A quasistatic evolution model for perfectly plastic plates derived by Γ-convergence - Elisa Davoli; Maria Giovanna Mora (Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 2013)
The subject of this paper is the rigorous derivation of a quasistatic evolution model for a linearly elastic–perfectly plastic thin plate. As the thickness of the plate tends to zero, we prove via Γ-convergence techniques that solutions to the three-dimensional quasistatic evolution problem of Prandtl–Reuss elastoplasticity converge to a quasistatic evolution of a suitable reduced model. In this limiting model the admissible displacements are of Kirchhoff–Love type and the stretching and bending components of the stress are coupled through a plastic flow rule. Some equivalent formulations of the limiting problem in rate form are derived, together with some two-dimensional characterizations for suitable choices of the data.
Keywords: -convergence; Perfect plasticity; Prandtl–Reuss plasticity; Quasistatic evolution; Rate-independent processes; Thin plates

Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants - Gianluigi Rozza; Phuong Huynh; Andrea Manzoni (Numerische Mathematik, 2013, 2013)
In this paper we review and we extend the reduced basis approximation and a posteriori error estimation for steady Stokes flows in a ffinely parametrized geometries, focusing on the role played by the Brezzi\\\'s and Babu ska\\\'s stability constants. The crucial ingredients of the methodology are a Galerkin projection onto a low-dimensional space of basis functions properly selected, an a ne parametric dependence enabling to perform competitive Off ine-Online splitting in the computational\\r\\nprocedure and a rigorous a posteriori error estimation on eld variables.\\r\\nThe combination of these three factors yields substantial computational savings which are at the basis of an e fficient model order reduction, ideally suited for real-time simulation and many-query contexts (e.g. optimization, control or parameter identi cation). In particular, in this work we focus on i) the stability of the reduced basis approximation based on the Brezzi\\\'s saddle point theory and the introduction of a supremizer operator on the pressure terms, ii) a rigorous a posteriori error estimation procedure for velocity and pressure elds based on the Babu ska\\\'s inf-sup constant (including residuals calculations), iii) the computation of a lower bound of the stability constant, and iv) di erent options for the reduced basis spaces construction. We present some illustrative results for both\\r\\ninterior and external steady Stokes flows in parametrized geometries representing two parametrized classical Poiseuille and Couette \\r\\nflows, a channel contraction and a simple flow control problem around a curved obstacle.
Keywords: parametrized Stokes equations

Reduced Basis Approximation for the Structural-Acoustic Design based on Energy Finite Element Analysis (RB-EFEA) - Denis Devaud; Gianluigi Rozza (CEMRACS 2013 - Modelling and simulation of complex systems: stochastic and deterministic approaches, 2013)

Reduced basis method for parametrized elliptic optimal control problems - Federico Negri; Gianluigi Rozza; Andrea Manzoni; Alfio Quarteroni (SIAM Journal on Scientific Computing, 2013)
We propose a suitable model reduction paradigm-the certified reduced basis method (RB)-for the rapid and reliable solution of parametrized optimal control problems governed by partial differential equations. In particular, we develop the methodology for parametrized quadratic optimization problems with elliptic equations as a constraint and infinite-dimensional control variable. First, we recast the optimal control problem in the framework of saddle-point problems in order to take advantage of the already developed RB theory for Stokes-type problems. Then, the usual ingredients of the RB methodology are called into play: a Galerkin projection onto a low-dimensional space of basis functions properly selected by an adaptive procedure; an affine parametric dependence enabling one to perform competitive offline-online splitting in the computational procedure; and an efficient and rigorous a posteriori error estimate on the state, control, and adjoint variables as well as on the cost functional. Finally, we address some numerical tests that confirm our theoretical results and show the efficiency of the proposed technique.

A Reduced Computational and Geometrical Framework for Inverse Problems in Haemodynamics - Toni Lassila; Andrea Manzoni; Alfio Quarteroni; Gianluigi Rozza (SISSA, 2013)

A reduced-order strategy for solving inverse Bayesian identification problems in physiological flows - Toni Lassila; Andrea Manzoni; Alfio Quarteroni; Gianluigi Rozza (SISSA, 2013)

Reduction Strategies for Shape Dependent Inverse Problems in Haemodynamics - Toni Lassila; Andrea Manzoni; Gianluigi Rozza (SISSA, 2013)

Remarks on the Moser–Trudinger inequality - Gabriele Mancini; Luca Battaglia (Advances in Nonlinear Analysis, 2013)
We extend the Moser-Trudinger inequality to any Euclidean domain satisfying Poincaré's inequality. We find out that the same equivalence does not hold in general for conformal metrics on the unit ball, showing counterexamples. We also study the existence of extremals for the Moser-Trudinger inequalities for unbounded domains, proving it for the infinite planar strip.

Self-adjoint extensions and stochastic completeness of the Laplace-Beltrami operator on conic and anticonic surfaces - Ugo Boscain; Dario Prandi (2013)

Semistability and Decorated Bundles - Andrea Pustetto (SISSA, 2013)
This thesis is devoted to the study of semistability condition of type t=(a,b,c,N) decorated bundles and sheaves in order to better understand and simplify it. We approach the problem in two different ways: on one side we “enclose” the above semistability condition between a stronger semistability condition (ε-semistability) and a weaker one (k-semistability), on the other side we try, and succeed for the case of a = 2, to bound the length of weighted filtrations on which one checks the semistability condition.
Keywords: Decorated sheaves, semistability, moduli space, Mehta-Ramanathan, maximal destabilizing subsheaf

Softly Constrained Films - Luca Giomi (SISSA, 2013)
The shape of materials is often subject to a number of geometric constraints\r\nthat limit the size of the system or fix the structure of its boundary. In soft\r\nand biological materials, however, these constraints are not always hard, but\r\nare due to other physical mechanisms that affect the overall force balance. A\r\ncapillary film spanning a flexible piece of wire or a cell anchored to a\r\ncompliant substrate by mean of adhesive contacts are examples of these softly\r\nconstrained systems in the macroscopic and microscopic world. In this article I\r\nreview some of the important mathematical and physical developments that\r\ncontributed to our understanding of shape formation in softly constrained films\r\nand their recent application to the mechanics of adherent cells.

Some models of crack growth in brittle materials - Simone Racca (SISSA, 2013)
Keywords: Quasi-static crack evolution

Some open problems - Andrei A. Agrachev (SISSA, 2013)
We discuss some challenging open problems in the geometric control theory and sub-Riemannian geometry.
Keywords: Geometry

Some remarks on the viscous approximation of crack growth - Giuliano Lazzaroni; Rodica Toader (Discrete Contin. Dyn. Syst. Ser. S, 2013)
We describe an existence result for quasistatic evolutions of cracks in antiplane elasticity obtained in [16] by a vanishing viscosity approach, with free (but regular enough) crack path. We underline in particular the motivations for the choice of the class of admissible cracks and of the dissipation potential. Moreover, we extend the result to a model with applied forces depending on time.
Keywords: Variational models

Some topics on Higgs bundles over projective varieties and their moduli spaces - Alessio Lo Giudice (SISSA, 2013)
In this thesis we study vector bundles on projective varieties and their moduli spaces. In Chapters 2, 3 and 4 we recall some basic notions as Higgs bundles, decorated bundles and generalized parabolic sheaves and introduce the problem we want to study. In chapter 5, we study Higgs bundles on nodal curves. After moving the problem on the normalization of the curve, starting from a Higgs bundle we obtain a generalized parabolic Higgs bundle. Using decorated bundles we are able to construct a projective moduli space which parametrizes equivalence classes of Higgs bundles on a nodal curve X. This chapter is an extract of a joint work with Andrea Pustetto Later on Chapter 6 is devoted to the study of holomorphic pairs (or twisted Higgs bundles) on elliptic curve. Holomorphic pairs were introduced by Nitsure and they are a natural generalization of the concept of Higgs bundles. In this Chapter we extend a result of E. Franco, O. Garc\'ia-Prada And P.E. Newstead valid for Higgs bundles to holomorphic pairs. Finally the last Chapter describes a joint work with Professor Ugo Bruzzo. We study Higgs bundles over varieties with nef tangent bundle. In particular generalizing a result of Nitsure we prove that if a Higgs bundle $(E,φ)$ over the variety X with nef tangent remains semisatble when pulled-back to any smooth curve then it discrimiant vanishes.
Keywords: Algebraic Geometry, Moduli spaces, Vector bundles

Spectra of random Hermitian matrices with a small-rank external source: the supercritical and subcritical regimes - Marco Bertola; Buckingham, R.; Lee, S. Y.; Pierce, V. (J. Stat. Phys., 2013)

The splitting theorem in non-smooth context - Nicola Gigli (2013)
We prove that an infinitesimally Hilbertian $CD(0,N)$ space containing a line splits as the product of $R$ and an infinitesimally Hilbertian $CD(0,N −1)$ space. By ‘infinitesimally Hilbertian’ we mean that the Sobolev space $W^{1,2}(X,d,m)$, which in general is a Banach space, is an Hilbert space. When coupled with a curvature-dimension bound, this condition is known to be stable with respect to measured Gromov-Hausdorff convergence.

