Strong asymptotics of the orthogonal polynomials with respect to a measure supported on the plane - Ferenc Balogh; Marco Bertola; Lee, Seung-Yeop; Kenneth McLaughlin (Comm. Pure Appl. Math., 2015)

A study of snake-like locomotion through the analysis of a flexible robot model - Giancarlo Cicconofri; Antonio DeSimone (Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2015)
We examine the problem of snake-like locomotion by studying a system consisting of a planar inextensible elastic rod with adjustable spontaneous curvature, which provides an internal actuation mechanism that mimics muscular action in a snake. Using a Cosserat model, we derive the equations of motion in two special cases: one in which the rod can only move along a prescribed curve, and one in which the rod is constrained to slide longitudinally without slipping laterally, but the path is not fixed a priori (free-path case). The second setting is inspired by undulatory locomotion of snakes on flat surfaces. The presence of constraints leads in both cases to non-standard boundary conditions that allow us to close and solve the equations of motion. The kinematics and dynamics of the system can be recovered from a one-dimensional equation, without any restrictive assumption on the followed trajectory or the actuation. We derive explicit formulae highlighting the role of spontaneous curvature in providing the driving force (and the steering, in the free-path case) needed for locomotion. We also provide analytical solutions for a special class of serpentine motions, which enable us to discuss the connection between observed trajectories, internal actuation and forces exchanged with the environment.

Supremizer stabilization of POD-Galerkin approximation of parametrized Navier-Stokes equations - F. Ballarin; Andrea Manzoni; Alfio Quarteroni; Gianluigi Rozza (Wiley, 2015)
In this work, we present a stable proper orthogonal decomposition–Galerkin approximation for parametrized steady incompressible Navier–Stokes equations with low Reynolds number.

Symmetry and localization in periodic crystals: triviality of Bloch bundles with a fermionic time-reversal symmetry - Domenico Monaco; Gianluca Panati (Acta Applicandae Mathematicae, vol. 137, Issue 1, 2015, pages: 185-203, 2015)
We describe some applications of group- and bundle-theoretic methods in solid state physics, showing how symmetries lead to a proof of the localization of electrons in gapped crystalline solids, as e.g. insulators and semiconductors. We shortly review the Bloch-Floquet decomposition of periodic operators, and the related concepts of Bloch frames and composite Wannier functions. We show that the latter are almost-exponentially localized if and only if there exists a smooth periodic Bloch frame, and that the obstruction to the latter condition is the triviality of a Hermitian vector bundle, called the Bloch bundle. The rôle of additional Z_2-symmetries, as time-reversal and space-reflection symmetry, is discussed, showing how time-reversal symmetry implies the triviality of the Bloch bundle, both in the bosonic and in the fermionic case. Moreover, the same Z_2-symmetry allows to define a finer notion of isomorphism and, consequently, to define new topological invariants, which agree with the indices introduced by Fu, Kane and Mele in the context of topological insulators.

Three-sphere low-Reynolds-number swimmer with a passive elastic arm - Alessandro Montino; Antonio DeSimone (Springer, 2015)
One of the simplest model swimmers at low Reynolds number is the three-sphere swimmer by Najafi and Golestanian. It consists of three spheres connected by two rods which change their lengths periodically in non-reciprocal fashion. Here we investigate a variant of this model in which one rod is periodically actuated while the other is replaced by an elastic spring. We show that the competition between the elastic restoring force and the hydrodynamic drag produces a delay in the response of the passive elastic arm with respect to the active one. This leads to non-reciprocal shape changes and self-propulsion. After formulating the equations of motion, we study their solutions qualitatively and numerically. The leading-order term of the solution is computed analytically. We then address questions of optimization with respect to both actuation frequency and swimmer's geometry. Our results can provide valuable conceptual guidance in the engineering of robotic microswimmers.

A topological join construction and the Toda system on compact surfaces of arbitrary genus - Aleks Jevnikar; Kallel, Sadok; Andrea Malchiodi (Analysis & PDE, 2015)

Translation and adaptation of Birman's paper "On the theory of self-adjoint extensions of positive definite operators" (1956) - Mikhail Khotyakov; Alessandro Michelangeli (SISSA, 2015)
This is an accurate translation from Russian and adaptation to the modern mathematical jargon of a classical paper by M. Sh. Birman published in 1956, which is still today central in the theory of self-adjoint extensions of semi-bounded operators, and for which yet no English version was available so far.

Universality Conjecture and Results for a Model of Several Coupled Positive-Definite Matrices - Marco Bertola; Thomas Bothner (Commun. Math. Phys., 2015)

Variational aspects of Liouville equations and systems - Aleks Jevnikar (SISSA, 2015)
Keywords: Toda system

Variational aspects of singular Liouville systems - Luca Battaglia (SISSA, 2015)
I studied singular Liouville systems on compact surfaces from a variational point of view. I gave sufficient and necessary conditions for the existence of globally minimizing solutions, then I found min-max solutions for some particular systems. Finally, I also gave some non-existence results.
Keywords: Variational methods, Liouville systems, Moser-Trudinger inequalities, min-max methods

Volume variation and heat kernel for affine control problems - Elisa Paoli (SISSA, 2015)
In this thesis we study two main problems. The first one is the small-time heat kernel expansion on the diagonal for second order hypoelliptic opeartors. We consider operators that can depend on a drift field and that satisfy only the weak Hörmander condition. In a first work we use perturbation techniques to determine the exact order of decay of the heat kernel, that depends on the Lie algebra generated by the fields involved in the hypoelliptic operator. We generalize in particular some results already obtained in the sub-Riemannian setting. In a second work we consider a model class of hypoelliptic operators and we characterize geometrically all the coefficients in the on-the diagonal asymptotics at the equilibrium points of the drift field. The class of operators that we consider contains the linear hypoelliptic operators with constant second order part on the Euclidean space. We describe the coefficients in terms only of the divergence of the drift field and of curvature-like invariants, related to the minimal cost of geodesics of the associated optimal control problem. In the second part of the thesis we consider the variation of a smooth volume along a geodesic. The structure of the manifold is induced by a quadratic Hamiltonian and the geodesic in described as the projection of the Hamiltonian flow. We find an expansion similar to the classical Riemannian one. It depends on the curvature operator associated to the Hamiltonian, on the symbol of the geodesic and on a new metric-measure invariant determined by the symbol of the geodesic and by the given volume.
Keywords: Heat kernel asymptotics

The wave equation on domains with cracks growing on a prescribed path: existence, uniqueness, and continuous dependence on the data - Gianni Dal Maso; Ilaria Lucardesi (2015)
Given a bounded open set $Ω \subset \mathbb R^d$ with Lipschitz boundary and an increasing family $Γ_t$, $t\in [0,T]$, of closed subsets of $Ω$, we analyze the scalar wave equation $\ddot{u} - div (A \nabla u) = f$ in the time varying cracked domains $Ω\setminusΓ_t$. Here we assume that the sets $Γ_t$ are contained into a prescribed $(d-1)$-manifold of class $C^2$. Our approach relies on a change of variables: recasting the problem on the reference configuration $Ω\setminus Γ_0$, we are led to consider a hyperbolic problem of the form $\ddot{v} - div (B\nabla v) + a · \nabla v - 2 b · \nabla \dot{v} = g$ in $Ω \setminus Γ_0$. Under suitable assumptions on the regularity of the change of variables that transforms $Ω\setminus Γ_t$ into $Ω\setminus Γ_0$, we prove existence and uniqueness of weak solutions for both formulations. Moreover, we provide an energy equality, which gives, as a by-product, the continuous dependence of the solutions with respect to the cracks.

An Abstract Nash–Moser Theorem and Quasi-Periodic Solutions for NLW and NLS on Compact Lie Groups and Homogeneous Manifolds - Massimiliano Berti; Livia Corsi; Michela Procesi (Springer, 2014)
We prove an abstract implicit function theorem with parameters for smooth operators defined on scales of sequence spaces, modeled for the search of quasi-periodic solutions of PDEs. The tame estimates required for the inverse linearised operators at each step of the iterative scheme are deduced via a multiscale inductive argument. The Cantor-like set of parameters where the solution exists is defined in a non inductive way. This formulation completely decouples the iterative scheme from the measure theoretical analysis of the parameters where the small divisors non-resonance conditions are verified. As an application, we deduce the existence of quasi-periodic solutions for forced NLW and NLS equations on any compact Lie group or manifold which is homogeneous with respect to a compact Lie group, extending previous results valid only for tori. A basic tool of harmonic analysis is the highest weight theory for the irreducible representations of compact Lie groups.

Achieving unanimous opinions in signed social networks - Claudio Altafini; Gabriele Lini (Institute of Electrical and Electronics Engineers Inc., 2014)
Being able to predict the outcome of an opinion forming process is an important problem in social network theory. However, even for linear dynamics, this becomes a difficult task as soon as non-cooperative interactions are taken into account. Such interactions are naturally modeled as negative weights on the adjacency matrix of the social network. In this paper we show how the Perron-Frobenius theorem can be used for this task also beyond its standard formulation for cooperative systems. In particular we show how it is possible to associate the achievement of unanimous opinions with the existence of invariant cones properly contained in the positive orthant. These cases correspond to signed adjacency matrices having the eventual positivity property, i.e., such that in sufficiently high powers all negative entries have disappeared. More generally, we show how for social networks the achievement of a, possibily non-unanimous, opinion can be associated to the existence of an invariant cone fully contained in one of the orthants of n.

Adaptive discontinuous Galerkin methods for nonstationary convection-diffusion problems - Andrea Cangiani; E.H. Georgoulis; Metcalfe, Stephen (IMA J. Numer. Anal., 2014)

Adler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras - Alberto De Sole; Victor G. Kac; Daniele Valeri (SISSA, 2014)
We put the Adler-Gelfand-Dickey approach to classical W-algebras in the framework of Poisson vertex algebras. We show how to recover the bi-Poisson structure of the KP hierarchy, together with its generalizations and reduction to the N-th KdV hierarchy, using the formal distribution calculus and the lambda-bracket formalism. We apply the Lenard-Magri scheme to prove integrability of the corresponding hierarchies. We also give a simple proof of a theorem of Kupershmidt and Wilson in this framework. Based on this approach, we generalize all these results to the matrix case. In particular, we find (non-local) bi-Poisson structures of the matrix KP and the matrix N-th KdV hierarchies, and we prove integrability of the N-th matrix KdV hierarchy.

Approximate Hermitian–Yang–Mills structures on semistable principal Higgs bundles - Ugo Bruzzo; Beatriz Grana-Otero (Springer, 2014)
We generalize the Hitchin-Kobayashi correspondence between semistability and the existence of approximate Hermitian-Yang-Mills structures to the case of principal Higgs bundles. We prove that a principal Higgs bundle on a compact Kaehler manifold, with structure group a connected linear algebraic reductive group, is semistable if and only if it admits an approximate Hermitian-Yang-Mills structure.

Approximate Hitchin-Kobayashi correspondence for Higgs G-bundles - Ugo Bruzzo; Beatriz Graña Otero (World Scientific Publishing, 2014)
We announce a result about the extension of the Hitchin-Kobayashi correspondence to principal Higgs bundles. A principal Higgs bundle on a compact Kähler manifold, with structure group a connected linear algebraic reductive group, is semistable if and only if it admits an approximate Hermitian-Yang-Mills structure.

Buckling dynamics of a solvent-stimulated stretched elastomeric sheet - Alessandro Lucantonio; Matthieu Roché; Paola Nardinocchi; Howard A. Stone (Royal Society of Chemistry, 2014)
When stretched uniaxially, a thin elastic sheet may exhibit buckling. The occurrence of buckling depends on the geometrical properties of the sheet and the magnitude of the applied strain. Here we show that an elastomeric sheet initially stable under uniaxial stretching can destabilize when exposed to a solvent that swells the elastomer. We demonstrate experimentally and computationally that the features of the buckling pattern depend on the magnitude of stretching, and this observation offers a new way for controlling the shape of a swollen homogeneous thin sheet.

Cauchy-Laguerre two-matrix model and the Meijer-G random point field - Marco Bertola; Gekhtman, M.; Szmigielski, J. (Comm. Math. Phys., 2014)

Classical W-algebras and generalized Drinfeld-Sokolov hierarchies for minimal and short nilpotents - Alberto De Sole; Victor G. Kac; Daniele Valeri (Communications in Mathematical Physics 331, nr. 2 (2014) 623-676, 2014)
We derive explicit formulas for lambda-brackets of the affine classical W-algebras attached to the minimal and short nilpotent elements of any simple Lie algebra g. This is used to compute explicitly the first non-trivial PDE of the corresponding intgerable generalized Drinfeld-Sokolov hierarchies. It turns out that a reduction of the equation corresponding to a short nilpotent is Svinolupov's equation attached to a simple Jordan algebra, while a reduction of the equation corresponding to a minimal nilpotent is an integrable Hamiltonian equation on 2h-3 functions, where h is the dual Coxeter number of g. In the case when g is sl_2 both these equations coincide with the KdV equation. In the case when g is not of type C_n, we associate to the minimal nilpotent element of g yet another generalized Drinfeld-Sokolov hierarchy.

Comparison between reduced basis and stochastic collocation methods for elliptic problems - Peng Chen; Alfio Quarteroni; Gianluigi Rozza (Springer, 2014)
The stochastic collocation method (Babuška et al. in SIAM J Numer Anal 45(3):1005-1034, 2007; Nobile et al. in SIAM J Numer Anal 46(5):2411-2442, 2008a; SIAM J Numer Anal 46(5):2309-2345, 2008b; Xiu and Hesthaven in SIAM J Sci Comput 27(3):1118-1139, 2005) has recently been applied to stochastic problems that can be transformed into parametric systems. Meanwhile, the reduced basis method (Maday et al. in Comptes Rendus Mathematique 335(3):289-294, 2002; Patera and Rozza in Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations Version 1.0. Copyright MIT, http://augustine.mit.edu, 2007; Rozza et al. in Arch Comput Methods Eng 15(3):229-275, 2008), primarily developed for solving parametric systems, has been recently used to deal with stochastic problems (Boyaval et al. in Comput Methods Appl Mech Eng 198(41-44):3187-3206, 2009; Arch Comput Methods Eng 17:435-454, 2010). In this work, we aim at comparing the performance of the two methods when applied to the solution of linear stochastic elliptic problems. Two important comparison criteria are considered: (1), convergence results of the approximation error; (2), computational costs for both offline construction and online evaluation. Numerical experiments are performed for problems from low dimensions O (1) to moderate dimensions O (10) and to high dimensions O (100). The main result stemming from our comparison is that the reduced basis method converges better in theory and faster in practice than the stochastic collocation method for smooth problems, and is more suitable for large scale and high dimensional stochastic problems when considering computational costs.

Comparison of a Modal Method and a Proper Orthogonal Decomposition approach for multi-group time-dependent reactor spatial kinetics - Alberto Sartori; Davide Baroli; Antonio Cammi; Davide Chiesa; Lelio Luzzi; Roberto R. Ponciroli; Ezio Previtali; Marco E. Ricotti; Gianluigi Rozza; Monica Sisti (Annals of Nuclear Energy, 2014)
In this paper, two modelling approaches based on a Modal Method (MM) and on the Proper Orthogonal Decomposition (POD) technique, for developing a control-oriented model of nuclear reactor spatial kinetics, are presented and compared. Both these methods allow developing neutronics description by means of a set of ordinary differential equations. The comparison of the outcomes provided by the two approaches focuses on the capability of evaluating the reactivity and the neutron flux shape in different reactor configurations, with reference to a TRIGA Mark II reactor. The results given by the POD-based approach are higher-fidelity with respect to the reference solution than those computed according to the MM-based approach, in particular when the perturbation concerns a reduced region of the core. If the perturbation is homogeneous throughout the core, the two approaches allow obtaining comparable accuracy results on the quantities of interest. As far as the computational burden is concerned, the POD approach ensures a better efficiency rather than direct Modal Method, thanks to the ability of performing a longer computation in the preprocessing that leads to a faster evaluation during the on-line phase.

