On the third critical speed for rotating Bose-Einstein condensates - Daniele Dimonte; Michele Correggi (Correggi, M., Dimonte, D., 2016. On the third critical speed for rotating Bose-Einstein condensates. J. Math. Phys. 57, 71901, 2016)
We study a two-dimensional rotating Bose-Einstein condensate confined by an anharmonic trap in the framework of the Gross-Pitaevskii theory. We consider a rapid rotation regime close to the transition to a giant vortex state. It was proven in Correggi et al. [J. Math. Phys. 53, 095203 (2012)] that such a transition occurs when the angular velocity is of order ε−4, with ε−2 denoting the coefficient of the nonlinear term in the Gross-Pitaevskii functional and ε ≪ 1 (Thomas-Fermi regime). In this paper, we identify a finite value Ωc such that if Ω = Ω0/ε4 with Ω0 > Ωc, the condensate is in the giant vortex phase. Under the same condition, we prove a refined energy asymptotics and an estimate of the winding number of any Gross-Pitaevskii minimizer.

Towards a gauge theory interpretation of the real topological string - Hayashi, Hirotaka; Nicolò Piazzalunga; Uranga, Angel M. (Phys. Rev. D, 2016)
We consider the real topological string on certain noncompact toric Calabi-Yau three-folds $\mathbb{X}$, in its physical realization describing an orientifold of type IIA on $\mathbb{X}$ with an O4-plane and a single D4-brane stuck on top. The orientifold can be regarded as a new kind of surface operator on the gauge theory with 8 supercharges arising from the singular geometry. We use the M-theory lift of this system to compute the real Gopakumar-Vafa invariants (describing wrapped M2-brane Bogomol’nyi-Prasad-Sommerfield (BPS) states) for diverse geometries. We show that the real topological string amplitudes pick up certain signs across flop transitions, in a well-defined pattern consistent with continuity of the real BPS invariants. We further give some preliminary proposals of an intrinsically gauge theoretical description of the effect of the surface operator in the gauge theory partition function.

t-Structures are Normal Torsion Theories - Domenico Fiorenza; Fosco Loregian (Applied Categorical Structures, 2016)
We characterize $t$-structures in stable ∞-categories as suitable quasicategorical factorization systems. More precisely we show that a $t$-structure $\mathcal{t}$ on a stable $∞$-category $\mathbb{C}$ is equivalent to a normal torsion theory $F$ on $\mathbb{C}$, i.e. to a factorization system $F = (\mathcal{ε}, \mathcal{M})$ where both classes satisfy the 3-for-2 cancellation property, and a certain compatibility with pullbacks/pushouts.

t-structures on stable (infinity,1)-categories - Fosco Loregian (SISSA, 2016)
The present work re-enacts the classical theory of t-structures reducing the classical definition coming from Algebraic Geometry to a rather primitive categorical gadget: suitable reflective factorization systems (defined in the work of Rosický, Tholen, and Cassidy-Hébert-Kelly), which we call "normal torsion theories" following. A relation between these two objects has previously been noticed by other authors, on the level of the triangulated homotopy categories of stable (infinity,1)-categories. The main achievement of the present thesis is to observe and prove that this relation exists genuinely when the definition is lifted to the higher-dimensional world where the notion of triangulated category comes from.
Keywords: category theory, higher category theory, factorization system, torsion theory, homological algebra, higher algebra

Two explorations in Dynamical Systems and Mechanics - Paolo Gidoni (SISSA, 2016)
This thesis contains the work done by Paolo Gidoni during the doctorate programme in Matematical Analysis at SISSA, under the supervision of A. Fonda and A. DeSimone. The thesis is composed of two parts: "Avoiding cones conditions and higher dimensional twist" and "Directional friction in bio-inspired locomotion".
Keywords: Poincaré-Birkhoff Theorem

Viscous approximation of quasistatic evolutions for a coupled elastoplastic-damage model - Vito Crismale; Giuliano Lazzaroni (Calculus of Variations and Partial Differential Equations, 2016)
Employing the technique of vanishing viscosity and time rescaling, we show the existence of quasistatic evolutions for elastoplastic materials with incomplete damage affecting both the elastic tensor and the plastic yield surface, in a softening framework and in small strain assumptions.

