\tilde B$ is not invertible. To deal with this case, we take as hypotheses conditions that were introduced by Kawashima and Shizuta relying on physically meaningful examples. We also introduce a new condition of block linear degeneracy. We prove that, if it is not satisfied, then pathological behaviours may occur.

Bubbles with prescribed mean curvature: the variational approach - Paolo Caldiroli; Roberta Musina (2009)
H-systems, prescribed mean curvature equation, blowup

The Cauchy two–matrix model - Marco Bertola; M Gekhtman; J Szmigielski (Comm. Math. Phys., 2009)

Characterization of the time course of changes of the evoked electrical activity in a model of a chemically-induced neuronal plasticity - Frederic D. Broccard; Silvia Pegoraro; Maria Elisabetta Ruaro; Claudio Altafini; Vincent Torre (BMC Research Notes (2009) 2:13, 2009)
BACKGROUND: Neuronal plasticity is initiated by transient elevations of neuronal networks activity leading to changes of synaptic properties and providing the basis for memory and learning 1. An increase of electrical activity can be caused by electrical stimulation 2 or by pharmacological manipulations: elevation of extracellular K+ 3, blockage of inhibitory pathways 4 or by an increase of second messengers intracellular concentrations 5. Neuronal plasticity is mediated by several biochemical pathways leading to the modulation of synaptic strength, density of ionic channels and morphological changes of neuronal arborisation 6. On a time scale of a few minutes, neuronal plasticity is mediated by local protein trafficking 7 while, in order to sustain modifications beyond 2-3 h, changes of gene expression are required 8. FINDINGS: In the present manuscript we analysed the time course of changes of the evoked electrical activity during neuronal plasticity and we correlated it with a transcriptional analysis of the underlying changes of gene expression. Our investigation shows that treatment for 30 min. with the GABAA receptor antagonist gabazine (GabT) causes a potentiation of the evoked electrical activity occurring 2-4 hours after GabT and the concomitant up-regulation of 342 genes. Inhibition of the ERK1/2 pathway reduced but did not abolish the potentiation of the evoked response caused by GabT. In fact not all the genes analysed were blocked by ERK1/2 inhibitors. CONCLUSION: These results are in agreement with the notion that neuronal plasticity is mediated by several distinct pathways working in unison.

Commuting difference operators, spinor bundles and the asymptotics of orthogonal polynomials with respect to varying complex weights - Marco Bertola; Mo, M. Y. (Adv. Math., 2009)

A connection between viscous profiles and singular ODEs - Stefano Bianchini; Laura Spinolo (Rend. Istit. Mat. Univ. Trieste 41 (2009) 35-41, 2009)

Controllability and simultaneous controllability of isospectral bilinear control systems on complex flag manifolds - Claudio Altafini (Systems Control Lett. 58 (2009) 213-216, 2009)
For isospectral bilinear control systems evolving on the so-called complex flag manifolds (i.e., on the orbits of the Hermitian matrices under unitary conjugation action) it is shown that controllability is almost always verified. Easy and generic sufficient conditions are provided. The result applies to the problem of density operator controllability of finite dimensional quantum mechanical systems. In addition, we show that systems having different drifts (corresponding for example to different Larmor frequencies) are simultaneously controllable by the same control field.

Controllability of the discrete-spectrum Schrodinger equation driven by an external field - Thomas Chambrion; Paolo Mason; Mario Sigalotti; Ugo Boscain (Ann. Inst. H. Poincare Anal. Non Lineaire 26 (2009) 329-349, 2009)
We prove approximate controllability of the bilinear Schrodinger equation in the case in which the uncontrolled Hamiltonian has discrete nonresonant\\nspectrum. The results that are obtained apply both to bounded or unbounded domains and to the case in which the control potential is bounded or unbounded. The method relies on finite-dimensional techniques applied to the\\nGalerkin approximations and permits, in addition, to get some controllability properties for the density matrix. Two examples are presented: the harmonic oscillator and the 3D well of potential controlled by suitable potentials.

Controllability on the group of diffeomorphisms - Andrei A. Agrachev; Marco Caponigro (Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009) 2503-2509, 2009)
Given a compact manifold M, we prove that any bracket generating family of vector fields on M, which is invariant under multiplication by smooth functions, generates the connected component of identity of the group of diffeomorphisms of M.

Convergence analysis of the mimetic finite difference method for elliptic problems - Andrea Cangiani; Manzini, Gianmarco; Russo, Alessandro (SIAM J. Numer. Anal., 2009)

On the convergence of viscous approximations after shock interactions - Alberto Bressan; Carlotta Donadello (Discrete Contin. Dyn. Syst. 23 (2009) 29-48, 2009)
We consider a piecewise smooth solution to a scalar conservation law, with possibly interacting shocks. We show that, after the interactions have taken place, vanishing viscosity approximations can still be represented by a regular expansion on smooth regions and by a singular perturbation expansion near the shocks, in terms of powers of the viscosity coefficient.