Conformal invariants from nodal sets. I. negative eigenvalues and curvature prescription - Rod R. Gover; Yaiza Canzani; Dmitry Jakobson; Raphaël Ponge; Andrea Malchiodi (Oxford University Press, 2014)
In this paper, we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators; more specifically, the Graham, Jenne, Mason, and Sparling (GJMS) operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a version in conformal geometry of Courant's Nodal Domain Theorem. We also show that on any manifold of dimension n≥3, there exist many metrics for which our invariants are nontrivial. We prove that the Yamabe operator can have an arbitrarily large number of negative eigenvalues on any manifold of dimension n≥3. We obtain similar results for some higher order GJMS operators on some Einstein and Heisenberg manifolds. We describe the invariants arising from the Yamabe and Paneitz operators associated to left-invariant metrics on Heisenberg manifolds. Finally, in Appendix, the second named author and Andrea Malchiodi study the Q-curvature prescription problems for noncritical Q-curvatures.

On conjugate times of LQ optimal control problems - Andrei A. Agrachev; Luca Rizzi; Pavel Silveira (Springer, 2014)
Motivated by the study of linear quadratic optimal control problems, we consider a dynamical system with a constant, quadratic Hamiltonian, and we characterize the number of conjugate times in terms of the spectrum of the Hamiltonian vector field $\vec{H}$. We prove the following dichotomy: the number of conjugate times is identically zero or grows to infinity. The latter case occurs if and only if $\vec{H}$ has at least one Jordan block of odd dimension corresponding to a purely imaginary eigenvalue. As a byproduct, we obtain bounds from below on the number of conjugate times contained in an interval in terms of the spectrum of $\vec{H}$.
Keywords: Optimal control, Lagrange Grassmannian, Conjugate point

A correction and an extension of Stampacchia's work on the geometric BVP - Giovanni Vidossich (Advanced Nonlinear Studies, 2014)
G. Stampacchia introduced the geometric boundary value problem for ODEs in his doctoral thesis and published four papers related to it. Here we point out that the proof of his last theorem on the subject is incorrect and we provide a substitute for it as well as a generalizations of some of his earlier results.

Coupled dynamic simulations of offshore wind turbines: influence of wave modelling on the fatigue load assesment - Marino, Enzo; Lugni, Claudio; Giovanni Stabile; Claudio Borri; Manuel, Lance (XIII Conference of the Italian Association for Wind Engineering (In-Vento 2014), 2014)

Coupled dynamic simulations of offshore wind turbines using linear, weakly and fully nonlinear wave models: the limitations of the second-order wave theory - Marino, Enzo; Lugni, Claudio; Giovanni Stabile; Claudio Borri (9th International Conference on Structural Dynamics (EURODYN 2014), 2014)

Crawling on directional surfaces - Paolo Gidoni; Giovanni Noselli; Antonio DeSimone (International Journal of Non-Linear Mechanics, 2014)
In this paper we study crawling locomotion based on directional frictional interactions, namely, frictional forces that are sensitive to the sign of the sliding velocity. Surface interactions of this type are common in biology, where they arise from the presence of inclined hairs or scales at the crawler/substrate interface, leading to low resistance when sliding ‘along the grain’, and high resistance when sliding ‘against the grain’. This asymmetry can be exploited for locomotion, in a way analogous to what is done in cross-country skiing (classic style, diagonal stride). We focus on a model system, namely, a continuous one-dimensional crawler and provide a detailed study of the motion resulting from several strategies of shape change. In particular, we provide explicit formulae for the displacements attainable with reciprocal extensions and contractions (breathing), or through the propagation of extension or contraction waves. We believe that our results will prove particularly helpful for the study of biological crawling motility and for the design of bio-mimetic crawling robots.
Keywords: Bio-mimetic micro-robots; Cell migration; Crawling motility; Directional surfaces; Self-propulsion

Critical points of the Moser-Trudinger functional on a disk - Andrea Malchiodi; Luca Martinazzi (European Mathematical Society, 2014)
On the 2-dimensional unit disk $B_1$ we study the Moser-Trudinger functional $$E(u)=\int_{B_1}(e^{u^2}-1)dx, u\in H^1_0(B_1)$$ and its restrictions to $M_Λ:=\{u \in H^1_0(B_1):\|u\|^2_{H^1_0}=Λ\}$ for $Λ>0$. We prove that if a sequence $u_k$ of positive critical points of $E|_{M_{Λ_k}}$ (for some $Λ_k>0$) blows up as $k→∞$, then $Λ_k→ 4π$, and $u_k→ 0$ weakly in $H^1_0(B_1)$ and strongly in $C^1_{\loc}(\bar B_1\setminus\{0\})$. Using this we also prove that when $Λ$ is large enough, then $E|_{M_Λ}$ has no positive critical point, complementing previous existence results by Carleson-Chang, M. Struwe and Lamm-Robert-Struwe.

The curvature of optimal control problems with applications to sub-Riemannian geometry - Luca Rizzi (SISSA, 2014)
Optimal control theory is an extension of the calculus of variations, and deals with the optimal behaviour of a system under a very general class of constraints. This field has been pioneered by the group of mathematicians led by Lev Pontryagin in the second half of the 50s and nowadays has countless applications to the real worlds (robotics, trains, aerospace, models for human behaviour, human vision, image reconstruction, quantum control, motion of self-propulsed micro-organism). In this thesis we introduce a novel definition of curvature for an optimal control problem. In particular it works for any sub-Riemannian and sub-Finsler structure. Related problems, such as comparison theorems for sub-Riemannian manifolds, LQ optimal control problem and Popp's volume and are also investigated.
Keywords: Sub-Riemannian geometry

Curvature-adapted remeshing of CAD surfaces - Franco Dassi; Andrea Mola; Hang Si (Procedia Engineering, 2014)
A common representation of surfaces with complicated topology and geometry is through composite parametric surfaces as is the case for most CAD modelers. A challenging problem is how to generate a mesh of such a surface that well approximates the geometry of the surface, preserves its topology and important geometric features, and contains nicely shaped elements. In this work, we present an optimization-based surface remeshing method that is able to satisfy many of these requirements simultaneously. This method is inspired by the recent work of Lévy and Bonneel (Proc. 21th International Meshing Roundtable, October 2012), which embeds a smooth surface into a high-dimensional space and remesh it uniformly in that embedding space. Our method works directly in the 3d spaces and uses an embedding space in R6 to evaluate mesh size and mesh quality. It generates a curvatureadapted anisotropic surface mesh that well represents the geometry of the surface with a low number of elements. We illustrate our approach through various examples.

Darboux Transformations and Random Point Processes - Marco Bertola; Mattia Cafasso (IMRN, 2014)

The decomposition of optimal transportation problems with convex cost - Mauro Bardelloni (SISSA, 2014)
Keywords: Optimal Transportation

The decomposition of optimal transportation problems with convex cost - Stefano Bianchini; Mauro Bardelloni (SISSA, 2014)

A density result for GSBD and its application to the approximation of brittle fracture energies - Flaviana Iurlano (Springer, 2014)
We present an approximation result for functions u: Ω → ℝ^n belonging to the space GSBD(Ω) ∩ L2(Ω, ℝn) with e(u) square integrable and Hn-1(Ju) finite. The approximating functions uk are piecewise continuous functions such that uk → u in (Formula Presented). As an application, we provide the extension to the vector-valued case of the Γ-convergence result in GSBV(Ω) proved by Ambrosio and Tortorelli (Commun Pure Appl Math 43:999-1036, 1990; Boll. Un. Mat. Ital. B (7) 6:105-123, 1992).

Dirac operators on noncommutative principal circle bundles - Andrzej Sitarz; Alessandro Zucca; Ludwik Dabrowski (World Scientific Publishing, 2014)
We study spectral triples over noncommutative principal U(1)-bundles of arbitrary dimension and a compatibility condition between the connection and the Dirac operator on the total space and on the base space of the bundle. Examples of low-dimensional noncommutative tori are analyzed in more detail and all connections found that are compatible with an admissible Dirac operator. Conversely, a family of new Dirac operators on the noncommutative tori, which arise from the base-space Dirac operator and a suitable connection is exhibited. These examples are extended to the theta-deformed principal U(1)-bundle S 3 θ → S2.

Dirac reduction for Poisson vertex algebras - Alberto De Sole; Victor G. Kac; Daniele Valeri (Communications in Mathematical Physics 331, nr. 3 (2014) 1155-1190, 2014)
We construct an analogue of Dirac's reduction for an arbitrary local or non-local Poisson bracket in the general setup of non-local Poisson vertex algebras. This leads to Dirac's reduction of an arbitrary non-local Poisson structure. We apply this construction to an example of a generalized Drinfeld-Sokolov hierarchy.

Discrete one-dimensional crawlers on viscous substrates: achievable net displacements and their energy cost - Giovanni Noselli; Amabile Tatone; Antonio DeSimone (Elsevier, 2014)
We study model one-dimensional crawlers, namely, model mechanical systems that can achieve self-propulsion by controlled shape changes of their body (extension or contraction of portions of the body), thanks to frictional interactions with a rigid substrate. We evaluate the achievable net displacement and the related energetic cost for self-propulsion by discrete crawlers (i.e., whose body is made of a discrete number of contractile or extensile segments) moving on substrates with either a Newtonian (linear) or a Bingham-type (stick-slip) rheology. Our analysis is aimed at constructing the basic building blocks towards an integrative, multi-scale description of crawling cell motility.
Keywords: Cell migration

Dislocations at the continuum scale: functional setting and variational properties - Riccardo Scala; Nicolas Van Goethem (2014)

Donagi–Markman cubic for the generalised Hitchin system - Ugo Bruzzo; Peter Dalakov (2014)
Donagi and Markman (1993) have shown that the infinitesimal period map for an algebraic completely integrable Hamiltonian system (ACIHS) is encoded in a section of the third symmetric power of the cotangent bundle to the base of the system. For the ordinary Hitchin system the cubic is given by a formula of Balduzzi and Pantev. We show that the Balduzzi–Pantev formula holds on maximal rank symplectic leaves of the G-generalised Hitchin system.
Keywords: Generalized Hitchin system, Donagi-Markman cubic, algebraically completely integrable systems, moduli space of Higgs G-bundles

Dynamics on a graph as the limit of the dynamics on a "fat graph" - Gianfausto Dell'Antonio; Alessandro Michelangeli (SISSA, 2014)
We discuss how the vertex boundary conditions for the dynamics of a quantum particle constrained on a graph emerge in the limit of the dynamics of a particle in a tubular region around the graph (\fat graph") when the transversal section of this region shrinks to zero. We give evidence of the fact that if the limit dynamics exists and is induced by the Laplacian on the graph with certain self-adjoint boundary conditions, such conditions are determined by the possible presence of a zero energy resonance on the fat graph. Pictorially, one may say that in the shrinking limit the resonance acts as a bridge connecting the boundary values at the vertex along the different rays.

Editorial - Ciro Ciliberto; Gianni Dal Maso; Pasquale Vetro (Springer, 2014)

An effective model for nematic liquid crystal composites with ferromagnetic inclusions - Maria Carme Calderer; Antonio DeSimone; Dmitry Golovaty; Alexander Panchenko (Society for Industrial and Applied Mathematics Publications, 2014)
Molecules of a nematic liquid crystal respond to an applied magnetic field by reorienting themselves in the direction of the field. Since the dielectric anisotropy of a nematic is small, it takes relatively large fields to elicit a significant liquid crystal response. The interaction may be enhanced in colloidal suspensions of ferromagnetic particles in a liquid crystalline matrix- ferronematics-as proposed by Brochard and de Gennes in 1970. The ability of these particles to align with the field and simultaneously cause reorientation of the nematic molecules greatly increases the magnetic response of the mixture. Essentially the particles provide an easy axis of magnetization that interacts with the liquid crystal via surface anchoring. We derive an expression for the effective energy of ferronematic in the dilute limit, that is, when the number of particles tends to infinity while their total volume fraction tends to zero. The total energy of the mixture is assumed to be the sum of the bulk elastic liquid crystal contribution, the anchoring energy of the liquid crystal on the surfaces of the particles, and the magnetic energy of interaction between the particles and the applied magnetic field. The homogenized limiting ferronematic energy is obtained rigorously using a variational approach. It generalizes formal expressions previously reported in the physical literature.

An efficient computational framework for reduced basis approximation and a posteriori error estimation of parametrized Navier-Stokes flows - Andrea Manzoni (2014)
We present the current Reduced Basis framework for the efficient numerical approximation of parametrized steady Navier-Stokes equations. We have extended the existing setting developed in the last decade (see e.g. [Deparis, Veroy & Patera, Quarteroni & Rozza] to more general affine and nonaffine parametrizations (such as volume-based techniques), to a simultaneous velocity-pressure error estimates and to a fully decoupled Offline/Online procedure in order to speedup the solution of the reduced-order problem. This is particularly suitable for real-time and many-query contexts, which are both part of our final goal. Furthermore, we present an efficient numerical implementation for treating nonlinear advection terms in a convenient way. A residual-based a posteriori error estimation with respect to a truth, full-order Finite Element approximation is provided for joint pressure/velocity errors, according to the Brezzi-Rappaz-Raviart stability theory. To do this, we take advantage of an extension of the Successive Constraint Method for the estimation of stability factors and of a suitable fixed-point algorithm for the approximation of Sobolev embedding constants. Finally, we present some numerical test cases, in order to show both the approximation properties and the computational efficiency of the derived framework.
Keywords: Reduced Basis Method, parametrized Navier-Stokes equations, steady incompressible fluids, a posteriori error estimation, approximation stability

Efficient geometrical parametrisation techniques of interfaces for reduced-order modelling: application to fluid–structure interaction coupling problems - Forti, D.; Gianluigi Rozza (International Journal of Computational Fluid Dynamics, 2014)
We present some recent advances and improvements in shape parametrisation techniques of interfaces for reduced-order modelling with special attention to fluid–structure interaction problems and the management of structural deformations, namely, to represent them into a low-dimensional space (by control points). This allows to reduce the computational effort, and to significantly simplify the (geometrical) deformation procedure, leading to more efficient and fast reduced-order modelling applications in this kind of problems. We propose an efficient methodology to select the geometrical control points for the radial basis functions based on a modal greedy algorithm to improve the computational efficiency in view of more complex fluid–structure applications in several fields. The examples provided deal with aeronautics and wind engineering.

Existence and uniqueness of the gradient flow of the Entropy in the space of probability measures - Stefano Bianchini; Alexander Dabrowski (EUT Edizioni Universita di Trieste, 2014)
After a brief introduction on gradient flows in metric spaces and on geodesically convex functionals, we give an account of the proof (following the outline of [3, 7]) of the existence and uniqueness of the gradient flow of the Entropy in the space of Borel probability measures over a compact geodesic metric space with Ricci curvature bounded from below.

Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds - Kuwert, Ernst; Andrea Mondino; Johannes Schygulla (Mathematische Annalen, 2014)
We study curvature functionals for immersed 2-spheres in a compact, three-dimensional Riemannian manifold $M$. Under the assumption that the sectional curvature $K^M$ is strictly positive, we prove the existence of a smooth immersion $f:{\mathbb{S}}^2 → M$ minimizing the $L^2$ integral of the second fundamental form. Assuming instead that $K^M ≤ 2 $ and that there is some point $\bar{x}\in M$ with scalar curvature $R^M(\bar{x})>6$, we obtain a smooth minimizer $f:{\mathbb{S}}^2 → M$ for the functional $\int \frac{1}{4}|H|^2+1$, where $H$ is the mean curvature.