Volume geodesic distortion and Ricci curvature for Hamiltonian dynamics - Andrei A. Agrachev; Davide Barilari; Elisa Paoli (arXiv preprint arXiv:1602.08745, 2016)

Young towers for product systems - Stefano Luzzatto; Marks Ruziboev (Discrete & Continuous Dynamical Systems - A, 2016)
We show that the direct product of maps with Young towers admits a Young tower whose return times decay at a rate which is bounded above by the slowest of the rates of decay of the return times of the component maps. An application of this result, together with other results in the literature, yields various statistical properties for the direct product of various classes of systems, including Lorenz-like maps, multimodal maps, piecewise $C^2$ interval maps with critical points and singularities, Hénon maps and partially hyperbolic systems.

Z2 Invariants of Topological Insulators as Geometric Obstructions - Domenico Fiorenza; Domenico Monaco; Gianluca Panati (Communications in Mathematical Physics, 2016)
We consider a gapped periodic quantum system with time-reversal symmetry of fermionic (or odd) type, i.e. the time-reversal operator squares to $-\mathbb{1}$. We investigate the existence of periodic and time-reversal invariant Bloch frames in dimensions 2 and 3. In 2d, the obstruction to the existence of such a frame is shown to be encoded in a $\mathbb{Z}_2$-valued topological invariant, which can be computed by a simple algorithm. We prove that the latter agrees with the Fu-Kane index. In 3d, instead, four $\mathbb{Z}_2$ invariants emerge from the construction, again related to the Fu-Kane-Mele indices. When no topological obstruction is present, we provide a constructive algorithm yielding explicitly a periodic and time-reversal invariant Bloch frame. The result is formulated in an abstract setting, so that it applies both to discrete models and to continuous ones.

An abstract Nash-Moser theorem and quasi-periodic solutions for NLW and NLS on compact Lie groups and homogeneous manifolds - Massimiliano Berti; Corsi, Livia; Procesi, Michela (Comm. Math. Phys., 2015)

Anisotropic mean curvature on facets and relations with capillarity - Stefano Amato; Lucia Tealdi; Giovanni Bellettini (de Gruyter, 2015)
We discuss the relations between the anisotropic calibrability of a facet F of a solid crystal E, and the capillary problem on a capillary tube with base F. When F is parallel to a facet of the Wulff shape, calibrability is equivalent to show the existence of an anisotropic subunitary vector field in $F, with suitable normal trace on the boundary of the facet, and with constant divergence equal to the anisotropic mean curvature of F. When the Wulff shape is a cylynder, assuming E convex at F, and F (strictly) calibrable, such a vector field is obtained by solving the capillary problem on F in absence of gravity and with zero contact angle. We show some examples of facets for which it is possible, even without the strict calibrability assumption, to build one of these vector fields. The construction provides, at least for convex facets of class C^{1,1}, the solution of the total variation flow starting at 1_F.

Asymptotics of orthogonal polynomials with complex varying quartic weight: global structure, critical point behavior and the first Painlevé equation - Marco Bertola; Alexander Tovbis (Constr. Approx., 2015)

Benchmarking the Immersed Finite Element Method for Fluid-Structure Interaction Problems - Roy Saswati; Luca Heltai; Francesco Costanzo (Computers and Mathematics with Applications 69 (2015) 1167–1188, 2015)
We present an implementation of a fully variational formulation of an immersed methods for fluid-structure interaction problems based on the finite element method. While typical implementation of immersed methods are characterized by the use of approximate Dirac delta distributions, fully variational formulations of the method do not require the use of said distributions. In our implementation the immersed solid is general in the sense that it is not required to have the same mass density and the same viscous response as the surrounding fluid. We assume that the immersed solid can be either viscoelastic of differential type or hyperelastic. Here we focus on the validation of the method via various benchmarks for fluid-structure interaction numerical schemes. This is the first time that the interaction of purely elastic compressible solids and an incompressible fluid is approached via an immersed method allowing a direct comparison with established benchmarks.