Cubic string boundary value problems and Cauchy biorthogonal polynomials - Marco Bertola; Gekhtman, M.; Szmigielski, J. (J. Phys. A, 2009)

Decoupling A and B model in open string theory: topological adventures in the world of tadpoles - Giulio Bonelli; Andrea Prudenziati; Alessandro Tanzini; Yang Jie (JHEP 06 (2009) 046, 2009)
In this paper we analyze the problem of tadpole cancellation in open topological strings. We prove that the inclusion of unorientable worldsheet diagrams guarantees a consistent decoupling of A and B model for open superstring amplitudes at all genera. This is proven by direct microscopic computation in Super Conformal Field Theory. For the B-model we explicitly calculate one loop amplitudes in terms of analytic Ray-Singer torsions of appropriate vector bundles and obtain that the decoupling corresponds to the cancellation of D-brane and orientifold charges. Local tadpole cancellation on the worldsheet then guarantees the decoupling at all loops. The holomorphic anomaly equations for open topological strings at one loop are also obtained and compared with the results of the Quillen formula.

Differential geometry of curves in Lagrange Grassmannians with given Young diagram - Igor Zelenko; Li Chengbo (Differential Geom. Appl. 27 (2009) 723-742, 2009)
Curves in Lagrange Grassmannians appear naturally in the intrinsic study of geometric structures on manifolds. By a smooth geometric structure on a manifold we mean any submanifold of its tangent bundle, transversal to the fibers. One can consider the time-optimal problem naturally associate with a geometric structure. The Pontryagin extremals of this optimal problem are integral curves of certain Hamiltonian system in the cotangent bundle. The dynamics of the fibers of the cotangent bundle w.r.t. this system along an extremal is described by certain curve in a Lagrange Grassmannian, called Jacobi curve of the extremal. Any symplectic invariant of the Jacobi curves produces the invariant of the original geometric structure. The basic characteristic of a curve in a Lagrange Grassmannian is its Young diagram. The number of boxes in its kth column is equal to the rank of the kth derivative of the curve (which is an appropriately defined linear mapping) at a generic point. We will describe the construction of the complete system of symplectic invariants for parameterized curves in a Lagrange Grassmannian with given Young diagram. It allows to develop in a unified way local differential geometry of very wide classes of geometric structures on manifolds, including both classical geometric structures such as Riemannian and Finslerian structures and less classical ones such as sub-Riemannian and sub-Finslerian structures, defined on nonholonomic distributions.

Discrete-to-continuum limits for strain-alignment-coupled systems: Magnetostrictive solids, ferroelectric crystals and nematic elastomers - Marco Cicalese; Antonio DeSimone; Caterina Ida Zeppieri (Netw. Heterog. Media 4 (2009) 667-708, 2009)
In the framework of linear elasticity, we study the limit of a class of discrete free energies modeling strain-alignment-coupled systems by a rigorous coarse-graining procedure, as the number of molecules diverges. We focus on three paradigmatic examples: magnetostrictive solids, ferroelectric crystals and nematic elastomers, obtaining in the limit three continuum models consistent with those commonly employed in the current literature. We also derive the correspondent macroscopic energies in the presence of displacement boundary conditions and of various kinds of applied external fields.

The Disintegration Theorem and Applications to Optimal Mass Transportation - Laura Caravenna (SISSA, 2009)

Equivariant cohomology and localization for Lie algebroids - Ugo Bruzzo; Lucio Cirio; Paolo Rossi; Vladimir Rubtsov (Funct. Anal. Appl. 43 (2009) 18-29, 2009)
Let M be a manifold carrying the action of a Lie group G, and A a Lie algebroid on M equipped with a compatible infinitesimal G-action. Out of these data we construct an equivariant Lie algebroid cohomology and prove for compact G a related localization formula. As an application we prove a Bott-type formula.

ERNEST: a toolbox for chemical reaction network theory - Nicola Soranzo; Claudio Altafini (Bioinformatics 25 (2009) 2853-2854, 2009)
Summary: ERNEST Reaction Network Equilibria Study Toolbox is a MATLAB package which, by checking various different criteria on the structure of a chemical reaction network, can exclude the multistationarity of the corresponding reaction system. The results obtained are independent of the rate constants of the reactions, and can be used for model discrimination.\\nAvailability and Implementation: The software, implemented in MATLAB, is available under the GNU GPL free software license from http://people.sissa.it/~altafini/papers/SoAl09/. It requires the MATLAB Optimization Toolbox.

Exact results for topological strings on resolved Yp,q singularities - Andrea Brini; Alessandro Tanzini (Comm. Math. Phys. 289 (2009) 205-252, 2009)

Existence of extremals for the Maz\\\'ya and for the Caffarelli-Kohn-Nirenberg inequalities - Roberta Musina (Nonlinear Anal. 70 (2009) 3002-3007, 2009)
This paper deals with some Sobolev-type inequalities with weights that were proved by Maz\\\'ya in 1980 and by Caffarelli-Kohn-Nirenberg in 1984.

An existence result for the Monge problem in R^n with norm cost - Laura Caravenna (2009)

On the extremality, uniqueness and optimality of transference plans - Stefano Bianchini; Laura Caravenna (Bull. Inst. Math. Acad. Sin. (N.S.) 4 (2009) 353-458, 2009)
We consider the following standard problems appearing in optimal mass transportation theory: when a transference plan is extremal; when a transference plan is the unique transference plan concentrated on a set A,; when a transference plan is optimal.