Existence of immersed spheres minimizing curvature functionals in non-compact 3-manifolds - Andrea Mondino; Johannes Schygulla (Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 2014)
We study curvature functionals for immersed 2-spheres in non-compact, three-dimensional Riemannian manifold $(M,h)$ without boundary. First, under the assumption that $(M,h)$ is the euclidean 3-space endowed with a semi-perturbed metric with perturbation small in $C^1$ norm and of compact support, we prove that if there is some point $\bar{x}\in M$ with scalar curvature $R^M(\bar{x})>0$ then there exists a smooth embedding $ f:\mathbb{S}^2 <span class="kwd" style="font-family: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; white-space: inherit; background-color: #eff0f1; font-size: 13px; margin: 0px; padding: 0px; border: 0px; font-stretch: inherit; line-height: inherit; vertical-align: baseline; box-sizing: inherit; color: #8a4a0b;">\hookrightarrow M$ minimizing the Willmore functional $\frac{1}{4}\int |H|^2$, where $H$ is the mean curvature. Second, assuming that $(M,h)$ is of bounded geometry (i.e. bounded sectional curvature and strictly positive injectivity radius) and asymptotically euclidean or hyperbolic we prove that if there is some point $\bar{x}\in M$ with scalar curvature $R^M(\bar{x})>6$ then there exists a smooth immersion $f:\mathbb{S}^2<span class="kwd" style="font-family: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; white-space: inherit; background-color: #eff0f1; font-size: 13px; margin: 0px; padding: 0px; border: 0px; font-stretch: inherit; line-height: inherit; vertical-align: baseline; box-sizing: inherit; color: #8a4a0b;">\hookrightarrow M$ minimizing the functional $\int (\frac{1}{2}|A|^2+1)$, where $A$ is the second fundamental form. Finally, adding the bound $K^M \leq 2$ to the last assumptions, we obtain a smooth minimizer $f:\mathbb{S}^2 <span class="kwd" style="font-family: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; white-space: inherit; background-color: #eff0f1; font-size: 13px; margin: 0px; padding: 0px; border: 0px; font-stretch: inherit; line-height: inherit; vertical-align: baseline; box-sizing: inherit; color: #8a4a0b;">\hookrightarrow M$ for the functional $\int \frac{1}{4}(|H|^2+1)$. The assumptions of the last two theorems are satisfied in a large class of 3-manifolds arising as spacelike timeslices solutions of the Einstein vacuum equation in case of null or negative cosmological constant.
Keywords: Direct methods in the calculus of variations; General Relativity; Geometric measure theory; second fundamental form; Willmore functional

Existence of integral m-varifolds minimizing $\int |A|^p $ and $\int |H|^p$ , p>m, in Riemannian manifolds - Andrea Mondino (Calculus of Variations and Partial Differential Equations, 2014)
We prove existence of integral rectifiable $m$-dimensional varifolds minimizing functionals of the type $\int |H|^p$ and $\int |A|^p$ in a given Riemannian $n$-dimensional manifold $(N,g)$, $2 ≤ m<n$ and $p>m$ under suitable assumptions on $N$ (in the end of the paper we give many examples of such ambient manifolds). To this aim we introduce the following new tools: some monotonicity formulas for varifolds in ${\mathbb{R }^S}$ involving $\int |H|^p$to avoid degeneracy of the minimizer, and a sort of isoperimetric inequality to bound the mass in terms of the mentioned functionals.

Finite dimensional Kadomtsev-Petviashvili τ-functions. I. Finite Grassmannians - Ferenc Balogh; Tiago Fonseca; John P. Harnad (American Institute of Physics Inc., 2014)
We study τ-functions of the Kadomtsev-Petviashvili hierarchy in terms of abelian group actions on finite dimensional Grassmannians, viewed as subquotients of the Hilbert space Grassmannians of Sato, Segal, and Wilson. A determinantal formula of Gekhtman and Kasman involving exponentials of finite dimensional matrices is shown to follow naturally from such reductions. All reduced flows of exponential type generated by matrices with arbitrary nondegenerate Jordan forms are derived, both in the Grassmannian setting and within the fermionic operator formalism. A slightly more general determinantal formula involving resolvents of the matrices generating the flow, valid on the big cell of the Grassmannian, is also derived. An explicit expression is deduced for the Plücker coordinates appearing as coefficients in the Schur function expansion of the τ-function.

A fully nonlinear potential model for ship hydrodynamics directly interfaced with CAD data structures - Andrea Mola; Luca Heltai; Antonio DeSimone (Proceedings of the 24th International Ocean and Polar Engineering Conference, Busan, 2014, 2014)
We present a model for ship hydrodynamics simulations currently under development at SISSA. The model employs potential flow theory and fully nonlinear free surface boundary conditions. The spatial discretization of the equations is performed by means of a collocation BEM. This gives rise to a Differential Algbraic Equations (DAE) system, solved using an implicit BDF scheme to time advance the solution. The model has been implemented into a C++ software able to automatically generate the computational grids from the CAD geometry of the hull. Numerical results on Kriso KCS and KVLCC2 hulls are presented and discussed.
Keywords: ship hydrodynamics

Fundamentals of Reduced Basis Method for problems governed by parametrized PDEs and applications - Gianluigi Rozza (Separated representations and PGD-based model reduction : fundamentals and applications, 2014)
In this chapter we consider Reduced Basis (RB) approximations of parametrized Partial Differential Equations (PDEs). The the idea behind RB is to decouple the generation and projection stages (Offline/Online computational procedures) of the approximation process in order to solve parametrized PDEs in a fast, inexpensive and reliable way. The RB method, especially applied to 3D problems, allows great computational savings with respect to the classical Galerkin Finite Element (FE) Method. The standard FE method is typically ill suited to (i) iterative contexts like optimization, sensitivity analysis and many-queries in general, and (ii) real time evaluation. We consider for simplicity coercive PDEs. We discuss all the steps to set up a RB approximation, either from an analytical and a numerical point of view. Then we present an application of the RB method to a steady thermal conductivity problem in heat transfer with emphasis on geometrical and physical parameters.
Keywords: reduced basis method, linear elasticity, heat transfer, error bounds, parametrized PDEs

Geometry and analysis of control-affine systems: motion planning, heat and Schrödinger evolution - Dario Prandi (SISSA, 2014)
This thesis is dedicated to two problems arising from geometric control theory, regarding control-affine systems $\dot q= f_0(q)+\sum_{j=1}^m u_j f_j(q)$, where $f_0$ is called the drift. In the first part we extend the concept of complexity of non-admissible trajectories, well understood for sub-Riemannian systems, to this more general case, and find asymptotic estimates. In order to do this, we also prove a result in the same spirit as the Ball-Box theorem for sub-Riemannian systems, in the context of control-affine systems equipped with the L1 cost. Then, in the second part of the thesis, we consider a family of 2-dimensional driftless control systems. For these, we study how the set where the control vector fields become collinear affects the diffusion dynamics. More precisely, we study whether solutions to the heat and Schrödinger equations associated with this Laplace-Beltrami operator are able to cross this singularity, and how its the presence affects the spectral properties of the operator, in particular under a magnetic Aharonov–Bohm-type perturbation.
Keywords: control theory

Global Structure of Admissible BV Solutions to Piecewise Genuinely Nonlinear, Strictly Hyperbolic Conservation Laws in One Space Dimension - Stefano Bianchini; Lei Yu (Taylor & Francis, 2014)
The paper describes the qualitative structure of an admissible BV solution to a strictly hyperbolic system of conservation laws whose characteristic families are piecewise genuinely nonlinear. More precisely, we prove that there are a countable set of points Θ and a countable family of Lipschitz curves Tscript such that outside Tscript ∪ Θ the solution is continuous, and for all points in Tscriptset minusΘ the solution has left and right limit. This extends the corresponding structural result in [7] for genuinely nonlinear systems. An application of this result is the stability of the wave structure of solution w.r.t. -convergence. The proof is based on the introduction of subdiscontinuities of a shock, whose behavior is qualitatively analogous to the discontinuities of the solution to genuinely nonlinear systems.

Hölder equivalence of the value function for control-affine systems - Dario Prandi (ESAIM: Control, Optimisation and Calculus of Variations, 2014)

Holomorphically symplectic varieties with Prym Lagrangian fibrations - Tommaso Matteini (SISSA, 2014)
The thesis presents a construction of singular holomorphically symplectic varieties as Lagrangian fibrations. They are relative compactified Prym varieties associated to curves on symplectic surfaces with an antisymplectic involution. They are identified with the fixed locus of a symplectic involution on singular moduli spaces of sheaves of dimension 1. An explicit example, giving a singular irreducible symplectic 6-fold without symplectic resolutions, is described for a K3 surface which is the double cover of a cubic surface. In the case of abelian surfaces, a variation of this construction is studied to get irreducible symplectic varieties: relative compactified 0-Prym varieties. A partial classification result is obtained for involutions without fixed points: either the 0-Prym variety is birational to an irreducible symplectic variety of K3[n]-type, or it does not admit symplectic resolutions.
Keywords: Holomorphically symplectic varieties

Homogenization of functional with linear growth in the context of A-quasiconvexity - Jose Matias; Marco Morandotti; Pedro M. Santos (SISSA, 2014)
This work deals with the homogenization of functionals with linear growth in the context of A-quasiconvexity. A representation theorem is proved, where the new integrand function is obtained by solving a cell problem where the coupling between homogenization and the A-free condition plays a crucial role. This result extends some previous work to the linear case, thus allowing for concentration effects.

Homology computation for a class of contact structures on T3 - Ali Maalaoui; Vittorio Martino (Springer, 2014)
We consider a family of tight contact forms on the three-dimensional torus and we compute the relative Contact Homology by using the variational theory of critical points at infinity. We will also show local stability.

$hp$-version discontinuous Galerkin methods on polygonal and polyhedral meshes - Andrea Cangiani; E.H. Georgoulis; Houston, Paul (Math. Models Methods Appl. Sci., 2014)

An improvement on geometrical parameterizations by transfinite maps - Jäggli, C.; Laura Iapichino; Gianluigi Rozza (Comptes Rendus Mathematique, 2014)
We present a method to generate a non-affine transfinite map from a given reference domain to a family of deformed domains. The map is a generalization of the Gordon-Hall transfinite interpolation approach. It is defined globally over the reference domain. Once we have computed some functions over the reference domain, the map can be generated by knowing the parametric expressions of the boundaries of the deformed domain. Being able to define a suitable map from a reference domain to a desired deformation is useful for the management of parameterized geometries.

Infinite-dimensional Frobenius manifolds underlying the Toda lattice hierarchy - Chaozhong Wu; Dafeng Zuo (Elsevier, 2014)
Following the approach of Carlet et al. (2011) [9], we construct a class of infinite-dimensional Frobenius manifolds underlying the Toda lattice hierarchy, which are defined on the space of pairs of meromorphic functions with possibly higher-order poles at the origin and at infinity. We also show a connection between these infinite-dimensional Frobenius manifolds and the finite-dimensional Frobenius manifolds on the orbit space of extended affine Weyl groups of type A defined by Dubrovin and Zhang.

Integrability of Dirac reduced bi-Hamiltonian equations - Alberto De Sole; Victor G. Kac; Daniele Valeri (SISSA, 2014)
First, we give a brief review of the theory of the Lenard-Magri scheme for a non-local bi-Poisson structure and of the theory of Dirac reduction. These theories are used in the remainder of the paper to prove integrability of three hierarchies of bi-Hamiltonian PDE's, obtained by Dirac reduction from some generalized Drinfeld-Sokolov hierarchies.

An irreducible symplectic orbifold of dimension 6 with a Lagrangian Prym fibration - Tommaso Matteini (2014)
A new example of an irreducible symplectic variety of dimension 6, with only finite quotient singularities, is described as a relative compactified Prymian of a family of genus 4 curves with involution. It is associated to a K3 surface which is a double cover of a cubic surface. It has a natural Lagrangian fibration in abelian 3-folds with polarization type (1,1,2). It does not admit any symplectic resolution.
Keywords: Irreducible symplectic variety, Lagrangian fibration, Prym variety, automorphism of symplectic varieties

On an isomonodromy deformation equation without the Painlevé property - Boris Dubrovin; Andrey Kapaev (Maik Nauka-Interperiodica Publishing, 2014)
We show that the fourth order nonlinear ODE which controls the pole dynamics in the general solution of equation $P_I^2$ compatible with the KdV equation exhibits two remarkable properties: 1) it governs the isomonodromy deformations of a $2×2$ matrix linear ODE with polynomial coefficients, and 2) it does not possesses the Painlev\'e property. We also study the properties of the Riemann--Hilbert problem associated to this ODE and find its large $t$ asymptotic solution for the physically interesting initial data.

KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation - P Baldi; Massimiliano Berti; Riccardo Montalto (Mathematische Annalen, 2014)
We prove the existence of small amplitude quasi-periodic solutions for quasi-linear and fully nonlinear forced perturbations of the linear Airy equation. For Hamiltonian or reversible nonlinearities we also prove their linear stability. The key analysis concerns the reducibility of the linearized operator at an approximate solution, which provides a sharp asymptotic expansion of its eigenvalues. For quasi-linear perturbations this cannot be directly obtained by a KAM iteration. Hence we first perform a regularization procedure, which conjugates the linearized operator to an operator with constant coefficients plus a bounded remainder. These transformations are obtained by changes of variables induced by diffeomorphisms of the torus and pseudo-differential operators. At this point we implement a Nash-Moser iteration (with second order Melnikov non-resonance conditions) which completes the reduction to constant coefficients. © 2014 Springer-Verlag Berlin Heidelberg.

KAM for quasi-linear and fully nonlinear perturbations of Airy and KdV equations - Riccardo Montalto (SISSA, 2014)

KAM for quasi-linear KdV - P Baldi; Massimiliano Berti; Riccardo Montalto (C. R. Math. Acad. Sci. Paris, 2014)
We prove the existence and stability of Cantor families of quasi-periodic, small-amplitude solutions of quasi-linear autonomous Hamiltonian perturbations of KdV.

KAM for Reversible Derivative Wave Equations - Massimiliano Berti; Luca Biasco; Michela Procesi (Arch. Ration. Mech. Anal., 2014)
We prove the existence of Cantor families of small amplitude, analytic, linearly stable quasi-periodic solutions of reversible derivative wave equations.

Laplace equation in a domain with a rectilinear crack: higher order derivatives of the energy with respect to the crack length - Gianni Dal Maso; Gianluca Orlando; Rodica Toader (SISSA, 2014)
We consider the weak solution of the Laplace equation in a planar domain with a straight crack, prescribing a homogeneous Neumann condition on the crack and a nonhomogeneous Dirichlet condition on the rest of the boundary. For every k we express the k-th derivative of the energy with respect to the crack length in terms of a finite number of coefficients of the asymptotic expansion of the solution near the crack tip and of a finite number of other parameters, which only depend on the shape of the domain.
Keywords: cracked domains, energy release rate, higher order derivatives, asymptotic expansion of solutions

Lecture notes on gradient flows and optimal transport - Sara Daneri; Giuseppe Savarè (Cambridge University Press, 2014)
We present a short overview on the strongest variational formulation for gradient flows of geodesically λ-convex functionals in metric spaces, with applications to diffusion equations in Wasserstein spaces of probability measures. These notes are based on a series of lectures given by the second author for the Summer School "Optimal transportation: Theory and applications" in Grenoble during the week of June 22-26, 2009.

Legendre duality on hypersurfaces in Kähler manifolds - Vittorio Martino (Walter de Gruyter and Co., 2014)
We give a sufficient condition on real strictly Levi-convex hypersurfaces M, embedded in four-dimensional Kähler manifolds V , such that Legendre duality can be performed. We consider the contact form onM whose kernel is the restriction of the holomorphic tangent space of V and show that if there exists a Legendrian Killing vector field v, then the dual form β(̇) := d(v, ̇) is a contact form on M with the same orientation than theta.