On the stability of continuous-discontinuous Galerkin methods for advection-diffusion-reaction problems - Andrea Cangiani; Chapman, John; E.H. Georgoulis; Jensen, Max (J. Sci. Comput., 2013)

Stabilization of Stochastic Quantum Dynamics via Open and Closed Loop Control - Claudio Altafini; Francesco Ticozzi; K. Nishio (IEEE Transactions on Automatic Control. Volume 58, Issue 1, 2013, Article number6228517, Pages 74-85, 2013)
In this paper, we investigate parametrization-free solutions of the problem of quantum pure state preparation and subspace stabilization by means of Hamiltonian control, continuous measurement, and quantum feedback, in the presence of a Markovian environment. In particular, we show that whenever suitable dissipative effects are induced either by the unmonitored environment, or by non-Hermitian measurements, there is no need for feedback, as open-loop time-invariant control is sufficient to achieve stabilization of the target set in probability. Constructive necessary and sufficient conditions on the form of the control Hamiltonian can be provided in this case. When time-invariant control is not sufficient, state stabilization can be attained by the addition of filtering-based feedback control

A stable and adaptive semi-Lagrangian potential model for unsteady and nonlinear ship-wave interactions - Andrea Mola; Luca Heltai; Antonio DeSimone (Engineering Analysis with Boundary Elements, 37(1):128 – 143, 2013., 2013)
We present an innovative numerical discretization of the equations of inviscid potential flow for the simulation of three dimensional unsteady and nonlinear water waves generated by a ship hull advancing in water. The equations of motion are written in a semi-Lagrangian framework, and the resulting integro-diff erential equations are discretized in space via an adaptive iso-parametric collocation Boundary Element Method, and in time via adaptive implicit Backward Di erentiation Formulas (BDF) with variable step and variable order. When the velocity of the advancing ship hull is non-negligible, the semi-Lagrangian formulation (also known as Arbitrary Lagrangian Eulerian formulation, or ALE) of the free surface equations contains dominant transport terms which are stabilized with a Streamwise Upwind Petrov-Galerkin (SUPG) method. The SUPG stabilization allows automatic and robust adaptation of the spatial discretization with unstructured quadrilateral grids. Preliminary results are presented where we compare our numerical model with experimental results on the case of a Wigley hull advancing in calm water with fi xed sink and trim.
Keywords: Unsteady ship-wave interaction

Stable determination of a body immersed in a fluid: the nonlinear stationary case - Andrea Ballerini (Applicable Analysis, 2013)
We consider the inverse problem of the detection of a single body, immersed in a bounded container filled with a fluid which obeys the stationary Navier–Stokes equations, from a single measurement of force and velocity on a portion of the boundary. We obtain an estimate of stability of log–log type.

Stochastic optimal robin boundary control problems of advection-dominated elliptic equations - Peng Chen; Alfio Quarteroni; Gianluigi Rozza (SIAM Journal on Numerical Analysis, 2013)
In this work we deal with a stochastic optimal Robin boundary control problem constrained by an advection-diffusion-reaction elliptic equation with advection-dominated term. We assume that the uncertainty comes from the advection field and consider a stochastic Robin boundary condition as control function. A stochastic saddle point system is formulated and proved to be equivalent to the first order optimality system for the optimal control problem, based on which we provide the existence and uniqueness of the optimal solution as well as some results on stochastic regularity with respect to the random variables. Stabilized finite element approximations in physical space and collocation approximations in stochastic space are applied to discretize the optimality system. A global error estimate in the product of physical space and stochastic space for the numerical approximation is derived. Illustrative numerical experiments are provided.

Strong asymptotics for Cauchy biorthogonal polynomials with application to the Cauchy two-matrix model - Marco Bertola; Gekhtman, M.; Szmigielski, J. (J. Math. Phys., 2013)

The structure and regularity of admissible BV solutions to hyperbolic conservation laws in one space dimension - Lei Yu (SISSA, 2013)
This thesis is devoted to the study of the qualitative properties of admissible BV solutions to the strictly hyperbolic conservation laws in one space dimension by using wave-front tracking approximation. This thesis consists of two parts: • SBV-like regularity of vanishing viscosity BV solutions to strict hyperbolic systems of conservation laws. • Global structure of admissible BV solutions to strict hyperbolic conservation laws.

Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions - Alberto Boscaggin; Fabio Zanolin (Discrete & Continuous Dynamical Systems - A, 2013)
We study the problem of existence and multiplicity of subharmonic solutions for a second order nonlinear ODE in presence of lower and upper solutions. We show how such additional information can be used to obtain more precise multiplicity results. Applications are given to pendulum type equations and to Ambrosetti-Prodi results for parameter dependent equations.
Keywords: lower and upper solutions; parameter dependent equations; Periodic solutions; Poincaré-Birkhoff twist theorem; subharmonic solutions

On Sudakov's type decomposition of transference plans with norm costs - Stefano Bianchini; Sara Daneri (SISSA, 2013)

Symplectic instanton bundles on P3 and 't Hooft instantons - Ugo Bruzzo; Dimitri Markushevich; Alexander Tikhomirov (arXiv:1312.5554 [math.AG], 2013)
We introduce the notion of tame symplectic instantons by excluding a kind of pathological monads and show that the locus $I^*_{n,r}$ of tame symplectic instantons is irreducible and has the expected dimension, equal to $4n(r+1)-r(2r+1)$. The proof is inherently based on a relation between the spaces $I^*_{n,r}$ and the moduli spaces of 't Hooft instantons.

Topology of moduli spaces of framed sheaves - Gharchia Abdellaoui (SISSA, 2013)
Keywords: Moduli spaces, framed sheaves, instantons

On the tritronquée solutions of P$_I^2$ - Tamara Grava; Andrey Kapaev; Christian Klein (SISSA, 2013)
For equation P$_I^2$, the second member in the P$_I$ hierarchy, we prove existence of various degenerate solutions depending on the complex parameter $t$ and evaluate the asymptotics in the complex $x$ plane for $|x|→∞$ and $t=o(x^{2/3})$. Using this result, we identify the most degenerate solutions $u^{(m)}(x,t)$, $\hat u^{(m)}(x,t)$, $m=0,…,6$, called {\em tritronqu\'ee}, describe the quasi-linear Stokes phenomenon and find the large $n$ asymptotics of the coefficients in a formal expansion of these solutions. We supplement our findings by a numerical study of the tritronqu\'ee solutions.

Universality for the focusing nonlinear Schrödinger equation at the gradient catastrophe point: rational breathers and poles of the \it Tritronquée solution to Painlevé I - Marco Bertola; Alexander Tovbis (Comm. Pure Appl. Math., 2013)

A variational Analysis of the Toda System on Compact Surfaces - Andrea Malchiodi; David Ruiz (Communications on Pure and Applied Mathematics, Volume 66, Issue 3, March 2013, Pages 332-371, 2013)
In this paper we consider the Toda system of equations on a compact surface. We will give existence results by using variational methods in a non coercive case. A key tool in our analysis is a new Moser-Trudinger type inequality under suitable conditions on the center of mass and the scale of concentration of the two components u_1, u_2.

A weighted reduced basis method for elliptic partial differential equations with random input data - Peng Chen; Alfio Quarteroni; Gianluigi Rozza (SIAM Journal on Numerical Analysis, 2013)
In this work we propose and analyze a weighted reduced basis method to solve elliptic partial differential equations (PDEs) with random input data. The PDEs are first transformed into a weighted parametric elliptic problem depending on a finite number of parameters. Distinctive importance of the solution at different values of the parameters is taken into account by assigning different weights to the samples in the greedy sampling procedure. A priori convergence analysis is carried out by constructive approximation of the exact solution with respect to the weighted parameters. Numerical examples are provided for the assessment of the advantages of the proposed method over the reduced basis method and the stochastic collocation method in both univariate and multivariate stochastic problems.

On 2-step, corank 2 nilpotent sub-Riemannian metrics - Davide Barilari; Ugo Boscain; Jean-Paul Gauthier (SIAM J. Control Optim., 50 (2012) 559–582, 2012)
In this paper we study the nilpotent 2-step, corank 2 sub-Riemannian metrics\\r\\nthat are nilpotent approximations of general sub-Riemannian metrics. We exhibit optimal syntheses for these problems. It turns out that in general the cut time is not equal to the first conjugate time but has a simple explicit expression. As a byproduct of this study we get some smoothness properties of the spherical Hausdorff measure in the case of a generic 6 dimensional, 2-step corank 2 sub-Riemannian metric.

Asymptotics of the s-perimeter as s →0 - Serena Dipierro; Alessio Figalli; Giampiero Palatucci; Enrico Valdinoci (Discrete Contin. Dyn. Syst. 33, nr.7 (2012): 2777-2790, 2012)
We deal with the asymptotic behavior of the $s$-perimeter of a set $E$ inside a domain $Ω$ as $s\searrow0$. We prove necessary and sufficient conditions for the existence of such limit, by also providing an explicit formulation in terms of the Lebesgue measure of $E$ and $Ω$. Moreover, we construct examples of sets for which the limit does not exist.