A bridging mechanism in the homogenisation of brittle composites with soft inclusions - Marco Barchiesi; Giuliano Lazzaroni; Caterina Ida Zeppieri (SISSA, 2015)
We provide a homogenisation result for the energy-functional associated with a purely brittle composite whose microstructure is characterised by soft periodic inclusions embedded in a stiffer matrix. We show that the two constituents as above can be suitably arranged on a microscopic scale ε to obtain, in the limit as ε tends to zero, a homogeneous macroscopic energy-functional explicitly depending on the opening of the crack.

Certified Reduced Basis Methods for Parametrized Partial Differential Equations - Jan S Hesthaven; Gianluigi Rozza; Benjamin Stamm (Springer Briefs in Mathematics, 2015)
This book provides a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations. Central aspects ranging from model construction, error estimation and computational efficiency to empirical interpolation methods are discussed in detail for coercive problems. More advanced aspects associated with time-dependent problems, non-compliant and non-coercive problems and applications with geometric variation are also discussed as examples.
Keywords: a posteriori error bounds; empirical interpolation; parametrized partial differential equations; reduced basis methods, greedy algorithms

A class of Hamiltonians for a three-particle fermionic system at unitarity - Michele Correggi; Gianfausto Dell'Antonio; Domenico Finco; Alessandro Michelangeli; Alessandro Teta (2015)
We consider a quantum mechanical three-particle system made of two identical fermions of mass one and a different particle of mass $m$, where each fermion interacts via a zero-range force with the different particle. In particular we study the unitary regime, i.e., the case of infinite two-body scattering length. The Hamiltonians describing the system are, by definition, self-adjoint extensions of the free Hamiltonian restricted on smooth functions vanishing at the two-body coincidence planes, i.e., where the positions of two interacting particles coincide. It is known that for $m$ larger than a critical value $m^* \simeq (13.607)^{-1}$ a self-adjoint and lower bounded Hamiltonian $H_0$ can be constructed, whose domain is characterized in terms of the standard point-interaction boundary condition at each coincidence plane. Here we prove that for $m\in(m^*,m^{**})$, where $m^{**}\simeq (8.62)^{-1}$, there is a further family of self-adjoint and lower bounded Hamiltonians $H_{0,β}$, $β \in \mathbb{R}$, describing the system. Using a quadratic form method, we give a rigorous construction of such Hamiltonians and we show that the elements of their domains satisfy a further boundary condition, characterizing the singular behavior when the positions of all the three particles coincide.
This SISSA preprint is composed of 29 pages and is recorded in PDF format

A compatible-incompatible decomposition of symmetric tensors in Lp with application to elasticity - Maggiani, Giovanni Battista; Riccardo Scala; Nicolas Van Goethem (Mathematical Methods in the Applied Sciences, 2015)
In this paper, we prove the Saint-Venant compatibility conditions in $L^p$ for $p\in(1,∞)$, in a simply connected domain of any space dimension. As a consequence, alternative, simple, and direct proofs of some classical Korn inequalities in Lp are provided. We also use the Helmholtz decomposition in $L^p$ to show that every symmetric tensor in a smooth domain can be decomposed in a compatible part, which is the symmetric part of a displacement gradient, and in an incompatible part, which is the incompatibility of a certain divergence-free tensor. Moreover, under a suitable Dirichlet boundary condition, this Beltrami-type decomposition is proved to be unique. This decomposition result has several applications, one of which being in dislocation models, where the incompatibility part is related to the dislocation density and where $1 < p < 2$. This justifies the need to generalize and prove these rather classical results in the Hilbertian case ($p = 2$), to the full range $p\in(1,∞)$. Copyright © 2015 John Wiley & Sons, Ltd.
Keywords: 35J58; 35Q74; compatibility conditions; elasticity; Korn inequality; strain decomposition; subclass74B05