Linearized plastic plate models as Γ-limits of 3D finite elastoplasticity - Elisa Davoli (ESAIM: Control, Optimisation and Calculus of Variations, 2014)
The subject of this paper is the rigorous derivation of reduced models for a thin plate by means of $Γ$-convergence, in the framework of finite plasticity. Denoting by $ε$ the thickness of the plate, we analyse the case where the scaling factor of the elasto-plastic energy per unit volume is of order $ε^{2 α -2}$, with $α ≥ 3$. According to the value of $α$, partially or fully linearized models are deduced, which correspond, in the absence of plastic deformation, to the Von Kármán plate theory and the linearized plate theory.

Lipschitz continuous viscosity solutions for a class of fully nonlinear equations on lie groups - Vittorio Martino; Annamaria Montanari (Springer, 2014)
In this paper, we prove existence and uniqueness of Lipschitz continuous viscosity solutions for Dirichlet problems involving a class a fully non-linear operators on Lie groups. In particular, we consider the elementary symmetric functions of the eigenvalues of the Hessian built with left-invariant vector fields.

Local and global minimality results for a nonlocal isoperimetric problem on R^N - Marco Bonacini; Riccardo Cristoferi (SIAM Journal on Mathematical Analysis, 2014)
We consider a nonlocal isoperimetric problem defined in the whole space R^N, whose nonlocal part is given by a Riesz potential with exponent $α\in(0, N-1)$. We show that critical configurations with positive second variation are local minimizers and satisfy a quantitative inequality with respect to the L^1-norm. This criterion provides the existence of a (explicitly determined) critical threshold determining the interval of volumes for which the ball is a local minimizer, and allows to address several global minimality issues.
Keywords: Nonlocal isoperimetric problem

Local behavior of fractional p-minimizers - Agnese Di Castro; Tuomo Kuusi; Giampiero Palatucci (SISSA, 2014)
We extend the De Giorgi-Nash Moser theory to nonlocal, possibly degerate integro-differential operators
Keywords: fractional Sobolev spaces

On local super-penalization of interior penalty discontinuous Galerkin methods - Andrea Cangiani; Chapman, John; E.H. Georgoulis; Jensen, Max (Int. J. Numer. Anal. Model., 2014)

On the Lp-differentiability of certain classes of functions - Giovanni Alberti; Stefano Bianchini; Gianluca Crippa (European Mathematical Society, 2014)
We prove the Lp-differentiability at almost every point for convolution products on ℝd of the form K*μ, where μ is bounded measure and K is a homogeneous kernel of degree 1-d. From this result we derive the Lp-differentiability for vector fields on R d whose curl and divergence are measures, and also for vector fields with bounded deformation.

Maximal generalized solution of eikonal equation - Sandro Zagatti (Elsevier, 2014)
We study the Dirichlet problem for the eikonal equation: 1/2 |∇u(x)|^2-a(x)=0 in Ω u(x)=(x) on Ω, without continuity assumptions on the map a(.). We find a class of maps a(.) contained in the space L∞(Ω) for which the problem admits a (maximal) generalized solution, providing a generalization of the notion of viscosity solution.

Minimal Liouville gravity correlation numbers from Douglas string equation - Alexander Belavin; Boris Dubrovin; Baur Mukhametzhanov (Springer, 2014)
We continue the study of $(q,p)$ Minimal Liouville Gravity with the help of Douglas string equation. We generalize the results of \cite{Moore:1991ir}, \cite{Belavin:2008kv}, where Lee-Yang series $(2,2s+1)$ was studied, to $(3,3s+p_0)$ Minimal Liouville Gravity, where $p_0=1,2$. We demonstrate that there exist such coordinates $τ_{m,n}$ on the space of the perturbed Minimal Liouville Gravity theories, in which the partition function of the theory is determined by the Douglas string equation. The coordinates $τ_{m,n}$ are related in a non-linear fashion to the natural coupling constants $λ_{m,n}$ of the perturbations of Minimal Lioville Gravity by the physical operators $O_{m,n}$. We find this relation from the requirement that the correlation numbers in Minimal Liouville Gravity must satisfy the conformal and fusion selection rules. After fixing this relation we compute three- and four-point correlation numbers when they are not zero. The results are in agreement with the direct calculations in Minimal Liouville Gravity available in the literature \cite{Goulian:1990qr}, \cite{Zamolodchikov:2005sj}, \cite{Belavin:2006ex}.

A model for crack growth with branching and kinking - Simone Racca (Asymptotic Analysis, 2014)
We study an evolution model for fractured elastic materials in the 2-dimensional case, for which the crack path is not assumed to be known a priori. We introduce some general assumptions on the structure of the fracture sets suitable to remove the restrictions on the regularity of the crack sets and to allow for kinking and branching to develop. In addition we define the front of the fracture and its velocity. By means of a time-discretization approach, we prove the existence of a continuous-time evolution that satisfies an energy inequality and a stability criterion. The energy balance also takes into account the energy dissipated at the front of the fracture. The stability criterion is stated in the framework of Griffith's theory, in terms of the energy release rate, when the crack grows at least at one point of its front.
Keywords: quasistatic crack evolution, branching, kinking, Griffith\\\'s criterion

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives - Toni Lassila; Andrea Manzoni; Alfio Quarteroni; Gianluigi Rozza (Springer, 2014)
This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities - which are mainly related either to nonlinear convection terms and/or some geometric variability - that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and-in the unsteady case - long-time stability of the reduced model. Moreover, we provide an extensive list of literature references.

A modular spectral triple for κ-Minkowski space - Marco Matassa (Elsevier, 2014)
We present a spectral triple for κ-Minkowski space in two dimensions. Starting from an algebra naturally associated to this space, a Hilbert space is built using a weight which is invariant under the κ-Poincaré algebra. The weight satisfies a KMS condition and its associated modular operator plays an important role in the construction. This forces us to introduce two ingredients which have a modular flavour: the first is a twisted commutator, used to obtain a boundedness condition for the Dirac operator, and the second is a weight replacing the usual operator trace, used to measure the growth of the resolvent of the Dirac operator. We show that, under some assumptions related to the symmetries and the classical limit, there is a unique Dirac operator and automorphism such that the twisted commutator is bounded. Then, using the weight mentioned above, we compute the spectral dimension associated to the spectral triple and find that is equal to the classical dimension. Finally we briefly discuss the introduction of a real structure.

A Moser-Trudinger inequality for the singular Toda system - Luca Battaglia; Andrea Malchiodi (Bull. Inst. Math. Acad. Sin., 2014)

M-theory interpretation of the real topological string - Nicolò Piazzalunga; Uranga, Angel M. (Journal of High Energy Physics, 2014)
We describe the type IIA physical realization of the unoriented topological string introduced by Walcher, describe its M-theory lift, and show that it allows to compute the open and unoriented topological amplitude in terms of one-loop diagram of BPS M2-brane states. This confirms and allows to generalize the conjectured BPS integer expansion of the topological amplitude. The M-theory lift of the orientifold is freely acting on the M-theory circle, so that integer multiplicities are a weighted version of the (equivariant subsector of the) original closed oriented Gopakumar-Vafa invariants. The M-theory lift also provides new perspective on the topological tadpole cancellation conditions. We finally comment on the M-theory version of other unoriented topological strings, and clarify certain misidentifications in earlier discussions in the literature.

N = 2 Quiver Gauge Theories on A-type ALE Spaces - Ugo Bruzzo; Francesco Sala; Richard J. Szabo (Springer, 2014)
We survey and compare recent approaches to the computation of the partition functions and correlators of chiral BPS observables in N = 2 gauge theories on ALE spaces based on quiver varieties and the minimal resolution Xk of the Ak-1 toric singularity C2/Zk, in light of their recently conjectured duality with two-dimensional coset conformal field theories. We review and elucidate the rigorous constructions of gauge theories for a particular family of ALE spaces, using their relation to the cohomology of moduli spaces of framed torsion-free sheaves on a suitable orbifold compactification of Xk. We extend these computations to generic N = 2 superconformal quiver gauge theories, obtaining in these instances new constraints on fractional instanton charges, a rigorous proof of the Nekrasov master formula, and new quantizations of Hitchin systems based on the underlying Seiberg–Witten geometry.

New results on Gamma-limits of integral functionals - Nadia Ansini; Gianni Dal Maso; Caterina Ida Zeppieri (Elsevier, 2014)
Keywords: Gamma-convergence

Non-commutative integration for spectral triples associated to quantum groups - Marco Matassa (SISSA, 2014)
This thesis is dedicated to the study of non-commutative integration, in the sense of spectral triples, for some non-commutative spaces associated to quantum groups.
Keywords: Non-commutative geometry

Nonsingular Isogeometric Boundary Element Method for Stokes Flows in 3D - Luca Heltai; Marino Arroyo; Antonio DeSimone (Elsevier, 2014)
Isogeometric analysis (IGA) is emerging as a technology bridging Computer Aided Geometric Design (CAGD), most commonly based on Non-Uniform Rational B-Splines (NURBS) surfaces, and engineering analysis. In finite element and boundary element isogeometric methods (FE-IGA and IGA-BEM), the NURBS basis functions that de- scribe the geometry define also the approximation spaces. In the FE-IGA approach, the surfaces generated by the CAGD tools need to be extended to volumetric descriptions, a major open problem in 3D. This additional passage can be avoided in principle when the partial differential equations to be solved admit a formulation in terms of bound- ary integral equations, leading to Boundary Element Isogeometric Analysis (IGA-BEM). The main advantages of such an approach are given by the dimensionality reduction of the problem (from volumetric-based to surface-based), by the fact that the interface with CAGD tools is direct, and by the possibility to treat exterior problems, where the computational domain is infinite. By contrast, these methods produce system matrices which are full, and require the integration of singular kernels. In this paper we address the second point and propose a nonsingular formulation of IGA-BEM for 3D Stokes flows, whose convergence is carefully tested numerically. Standard Gaussian quadrature rules suffice to integrate the boundary integral equations, and carefully chosen known exact solutions of the interior Stokes problem are used to correct the resulting matrices, extending the work by Klaseboer et al. [27] to IGA-BEM.
Keywords: Isogeometric Analysis

Pfaffian representations of cubic surfaces - Fabio Tanturri (Springer, 2014)
Let K be a field of characteristic zero. We describe an algorithm which requires a homogeneous polynomial F of degree three in K[x0,x1,x2,x3] and a zero a of F in P3 K and ensures a linear Pfaffian representation of V(F) with entries in K[x0,x1,x2,x3], under mild assumptions on F and a. We use this result to give an explicit construction of (and to prove the existence of) a linear Pfaffian representation of V (F), with entries in K′[x0,x1,x2,x3], being K′ an algebraic extension of K of degree at most six. An explicit example of such a construction is given.

Potential Model for Ship Hydrodynamics Simulations Directly Interfaced with CAD Data Structures - Andrea Mola; Luca Heltai; Antonio DeSimone; Massimiliano Berti (The 24th International Ocean and Polar Engineering Conference, 2014)

Pseudo-automorphisms of positive entropy on the blowups of products of projective spaces - Fabio Perroni; Deqi Zhang (Springer, 2014)
We use a concise method to construct pseudo-automorphisms fn of the first dynamical degree d1(fn) > 1 on the blowups of the projective n-space for all n ≥ 2 and more generally on the blowups of products of projective spaces. These fn, for n=3 have positive entropy, and for n≥ 4 seem to be the first examples of pseudo-automorphisms with d1(fn) > 1 (and of non-product type) on rational varieties of higher dimensions.

On a quadratic functional for scalar conservation laws - Stefano Bianchini; Stefano Modena (Journal of Hyperbolic Differential Equations, 2014)
We prove a quadratic interaction estimate for approximate solutions to scalar conservation laws obtained by the wavefront tracking approximation or the Glimm scheme. This quadratic estimate has been used in the literature to prove the convergence rate of the Glimm scheme.

Quadratic interaction functional for systems of conservation laws: a case study - Stefano Bianchini; Stefano Modena (Bulletin of the Institute of Mathematics of Academia Sinica (New Series), 2014)

Quantum dimension and quantum projective spaces - Marco Matassa (Institute of Mathematics, 2014)
We show that the family of spectral triples for quantum projective spaces introduced by D'Andrea and Dbrowski, which have spectral dimension equal to zero, can be reconsidered as modular spectral triples by taking into account the action of the element K2por its inverse. The spectral dimension computed in this sense coincides with the dimension of the classical projective spaces. The connection with the well known notion of quantum dimension of quantum group theory is pointed out.

Quantum gauge symmetries in noncommutative geometry - Jyotishman Bhowmick; Francesco D'Andrea; Biswarup Krishna Das; Ludwik Dabrowski (European Mathematical Society Publishing House, 2014)
We discuss generalizations of the notion of i) the group of unitary elements of a (real or complex) finite-dimensional C*-algebra, ii) gauge transformations and iii) (real) automorphisms in the framework of compact quantum group theory and spectral triples. The quantum analogue of these groups are defined as universal (initial) objects in some natural categories. After proving the existence of the universal objects, we discuss several examples that are of interest to physics, as they appear in the noncommutative geometry approach to particle physics: in particular, the C*-algebras M n(R), Mn(C) and Mn(H), describing the finite noncommutative space of the Einstein-Yang-Mills systems, and the algebras A F = C H M3 (C) and Aev = H H M4(C), that appear in Chamseddine-Connes derivation of the Standard Model of particle physics coupled to gravity. As a byproduct, we identify a "free" version of the symplectic group Sp.n/ (quaternionic unitary group).

Quasi-static crack growth in hydraulic fracture - Stefano Almi; Gianni Dal Maso; Rodica Toader (Nonlinear Analysis, 2014)
We present a variational model for the quasi-static crack growth in hydraulic fracture in the framework of the energy formulation of rate-independent processes. The cracks are assumed to lie on a prescribed plane and to satisfy a very weak regularity assumption.

Quasistatic Evolution in Perfect Plasticity as Limit of Dynamic Processes - Gianni Dal Maso; Riccardo Scala (Journal of Dynamics and Differential Equations, 2014)
We introduce a model of dynamic visco-elasto-plastic evolution in the linearly elastic regime and prove an existence and uniqueness result. Then we study the limit of (a rescaled version of) the solutions when the data vary slowly. We prove that they converge, up to a subsequence, to a quasistatic evolution in perfect plasticity.

Quasistatic evolution models for thin plates arising as low energy Γ-limits of finite plasticity - Elisa Davoli (Mathematical Models and Methods in Applied Sciences, 2014)
In this paper we deduce by $Γ$-convergence some partially and fully linearized quasistatic evolution models for thin plates, in the framework of finite plasticity. Denoting by $ε$ the thickness of the plate, we study the case where the scaling factor of the elasto-plastic energy is of order $ε^{2 α -2}$, with $α≥ 3$. These scalings of the energy lead, in the absence of plastic dissipation, to the Von Kármán and linearized Von Kármán functionals for thin plates. We show that solutions to the three-dimensional quasistatic evolution problems converge, as the thickness of the plate tends to zero, to a quasistatic evolution associated to a suitable reduced model depending on $α$.

Rate-independent damage in thermo-viscoelastic materials with inertia - Giuliano Lazzaroni; Riccarda Rossi; Marita Thomas; Rodica Toader (SISSA, 2014)
We present a model for rate-independent, unidirectional, partial damage in visco-elastic materials with inertia and thermal effects. The damage process is modeled by means of an internal variable, governed by a rate-independent flow rule. The heat equation and the momentum balance for the displacements are coupled in a highly nonlinear way. Our assumptions on the corresponding energy functional also comprise the case of the Ambrosio-Tortorelli phase-field model (without passage to the brittle limit). We discuss a suitable weak formulation and prove an existence theorem obtained with the aid of a (partially) decoupled time-discrete scheme and variational convergence methods. We also carry out the asymptotic analysis for vanishing viscosity and inertia and obtain a fully rate-independent limit model for displacements and damage, which is Independent of temperature.