On the behaviour of flexible retaining walls under seismic actions - Riccardo Conti; G.S.P. Madabhushi; Giulia M.B. Viggiani (Geotechnique, Volume 62, Issue 12, December 2012, Pages 1081-1094, 2012)
This paper describes an experimental investigation of the behaviour of embedded retaining walls under seismic actions. Nine centrifuge tests were carried out on reduced-scale models of pairs of retaining walls in dry sand, either cantilevered or with one level of props near the top. The experimental data indicate that, for maximum accelerations that are smaller than the critical limit equilibrium value, the retaining walls experience significant permanent displacements under increasing structural loads, whereas for larger accelerations the walls rotate under constant internal forces. The critical acceleration at which the walls start to rotate increases with increasing maximum acceleration. No significant displacements are measured if the current earthquake is less severe than earthquakes previously experienced by the wall. The increase of critical acceleration is explained in terms of redistribution of earth pressures and progressive mobilisation of the passive strength in front of the wall. The experimental data for cantilevered retaining walls indicate that the permanent displacements of the wall can be reasonably predicted adopting a Newmark-type calculation with a critical acceleration that is a fraction of the limit equilibrium value.
Keywords: Centrifuge modelling

Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty - Toni Lassila; Andrea Manzoni; Alfio Quarteroni; Gianluigi Rozza (Mathematical Modelling and Numerical Analysis, in press, 2012-13, 2012)
We review the optimal design of an arterial bypass graft following either a (i) boundary optimal control approach, or a (ii) shape optimization formulation. The main focus is quantifying and treating the uncertainty in the residual flow when the hosting artery is not completely occluded,\\r\\nfor which the worst-case in terms of recirculation e ffects is inferred to correspond to a strong ori fice flow through near-complete occlusion. A worst-case optimal control approach is applied to the steady\\r\\nNavier-Stokes equations in 2D to identify an anastomosis angle and a cu ed shape that are robust with respect to a possible range of residual \\r\\nflows. We also consider a reduced order modelling framework\\r\\nbased on reduced basis methods in order to make the robust design problem computationally feasible. The results obtained in 2D are compared with simulations in a 3D geometry but without model\\r\\nreduction or the robust framework.
Keywords: shape optimization

On a class of vector fields with discontinuity of divide-by-zero type and its applications - Roberta Ghezzi; Alexey O. Remizov (Journal of dynamical and control systems, 2012)
We study phase portraits and singular points of vector fields of a special type, that is, vector fields whose components are fractions with a common denominator vanishing on a smooth regular hypersurface in the phase space. We assume also some additional conditions, which are fulfilled, for instance, if the vector field is divergence-free. This problem is motivated by a large number of applications. In this paper, we consider three of them in the framework of differential geometry: singularities of geodesic flows in various singular metrics on surfaces.

Classical double, R-operators, and negative flows of integrable hierarchies - Boris Dubrovin; Taras V. Skrypnyk (Theoretical and Mathematical Physics. Volume 172, Issue 1, July 2012, Pages 911-931, 2012)
Using the classical double G of a Lie algebra g equipped with the classical R-operator, we define two sets of functions commuting with respect to the initial Lie–Poisson bracket on g and its extensions. We consider examples of Lie algebras g with the “Adler–Kostant–Symes” R-operators and the two corresponding sets of mutually commuting functions in detail. Using the constructed commutative Hamiltonian flows on different extensions of g, we obtain zero-curvature equations with g-valued U–V pairs. The so-called negative flows of soliton hierarchies are among such equations. We illustrate the proposed approach with examples of two-dimensional Abelian and non-Abelian Toda field equations.

A Codazzi-like equation and the singular set for C1 smooth surfaces in the Heisenberg group. - Andrea Malchiodi; Paul Yang; Jih-Hsin Cheng; JennFang Hwang (Journal fur die Reine und Angewandte Mathematik, Issue 671, October 2012, Pages 131-198, 2012)
In this paper, we study the structure of the singular set for a C 1 smooth surface in the 3-dimensional Heisenberg group ℍ 1. We discover a Codazzi-like equation for the p-area element along the characteristic curves on the surface. Information obtained from this ordinary differential equation helps us to analyze the local configuration of the singular set and the characteristic curves. In particular, we can estimate the size and obtain the regularity of the singular set. We understand the global structure of the singular set through a Hopf-type index theorem. We also justify the Codazzi-like equation by proving a fundamental theorem for local surfaces in ℍ 1

Computing optimal strokes for low reynolds number swimmers - Antonio DeSimone; Luca Heltai; François Alouges; Lefebvre-Lepot Aline (Natural locomotion in fluids and on surfaces : swimming, flying, and sliding / editors Stephen Childress, Anette Hosoi, William W. Schultz, and Z. Jane Wang, editors,, 2012)
We discuss connections between low-Reynolds-number swimming and geometric control theory, and present a general algorithm for the numerical computation of energetically optimal strokes. As an illustration of our approach, we show computed motility maps and optimal strokes for two model swimmers.
Keywords: Numerical analysis.

Concentration on circles for nonlinear Schrödinger–Poisson systems with unbounded potentials vanishing at infinity - Bonheure, Denis; Di Cosmo, Jonathan; Mercuri, Carlo (Communications in Contemporary Mathematics, 2012)
The present paper is devoted to weighted nonlinear Schrödinger–Poisson systems with potentials possibly unbounded and vanishing at infinity. Using a purely variational approach, we prove the existence of solutions concentrating on a circle.

Conservation of Geometric Structures for Non-Homogeneous Inviscid Incompressible Fluids - Francesco Fanelli (Communications in Partial Differential Equations, 2012)
In this article we get a result on propagation of geometric properties for solutions of the non-homogeneous incompressible Euler system in any dimension N ≥ 2. In particular, we investigate conservation of striated and conormal regularity, which generalize the 2-D structure of vortex patches. The results we get are only local in time, even for N = 2; however, we provide an explicit lower bound for the lifespan of the solution. In the case of physical dimension N = 2 or 3, we investigate also propagation of Hölder regularity in the interior of a bounded domain.

Convergence of equilibria of thin elastic plates under physical growth conditions for the energy density - Maria Giovanna Mora; Lucia Scardia (Elsevier, 2012)
The asymptotic behaviour of the equilibrium configurations of a thin elastic plate is studied, as the thickness $h$ of the plate goes to zero. More precisely, it is shown that critical points of the nonlinear elastic functional $\mathcal E^h$, whose energies (per unit thickness) are bounded by $Ch^4$, converge to critical points of the $Γ$-limit of $h^{-4}\mathcal E^h$. This is proved under the physical assumption that the energy density $W(F)$ blows up as $\det F→0$.

Convex pencils of real quadratic forms - Antonio Lerario (Discrete and Computational Geometry, Volume 48, Issue 4, December 2012, Pages 1025-1047, 2012)
We study the topology of the set X of the solutions of a system of two quadratic inequalities in the real projective space RP^n (e.g. X is the intersection of two real quadrics). We give explicit formulae for its Betti numbers and for those of its double cover in the sphere S^n; we also give similar formulae for level sets of homogeneous quadratic maps to the plane. We discuss some applications of these results, especially in classical convexity theory. We prove the sharp bound b(X)≤ 2n for the total Betti number of X; we show that for odd n this bound is attained only by a singular X. In the nondegenerate case we also prove the bound on each specific Betti number b_k(X)≤ 2(k+2).

Crawling motility through the analysis of model locomotors: two case studies - Antonio DeSimone; Amabile Tatone (The European Physical Journal E, Volume 35, Issue 9, September 2012, Article number85, 2012)
We study model locomotors on a substrate, which derive their propulsive capabilities from the tangential (viscous or frictional) resistance offered by the substrate. Our aim is to develop new tools and insight for future studies of cellular motility by crawling and of collective bacterial motion. The purely viscous case (worm) is relevant for cellular motility by crawling of individual cells. We re-examine some recent results on snail locomotion in order to assess the role of finely regulated adhesion mechanisms in crawling motility. Our main conclusion is that such regulation, although well documented in several biological systems, is not indispensable to accomplish locomotion driven by internal deformations, provided that the crawler may execute sufficiently large body deformations. Thus, there is no snail theorem. Namely, the crawling analog of the scallop theorem of low Reynolds number hydrodynamics does not hold for snail-like crawlers. The frictional case is obtained by assuming that the viscous coefficient governing tangential resistance forces, which act parallel and in the direction opposite to the velocity of the point to which they are applied, depends on the normal force acting at that point. We combine these surface interactions with inertial effects in order to investigate the mechanisms governing the motility of a bristle-robot. This model locomotor is easily manufactured and has been proposed as an effective tool to replicate and study collective bacterial motility.

On the critical behavior in nonlinear evolutionary PDEs with small viscocity - Boris Dubrovin; Maria Elaeva (Russian Journal of Mathematical Physics. Volume 19, Issue 4, December 2012, Pages 449-460, 2012)
We address the problem of general dissipative regularization of the quasilinear transport equation. We argue that the local behavior of solutions to the regularized equation near the point of gradient catastrophe for the transport equation is described by the logarithmic derivative of the Pearcey function, a statement generalizing the result of A.M.Il\\\'in \\\\cite{ilin}. We provide some analytic arguments supporting such conjecture and test it numerically.

Decompositions of large-scale biological systems based on dynamical properties - Nicola Soranzo; Fahimeh Ramezani; Giovanni Iacono; Claudio Altafini (Bioinformatics (Oxford, England). 2012 Jan; 28(1):76-83, 2012)
MOTIVATION: Given a large-scale biological network represented as an influence graph, in this article we investigate possible decompositions of the network aimed at highlighting specific dynamical properties.\\r\\nRESULTS: The first decomposition we study consists in finding a maximal directed acyclic subgraph of the network, which dynamically corresponds to searching for a maximal open-loop subsystem of the given system. Another dynamical property investigated is strong monotonicity. We propose two methods to deal with this property, both aimed at decomposing the system into strongly monotone subsystems, but with different structural characteristics: one method tends to produce a single large strongly monotone component, while the other typically generates a set of smaller disjoint strongly monotone subsystems.\\r\\nAVAILABILITY: Original heuristics for the methods investigated are described in the article.