Complexity of Control-Affine Motion Planning - Jean, F.; Dario Prandi (SIAM Journal on Control and Optimization, 2015)
In this paper we study the complexity of the motion planning problem for control-affine systems. Such complexities are already defined and rather well understood in the particular case of nonholonomic (or sub-Riemannian) systems. Our aim is to generalize these notions and results to systems with a drift. Accordingly, we present various definitions of complexity, as functions of the curve that is approximated, and of the precision of the approximation. Due to the lack of time-rescaling invariance of these systems, we consider geometric and parametrized curves separately. Then, we give some asymptotic estimates for these quantities. As a byproduct, we are able to treat the long time local controllability problem, giving quantitative estimates on the cost of stabilizing the system near a nonequilibrium point of the drift.

Constrained BV functions on double coverings for Plateau's type problems - Stefano Amato; Giovanni Bellettini; Maurizio Paolini (Adv. Calc. Var., 2015)
We link Brakke's "soap films" covering construction with the theory of finite perimeter sets, in order to study Plateau's problem without fixing a priori the topology of the solution. The minimization is set up in the class of $BV$ functions defined on a double covering space of the complement of an $(n − 2)$-dimensional smooth compact manifold $S$ without boundary. The main novelty of our approach stands in the presence of a suitable constraint on the fibers, which couples together the covering sheets. The model allows to avoid all issues concerning the presence of the boundary $S$. The constraint is lifted in a natural way to Sobolev spaces, allowing also an approach based on $Γ$-convergence theory.

Convergence rate of the Glimm scheme - Stefano Modena; Stefano Bianchini (Bulletin of the Institute of Mathematics of Academia Sinica (New Series), 2015)

Convex combinations of low eigenvalues, Fraenkel asymmetries and attainable sets - Dario Mazzoleni; Davide Zucco (SISSA, 2015)
We consider the problem of minimizing convex combinations of the first two eigenvalues of the Dirichlet-Laplacian among open set of $R^N$ of fixed measure. We show that, by purely elementary arguments, based on the minimality condition, it is possible to obtain informations on the geometry of the minimizers of convex combinations: we study, in particular, when these minimizers are no longer convex, and the optimality of balls. As an application of our results we study the boundary of the attainable set for the Dirichlet spectrum. Our techniques involve symmetry results à la Serrin, explicit constants in quantitative inequalities, as well as a purely geometrical problem: the minimization of the Fraenkel 2-asymmetry among convex sets of fixed measure.

Deal2lkit: a Toolkit Library for High Performance Programming in deal.II - Alberto Sartori; Nicola Giuliani; Mauro Bardelloni; Luca Heltai (SISSA, 2015)
We present version 1.0.0 of the deal2lkit (deal.II ToolKit) library. deal2lkit is a collection of modules and classes for the general purpose finite element library deal.II. Its principal aim is to provide a high level interface, controlled via parameter files, for those steps that are common in all finite element programs: mesh generation, selection of the finite element type, application of boundary conditions and many others. Each module can be used as a building block independently on the others, and can be integrated in existing finite element codes based on deal.II, drastically reducing the size of programs, rendering their use automatically parametrised, and reducing the overall time-to-market of finite element programming. Moreover, deal2lkit features interfaces with the SUNDIALS (SUite of Nonlinear and DIfferential/ALgebraic equation Solvers) and ASSIMP (Open Asset Import Library) libraries. Some examples are provided which show the aim and scopes of deal2lkit. The deal2lkit library is released under the GNU Lesser General Public License (LGPL) and can be retrieved from the deal2lkit repository https://github.com/mathLab/deal2lkit.