Rational curves and instantons on the Fano threefold Y_5 - Giangiacomo Sanna (arXiv preprint, 2014)
This thesis is an investigation of the moduli spaces of instanton bundles on the Fano threefold Y_5 (a linear section of Gr(2,5)). It contains new proofs of classical facts about lines, conics and cubics on Y_5, and about linear sections of Y_5. The main original results are a Grauert-Mülich theorem for the splitting type of instantons on conics, a bound to the splitting type of instantons on lines and an SL_2-equivariant description of the moduli space in charge 2 and 3. Using these results we prove the existence of a unique SL_2-equivariant instanton of minimal charge and we show that for all instantons of charge 2 the divisor of jumping lines is smooth. In charge 3, we provide examples of instantons with reducible divisor of jumping lines. Finally, we construct a natural compactification for the moduli space of instantons of charge 3, together with a small resolution of singularities for it.
Keywords: Moduli space of vector bundles

Reduced basis method for the Stokes equations in decomposable domains using greedy optimization - Laura Iapichino; Alfio Quarteroni; Gianluigi Rozza; Volkwein, Stefan (ECMI 2014 proceedings, 2014)

Reduced Order Methods for Modeling and Computational Reduction - Alfio Quarteroni; Gianluigi Rozza (MS&A, 2014)
This monograph addresses the state of the art of reduced order methods for modeling and computational reduction of complex parametrized systems, governed by ordinary and/or partial differential equations, with a special emphasis on real time computing techniques and applications in computational mechanics, bioengineering and computer graphics. Several topics are covered, including: design, optimization, and control theory in real-time with applications in engineering; data assimilation, geometry registration, and parameter estimation with special attention to real-time computing in biomedical engineering and computational physics; real-time visualization of physics-based simulations in computer science; the treatment of high-dimensional problems in state space, physical space, or parameter space; the interactions between different model reduction and dimensionality reduction approaches; the development of general error estimation frameworks which take into account both model and discretization effects. This book is primarily addressed to computational scientists interested in computational reduction techniques for large scale differential problems.
Keywords: reduced order methods, MOR, ROM, POD, RB, greedy, CFD, Numerical Analysis

A reduced order model for multi-group time-dependent parametrized reactor spatial kinetics - Alberto Sartori; Davide Baroli; Antonio Cammi; Lelio Luzzi; Gianluigi Rozza (22nd International Conference on Nuclear Engineering ICONE22, 2014)
In this work, a Reduced Order Model (ROM) for multigroup time-dependent parametrized reactor spatial kinetics is presented. The Reduced Basis method (built upon a high-fidelity "truth" finite element approximation) has been applied to model the neutronics behavior of a parametrized system composed by a control rod surrounded by fissile material. The neutron kinetics has been described by means of a parametrized multi-group diffusion equation where the height of the control rod (i.e., how much the rod is inserted) plays the role of the varying parameter. In order to model a continuous movement of the rod, a piecewise affine transformation based on subdomain division has been implemented. The proposed ROM is capable to efficiently reproduce the neutron flux distribution allowing to take into account the spatial effects induced by the movement of the control rod with a computational speed-up of 30000 times, with respect to the "truth" model.

Reduction on characteristics for continuous of a scalar balance law - Giovanni Alberti; Stefano Bianchini; Laura Caravenna (AIMS Series on Applied Mathematics, vol. 8 (2014): 399 - 406, 2014)
Keywords: Method of characteristics

A Review of the Sixth Painlevé Equation - Davide Guzzetti (Springer, 2014)
For the Painlevé VI transcendents, we provide a unitary description of the critical behaviours, the connection formulae, their complete tabulation, and the asymptotic distribution of poles close to a critical point.

A robotic crawler exploiting directional frictional interactions: experiments, numerics, and derivation of a reduced model - Giovanni Noselli; Antonio DeSimone (Proceedings of the Royal Society A 470, 20140333 (2014), 2014)
We present experimental and numerical results for a model crawler which is able to extract net positional changes from reciprocal shape changes, i.e. ‘breathing-like’ deformations, thanks to directional, frictional interactions with a textured solid substrate, mediated by flexible inclined feet. We also present a simple reduced model that captures the essential features of the kinematics and energetics of the gait, and compare its predictions with the results from experiments and from numerical simulations.

SBV Regularity of Systems of Conservation Laws and Hamilton–Jacobi Equations - Stefano Bianchini (Springer, 2014)
We review the SBV regularity for solutions to hyperbolic systems of conservation laws and Hamilton-Jacobi equations. We give an overview of the techniques involved in the proof, and a collection of related problems concludes the paper.

Second Order Asymptotic Development for the Anisotropic Cahn-Hilliard Functional - Gianni Dal Maso; Irene Fonseca; Giovanni Leoni (SISSA, 2014)
The asymptotic behavior of an anisotropic Cahn-Hilliard functional with prescribed mass and Dirichlet boundary condition is studied when the parameter $ε$ that determines the width of the transition layers tends to zero. The double-well potential is assumed to be even and equal to $|s-1|^β$ near $s=1$, with $1<β<2$. The first order term in the asymptotic development by $Γ$-convergence is well-known, and is related to a suitable anisotropic perimeter of the interface. Here it is shown that, under these assumptions, the second order term is zero, which gives an estimate on the rate of convergence of the minimum values.
Keywords: Gamma-convergence, Cahn-Hilliard functional, phase transitions

Semiclassical limit of focusing NLS for a family of square barrier initial data - Robert Jenkins; Kenneth McLaughlin (Wiley Periodicals, 2014)
The small dispersion limit of the focusing nonlinear Schrödinger equation (NLS) exhibits a rich structure of sharply separated regions exhibiting disparate rapid oscillations at microscopic scales. The non-self-adjoint scattering problem and ill-posed limiting Whitham equations associated to focusing NLS make rigorous asymptotic results difficult. Previous studies have focused on special classes of analytic initial data for which the limiting elliptic Whitham equations are wellposed. In this paper we consider another exactly solvable family of initial data,the family of square barriers,ψ 0(x) = qχ[-L,L] for real amplitudes q. Using Riemann-Hilbert techniques, we obtain rigorous pointwise asymptotics for the semiclassical limit of focusing NLS globally in space and up to an O(1) maximal time. In particular, we show that the discontinuities in our initial data regularize by the immediate generation of genus-one oscillations emitted into the support of the initial data. To the best of our knowledge, this is the first case in which the genus structure of the semiclassical asymptotics for focusing NLS have been calculated for nonanalytic initial data.

Shape control of active surfaces inspired by the movement of euglenids - Marino Arroyo; Antonio DeSimone (Elsevier, 2014)
We examine a novel mechanism for active surface morphing inspired by the cell body deformations of euglenids. Actuation is accomplished through in-plane simple shear along prescribed slip lines decorating the surface. Under general non-uniform actuation, such local deformation produces Gaussian curvature, and therefore leads to shape changes. Geometrically, a deformation that realizes the prescribed local shear is an isometric embedding. We explore the possibilities and limitations of this bio-inspired shape morphing mechanism, by first characterizing isometric embeddings under axisymmetry, understanding the limits of embeddability, and studying in detail the accessibility of surfaces of zero and constant curvature. Modeling mechanically the active surface as a non-Euclidean plate (NEP), we further examine the mechanism beyond the geometric singularities arising from embeddability, where mechanics and buckling play a decisive role. We also propose a non-axisymmetric actuation strategy to accomplish large amplitude bending and twisting motions of elongated cylindrical surfaces. Besides helping understand how euglenids delicately control their shape, our results may provide the background to engineer soft machines.

Shape Optimization by Free-Form Deformation: Existence Results and Numerical Solution for Stokes Flows - F. Ballarin; Andrea Manzoni; Gianluigi Rozza; Sandro Salsa (Springer, 2014)
Shape optimization problems governed by PDEs result from many applications in computational fluid dynamics. These problems usually entail very large computational costs and require also a suitable approach for representing and deforming efficiently the shape of the underlying geometry, as well as for computing the shape gradient of the cost functional to be minimized. Several approaches based on the displacement of a set of control points have been developed in the last decades, such as the so-called free-form deformations. In this paper we present a new theoretical result which allows to recast free-form deformations into the general class of perturbation of identity maps, and to guarantee the compactness of the set of admissible shapes. Moreover, we address both a general optimization framework based on the continuous shape gradient and a numerical procedure for solving efficiently three-dimensional optimal design problems. This framework is applied to the optimal design of immersed bodies in Stokes flows, for which we consider the numerical solution of a benchmark case study from literature.

Singular Value Decomposition of a Finite Hilbert Transform Defined on Several Intervals and the Interior Problem of Tomography: The Riemann-Hilbert Problem Approach - Marco Bertola; Alexander Katsevich; Alexander Tovbis (Comm. Pure Appl. Math., 2014)

Six-dimensional supersymmetric gauge theories, quantum cohomology of instanton moduli spaces and gl(N) Quantum Intermediate Long Wave Hydrodynamics - Giulio Bonelli; Antonio Sciarappa; Alessandro Tanzini; Petr Vasko (Springer, 2014)
We show that the exact partition function of U(N) six-dimensional gauge theory with eight supercharges on C^2 x S^2 provides the quantization of the integrable system of hydrodynamic type known as gl(N) periodic Intermediate Long Wave (ILW). We characterize this system as the hydrodynamic limit of elliptic Calogero-Moser integrable system. We compute the Bethe equations from the effective gauged linear sigma model on S^2 with target space the ADHM instanton moduli space, whose mirror computes the Yang-Yang function of gl(N) ILW. The quantum Hamiltonians are given by the local chiral ring observables of the six-dimensional gauge theory. As particular cases, these provide the gl(N) Benjamin-Ono and Korteweg-de Vries quantum Hamiltonians. In the four dimensional limit, we identify the local chiral ring observables with the conserved charges of Heisenberg plus W_N algebrae, thus providing a gauge theoretical proof of AGT correspondence.

Smooth approximation of bi-Lipschitz orientation-preserving homeomorphisms - Sara Daneri; Aldo Pratelli (Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 2014)
We show that a planar bi-Lipschitz orientation-preserving homeomorphism can be approximated in the W1,p norm, together with its inverse, with an orientation-preserving homeomorphism which is piecewise affine or smooth.

Some remarks on a model for rate-independent damage in thermo-visco-elastodynamics - Giuliano Lazzaroni; Riccarda Rossi; Marita Thomas; Rodica Toader (SISSA, 2014)
This note deals with the analysis of a model for partial damage, where the rateindependent, unidirectional flow rule for the damage variable is coupled with the rate-dependent heat equation, and with the momentum balance featuring inertia and viscosity according to Kelvin-Voigt rheology. The results presented here combine the approach from Roubicek [1] with the methods from Lazzaroni/Rossi/Thomas/Toader [2] and extend the analysis to the setting of inhomogeneous time-dependent Dirichlet data.

Some remarks on the seismic behaviour of embedded cantilevered retaining walls - Riccardo Conti; F. Burali D'Arezzo; Giulia M.B. Viggiani (Thomas Telford, 2014)
This paper is a numerical investigation of the physical phenomena that control the dynamic behaviour of embedded cantilevered retaining walls. Recent experimental observations obtained from centrifuge tests have shown that embedded cantilevered retaining walls experience permanent displacements even before the acceleration reaches its critical value, corresponding to full mobilisation of the soil strength. The motivation for this work stems from the need to incorporate these observations in simplified design procedures. A parametric study was carried out on a pair of embedded cantilevered walls in dry sand, subjected to real earthquakes scaled at different values of the maximum acceleration. The results of these analyses indicate that, for the geotechnical design of the wall, the equivalent acceleration to be used in pseudo-static calculations can be related to the maximum displacement that the structure can sustain, and can be larger than the maximum acceleration expected at the site. For the structural design of the wall, it is suggested that the maximum bending moments of the wall can be computed using a realistic distribution of contact stress and a conservative value of the pseudo-static acceleration, taking into account two-dimensional amplification effects near the walls.

Spontaneous division and motility in active nematic droplets - Luca Giomi; Antonio DeSimone (American Physical Society, 2014)
We investigate the mechanics of an active droplet endowed with internal nematic order and surrounded by an isotropic Newtonian fluid. Using numerical simulations we demonstrate that, due to the interplay between the active stresses and the defective geometry of the nematic director, this system exhibits two of the fundamental functions of living cells: spontaneous division and motility, by means of self-generated hydrodynamic flows. These behaviors can be selectively activated by controlling a single physical parameter, namely, an active variant of the capillary number.

Stability of equilibrium configurations for elastic films in two and three dimensions - Marco Bonacini (Advances in Calculus of Variations, 2014)
We establish a local minimality sufficiency criterion, based on the strict positivity of the second variation, in the context of a variational model for the epitaxial growth of elastic films. Our result holds also in the three-dimensional case and for a general class of nonlinear elastic energies. Applications to the study of the local minimality of flat morphologies are also shown.
Keywords: Epitaxially strained elastic films

Stabilized reduced basis method for parametrized advection-diffusion PDEs - Pacciarini, P.; Gianluigi Rozza (Computer Methods in Applied Mechanics and Engineering, 2014)
In this work, we propose viable and efficient strategies for the stabilization of the reduced basis approximation of an advection dominated problem. In particular, we investigate the combination of a classic stabilization method (SUPG) with the Offline-Online structure of the RB method. We explain why the stabilization is needed in both stages and we identify, analytically and numerically, which are the drawbacks of a stabilization performed only during the construction of the reduced basis (i.e. only in the Offline stage). We carry out numerical tests to assess the performances of the ``double'' stabilization both in steady and unsteady problems, also related to heat transfer phenomena.

Stabilized reduced basis method for parametrized scalar advection-diffusion problems at higher Péclet number: Roles of the boundary layers and inner fronts - Pacciarini, P.; Gianluigi Rozza (11th World Congress on Computational Mechanics, WCCM 2014, 5th European Conference on Computational Mechanics, ECCM 2014 and 6th European Conference on Computational Fluid Dynamics, ECFD 2014, 2014)
Advection-dominated problems, which arise in many engineering situations, often require a fast and reliable approximation of the solution given some parameters as inputs. In this work we want to investigate the coupling of the reduced basis method - which guarantees rapidity and reliability - with some classical stabilization techiques to deal with the advection-dominated condition. We provide a numerical extension of the results presented in [1], focusing in particular on problems with curved boundary layers and inner fronts whose direction depends on the parameter.

Steady nearly incompressible vector elds in 2D: chain rule and renormalization - Stefano Bianchini; N.A. Gusev (SISSA, 2014)

The stringy instanton partition function - Giulio Bonelli; Antonio Sciarappa; Alessandro Tanzini; Petr Vasko (Springer, 2014)
We perform an exact computation of the gauged linear sigma model associated to a D1-D5 brane system on a resolved A_1 singularity. This is accomplished via supersymmetric localization on the blown-up two-sphere. We show that in the blow-down limit C^2/Z_2 the partition function reduces to the Nekrasov partition function evaluating the equivariant volume of the instanton moduli space. For finite radius we obtain a tower of world-sheet instanton corrections, that we identify with the equivariant Gromov-Witten invariants of the ADHM moduli space. We show that these corrections can be encoded in a deformation of the Seiberg-Witten prepotential. From the mathematical viewpoint, the D1-D5 system under study displays a twofold nature: the D1-branes viewpoint captures the equivariant quantum cohomology of the ADHM instanton moduli space in the Givental formalism, and the D5-branes viewpoint is related to higher rank equivariant Donaldson-Thomas invariants of P^1 x C^2.

Structure of classical (finite and affine) W-algebras - Alberto De Sole; Victor G. Kac; Daniele Valeri (SISSA, 2014)
First, we derive an explicit formula for the Poisson bracket of the classical finite W-algebra W^{fin}(g,f), the algebra of polynomial functions on the Slodowy slice associated to a simple Lie algebra g and its nilpotent element f. On the other hand, we produce an explicit set of generators and we derive an explicit formula for the Poisson vertex algebra structure of the classical affine W-algebra W(g,f). As an immediate consequence, we obtain a Poisson algebra isomorphism between W^{fin}(g,f) and the Zhu algebra of W(g,f). We also study the generalized Miura map for classical W-algebras.