Deformed Lorentz symmetry and relative locality in a curved/expanding spacetime - Giovanni Amelino-Camelia; Antonino Marciano; Marco Matassa; Giacomo Rosati (Phys. Rev. D 86 (2012) 124035, 2012)
The interest of part of the quantum-gravity community in the possibility of\r\nPlanck-scale-deformed Lorentz symmetry is also fueled by the opportunities for testing the relevant scenarios with analyses, from a signal-propagation perspective, of observations of bursts of particles from cosmological distances. In this respect the fact that so far the implications of deformed Lorentz symmetry have been investigated only for flat (Minkowskian) spacetimes represents a very significant limitation, since for propagation over cosmological distances the curvature/expansion of spacetime is evidently tangible. We here provide a significant step toward filling this gap by exhibiting an explicit example of Planck-scale-deformed relativistic symmetries of a spacetime with constant rate of expansion (deSitterian). Technically we obtain the first ever example of a relativistic theory of worldlines of particles with 3 nontrivial relativistic invariants: a large speed scale (\"speed-of-light scale\"), a large distance scale (inverse of the \"expansion-rate scale\"), and a large momentum scale (\"Planck scale\"). We address some of the challenges that had obstructed success for previous attempts by exploiting the recent understanding of the connection between deformed Lorentz symmetry and relativity of spacetime locality. We also offer a preliminary analysis of the differences between the scenario we here propose and the most studied scenario for broken (rather than deformed) Lorentz symmetry in expanding spacetimes.
Keywords: Doubly special relativity

Detection of transcriptional triggers in the dynamics of microbial growth: application to the respiratory-versatile bacterium Shewanella oneidensis - Q Beg; Mattia Zampieri; N Klitgord; S Collins; M Serres; Daniel Segrè; Claudio Altafini (Nucleic Acids Research, Volume 40, Issue 15, August 2012, Pages 7132-7149, 2012)
The capacity of microorganisms to respond to variable external conditions requires a coordination of environment-sensing mechanisms and decisionmaking regulatory circuits. Here, we seek to understand the interplay between these two processes by combining high-throughput measurement of time-dependent mRNA profiles with a novel computational approach that searches for key genetic triggers of transcriptional changes. Our approach helped us understand the regulatory strategies of a respiratorily versatile bacterium with promising bioenergy and bioremediation applications, Shewanella oneidensis, in minimal and rich media. By comparing expression profiles across these two conditions, we unveiled components of the transcriptional program that depend mainly on the growth phase. Conversely, by integrating our time-dependent data with a previously available large compendium of static perturbation responses, we identified transcriptional changes that cannot be explained solely by internal network dynamics, but are rather triggered by specific genes acting as key mediators of an environment-dependent response. These transcriptional triggers include known and novel regulators that respond to carbon, nitrogen and oxygen limitation. Our analysis suggests a sequence of physiological responses, including a coupling between nitrogen depletion and glycogen storage, partially recapitulated through dynamic flux balance analysis, and experimentally confirmed by metabolite measurements. Our approach is broadly applicable to other systems

A dynamical feedback model for adaptation in the olfactory transduction pathway - Giovanna De Palo; Anna Boccaccio; Andrew Miri; Anna Menini; Claudio Altafini (Biophysical Journal. Volume 102, Issue 12, 20 June 2012, Pages 2677-2686, 2012)
Olfactory transduction exhibits two distinct types of adaptation, which we denote multipulse and step adaptation. In terms of measured transduction current, multipulse adaptation appears as a decrease in the amplitude of the second of two consecutive responses when the olfactory neuron is stimulated with two brief pulses. Step adaptation occurs in response to a sustained steplike stimulation and is characterized by a return to a steady-state current amplitude close to the prestimulus value, after a transient peak. In this article, we formulate a dynamical model of the olfactory transduction pathway, which includes the kinetics of the CNG channels, the concentration of Ca ions flowing through them, and the Ca-complexes responsible for the regulation. Based on this model, a common dynamical explanation for the two types of adaptation is suggested. We show that both forms of adaptation can be well described using different time constants for the kinetics of Ca ions (faster) and the kinetics of the feedback mechanisms (slower). The model is validated on experimental data collected in voltage-clamp conditions using different techniques and animal species.

Dynamics of opinion forming in structurally balanced social networks - Claudio Altafini (PloS one. 2012 ; 7(6):e38135, 2012)
A structurally balanced social network is a social community that splits into two antagonistic factions (typical example being a two-party political system). The process of opinion forming on such a community is most often highly predictable, with polarized opinions reflecting the bipartition of the network. The aim of this paper is to suggest a class of dynamical systems, called monotone systems, as natural models for the dynamics of opinion forming on structurally balanced social networks. The high predictability of the outcome of a decision process is explained in terms of the order-preserving character of the solutions of this class of dynamical systems. If we represent a social network as a signed graph in which individuals are the nodes and the signs of the edges represent friendly or hostile relationships, then the property of structural balance corresponds to the social community being splittable into two antagonistic factions, each containing only friends.

Exploring the low-energy landscape of large-scale signed social networks - Giuseppe Facchetti; Giovanni Iacono; Claudio Altafini (Physical Review E - Statistical, Nonlinear, and Soft Matter Physics. Volume 86, Issue 3, 26 September 2012, Article number036116, 2012)
Analogously to a spin glass, a large-scale signed social network is characterized by the presence of disorder, expressed in this context (and in the social network literature) by the concept of structural balance. If, as we have recently shown, the signed social networks currently available have a limited amount of true disorder (or frustration), it is also interesting to investigate how this frustration is organized, by exploring the landscape of near-optimal structural balance. What we obtain in this paper is that while one of the networks analyzed shows a unique valley of minima, and a funneled landscape that gradually and smoothly worsens as we move away from the optimum, another network shows instead several distinct valleys of optimal or near-optimal structural balance, separated by energy barriers determined by internally balanced subcommunities of users, a phenomenon similar to the replica-symmetry breaking of spin glasses. Multiple, essentially isoenergetic, arrangements of these communities are possible. Passing from one valley to another requires one to destroy the internal arrangement of these balanced subcommunities and then to reform it again. It is essentially this process of breaking the internal balance of the subcommunities which gives rise to the energy barriers.

A formula for Popp\'s volume in sub-Riemannian geometry - Luca Rizzi; Davide Barilari (Analysis and Geometry in Metric Spaces, vol. 1 (2012), pages : 42-57, 2012)
For an equiregular sub-Riemannian manifold M, Popp\'s volume is a smooth\r\nvolume which is canonically associated with the sub-Riemannian structure, and\r\nit is a natural generalization of the Riemannian one. In this paper we prove a\r\ngeneral formula for Popp\'s volume, written in terms of a frame adapted to the\r\nsub-Riemannian distribution. As a first application of this result, we prove an\r\nexplicit formula for the canonical sub-Laplacian, namely the one associated\r\nwith Popp\'s volume. Finally, we discuss sub-Riemannian isometries, and we prove\r\nthat they preserve Popp\'s volume. We also show that, under some hypotheses on\r\nthe action of the isometry group of M, Popp\'s volume is essentially the unique\r\nvolume with such a property.
Keywords: subriemannian, volume, Popp, control

Fredholm determinants and pole-free solutions to the noncommutative Painlevé II equation - Marco Bertola; Mattia Cafasso (Comm. Math. Phys., 2012)

Frobenius manifold for the dispersionless Kadomtsev-Petviashvili equation - Andrea Raimondo (Communications in Mathematical Physics 311 (2012) 557-594, 2012)
We consider a Frobenius structure associated with the dispersionless\\r\\nKadomtsev-Petviashvili equation. This is done, essentially, by applying a\\r\\ncontinuous analogue of the finite dimensional theory in the space of Schwartz\\r\\nfunctions on the line. The potential of the Frobenius manifold is found to be a\\r\\nlogarithmic potential with quadratic external field. Following the construction\\r\\nof the principal hierarchy, we construct a set of infinitely many commuting\\r\\nflows, which extends the classical dKP hierarchy.

A Fully Coupled Immersed Finite Element Method for Fluid Structure Interaction via the Deal.II Library - Luca Heltai; Saswati Roy; Francesco Costanzo (SISSA, 2012)
We present the implementation of a solution scheme for fluid-structure\\r\\ninteraction problems via the finite element software library deal.II. The\\r\\nsolution scheme is an immersed finite element method in which two independent discretizations are used for the fluid and immersed deformable body. In this type of formulation the support of the equations of motion of the fluid is extended to cover the union of the solid and fluid domains. The equations of motion over the extended solution domain govern the flow of a fluid under the action of a body force field. This body force field informs the fluid of the presence of the immersed solid. The velocity field of the immersed solid is the restriction over the immersed domain of the velocity field in the extended equations of motion. The focus of this paper is to show how the determination of the motion of the immersed domain is carried out in practice. We show that our implementation is general, that is, it is not dependent on a specific choice of the finite element spaces over the immersed solid and the extended fluid domains. We present some preliminary results concerning the accuracy of the proposed method.
Keywords: Finite Element Method; Immersed Boundary Method; Immersed Finite Element Method

Gamma-convergence and H-convergence of linear elliptic operators - Nadia Ansini; Gianni Dal Maso; Caterina Ida Zeppieri (Journal de Mathématiques Pures et Appliquées, Available online 12 September 2012, 2012)
Keywords: Linear elliptic operators

Gauge Theories on ALE Space and Super Liouville Correlation Functions - Giulio Bonelli; Kazunobu Maruyoshi; Alessandro Tanzini (Lett. Math. Phys. 101 (2012) 103-124, 2012)
We present a relation between N=2 quiver gauge theories on the ALE space O_{P^1}(-2) and correlators of N=1 super Liouville conformal field theory, providing checks in the case of punctured spheres and tori. We derive a blow-up formula for the full Nekrasov partition function and show that, up to a U(1) factor, the N=2^* instanton partition function is given by the product of the character of \\\\hat{SU}(2)_2 times the super Virasoro conformal block on the torus with one puncture.

A general method for the existence of periodic solutions of differential systems in the plane - Alessandro Fonda; Andrea Sfecci (Journal of Differential Equations, 2012)
We propose a general method to prove the existence of periodic solutions for planar systems of ordinary differential equations, which can be used in many different circumstances. Applications are given to some nonresonant cases, even for systems with superlinear growth in some direction, or with a singularity. Systems “at resonance” are also considered, provided a Landesman–Lazer type of condition is assumed.
Keywords: Nonlinear dynamics; Periodic solutions

Generalized reduced basis methods and n-width estimates for the approximation of the solution manifold of parametric PDEs - Toni Lassila; Andrea Manzoni; Alfio Quarteroni; Gianluigi Rozza (Springer, Indam Series, Vol. 4, 2012, 2012)
The set of solutions of a parameter-dependent linear partial di fferential equation with smooth coe fficients typically forms a compact manifold in a Hilbert space. In this paper we review the generalized reduced basis method as a fast computational tool for the uniform approximation of the solution manifold. We focus on operators showing an affi ne parametric dependence, expressed as a linear combination of parameter-independent operators through some smooth, parameter-dependent scalar functions. In the case that the parameter-dependent operator has a dominant term in its affi ne expansion, one can prove the existence of exponentially convergent uniform approximation spaces for the entire solution manifold. These spaces can be constructed without any assumptions on the parametric regularity of the manifold \\r\\nonly spatial regularity of the solutions is required. The exponential convergence rate is then inherited by the generalized reduced basis method. We provide a numerical example related to parametrized elliptic\\r\\nequations con rming the predicted convergence rates.
Keywords: solution manifold

On the genus two free energies for semisimple Frobenius manifolds - Boris Dubrovin; Si-Qi Liu; Youjin Zhang (Russian Journal of Mathematical Physics. Volume 19, Issue 3, September 2012, Pages 273-298, 2012)
We represent the genus two free energy of an arbitrary semisimple Frobenius\\r\\nmanifold as a sum of contributions associated with dual graphs of certain\\r\\nstable algebraic curves of genus two plus the so-called \\\"genus two G-function\\\".\\r\\nConjecturally the genus two G-function vanishes for a series of important\\r\\nexamples of Frobenius manifolds associated with simple singularities as well as\\r\\nfor ${\\\\bf P}^1$-orbifolds with positive Euler characteristics. We explain the\\r\\nreasons for such Conjecture and prove it in certain particular cases.

On the Hausdorff volume in sub-Riemannian geometry - Andrei A. Agrachev; Davide Barilari; Ugo Boscain (Calculus of Variations and Partial Differential Equations. Volume 43, Issue 3-4, March 2012, Pages 355-388, 2012)
For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative\r\nof the spherical Hausdorff measure with respect to a smooth volume. We prove\r\nthat this is the volume of the unit ball in the nilpotent approximation and it\r\nis always a continuous function. We then prove that up to dimension 4 it is\r\nsmooth, while starting from dimension 5, in corank 1 case, it is C^3 (and C^4\r\non every smooth curve) but in general not C^5. These results answer to a\r\nquestion addressed by Montgomery about the relation between two intrinsic\r\nvolumes that can be defined in a sub-Riemannian manifold, namely the Popp and\r\nthe Hausdorff volume. If the nilpotent approximation depends on the point (that\r\nmay happen starting from dimension 5), then they are not proportional, in\r\ngeneral.

Hybridization in nanostructured DNA monolayers probed by AFM: theory versus experiment - Alessandro Bosco; Fouzia Bano; Pietro Parisse; Loredana Casalis; Antonio DeSimone; Cristian Micheletti (Nanoscale. 2012 Mar; 4(5):1734-41, 2012)
Nanografted monolayers (NAMs) of DNA show novel physico-chemical properties that make them ideally suited for advanced biosensing applications. In comparison with alternative solid-phase techniques for diagnostic DNA detection, NAMs have the advantage of combining a small size with a high homogeneity of the DNA surface coverage. These two properties favour the extreme miniaturization and ultrasensitivity in high-throughput biosensing devices. The systematic use of NAMs for quantitative DNA (and protein) detection has so far suffered from the lack of a control on key fabrication parameters, such as the ss- or ds-DNA surface coverage. Here we report on a combined experimental-computational study that allows us to estimate the surface density of the grafted DNA by analyzing the sample mechanical response, that is the DNA patch height vs. applied tip load curves. It is shown that the same analysis scheme can be used to detect the occurrence of hybridization with complementary strands in solution and estimate its efficiency. Thanks to these quantitative relationships it is possible to use a single AFM-based setup to: (i) fabricate a DNA NAM, (ii) control the DNA surface coverage, and (iii) characterize its level of hybridization helping the design of NAMs with pre-determined fabrication parameters.

Introduction to Riemannian and sub-Riemannian geometry - Andrei A. Agrachev; Davide Barilari; Ugo Boscain (SISSA, 2012)

The KdV hierarchy: universality and a Painleve transcendent - Tom Claeys; Tamara Grava (International Mathematics Research Notices, vol. 22 (2012) , page 5063-5099, 2012)
We study the Cauchy problem for the Korteweg-de Vries (KdV) hierarchy in the small dispersion limit where $\e→ 0$. For negative analytic initial data with a single negative hump, we prove that for small times, the solution is approximated by the solution to the hyperbolic transport equation which corresponds to $\e=0$. Near the time of gradient catastrophe for the transport equation, we show that the solution to the KdV hierarchy is approximated by a particular Painlev\'e transcendent. This supports Dubrovins universality conjecture concerning the critical behavior of Hamiltonian perturbations of hyperbolic equations. We use the Riemann-Hilbert approach to prove our results.
Keywords: Small-Dispersion limit

Linear elasticity obtained from finite elasticity by Gamma-convergence under weak coerciveness conditions - Virginia Agostiniani; Gianni Dal Maso; Antonio DeSimone (Ann. Inst. H. Poincare Anal. Non Lineaire, 2012)
The energy functional of linear elasticity is obtained as G-limit of suitable rescalings of the energies of finite elasticity...
Keywords: Nonlinear elasticity

On localization in holomorphic equivariant cohomology - Ugo Bruzzo; Vladimir Rubtsov (Central European Journal of Mathematics, Volume 10, Issue 4, August 2012, Pages 1442-1454, 2012)
We prove a localization formula for a "holomorphic equivariant cohomology" attached to the Atiyah algebroid of an equivariant holomorphic vector bundle. This generalizes Feng-Ma, Carrell-Liebermann, Baum-Bott and K. Liu's localization formulas.
Keywords: Lie algebroids

On the location of poles for the Ablowitz-Segur family of solutions to the second Painlevé equation - Marco Bertola (Nonlinearity, 2012)

Mathematical and numerical modeling of liquid crystal elastomer phase transition and deformation - Mariarita De Luca; Antonio DeSimone (Materials Research Society Symposium Proceedings. Volume 1403, 2012, Pages 125-130, 2012)
Liquid crystal (in particular, nematic) elastomers consist of cross-linked flexible polymer chains with embedded stiff rod molecules that allow them to behave as a rubber and a liquid crystal. Nematic elastomers are characterized by a phase transition from isotropic to nematic past a temperature threshold. They behave as rubber at high temperature and show nematic behavior below the temperature threshold. Such transition is reversible. While in the nematic phase, the rod molecules are aligned along the direction of the "nematic director". This molecular rearrangement induces a stretch in the polymer chains and hence macroscopic spontaneous deformations. The coupling between nematic order parameter and deformation gives rise to interesting phenomena with a potential for new interesting applications. In the biological field, the ability to considerably change their length makes them very promising as artificial muscles actuators. Their tunable optical properties make them suitable, for example, as lenses for new imaging systems. We present a mathematical model able to describe the behavior of nematic elastomers and numerical simulations reproducing such peculiar behavior. We use a geometrically linear version of the Warner and Terentjev model [1] and consider cooling experiments and stretching experiments in the direction perpendicular to the one of the director at cross-linking.
Keywords: Artificial muscle

Modeling and control of quantum systems: An introduction - Claudio Altafini; Francesco Ticozzi (IEEE Transactions on Automatic Control. Volume 57, Issue 8, 2012, Article number6189035, Pages 1898-1917, 2012)
The scope of this work is to provide a self-contained introduction to a selection of basic theoretical aspects in the modeling and control of quantum mechanical systems, as well as a brief survey on the main approaches to control synthesis. While part of the existing theory, especially in the open-loop setting, stems directly from classical control theory (most notably geometric control and optimal control), a number of tools specifically tailored for quantum systems have been developed since the 1980s, in order to take into account their distinctive features: the probabilistic nature of atomic-scale physical systems, the effect of dissipation and the irreversible character of the measurements have all proved to be critical in feedback-design problems. The relevant dynamical models for both closed and open quantum systems are presented, along with the main results on their controllability and stability. A brief review of several currently available control design methods is meant to provide the interested reader with a roadmap for further studies

Moduli of symplectic instanton vector bundles of higher rank on projective space $\\mathbbP^3$ - Ugo Bruzzo; Dimitri Markushevich; Alexander Tikhomirov (Central European Journal of Mathematics 10, nr. 4 (2012) 1232, 2012)
Symplectic instanton vector bundles on the projective space $\\mathbb{P}^3$ constitute a natural generalization of mathematical instantons of rank 2. We study the moduli space $I_{n,r}$ of rank-$2r$ symplectic instanton vector bundles on $\\mathbb{P}^3$ with $r\\ge2$ and second Chern class $n\\ge r,\\ n\\equiv r({\\rm mod}2)$. We give an explicit construction of an irreducible component $I^*_{n,r}$ of this space for each such value of $n$ and show that $I^*_{n,r}$ has the expected dimension $4n(r+1)-r(2r+1)$.

Moduli spaces of noncommutative instantons: gauging away noncommutative parameters - Simon Brain; Giovanni Landi (Quarterly Journal of Mathematics (2012) 63 (1): 41-86, 2012)
Using the theory of noncommutative geometry in a braided monoidal category, we improve upon a previous construction of noncommutative families of instantons of arbitrary charge on the deformed sphere S^4_\\\\theta. We formulate a notion of noncommutative parameter spaces for families of instantons and we explore what it means for such families to be gauge equivalent, as well as showing how to remove gauge parameters using a noncommutative quotient construction. Although the parameter spaces are a priori noncommutative, we show that one may always recover a classical parameter space by making an appropriate choice of gauge transformation.