Structure of entropy solutions to general scalar conservation laws in one space dimension - Stefano Bianchini; Lei Yu (Journal of Mathematical Analysis and Applications, 2014)

Swelling dynamics of a thin elastomeric sheet under uniaxial pre-stretch - Alessandro Lucantonio; Paola Nardinocchi; Howard A. Stone (American Institute of Physics, 2014)
It has been demonstrated experimentally that pre-stretch affects the swelling of an elastomeric membrane when it is exposed to a solvent. We study theoretically the one-dimensional swelling of a pre-stretched thin elastomeric sheet, bonded to an impermeable rigid substrate, to quantify the influence of pre-stretch. We show that the solvent uptake increases when pre-stretch increases, both at equilibrium and during the swelling transient, where it exhibits two different scaling regimes. The coupling between the solvent uptake and pre-stretch may be practically exploited to design soft actuators where the swelling-induced deformations can be controlled by varying the pre-stretch.

Swelling-induced and controlled curving in layered gel beams - Alessandro Lucantonio; Paola Nardinocchi; Matteo Pezzulla (Royal Society of London, 2014)
We describe swelling-driven curving in originally straight and non-homogeneous beams. We present and verify a structural model of swollen beams, based on a new point of view adopted to describe swelling-induced deformation processes in bilayered gel beams, that is based on the split of the swelling-induced deformation of the beam at equilibrium into two components, both depending on the elastic properties of the gel. The method allows us to: (i) determine beam stretching and curving, once assigned the characteristics of the solvent bath and of the non-homogeneous beam, and (ii) estimate the characteristics of non-homogeneous flat gel beams in such a way as to obtain, under free-swelling conditions, three-dimensional shapes. The study was pursued by means of analytical, semi-analytical and numerical tools; excellent agreement of the outcomes of the different techniques was found, thus confirming the strength of the method.

Topological Invariants of Eigenvalue Intersections and Decrease of Wannier Functions in Graphene - Domenico Monaco; Gianluca Panati (J. Stat. Phys 155 (2014) 1027-1071, 2014)
We investigate the asymptotic decrease of the Wannier functions for the valence and conduction band of graphene, both in the monolayer and the multilayer case. Since the decrease of the Wannier functions is characterised by the structure of the Bloch eigenspaces around the Dirac points, we introduce a geometric invariant of the family of eigenspaces, baptised eigenspace vorticity. We compare it with the pseudospin winding number. For every value n∈Z of the eigenspace vorticity, we exhibit a canonical model for the local topology of the eigenspaces. With the help of these canonical models, we show that the single band Wannier function w satisfies |w(x)|≤const |x|^{−2} as |x|→∞, both in monolayer and bilayer graphene.
Keywords: Wannier functions, Bloch bundles, conical intersections, eigenspace vorticity, pseudospin winding number, graphene

The topology of a subspace of the Legendrian curves on a closed contact 3-manifold - Ali Maalaoui; Vittorio Martino (Advanced Nonlinear Studies, 2014)
In this paper we study a subspace of the space of Legendrian loops and we show that the injection of this space into the full loop space is an S 1-equivariant homotopy equivalence. This space can be also seen as the space of zero Maslov index Legendrian loops and it shows up as a suitable space of variations in contact form geometry.

A uniqueness result for the continuity equation in two dimensions: dedicated to constantine dafermos on the occasion of his 70th birthday - Giovanni Alberti; Stefano Bianchini; Gianluca Crippa (European Mathematical Society; Springer Verlag, 2014)
We characterize the autonomous, divergence-free vector fields b on the plane such that the Cauchy problem for the continuity equation ∂tu +div(bu) = 0 admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential f associated to b. As a corollary we obtain uniqueness under the assumption that the curl of b is a measure. This result can be extended to certain nonautonomous vector fields b with bounded divergence.

A variational approach to statics and dynamics of elasto-plastic systems - Riccardo Scala (SISSA, 2014)
We prove some existence results for dynamic evolutions in elasto-plasticity and delamination. We study the limit as the data vary very slowly and prove convergence results to quasistatic evolutions. We model dislocations by mean of currents, we introduce the space of deformations in the presence of dislocations and study the graphs of these maps. We prove existence results for minimum problems. We study the properties of minimizers.
Keywords: delamination

A variational model for the quasi-static growth of fractional dimensional brittle fractures - Simone Racca; Rodica Toader (European Mathematical Society, 2014)
We propose a variational model for the irreversible quasi-static evolution of brittle fractures having fractional Hausdorff dimension in the setting of two-dimensional antiplane and plane elasticity. The evolution along such irregular crack paths can be obtained as $Γ$-limit of evolutions along one-dimensional cracks when the fracture toughness tends to zero.
Keywords: Variational models

Vortex Partition Functions, Wall Crossing and Equivariant Gromov–Witten Invariants - Giulio Bonelli; Antonio Sciarappa; Alessandro Tanzini; Petr Vasko (Springer, 2014)
In this paper we identify the problem of equivariant vortex counting in a (2,2) supersymmetric two dimensional quiver gauged linear sigma model with that of computing the equivariant Gromov–Witten invariants of the GIT quotient target space determined by the quiver. We provide new contour integral formulae for the I and J-functions encoding the equivariant quantum cohomology of the target space. Its chamber structure is shown to be encoded in the analytical properties of the integrand. This is explained both via general arguments and by checking several key cases. We show how several results in equivariant Gromov–Witten theory follow just by deforming the integration contour. In particular, we apply our formalism to compute Gromov–Witten invariants of the C3/Zn orbifold, of the Uhlembeck (partial) compactification of the moduli space of instantons on C2, and of An and Dn singularities both in the orbifold and resolved phases. Moreover, we analyse dualities of quantum cohomology rings of holomorphic vector bundles over Grassmannians, which are relevant to BPS Wilson loop algebrae.

A weighted empirical interpolation method: A priori convergence analysis and applications - Peng Chen; Alfio Quarteroni; Gianluigi Rozza (EDP Sciences, 2014)
We extend the classical empirical interpolation method [M. Barrault, Y. Maday, N.C. Nguyen and A.T. Patera, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. Compt. Rend. Math. Anal. Num. 339 (2004) 667-672] to a weighted empirical interpolation method in order to approximate nonlinear parametric functions with weighted parameters, e.g. random variables obeying various probability distributions. A priori convergence analysis is provided for the proposed method and the error bound by Kolmogorov N-width is improved from the recent work [Y. Maday, N.C. Nguyen, A.T. Patera and G.S.H. Pau, A general, multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8 (2009) 383-404]. We apply our method to geometric Brownian motion, exponential Karhunen-Loève expansion and reduced basis approximation of non-affine stochastic elliptic equations. We demonstrate its improved accuracy and efficiency over the empirical interpolation method, as well as sparse grid stochastic collocation method.

Weighted quantile correlation test for the logistic family - Ferenc Balogh; Éva Krauczi (University of Szeged, 2014)
We summarize the results of investigating the asymptotic behavior of the weighted quantile correlation tests for the location-scale family associated to the logistic distribution. Explicit representations of the limiting distribution are given in terms of integrals of weighted Brownian bridges or alternatively as infinite series of independent Gaussian random variables. The power of this test and the test for the location logistic family against some alternatives are demonstrated by numerical simulations.

Where best to place a Dirichlet condition in an anisotropic membrane? - Paolo Tilli; Davide Zucco (SISSA, 2014)
We study a shape optimization problem for the first eigenvalue of an elliptic operator in divergence form, with non constant coefficients, over a fixed domain $Ω$. Dirichlet conditions are imposed along $\partialΩ$ and, in addition, along a set $Σ$ of prescribed length ($1$-dimensional Hausdorff measure). We look for the best shape and position for the supplementary Dirichlet region $Σ$ in order to maximize the first eigenvalue. We characterize the limit distribution of the optimal sets, as their prescribed length tends to infinity, via $Γ$-convergence.

Zeros of Large Degree Vorob'ev-Yablonski Polynomials via a Hankel Determinant Identity - Marco Bertola; Thomas Bothner (International Mathematics Research Notices, 2014)

Ambrosio-Tortorelli approximation of cohesive fracture models in linearized elasticity - Matteo Focardi; Flaviana Iurlano (SISSA, 2013)
We provide an approximation result in the sense of $\Gamma$-convergence for cohesive fracture energies of the form \[ \int_\Omega \mathscr{Q}_1(e(u))\,dx+a\,\mathcal{H}^{n-1}(J_u)+b\,\int_{J_u}\mathscr{Q}_0^{1/2}([u]\odot\nu_u)\,d\mathcal{H}^{n-1}, \] where $\Omega\subset{\mathbb R}^n$ is a bounded open set with Lipschitz boundary, $\mathscr{Q}_0$ and $\mathscr{Q}_1$ are coercive quadratic forms on ${\mathbb M}^{n\times n}_{sym}$, $a,\,b$ are positive constants, and $u$ runs in the space of fields $SBD^2(\Omega)$ , i.e., it's a special field with bounded deformation such that its symmetric gradient $e(u)$ is square integrable, and its jump set $J_u$ has finite $(n-1)$-Hausdorff measure in ${\mathbb R}^n$. The approximation is performed by means of Ambrosio-Tortorelli type elliptic regularizations, the prototype example being \[ \int_\Omega\Big(v|e(u)|^2+\frac{(1-v)^2}{\varepsilon}+{\gamma\,\varepsilon}|\nabla v|^2\Big)\,dx, \] where $(u,v)\in H^1(\Omega,{\mathbb R}^n){\times} H^1(\Omega)$, $\varepsilon\leq v\leq 1$ and $\gamma>0$.
Keywords: Functions of bounded deformation

Analytical validation of a continuum model for epitaxial growth with elasticity on vicinal surfaces - Gianni Dal Maso; Irene Fonseca; Giovanni Leoni (Springer, 2013)
In this paper it is shown existence of weak solutions of a variational inequality derived from the continuum model introduced by Xiang [7, formula (3.62)] (see also the work of Xiang and E [8] and Xu and Xiang [9]) to describe the self-organization of terraces and steps driven by misfit elasticity between a film and a substrate in heteroepitaxial growth. This model is obtained as a continuum limit of discrete theories of Duport, Politi, and Villain [3] and Tersoff, Phang, Zhang, and Lagally[6].
Keywords: singular nonlinear parabolic equations, Hilbert transform, thin films

An Approximation Result for Generalised Functions of Bounded Deformation and Applications to Damage Problems - Flaviana Iurlano (SISSA, 2013)
Keywords: Functions of bounded deformation

Asymptotics of the first Laplace eigenvalue with Dirichlet regions of prescribed length - Paolo Tilli; Davide Zucco (Society for Industrial and Applied Mathematics, 2013)
We consider the problem of maximizing the first eigenvalue of the $p$-Laplacian (possibly with nonconstant coefficients) over a fixed domain $Ω$, with Dirichlet conditions along $\partialΩ$ and along a supplementary set $Σ$, which is the unknown of the optimization problem. The set $Σ$, which plays the role of a supplementary stiffening rib for a membrane $Ω$, is a compact connected set (e.g., a curve or a connected system of curves) that can be placed anywhere in $\overline{Ω}$ and is subject to the constraint of an upper bound $L$ to its total length (one-dimensional Hausdorff measure). This upper bound prevents $Σ$ from spreading throughout $Ω$ and makes the problem well-posed. We investigate the behavior of optimal sets $Σ_L$ as $L→∞$ via $Γ$-convergence, and we explicitly construct certain asymptotically optimal configurations. We also study the behavior as $p→∞$ with $L$ fixed, finding connections with maximum-distance problems related to the principal frequency of the $∞$-Laplacian.

Attainment results for nematic elastomers - Virginia Agostiniani; Gianni Dal Maso; Antonio DeSimone (SISSA, 2013)
We consider a class of non-quasiconvex frame indifferent energy densities which includes Ogden-type energy densities for nematic elastomers. For the corresponding geometrically linear problem we provide an explicit minimizer of the energy functional satisfying a nontrivial boundary condition. Other attainment results, both for the nonlinear and the linearized model, are obtained by using the theory of convex integration introduced by Mueller and Sverak in the context of crystalline solids.

Basic principles of virtual element methods - Beirão da Veiga, L.; Brezzi, F.; Andrea Cangiani; Manzini, G.; Marini, L. D.; Russo, A. (Math. Models Methods Appl. Sci., 2013)

Biregular and Birational Geometry of Algebraic Varieties - Alex Massarenti (SISSA, 2013)
Every area of mathematics is characterized by a guiding problem. In algebraic geometry such problem is the classification of algebraic varieties. In its strongest form it means to classify varieties up to biregular morphisms. However, birationally equivalent varieties share many interesting properties. Therefore for any birational equivalence class it is natural to work out a variety, which is the simplest in a suitable sense, and then study these varieties. This is the aim of birational geometry. In the first part of this thesis we deal with the biregular geometry of moduli spaces of curves, and in particular with their biregular automorphisms. However, in doing this we will consider some aspects of their birational geometry. The second part is devoted to the birational geometry of varieties of sums of powers and to some related problems which will lead us to computational geometry and geometric complexity theory.
Keywords: Moduli spaces of curves, automorphisms, Hassett's moduli spaces, varieties of sums of powers

Classical W-algebras and generalized Drinfeld-Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras - Alberto De Sole; Victor G. Kac; Daniele Valeri (Communications in Mathematical Physics 323, nr. 2 (2013) 663-711, 2013)
We provide a description of the Drinfeld-Sokolov Hamiltonian reduction for the construction of classical W-algebras within the framework of Poisson vertex algebras. In this context, the gauge group action on the phase space is translated in terms of (the exponential of) a Lie conformal algebra action on the space of functions. Following the ideas of Drinfeld and Sokolov, we then establish under certain sufficient conditions the applicability of the Lenard-Magri scheme of integrability and the existence of the corresponding integrable hierarchy of bi-Hamiltonian equations.

A combination between the reduced basis method and the ANOVA expansion: On the computation of sensitivity indices - Denis Devaud; Andrea Manzoni; Gianluigi Rozza (Comptes Rendus Mathematique. Volume 351, Issue 15-16, August 2013, Pages 593-598, 2013)
We consider a method to efficiently evaluate in a real-time context an output based on the numerical solution of a partial differential equation depending on a large number of parameters. We state a result allowing to improve the computational performance of a three-step RB-ANOVA-RB method. This is a combination of the reduced basis (RB) method and the analysis of variations (ANOVA) expansion, aiming at compressing the parameter space without affecting the accuracy of the output. The idea of this method is to compute a first (coarse) RB approximation of the output of interest involving all the parameter components, but with a large tolerance on the a posteriori error estimate; then, we evaluate the ANOVA expansion of the output and freeze the least important parameter components; finally, considering a restricted model involving just the retained parameter components, we compute a second (fine) RB approximation with a smaller tolerance on the a posteriori error estimate. The fine RB approximation entails lower computational costs than the coarse one, because of the reduction of parameter dimensionality. Our result provides a criterion to avoid the computation of those terms in the ANOVA expansion that are related to the interaction between parameters in the bilinear form, thus making the RB-ANOVA-RB procedure computationally more feasible.
Keywords: Partial differential equations

Common dynamical features of sensory adaptation in photoreceptors and olfactory sensory neurons. - Giovanna De Palo; Giuseppe Facchetti; Monica Mazzolini; Anna Menini; Vincent Torre; Claudio Altafini (Nature. Scientific Reports 3, Article number: 1251, Published : 13 February 2013, 2013)
Sensory systems adapt, i.e., they adjust their sensitivity to external stimuli according to the ambient level. In this paper we show that single cell electrophysiological responses of vertebrate olfactory receptors and of photoreceptors to different input protocols exhibit several common features related to adaptation, and that these features can be used to investigate the dynamical structure of the feedback regulation responsible for the adaptation. In particular, we point out that two different forms of adaptation can be observed, in response to steps and to pairs of pulses. These two forms of adaptation appear to be in a dynamical trade-off: the more adaptation to a step is close to perfect, the slower is the recovery in adaptation to pulse pairs and viceversa. Neither of the two forms is explained by the dynamical models currently used to describe adaptation, such as the integral feedback model.