The Monge problem in Wiener space - Fabio Cavalletti (Calculus of Variations and Partial Differential Equations, 2012)
We address the Monge problem in the abstract Wiener space and we give an existence result provided both marginal measures are absolutely continuous with respect to the infinite dimensional Gaussian measure γ.

Nonlinear thin-walled beams with a rectangular cross-section-Part I - Lorenzo Freddi; Maria Giovanna Mora; Roberto Paroni (Math. Models Methods Appl. Sci. 22, 1150016 (2012), 2012)
Our aim is to rigorously derive a hierarchy of one-dimensional models for thin-walled beams with rectangular cross-section, starting from three-dimensional nonlinear elasticity. The different limit models are distinguished by the different scaling of the elastic energy and of the ratio between the sides of the cross-section. In this paper we report the first part of our results.

A nonresonance condition for radial solutions of a nonlinear Neumann elliptic problem - Andrea Sfecci (Nonlinear Analysis: Theory, Methods & Applications, 2012)
We prove an existence result for radial solutions of a Neumann elliptic problem whose nonlinearity asymptotically lies between the first two eigenvalues. To this aim, we introduce an alternative nonresonance condition with respect to the second eigenvalue which, in the scalar case, generalizes the classical one, in the spirit of Fonda et al. (1991) [2]. Our approach also applies for nonlinearities which do not necessarily satisfy a subcritical growth assumption.
Keywords: Neumann problem; Nonresonance; Radial solutions; Time-map

Non-uniqueness results for critical metrics of regularized determinants in four dimensions - Matthew Gursky; Andrea Malchiodi (Communications in Mathematical Physics, Volume 315, Issue 1, September 2012, Pages 1-37, 2012)
The regularized determinant of the Paneitz operator arises in quantum gravity (see Connes 1994, IV.4.$γ$). An explicit formula for the relative determinant of two conformally related metrics was computed by Branson in Branson (1996). A similar formula holds for Cheeger's half-torsion, which plays a role in self-dual field theory (see Juhl, 2009), and is defined in terms of regularized determinants of the Hodge laplacian on $p$-forms ($p < n/2$). In this article we show that the corresponding actions are unbounded (above and below) on any conformal four-manifold. We also show that the conformal class of the round sphere admits a second solution which is not given by the pull-back of the round metric by a conformal map, thus violating uniqueness up to gauge equivalence. These results differ from the properties of the determinant of the conformal Laplacian established in Chang and Yang (1995), Branson, Chang, and Yang (1992), and Gursky (1997). We also study entire solutions of the Euler-Lagrange equation of $\log \det P$ and the half-torsion $τ_h$ on $\mathbb{R}^4 \setminus {0}$, and show the existence of two families of periodic solutions. One of these families includes Delaunay-type solutions.

Numerical modelling of installation effects for diaphragm walls in sand - Riccardo Conti; Luca de Sanctis; Giulia M.B. Viggiani (Acta Geotechnica, Volume 7, Issue 3, September 2012, Pages 219-237, 2012)
The scopes of this work are to study the mechanisms of load transfer and the deformations of the ground during slurry trenching and concreting in dry sand and to evaluate their effects on service structural loads, wall deflections and ground displacements behind the wall caused by subsequent excavation. A series of three-dimensional finite element analyses was carried out modelling the installation of diaphragm walls consisting of panels of different length. The soil was modelled as either linearly elastic-perfectly plastic or incrementally non-linear (hypoplastic) with elastic strain range. Plane strain analyses of diaphragm walls of identical cross section were also carried out in which wall installation was either modelled or the wall was wished in place (WIP). The analyses predict ground movements consistent with the experimental observations both in magnitude and trend. The results also show that the maximum horizontal wall deflections and structural loads reduce with increasing panel aspect ratio towards a minimum which is about twice the value computed for WIP analyses. Panel aspect ratios should be larger than about three to take advantage of the three-dimensional effects. The pattern and magnitude of surface vertical displacements obtained from linearly elastic-perfectly plastic analyses, no matter whether three- or two-dimensional, are unrealistic.
Keywords: Constitutive relations

Numerical study of the small dispersion limit of the Korteweg-de Vries equation and asymptotic solutions - Tamara Grava; Christian Klein (Physica D 241, nr. 23-24 (2012): 2246-2264, 2012)
We study numerically the small dispersion limit for the Korteweg-de Vries (KdV) equation $u_t+6uu_x+ε^{2}u_{xxx}=0$ for $ε\ll1$ and give a quantitative comparison of the numerical solution with various asymptotic formulae for small $ε$ in the whole $(x,t)$-plane. The matching of the asymptotic solutions is studied numerically.
Keywords: Korteweg-de Vries equation

Ogden-type energies for nematic elastomers - Virginia Agostiniani; Antonio DeSimone (International Journal of Non-Linear mechanics, 2012)
Ogden-type extensions of the free-energy densities currently used to model the mechanical behavior of nematic elastomers are proposed and analyzed. Based on a multiplicative decomposition of the deformation gradient into an elastic and a spontaneous or remanent part, they provide a suitable framework to study the stiffening response at high imposed stretches. Geometrically linear versions of the models (Taylor expansions at order two) are provided and discussed. These small strain theories provide a clear illustration of the geometric structure of the underlying energy landscape (the energy grows quadratically with the distance from a non-convex set of spontaneous strains or energy wells). The comparison between small strain and finite deformation theories may also be useful in the opposite direction, inspiring finite deformation generalizations of small strain theories currently used in the mechanics of active and phase-transforming materials. The energy well structure makes the free-energy densities non-convex. Explicit quasi-convex envelopes are provided, and applied to compute the stiffening response of a specimen tested in plane strain extension experiments (pure shear).
Keywords: Nonlinear elasticity

One-signed harmonic solutions and sign-changing subharmonic solutions to scalar second order differential equations - Alberto Boscaggin (Advanced Nonlinear Studies, 2012)

Optimal Transport with Branching Distance Costs and the Obstacle Problem - Fabio Cavalletti (SIAM Journal on Mathematical Analysis, 2012)

Pairs of positive periodic solutions of second order nonlinear equations with indefinite weight - Alberto Boscaggin; Fabio Zanolin (Journal of Differential Equations, 2012)
We study the problem of the existence and multiplicity of positive periodic solutions to the scalar ODEu″+λa(t)g(u)=0,λ>0, where g(x) is a positive function on R+, superlinear at zero and sublinear at infinity, and a(t) is a T-periodic and sign indefinite weight with negative mean value. We first show the nonexistence of solutions for some classes of nonlinearities g(x) when λ is small. Then, using critical point theory, we prove the existence of at least two positive T-periodic solutions for λ large. Some examples are also provided.
Keywords: Critical points; Necessary conditions; Pairs of positive solutions; Periodic solutions

Periodic solutions of a system of coupled oscillators with one-sided superlinear retraction forces - Alessandro Fonda; Andrea Sfecci (Differential Integral Equations, 2012)

Periodic solutions to superlinear planar Hamiltonian systems - Alberto Boscaggin (Portugaliae Mathematica, 2012)
We prove the existence of infinitely many periodic (harmonic and subharmonic) solutions to planar Hamiltonian systems satisfying a suitable superlinearity condition at infinity. The proof relies on the Poincare-Birkhoff fixed point theorem.

Poles Distribution of PVI Transcendents close to a Critical Point (summer 2011) - Davide Guzzetti (Physica D: Nonlinear Phenomena, Volume 241, Issue 23-24, 1 December 2012, Pages 2188-2203, 2012)
The distribution of the poles of Painlevé VI transcendents associated to semi-simple Frobenius manifolds is determined close to a critical point. It is shown that the poles accumulate at the critical point,asymptotically along two rays. As an example, the Frobenius manifold given by the quantum cohomology of CP2 is considered. The general PVI is also considered.
Keywords: Painleve' equations

Positive periodic solutions of second order nonlinear equations with indefinite weight: Multiplicity results and complex dynamics - Alberto Boscaggin; Fabio Zanolin (Journal of Differential Equations, 2012)
We prove the existence of a pair of positive T-periodic solutions as well as the existence of positive subharmonic solutions of any order and the presence of chaotic-like dynamics for the scalar second order ODEu″+aλ,μ(t)g(u)=0, where g(x) is a positive function on R+, superlinear at zero and sublinear at infinity, and aλ,μ(t) is a T-periodic and sign indefinite weight of the form λa+(t)−μa−(t), with λ,μ>0 and large.
Keywords: Complex dynamics; Poincaré map; Positive periodic solutions; Subharmonics

Predicting and characterizing selective multiple drug treatments for metabolic diseases and cancer. - Giuseppe Facchetti; Claudio Altafini; Mattia Zampieri (BMC Systems Biology. 29 August 2012, Page 115, 2012)
Background: In the field of drug discovery, assessing the potential of multidrug therapies is a difficult task because of the combinatorial complexity (both theoretical and experimental) and because of the requirements on the selectivity of the therapy. To cope with this problem, we have developed a novel method for the systematic in silico investigation of synergistic effects of currently available drugs on genome-scale metabolic networks. The algorithm finds the optimal combination of drugs which guarantees the inhibition of an objective function, while minimizing the side effect on the overall network. Results: Two different applications are considered: finding drug synergisms for human metabolic diseases (like diabetes, obesity and hypertension) and finding antitumoral drug combinations with minimal side effect on the normal human metabolism.The results we obtain are consistent with some of the available therapeutic indications and predict some new multiple drug treatments.A cluster analysis on all possible interactions among the currently available drugs indicates a limited variety on the metabolic targets for the approved drugs. Conclusion: The in silico prediction of drug synergism can represent an important tool for the repurposing of drug in a realistic perspective which considers also the selectivty of the therapy. Moreover, for a more profitable exploitation of drug-drug interactions, also drugs which show a too low efficacy but which have a non-common mechanism of action, can be reconsider as potential ingredients of new multicompound therapeutic indications.Needless to say the clues provided by a computational study like ours need in any case to be thoroughly evaluated experimentally.