A comparative study about the effects of linear, weakly and fully nonlinear wave models on the dynamic response of offshore wind turbines - Marino, Enzo; Giovanni Stabile; Claudio Borri; Lugni, Claudio (Research and Applications in Structural Engineering, Mechanics and Computation, 2013)

Concentration of solutions for a singularly perturbed mixed problem in non-smooth domains - Serena Dipierro (Journal of Differential Equations, 2013)
We consider a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions in a bounded domain $Ω \subset \mathbb{R}^n$ whose boundary has an $(n−2)$-dimensional singularity. Assuming $1<p<\frac{n+2}{n−2}$, we prove that, under suitable geometric conditions on the boundary of the domain, there exist solutions which approach the intersection of the Neumann and the Dirichlet parts as the singular perturbation parameter tends to zero.
Keywords: Finite-dimensional reductions; Local inversion; Singularly perturbed elliptic problems

The Conformal Willmore Functional: A Perturbative Approach - Andrea Mondino (Journal of Geometric Analysis, 2013)
The conformal Willmore functional (which is conformal invariant in general Riemannian manifolds $(M,g)$ is studied with a perturbative method: the Lyapunov–Schmidt reduction. Existence of critical points is shown in ambient manifolds $(\mathbb{R}^3,g_ε)$ – where $g_ε$ is a metric close and asymptotic to the Euclidean one. With the same technique a non-existence result is proved in general Riemannian manifolds $(M,g)$ of dimension three.

Connected Sum Construction for σk-Yamabe Metrics - Giovanni Catino; Lorenzo Mazzieri (Journal of Geometric Analysis 23, nr.2 (2013), pages 812-854, 2013)
In this paper we produce families of Riemannian metrics with positive constant $σ_k$-curvature equal to $2^{-k} {n \choose k}$ by performing the connected sum of two given compact {\em non degenerate} $n$--dimensional solutions $(M_1,g_1)$ and $(M_2,g_2)$ of the (positive) $σ_k$-Yamabe problem, provided $2 ≤ 2k < n$. The problem is equivalent to solve a second order fully nonlinear elliptic equation.

Crawlers in viscous environments: linear vs nonlinear rheology - Antonio DeSimone; Federica Guarnieri; Giovanni Noselli; Amabile Tatone (International Journal of Non-Linear Mechanics 56, 142-147 (2013), 2013)
We study model self-propelled crawlers which derive their propulsive capabilities from the tangential resistance to motion offered by the environment. Two types of relationships between tangential forces and slip velocities are considered: a linear, Newtonian one and a nonlinear one of Bingham-type. Different behaviors result from the two different rheologies. These differences and their implications in terms of motility performance are discussed. Our aim is to develop new tools and insight for future studies of cell motility by crawling.

On critical behaviour in systems of Hamiltonian partial differential equations - Boris Dubrovin; Tamara Grava; Christian Klein; Antonio Moro (SISSA, 2013)
We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlev\'e-I (P$_I$) equation or its fourth order analogue P$_I^2$. As concrete examples we discuss nonlinear Schr\"odinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture.

The curvature: a variational approach - Andrei A. Agrachev; Davide Barilari; Luca Rizzi (SISSA, 2013)
The curvature discussed in this paper is a rather far going generalization of the Riemannian sectional curvature. We define it for a wide class of optimal control problems: a unified framework including geometric structures such as Riemannian, sub-Riemannian, Finsler and sub-Finsler structures; a special attention is paid to the sub-Riemannian (or Carnot-Caratheodory) metric spaces. Our construction of the curvature is direct and naive, and it is similar to the original approach of Riemann. Surprisingly, it works in a very general setting and, in particular, for all sub-Riemannian spaces.
Keywords: Crurvature, subriemannian metric, optimal control problem

Curved noncommutative torus and Gauss--Bonnet - Ludwik Dabrowski; Andrzej Sitarz (Journal of Mathematical Physics. Volume 54, Issue 1, 22 January 2013, Article number 013518, 2013)
We study perturbations of the flat geometry of the noncommutative two-dimensional torus T^2_θ (with irrational θ). They are described by spectral triples (A_θ, \H, D), with the Dirac operator D, which is a differential operator with coefficients in the commutant of the (smooth) algebra A_θ of T_θ. We show, up to the second order in perturbation, that the zeta-function at 0 vanishes and so the Gauss-Bonnet theorem holds. We also calculate first two terms of the perturbative expansion of the corresponding local scalar curvature.
Keywords: Geometry

The deal.II Library, Version 8.1 - Wolfgang Bangerth; Timo Heister; Luca Heltai; G. Kanschat; Martin Kronbichler; Matthias Maier; Bruno Turcksin; T. D. Young (SISSA, 2013)
This paper provides an overview of the new features of the finite element library deal.II version 8.0.

Defect annihilation and proliferation in active nematics - Luca Giomi; Mark J. Bowick; Xu Ma; M. Cristina Marchetti (SISSA, 2013)
Liquid crystals inevitably possess topological defect excitations generated\r\nthrough boundary conditions, applied fields or in quenches to the ordered\r\nphase. In equilibrium pairs of defects coarsen and annihilate as the uniform\r\nground state is approached. Here we show that defects in active liquid crystals\r\nexhibit profoundly different behavior, depending on the degree of activity and\r\nits contractile or extensile character. While contractile systems enhance the\r\nannihilation dynamics of passive systems, extensile systems act to drive\r\ndefects apart so that they swarm around in the manner of topologically\r\nwell-characterized self-propelled particles. We develop a simple analytical\r\nmodel for the defect dynamics which reproduces the key features of both the\r\nnumerical solutions and recent experiments on microtuble-kinesin assemblies.

On deformations of multidimensional Poisson brackets of hydrodynamic type - Matteo Casati (SISSA, 2013)
The theory of Poisson Vertex Algebras (PVAs) is a good framework to treat Hamiltonian partial differential equations. A PVA consist of a pair $(\mathcal{A},\{·_{λ}·\})$ of a differential algebra $\mathcal{A}$ and a bilinear operation called the $λ$-bracket. We extend the definition to the class of algebras $\mathcal{A}$ endowed with $d≥ 1$ commuting derivations. We call this structure a multidimensional PVA: it is a suitable setting to the study of deformations of the Poisson bracket of hydrodynamic type associated to the Euler's equation of motion of $d$-dimensional incompressible fluids. We prove that for $d=2$ all the first order deformations of such class of Poisson brackets are trivial.
Keywords: Hamiltonian operator

On the desingularization of Kahler orbifolds with constant scalar curvature - Riccardo Lena (SISSA, 2013)

Dirac operator on spinors and diffeomorphisms - Ludwik Dabrowski; Giacomo Dossena (Classical and Quantum Gravity. Volume 30, Issue 1, 7 January 2013, Article number 015006, 2013)
The issue of general covariance of spinors and related objects is reconsidered. Given an oriented manifold $M$, to each spin structure $\sigma$ and Riemannian metric $g$ there is associated a space $S_{\sigma, g}$ of spinor fields on $M$ and a Hilbert space $\HH_{\sigma, g}= L^2(S_{\sigma, g},\vol{M}{g})$ of $L^2$-spinors of $S_{\sigma, g}$. The group $\diff{M}$ of orientation-preserving diffeomorphisms of $M$ acts both on $g$ (by pullback) and on $[\sigma]$ (by a suitably defined pullback $f^*\sigma$). Any $f\in \diff{M}$ lifts in exactly two ways to a unitary operator $U$ from $\HH_{\sigma, g} $ to $\HH_{f^*\sigma,f^*g}$. The canonically defined Dirac operator is shown to be equivariant with respect to the action of $U$, so in particular its spectrum is invariant under the diffeomorphisms.
Keywords: gravity

Discontinuous Galerkin methods for mass transfer through semipermeable membranes - Andrea Cangiani; E.H. Georgoulis; Jensen, Max (SIAM J. Numer. Anal., 2013)

Dislocation dynamics in crystals: a macroscopic theory in a fractional Laplace setting - Serena Dipierro; Giampiero Palatucci; Enrico Valdinoci (SISSA, 2013)
We consider an evolution equation arising in the Peierls-Nabarro model for crystal dislocation. We study the evolution of such dislocation function and show that, at a macroscopic scale, the dislocations have the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium. These dislocation points evolve according to the external stress and an interior repulsive potential.
Keywords: nonlocal Allen-Cahn equation

Early phase of plasticity-related gene regulation and SRF dependent transcription in the hippocampus - Giovanni Iacono; Claudio Altafini; Vincent Torre (PloS one. Volume 8, Issue 7, July 2013 : e68078, 2013)
Hippocampal organotypic cultures are a highly reliable in vitro model for studying neuroplasticity: in this paper, we analyze the early phase of the transcriptional response induced by a 20 µM gabazine treatment (GabT), a GABA-Ar antagonist, by using Affymetrix oligonucleotide microarray, RT-PCR based time-course and chromatin-immuno-precipitation. The transcriptome profiling revealed that the pool of genes up-regulated by GabT, besides being strongly related to the regulation of growth and synaptic transmission, is also endowed with neuro-protective and pro-survival properties. By using RT-PCR, we quantified a time-course of the transient expression for 33 of the highest up-regulated genes, with an average sampling rate of 10 minutes and covering the time interval [10:90] minutes. The cluster analysis of the time-course disclosed the existence of three different dynamical patterns, one of which proved, in a statistical analysis based on results from previous works, to be significantly related with SRF-dependent regulation (p-value<0.05). The chromatin immunoprecipitation (chip) assay confirmed the rich presence of working CArG boxes in the genes belonging to the latter dynamical pattern and therefore validated the statistical analysis. Furthermore, an in silico analysis of the promoters revealed the presence of additional conserved CArG boxes upstream of the genes Nr4a1 and Rgs2. The chip assay confirmed a significant SRF signal in the Nr4a1 CArG box but not in the Rgs2 CArG box.

Epitaxially strained elastic films: the case of anisotropic surface energies - Marco Bonacini (ESAIM Control. Optim. Calc. Var. 19 (2013) 167-189, 2013)
In the context of a variational model for the epitaxial growth of strained elastic films, we study the effects of the presence of anisotropic surface energies in the determination of equilibrium configurations. We show that the threshold effect that describes the stability of flat morphologies in the isotropic case remains valid for weak anisotropies, but is no longer present in the case of highly anisotropic surface energies, where we show that the flat configuration is always a local minimizer of the total energy. The main tool used to obtain these results is a minimality criterion based on the positivity of the second variation.

Equilibrium measures for a class of potentials with discrete rotational symmetries - Ferenc Balogh; Dario Merzi (SISSA, 2013)
In this note the logarithmic energy problem with external potential $|z|^{2n}+tz^d+\bar{t}\bar{z}^d$ is considered in the complex plane, where $n$ and $d$ are positive integers satisfying $d≤ 2n$. Exploiting the discrete rotational invariance of the potential, a simple symmetry reduction procedure is used to calculate the equilibrium measure for all admissible values of $n,d$ and $t$. It is shown that, for fixed $n$ and $d$, there is a critical value $|t|=t_{cr}$ such that the support of the equilibrium measure is simply connected for $|t|<t_{cr}$ and has $d$ connected components for $|t|>t_{cr}$.

Existence and stability of quasi-periodic solutions for derivative wave equations - Massimiliano Berti; Luca Biasco; Michela Procesi (Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni, 2013)
In this note we present the new KAM result in [3] which proves the existence of Cantor families of small amplitude, analytic, quasi-periodic solutions of derivative wave equations, with zero Lyapunov exponents and whose linearized equation is reducible to constant coefficients. In turn, this result is derived by an abstract KAM theorem for infinite dimensional reversible dynamical systems*.
Keywords: Constant coefficients; Dynamical systems; Existence and stability; Infinite dimensional; KAM for PDEs; Linearized equations; Lyapunov exponent; Lyapunov methods; Quasi-periodic solution; Small divisors; Wave equations

Existence and symmetry results for a Schrodinger type problem involving the fractional Laplacian - Serena Dipierro; Giampiero Palatucci; Enrico Valdinoci (Le Matematiche (Catania), Vol. 68 (2013), no. 1: 201-216, 2013)
This paper deals with the following class of nonlocal Schr\"odinger equations $$ \displaystyle (-Δ)^s u + u = |u|^{p-1}u \ \ \text{in} \ \mathbb{R}^N, \quad \text{for} \ s\in (0,1). $$ We prove existence and symmetry results for the solutions $u$ in the fractional Sobolev space $H^s(\mathbb{R}^N)$. Our results are in clear accordance with those for the classical local counterpart, that is when $s=1$.

An existence result for the mean-field equation on compact surfaces in a doubly supercritical regime - Aleks Jevnikar (Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2013)

Expanded degenerations and pairs - Dan Abramovich; Charles Cadman; Barbara Fantechi; Jonathan Wise (Communications in Algebra. Volume 41, Issue 6, May 2013, Pages 2346-2386, 2013)
Since Jun Li's original definition, several other definitions of expanded pairs and expanded degenerations have appeared in the literature. We explain how these definitions are related and introduce several new variants and perspectives. Among these are the twisted expansions used by Abramovich and Fantechi as a basis for orbifold techniques in degeneation formulas.
Keywords: Expanded pairs

Fields of bounded deformation for mesoscopic dislocations - Nicolas Van Goethem (SISSA, 2013)
In this paper we discuss the consequences of the distributional approach to dislocations in terms of the mathematical properties\\r\\nof the auxiliary model fields such as displacement and displacement gradient which are obtained directly from \\r\\nthe main model field here considered as the linear strain. We show that these fields cannot be introduced rigourously without \\r\\nthe introduction of gauge fields, or equivalently, without cuts in the Riemann foliation associated to the dislocated crystal.\\r\\nIn a second step we show that the space of bounded deformations follows from the distributional approach in a natural way and \\r\\ndiscuss the reasons why it is adequate to model dislocations. The case of dislocation clusters is also addressed, as it represents an important issue in industrial crystal growth while from a mathematical point of view, peculiar phenomena might appear at the set of accumulation points. \\r\\nThe elastic-plastic decomposition of the strain within this approach is also given a precise meaning.

Fracture models as Gamma-limits of damage models - Gianni Dal Maso; Flaviana Iurlano (Communications on Pure and Applied Analysis 12 (2013) 1657-1686, 2013)
We analyze the asymptotic behavior of a variational model for damaged elastic materials. This model depends on two small parameters, which govern the width of the damaged regions and the minimum elasticity constant attained in the damaged regions. When these parameters tend to zero, we find that the corresponding functionals Gamma-converge to a functional related to fracture mechanics. The corresponding problem is brittle or cohesive, depending on the asymptotic ratio of the two parameters.

Framed sheaves on projective stacks - Ugo Bruzzo; Francesco Sala (2013)
Given a normal projective irreducible stack $\mathscr X$ over an algebraically closed field of characteristic zero we consider {\em framed sheaves} on $\mathscr X$, i.e., pairs $(\mathcal E,φ_{\mathcal E})$, where $\mathcal E$ is a coherent sheaf on $\mathscr X$ and $φ_{\mathcal E}$ is a morphism from $\mathcal E$ to a fixed coherent sheaf $\mathcal F$. After introducing a suitable notion of (semi)stability, we construct a projective scheme, which is a moduli space for semistable framed sheaves with fixed Hilbert polynomial, and an open subset of it, which is a fine moduli space for stable framed sheaves. If $\mathscr X$ is a projective irreducible orbifold of dimension two and $\mathcal F$ a locally free sheaf on a smooth divisor $\mathscr D\subset \mathscr X$ satisfying certain conditions, we consider {\em $(\mathscr{D}, \mathcal{F})$-framed sheaves}, i.e., framed sheaves $(\mathcal E,φ_{\mathcal E})$ with $\mathcal E$ a torsion-free sheaf which is locally free in a neighborhood of $\mathscr D$, and ${φ_{\mathcal{E}}}_{\vert \mathscr{D}}$ an isomorphism. These pairs are $μ$-stable for a suitable choice of a parameter entering the (semi)stability condition, and of the polarization of $\mathscr X$. This implies the existence of a fine moduli space parameterizing isomorphism classes of $(\mathscr{D}, \mathcal{F})$-framed sheaves on $\mathscr{X}$ with fixed Hilbert polynomial, which is a quasi-projective scheme. In an appendix we develop the example of stacky Hirzebruch surfaces. This is the first paper of a project aimed to provide an algebro-geometric approach to the study of gauge theories on a wide class of 4-dimensional Riemannian manifolds by means of framed sheaves on ``stacky" compactifications of them. In particular, in a subsequent paper \cite{art:bruzzopedrinisalaszabo2013} these results are used to study gauge theories on ALE spaces of type $A_k$.

Free Form Deformation Techniques Applied to 3D Shape Optimization Problems - Anwar Koshakji; Alfio Quarteroni; Gianluigi Rozza (Communications in Applied and Industrial Mathematics, 2013)
The purpose of this work is to analyse and study an efficient parametrization technique for a 3D shape optimization problem. After a brief review of the techniques and approaches already available in literature, we recall the Free Form Deformation parametrization, a technique which proved to be efficient and at the same time versatile, allowing to manage complex shapes even with few parameters. We tested and studied the FFD technique by establishing a path, from the geometry definition, to the method implementation, and finally to the simulation and to the optimization of the shape. In particular, we have studied a bulb and a rudder of a race sailing boat as model applications, where we have tested a complete procedure from Computer-Aided-Design to build the geometrical model to discretization and mesh generation.

The gap probabilities of the tacnode, Pearcey and Airy point processes, their mutual relationship and evaluation - Marco Bertola; Mattia Cafasso (Random Matrices: Theory and Applications, 2013)

Generalized Sturm-Liouville boundary conditions for first order differential systems in the plane - Alessandro Fonda; Maurizio Garrione (Topol. Methods Nonlinear Anal., 2013)
We study asymptotically positively homogeneous first order systems in the plane, with boundary conditions which are positively homogeneous, as well. Defining a generalized concept of Fučík spectrum which extends the usual one for the scalar second order equation, we prove existence and multiplicity of solutions. In this way, on one hand we extend to the plane some known results for scalar second order equations (with Dirichlet, Neumann or Sturm-Liouville boundary conditions), while, on the other hand, we investigate some other kinds of boundary value problems, where the boundary points are chosen on a polygonal line, or in a cone. Our proofs rely on the shooting method.

Genus stabilization for moduli of curves with symmetries - Fabrizio Catanese; Michael Lönne; Fabio Perroni (SISSA, 2013)
In a previous paper, arXiv:1206.5498, we introduced a new homological\r\ninvariant $\\e$ for the faithful action of a finite group G on an algebraic\r\ncurve.\r\n We show here that the moduli space of curves admitting a faithful action of a\r\nfinite group G with a fixed homological invariant $\\e$, if the genus g\' of the\r\nquotient curve is sufficiently large, is irreducible (and non empty iff the\r\nclass satisfies the condition which we define as \'admissibility\'). In the\r\nunramified case, a similar result had been proven by Dunfield and Thurston\r\nusing the classical invariant in the second homology group of G, H_2(G, \\ZZ).\r\n We achieve our result showing that the stable classes are in bijection with\r\nthe set of admissible classes $\\e$.
Keywords: group actions; mapping class group; Moduli space of curves; Teichmüller space

Implementation of the continuous-discontinuous Galerkin finite element method - Andrea Cangiani; Chapman, J.; E.H. Georgoulis; Jensen, M. (Numerical mathematics and advanced applications 2011, 2013)

An improved geometric inequality via vanishing moments, with applications to singular Liouville equations - Mauro Bardelloni; Andrea Malchiodi (Communications in Mathematical Physics 322, nr.2 (2013): 415-452, 2013)

Inversion formulae for the $\romancosh$-weighted Hilbert transform - Marco Bertola; Katsevich, A.; Alexander Tovbis (Proc. Amer. Math. Soc., 2013)

KAM theory for the Hamiltonian derivative wave equation - Massimiliano Berti; Luca Biasco; Michela Procesi (Annales Scientifiques de l'Ecole Normale Superieure, 2013)
We prove an infinite dimensional KAM theorem which implies the existence of Can- tor families of small-amplitude, reducible, elliptic, analytic, invariant tori of Hamiltonian derivative wave equations. © 2013 Société Mathématique de France.

Lipschitz Classification of Almost-Riemannian Distances on Compact Oriented Surfaces - Ugo Boscain; Grégoire Charlot; Roberta Ghezzi; Mario Sigalotti (Journal of Geometric Analysis, 2013)
Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. We consider the Carnot–Carathéodory distance canonically associated with an almost-Riemannian structure and study the problem of Lipschitz equivalence between two such distances on the same compact oriented surface. We analyze the generic case, allowing in particular for the presence of tangency points, i.e., points where two generators of the distribution and their Lie bracket are linearly dependent. The main result of the paper provides a characterization of the Lipschitz equivalence class of an almost-Riemannian distance in terms of a labeled graph associated with it.

Macroscopic contact angle and liquid drops on rough solid surfaces via homogenization and numerical simulations - Cacace, S.; Antonin Chambolle; Antonio DeSimone; Livio Fedeli (ESAIM: Mathematical Modelling and Numerical Analysis, 2013)

Minimal partitions and image classification using a gradient-free perimeter approximation - Samuel Amstutz; Nicolas Van Goethem; Antonio André Novotny (SISSA, 2013)
In this paper a new mathematically-founded method for the optimal partitioning of domains, with applications to the classification of greyscale and color images, is proposed. Since optimal partition problems are in general ill-posed, some regularization strategy is required. Here we regularize by a non-standard approximation of the total interface length, which does not involve the gradient of approximate characteristic functions, in contrast to the classical Modica-Mortola approximation. Instead, it involves a system of uncoupled linear partial differential equations and nevertheless shows $Γ$-convergence properties in appropriate function spaces. This approach leads to an alternating algorithm that ensures a decrease of the objective function at each iteration, and which always provides a partition, even during the iterations. The efficiency of this algorithm is illustrated by various numerical examples. Among them we consider binary and multilabel minimal partition problems including supervised or automatic image classification, inpainting, texture pattern identification and deblurring.
Keywords: Image classification, deblurring, optimal partitions, perimeter approximation

Minimality and stability results for a class of free-discontinuity and nonlocal isoperimetric problems - Marco Bonacini (SISSA, 2013)
Keywords: free-discontinuity problems

Monads for framed sheaves on Hirzebruch surfaces - Claudio Bartocci; Ugo Bruzzo; Claudio L.S. Rava (2013)
We define monads for framed torsion-free sheaves on Hirzebruch surfaces and use them to construct moduli spaces for these objects. These moduli spaces are smooth algebraic varieties, and we show that they are fine by constructing a universal monad.
Keywords: Monads, framed sheaves, Hirzebruch surfaces

The Monge Problem for Distance Cost in Geodesic Spaces - Stefano Bianchini; Fabio Cavalletti (Communications in Mathematical Physics, 2013)
We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dLis a geodesic Borel distance which makes (X, dL) a non branching geodesic space. We show that under the assumption that geodesics are d-continuous and locally compact, we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce two assumptions on the transport problem π which imply that the conditional probabilities of the first marginal on each geodesic are continuous or absolutely continuous w.r.t. the 1-dimensional Hausdorff distance induced by dL. It is known that this regularity is sufficient for the construction of a transport map. We study also the dynamics of transport along the geodesic, the stability of our conditions and show that in this setting dL-cyclical monotonicity is not sufficient for optimality.

Multiplicity result for a nonhomogeneous Yamabe type equation involving the Kohn Laplacian - Ali Maalaoui; Vittorio Martino (Journal of Mathematical Analysis and Applications. Volume 399, Issue 1, 1 March 2013, Pages 333-339, 2013)
Keywords: CR-Yamabe

N=2 gauge theories on toric singularities, blow-up formulae and W-algebrae - Giulio Bonelli; Kazunobu Maruyoshi; Alessandro Tanzini; Futoshi Yagi (SISSA, 2013)
We compute the Nekrasov partition function of gauge theories on the\r\n(resolved) toric singularities C^2/\\Gamma in terms of blow-up formulae. We\r\ndiscuss the expansion of the partition function in the \\epsilon_1,\\epsilon_2\r\n\\to 0 limit along with its modular properties and how to derive them from the\r\nM-theory perspective. On the two-dimensional conformal field theory side, our\r\nresults can be interpreted in terms of representations of the direct sum of\r\nHeisenberg plus W_N-algebrae with suitable central charges, which can be\r\ncomputed from the fan of the resolved toric variety.We provide a check of this\r\ncorrespondence by computing the central charge of the two-dimensional theory\r\nfrom the anomaly polynomial of M5-brane theory. Upon using the AGT\r\ncorrespondence our results provide a candidate for the conformal blocks and\r\nthree-point functions of a class of the two-dimensional CFTs which includes\r\nparafermionic theories.

A New Quadratic Potential for Scalar Conservation Laws - Stefano Bianchini; Stefano Modena (Oberwolfach Reports, 2013)

Nonabelian Lie algebroid extensions - Ugo Bruzzo; Igor Mencattini; Pietro Tortella; Vladimir Rubtsov (2013)
We classify nonabelian extensions of Lie algebroids in the holomorphic or algebraic category, and introduce and study a spectral sequence that one can attach to any such extension and generalizes the Hochschild-Serre spectral sequence associated to an ideal in a Lie algebra. We compute the differentials of the spectral sequence up to $d_2$
Keywords: Lie algebroids, nonabelian extensions, spectral sequences

Noncommutative circle bundles and new Dirac operators - Ludwik Dabrowski; Andrzej Sitarz (Communications in Mathematical Physics. Volume 318, Issue 1, 2013, Pages 111-130, 2013)
We study spectral triples over noncommutative principal U(1) bundles. Basing on the classical situation and the abstract algebraic approach, we propose an operatorial definition for a connection and compatibility between the connection and the Dirac operator on the total space and on the base space of the bundle. We analyze in details the example of the noncommutative three-torus viewed as a U(1) bundle over the noncommutative two-torus and find all connections compatible with an admissible Dirac operator. Conversely, we find a family of new Dirac operators on the noncommutative tori, which arise from the base-space Dirac operator and a suitable connection.
Keywords: Quantum principal bundles

The nonlinear multidomain model: a new formal asymptotic analysis. - Stefano Amato; Giovanni Bellettini; Maurizio Paolini (Geometry Partial Differential Equations – proceedings, CRM Series (15), 2013., 2013)
We study the asymptotic analysis of a singularly perturbed weakly parabolic system of m- equations of anisotropic reaction-diffusion type. Our main result formally shows that solutions to the system approximate a geometric motion of a hypersurface by anisotropic mean curvature. The anisotropy, supposed to be uniformly convex, is explicit and turns out to be the dual of the star-shaped combination of the m original anisotropies.
Keywords: bidomain model, anisotropic mean curvature, star-shaped combination

A note on KAM theory for quasi-linear and fully nonlinear forced KdV - P Baldi; Massimiliano Berti; Riccardo Montalto (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 24 (2013), no. 3: 437–450, 2013)
We present the recent results in [3] concerning quasi-periodic solutions for quasi-linear and fully nonlinear forced perturbations of KdV equations. For Hamiltonian or reversible nonlinearities the solutions are linearly stable. The proofs are based on a combination of di erent ideas and techniques: (i) a Nash-Moser iterative scheme in Sobolev scales. (ii) A regularization procedure, which conjugates the linearized operator to a di erential operator with constant coe cients plus a bounded remainder. These transformations are obtained by changes of variables induced by di eomorphisms of the torus and pseudo-di erential operators. (iii) A reducibility KAM scheme, which completes the reduction to constant coe cients of the linearized operator, providing a sharp asymptotic expansion of the perturbed eigenvalues.
Keywords: KAM for PDEs

A note on non-homogeneous hyperbolic operators with low-regularity coefficients - Ferruccio Colombini; Francesco Fanelli (2013)
In this paper we obtain an energy estimate for a complete strictly hyperbolic operator with second order coefficients satisfying a log-Zygmund-continuity condition with respect to $t$, uniformly with respect to $x$, and a log-Lipschitz-continuity condition with respect to $x$, uniformly with respect to $t$.

One-dimensional swimmers in viscous fluids: dynamics, controllability, and existence of optimal controls - Gianni Dal Maso; Antonio DeSimone; Marco Morandotti (SISSA, 2013)
In this paper we study a mathematical model of one-dimensional swimmers performing a planar motion while fully immersed in a viscous fluid. The swimmers are assumed to be of small size, and all inertial effects are neglected. Hydrodynamic interactions are treated in a simplified way, using the local drag approximation of resistive force theory. We prove existence and uniqueness of the solution of the equations of motion driven by shape changes of the swimmer. Moreover, we prove a controllability result showing that given any pair of initial and final states, there exists a history of shape changes such that the resulting motion takes the swimmer from the initial to the final state. We give a constructive proof, based on the composition of elementary maneuvers (straightening and its inverse, rotation, translation), each of which represents the solution of an interesting motion planning problem. Finally, we prove the existence of solutions for the optimal control problem of finding, among the histories of shape changes taking the swimmer from an initial to a final state, the one of minimal energetic cost.

Pairs of nodal solutions for a class of nonlinear problems with one-sided growth conditions - Alberto Boscaggin; Fabio Zanolin (Advanced Nonlinear Studies, 2013)

Periodic bouncing solutions for nonlinear impact oscillators - Alessandro Fonda; Andrea Sfecci (Advanced Nonlinear Studies, 2013)

Planar Hamiltonian systems at resonance: the Ahmad–Lazer–Paul condition - Alberto Boscaggin; Maurizio Garrione (Nonlinear Differential Equations and Applications NoDEA, 2013)
We consider the planar Hamiltonian system\$\$Ju^{\backslashprime} = \backslashnabla F(u) + \backslashnabla_u R(t,u), \backslashquad t \backslashin [0,T], \backslash,u \backslashin \backslashmathbb{R}^2,\$\$with F(u) positive and positively 2-homogeneous and \$\${\backslashnabla_{u}R(t, u)}\$\$sublinear in u. By means of an Ahmad-Lazer-Paul type condition, we prove the existence of a T-periodic solution when the system is at resonance. The proof exploits a symplectic change of coordinates which transforms the problem into a perturbation of a linear one. The relationship with the Landesman–Lazer condition is analyzed, as well.

Quadratic cohomology - Andrei A. Agrachev (SISSA, 2013)
We study homological invariants of smooth families of real quadratic forms as\r\na step towards a \"Lagrange multipliers rule in the large\" that intends to\r\ndescribe topology of smooth vector functions in terms of scalar Lagrange\r\nfunctions.

Quasi-periodic solutions with Sobolev regularity of NLS on Td with a multiplicative potential - Massimiliano Berti; Philippe Bolle (Journal of the European Mathematical Society, 2013)
We prove the existence of quasi-periodic solutions for Schrödinger equations with a multiplicative potential on Td , d ≥ 1, finitely differentiable nonlinearities, and tangential frequencies constrained along a pre-assigned direction. The solutions have only Sobolev regularity both in time and space. If the nonlinearity and the potential are C∞ then the solutions are C∞. The proofs are based on an improved Nash-Moser iterative scheme, which assumes the weakest tame estimates for the inverse linearized operators ("Green functions") along scales of Sobolev spaces. The key off-diagonal decay estimates of the Green functions are proved via a new multiscale inductive analysis. The main novelty concerns the measure and "complexity" estimates. © European Mathematical Society 2013.