Quasistatic evolution for Cam-Clay plasticity: properties of the viscosity solution - Gianni Dal Maso; Antonio DeSimone; Francesco Solombrino (Calculus of variations and partial differential equations 44 (2012) 495-541, 2012)
Cam-Clay plasticity is a well established model for the description of the mechanics of fine grained soils. As solutions can develop discontinuities in time, a weak notion of solution, in terms of a rescaled time s , has been proposed in [8] to give a meaning to this discontinuous evolution. In this paper we first prove that this rescaled evolution satisfies the flow-rule for the rate of plastic strain, in a suitable measure-theoretical sense. In the second part of the paper we consider the behavior of the evolution in terms of the original time variable t . We prove that the unrescaled solution satisfies an energy-dissipation balance and an evolution law for the internal variable, which can be expressed in terms of integrals depending only on the original time. Both these integral identities contain terms concentrated on the jump times, whose size can only be determined by looking at the rescaled formulation.

Quasistatic evolution in non-associative plasticity - the cap models - Jean-Francois Babadjian; Gilles A. Francfort; Maria Giovanna Mora (SIAM Journal on Mathematical Analysis 44, nr. 1 (2012) 245-292, 2012)
Non-associative elasto-plasticity is the working model of plasticity for soil and rocks mechanics. Yet, it is usually viewed as non-variational. In this work, we prove a contrario the existence of a variational evolution for such a model under a natural capping assumption on the hydrostatic stresses and a less natural mollification of the stress admissibility constraint. The obtained elasto-plastic evolution is expressed for times that are conveniently rescaled.
Keywords: Elasto-plasticity

Reduction strategies for PDE-constrained oprimization problems in Haemodynamics - Gianluigi Rozza; Andrea Manzoni; Federico Negri (European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012) J. Eberhardsteiner et.al. (eds.), Vienna, Austria, 10-14 sept. 2012, 2012)
Solving optimal control problems for many different scenarios obtained by varying a set of parameters in the state system is a computationally extensive task. In this paper we present a new reduced framework for the formulation, the analysis and the numerical solution of parametrized PDE-constrained optimization problems. This framework is based on a suitable saddle-point formulation of the optimal control problem and exploits the reduced basis method for the rapid and reliable solution of parametrized PDEs, leading to a relevant computational reduction with respect to traditional discretization techniques such as the finite element method. This allows a very efficient evaluation of state solutions and cost functionals, leading to an effective solution of repeated optimal control problems, even on domains of variable shape, for which a further (geometrical) reduction is pursued, relying on flexible shape parametrization techniques. This setting is applied to the solution of two problems arising from haemodynamics, dealing with both data reconstruction and data assimilation over domains of variable shape,\\r\\nwhich can be recast in a common PDE-constrained optimization formulation.
Keywords: inverse problems

Resonance at the first eigenvalue for first-order systems in the plane: vanishing Hamiltonians and the Landesman-Lazer condition - Maurizio Garrione (Differential Integral Equations, 2012)

Reverse engineering the euglenoid movement - Marino Arroyo; Luca Heltai; Daniel Millán; Antonio DeSimone (Proceedings of the National Academy of Sciences of the United States of America. Volume 109, Issue 44, 30 October 2012, Pages 17874-17879, 2012)
Euglenids exhibit an unconventional motility strategy amongst unicellular eukaryotes, consisting of large-amplitude highly concerted deformations of the entire body (euglenoid movement or metaboly). A plastic cell envelope called pellicle mediates these deformations. Unlike ciliary or flagellar motility, the biophysics of this mode is not well understood, including its efficiency and molecular machinery. We quantitatively examine video recordings of four euglenids executing such motions with statistical learning methods. This analysis reveals strokes of high uniformity in shape and pace. We then interpret the observations in the light of a theory for the pellicle kinematics, providing a precise understanding of the link between local actuation by pellicle shear and shape control. We systematically understand common observations, such as the helical conformations of the pellicle, and identify previously unnoticed features of metaboly. While two of our euglenids execute their stroke at constant body volume, the other two exhibit deviations of about 20% from their average volume, challenging current models of low Reynolds number locomotion. We find that the active pellicle shear deformations causing shape changes can reach 340%, and estimate the velocity of the molecular motors. Moreover, we find that metaboly accomplishes locomotion at hydrodynamic efficiencies comparable to those of ciliates and flagellates. Our results suggest new quantitative experiments, provide insight into the evolutionary history of euglenids, and suggest that the pellicle may serve as a model for engineered active surfaces with applications in microfluidics.
Keywords: microswimmers

A Review on The Sixth Painlevé Equation - Davide Guzzetti (SISSA, 2012)
For the Painlev\\\'e 6 transcendents, we provide a unitary description of the\r\ncritical behaviours, the connection formulae, their complete tabulation, and\r\nthe asymptotic distribution of the poles close to a critical point.
Keywords: Painlevé equation

Riemann–Hilbert approach to multi-time processes: The Airy and the Pearcey cases - Marco Bertola; Mattia Cafasso (Physica D: Nonlinear Phenomena, 2012)
Keywords: Integrable kernels

On robust Lie-algebraic stability conditions for switched linear systems - Andrei A. Agrachev; Yurij Baryshnikov; Daniel Liberzon (Systems and Control Letters. Volume 61, Issue 2, February 2012, Pages 347-353, 2012)
This paper presents new sufficient conditions for exponential stability of switched linear systems under arbitrary switching, which involve the commutators (Lie brackets) among the given matrices generating the switched system. The main novelty feature of these stability criteria is that, unlike their earlier counterparts, they are robust with respect to small perturbations of the system parameters.

SBV regularity for genuinely nonlinear, strictly hyperbolic systems of conservation laws in one space dimension - Stefano Bianchini; Laura Caravenna (Communications in Mathematical Physics 313 (2012) 1-33, 2012)

SBV regularity for Hamilton-Jacobi equations with Hamiltonian depending on (t,x) - Stefano Bianchini; Daniela Tonon (Siam Journal on Mathematical Analysis, 2012)

SBV regularity of genuinely nonlinear hyperbolic systems of conservation laws in one space dimension - Stefano Bianchini (Acta Mathematica Scientia, Volume 32, Issue 1, January 2012, Pages 380-388, 2012)
The problem of the presence of Cantor part in the derivative of a solution to a hyperbolic system of conservation laws is considered. An overview of the techniques involved in the proof is given, and a collection of related problems concludes the paper. Key words hyperbolic systems; conservation laws; SBV; regularity
Keywords: Hyperbolic systems

SBV-like regularity for general hyperbolic systems of conservation laws in one space dimension - Stefano Bianchini; Lei Yu (Rend. Istit. Mat. Univ. Trieste, 2012)

SBV-like regularity for Hamilton-Jacobi equations with a convex Hamiltonian - Stefano Bianchini; Daniela Tonon (Journal of Mathematical Analysis and Applications, 2012)

Second order approximations of quasistatic evolution problems in finite dimension - Virginia Agostiniani (Discrete & Continuous Dynamical Systems - A, 2012)
In this paper, we study the limit, as ε goes to zero, of a particular solution of the equation $ε^2A\ddot u^ε(t)+εB\dot u^ε(t)+\nabla_xf(t,u^ε(t))=0$, where $f(t,x)$ is a potential satisfying suitable coerciveness conditions. The limit $u(t)$ of $u^ε(t)$ is piece-wise continuous and verifies $\nabla_xf(t,u(t))=0$. Moreover, certain jump conditions characterize the behaviour of $u(t)$ at the discontinuity times. The same limit behaviour is obtained by considering a different approximation scheme based on time discretization and on the solutions of suitable autonomous systems.
Keywords: discrete approximations; perturbation methods; saddle-node bifurcation; Singular perturbations; vanishing viscosity

Self-propelled micro-swimmers in a Brinkman fluid - Marco Morandotti (Journal of Biological Dynamics, 2012)
We prove an existence, uniqueness, and regularity result for the motion of a self-propelled micro-swimmer in a particulate viscous medium, modelled as a Brinkman fluid. A suitable functional setting is introduced to solve the Brinkman system for the velocity field and the pressure of the fluid by variational techniques. The equations of motion are written by imposing a self-propulsion constraint, thus allowing the viscous forces and torques to be the only ones acting on the swimmer. From an infinite-dimensional control on the shape of the swimmer, a system of six ordinary differential equations for the spatial position and the orientation of the swimmer is obtained. This is dealt with standard techniques for ordinary differential equations, once the coefficients are proved to be measurable and bounded. The main result turns out to extend an analogous result previously obtained for the Stokes system.

Simulation-based uncertainty quantification of human arterial network hemodynamics - Peng Chen; Alfio Quarteroni; Gianluigi Rozza (International Journal Numerical Methods Biomedical Engineering, 2012)
This work aims at identifying and quantifying uncertainties from various sources in human cardiovascular\r\nsystem based on stochastic simulation of a one dimensional arterial network. A general analysis of\r\ndifferent uncertainties and probability characterization with log-normal distribution of these uncertainties\r\nis introduced. Deriving from a deterministic one dimensional fluid structure interaction model, we establish\r\nthe stochastic model as a coupled hyperbolic system incorporated with parametric uncertainties to describe\r\nthe blood flow and pressure wave propagation in the arterial network. By applying a stochastic collocation\r\nmethod with sparse grid technique, we study systemically the statistics and sensitivity of the solution with\r\nrespect to many different uncertainties in a relatively complete arterial network with potential physiological\r\nand pathological implications for the first time.
Keywords: uncertainty quantification, mathematical modelling of the cardiovascular system, fluid-structure interaction

Sobolev quasi-periodic solutions of multidimensional wave equations with a multiplicative potential - Massimiliano Berti; Philippe Bolle (Nonlinearity, 2012)
We prove the existence of quasi-periodic solutions for wave equations with a multiplicative potential on T d , d ≥ 1, and finitely differentiable nonlinearities, quasi-periodically forced in time. The only external parameter is the length of the frequency vector. The solutions have Sobolev regularity both in time and space. The proof is based on a Nash-Moser iterative scheme as in [5]. The key tame estimates for the inverse linearized operators are obtained by a multiscale inductive argument, which is more difficult than for NLS due to the dispersion relation of the wave equation. We prove the 'separation properties' of the small divisors assuming weaker non-resonance conditions than in [11]. © 2012 IOP Publishing Ltd.

Solving the Sixth Painlevé Equation: Towards the Classification of all the Critical Behaviors and the Connection Formulae - Davide Guzzetti (Int Math Res Notices (2012) 2012 (6): 1352-1413, 2012)
The critical behavior of a three real parameter class of solutions of the\\r\\nsixth Painlev\\\\\\\'e equation is computed, and parametrized in terms of monodromy\\r\\ndata of the associated $2\\\\times 2$ matrix linear Fuchsian system of ODE. The\\r\\nclass may contain solutions with poles accumulating at the critical point. The\\r\\nstudy of this class closes a gap in the description of the transcendents in one\\r\\nto one correspondence with the monodromy data. These transcendents are reviewed in the paper. Some formulas that relate the monodromy data to the critical behaviors of the four real (two complex) parameter class of solutions are\\r\\nmissing in the literature, so they are computed here. A computational procedure to write the full expansion of the four and three real parameter class of solutions is proposed.

Some applications of the SBV Regularity Theorem for entropy solutions of 1D scalar conservation laws to ConvectionTheory and sticky particles - Daniela Tonon (Riv. Mat. Univ. Parma, 2012)
We show how it is possible to apply the SBV Regularity Theorem for entropy solutions of one-dimensional scalar conservation laws, proved by Ambrosio and De Lellis, to Convection Theory and sticky particles. In the multi-dimensional case we present a counterexample which prevent us from using the same approach.

Some aspects of spinors – classical and noncommutative - Giacomo Dossena (SISSA, 2012)

Some remarks on quantum mechanics - Gianfausto Dell'Antonio (International Journal of Geometric Methods in Modern Physics, Volume 9, Issue 2, March 2012, Article number1260018, 2012)
We discuss the similarities and differences between the formalism of Hamiltonian Classical Mechanics and of Quantum Mechanics and exemplify the differences through an analysis of tracks in a cloud chamber.
Keywords: Quantum mechanics

Spectra of random Hermitian matrices with a small-rank external source: the critical and near-critical regimes - Marco Bertola; Buckingham, R.; Lee, S. Y.; Pierce, V. (J. Stat. Phys., 2012)

Stability for a System of N Fermions Plus a Different Particle with Zero-Range Interactions - Michele Correggi; Gianfausto Dell'Antonio; Domenico Finco; Alessandro Michelangeli; Alessandro Teta (Rev. Math. Phys. 24 (2012), 1250017, 2012)
We study the stability problem for a non-relativistic quantum system in\\r\\ndimension three composed by $ N \\\\geq 2 $ identical fermions, with unit mass,\\r\\ninteracting with a different particle, with mass $ m $, via a zero-range\\r\\ninteraction of strength $ \\\\alpha \\\\in \\\\R $. We construct the corresponding\\r\\nrenormalised quadratic (or energy) form $ \\\\form $ and the so-called\\r\\nSkornyakov-Ter-Martirosyan symmetric extension $ H_{\\\\alpha} $, which is the\\r\\nnatural candidate as Hamiltonian of the system. We find a value of the mass $\\r\\nm^*(N) $ such that for $ m > m^*(N)$ the form $ \\\\form $ is closed and bounded from below. As a consequence, $ \\\\form $ defines a unique self-adjoint and bounded from below extension of $ H_{\\\\alpha}$ and therefore the system is stable. On the other hand, we also show that the form $ \\\\form $ is unbounded from below for $ m < m^*(2)$. In analogy with the well-known bosonic case, this suggests that the system is unstable for $ m < m^*(2)$ and the so-called Thomas effect occurs.

A stable semi-lagrangian potential method for the simulation of ship interaction with unsteady and nonlinear waves - Andrea Mola; Luca Heltai; Antonio DeSimone (17th Int. Conf. Ships Shipp. Res., 2012)

Sub-Riemannian structures on 3D Lie groups - Andrei A. Agrachev; Davide Barilari (Journal of Dynamical and Control Systems. Volume 18, Issue 1, January 2012, Pages 21-44, 2012)
We give a complete classification of left-invariant sub-Riemannian structures on three dimensional Lie groups in terms of the basic differential invariants. As a corollary we explicitly find a sub-Riemannian isometry between the nonisomorphic Lie groups $SL(2)$ and $A^+(\mathbb{R})× S^1$, where $A^+(\mathbb{R})$ denotes the group of orientation preserving affine maps on the real line.

Systems of Quadratic Inequalities - Andrei A. Agrachev; Antonio Lerario (Proceedings of the London Mathematical Society, Volume 105, Issue 3, September 2012, Pages 622-660, 2012)
We present a spectral sequence which efficiently computes Betti numbers of a closed semi-algebraic subset of RP^n defined by a system of quadratic inequalities and the image of the homology homomorphism induced by the inclusion of this subset in RP^n. We do not restrict ourselves to the term E_2 of the spectral sequence and give a simple explicit formula for the differential d_2.

Tabulation of Painlevé 6 transcendents - Davide Guzzetti (Nonlinearity, Volume 25, Issue 12, December 2012, Pages 3235-3276, 2012)
The critical and asymptotic behaviors of solutions of the sixth Painlev'e equation PVI, obtained in the framework of the monodromy preserving deformation method, and their explicit parametrization in terms of monodromy data, are tabulated.

Thermodynamic phase transitions and shock singularities - Giuseppe De Nittis; Antonio Moro (Proc. R. Soc. A 8 March 2012 vol. 468 no. 2139 701-719, 2012)
We show that under rather general assumptions on the form of the entropy\\r\\nfunction, the energy balance equation for a system in thermodynamic equilibrium is equivalent to a set of nonlinear equations of hydrodynamic type. This set of equations is integrable via the method of the characteristics and it provides the equation of state for the gas. The shock wave catastrophe set identifies the phase transition. A family of explicitly solvable models of\\r\\nnon-hydrodynamic type such as the classical plasma and the ideal Bose gas are\\r\\nalso discussed.

Topological sensitivity analysis for high order elliptic operators - Samuel Amstutz; Antonio André Novotny; Nicolas Van Goethem (SISSA, 2012)
The topological derivative is defined as the first term of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of a singular domain perturbation. It has applications in many different fields such as shape and topology optimization, inverse problems, image processing and mechanical modeling including synthesis and/or optimal design of microstructures, fracture mechanics sensitivity analysis and damage evolution modeling. The topological derivative has been fully developed for a wide range of second order differential operators. In this paper we deal with the topological asymptotic expansion of a class of shape functionals associated with elliptic differential operators of order 2m, m>=1. The general structure of the polarization tensor is derived and the concept of degenerate polarization tensor is introduced. We provide full mathematical justifications for the derived formulas, including precise estimates of remainders.
Keywords: Topological derivative, Elliptic operators, Polarization tensor

Variational implementation of immersed finite element methods - Luca Heltai; Francesco Costanzo (Computer Methods in Applied Mechanics and Engineering. Volume 229-232, 1 July 2012, Pages 110-127, 2012)
Dirac-delta distributions are often crucial components of the solid-fluid coupling operators in immersed solution methods for fluid-structure interaction (FSI) problems. This is certainly so for methods like the Immersed Boundary Method (IBM) or the Immersed Finite Element Method (IFEM), where Dirac-delta distributions are approximated via smooth functions. By contrast, a truly variational formulation of immersed methods does not require the use of Dirac-delta distributions, either formally or practically. This has been shown in the Finite Element Immersed Boundary Method (FEIBM), where the variational structure of the problem is exploited to avoid Dirac-delta distributions at both the continuous and the discrete level. In this paper, we generalize the FEIBM to the case where an incompressible Newtonian fluid interacts with a general hyperelastic solid. Specifically, we allow (i) the mass density to be different in the solid and the fluid, (ii) the solid to be either viscoelastic of differential type or purely elastic, and (iii) the solid to be and either compressible or incompressible. At the continuous level, our variational formulation combines the natural stability estimates of the fluid and elasticity problems. In immersed methods, such stability estimates do not transfer to the discrete level automatically due to the non- matching nature of the finite dimensional spaces involved in the discretization. After presenting our general mathematical framework for the solution of FSI problems, we focus in detail on the construction of natural interpolation operators between the fluid and the solid discrete spaces, which guarantee semi-discrete stability estimates and strong consistency of our spatial discretization.
Keywords: Turbulent flow

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.

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.

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.

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.

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.\\\"
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.

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.

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.

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.

Generalized Ricci Curvature Bounds for Three Dimensional Contact Subriemannian manifolds - Andrei A. Agrachev; Paul Lee (SISSA, 2011)

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).

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.

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.

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.

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.

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.

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