The deal.II Library, Version 8.2 - Wolfgang Bangerth; Timo Heister; Luca Heltai; G. Kanschat; Martin Kronbichler; Matthias Maier; Bruno Turcksin; T. D. Young (Archive of Numerical Software, vol. 3, n. 100, (2015), pages : 1-8, 2015)
This paper provides an overview of the new features of the finite element library deal.II version 8.2

Decay of correlations for invertible maps with non-Hölder observables - Marks Ruziboev (Dynamical Systems, 2015)
An invertible dynamical system with some hyperbolic structure is considered. Upper estimates for the correlations of continuous observables are given in terms of modulus of continuity. The result is applied to certain Hénon maps and Solenoid maps with intermittency.

A degeneration of two-phase solutions of the focusing nonlinear Schrödinger equation via Riemann-Hilbert problems - Marco Bertola; Giavedoni, Pietro (J. Math. Phys., 2015)

Dispersive deformations of the Hamiltonian structure of Euler's equations - Matteo Casati (2015)
Euler's equations for a two-dimensional system can be written in Hamiltonian form, where the Poisson bracket is the Lie-Poisson bracket associated to the Lie algebra of divergence free vector fields. We show how to derive the Poisson brackets of 2d hydrodynamics of ideal fluids as a reduction from the one associated to the full algebra of vector fields. Motivated by some recent results about the deformations of Lie-Poisson brackets of vector fields, we study the dispersive deformations of the Poisson brackets of Euler's equation and show that, up to the second order, they are trivial.

Dynamics of screw dislocations: a generalised minimising-movements scheme approach - Giovanni A. Bonaschi; Patrick Van Meurs; Marco Morandotti (SISSA, 2015)
The gradient flow structure of the model introduced in [CG99] for the dynamics of screw dislocations is investigated by means of a generalised minimising-movements scheme approach. The assumption of a finite number of available glide directions, together with the "maximal dissipation criterion" that governs the equations of motion, results into solving a differential inclusion rather than an ODE. This paper addresses how the model in [CG99] is connected to a time-discrete evolution scheme which explicitly confines dislocations to move each time step along a single glide direction. It is proved that the time-continuous model in [CG99] is the limit of these time-discrete minimising-movement schemes when the time step converges to 0. The study presented here is a first step towards a generalization of the setting in [AGS08, Chap. 2 and 3] that allows for dissipations which cannot be described by a metric.

Existence and multiplicity result for the singular Toda system - Luca Battaglia (Journal of Mathematical Analysis and Applications, 2015)
We consider the Toda system on a compact surface (Σ,g)−Δu1=2ρ1(h1eu1∫Σh1eu1dVg−1)−ρ2(h2eu2∫Σh2eu2dVg−1)−4π∑j=1Jα1j(δpj−1),−Δu2=2ρ2(h2eu2∫Σh2eu2dVg−1)−ρ1(h1eu1∫Σh1eu1dVg−1)−4π∑j=1Jα2j(δpj−1), where hi are smooth positive functions, ρi are positive real parameters, pj are given points on Σ and αij are numbers greater than −1. We give existence and multiplicity results, using variational and Morse-theoretical methods. It is the first existence result when some of the αij's are allowed to be negative."
Keywords: Existence result; Liouville-type equations; Multiplicity result; PDEs on compact surfaces; Toda system

Existence for constrained dynamic Griffith fracture with a weak maximal dissipation condition - Gianni Dal Maso; Cristopher J. Larsen; Rodica Toader (2015)
There are very few existence results for fracture evolution, outside of globally minimizing quasi-static evolutions. Dynamic evolutions are particularly problematic, due to the difficulty of showing energy balance, as well as of showing that solutions obey a maximal dissipation condition, or some similar condition that prevents stationary cracks from always being solutions. Here we introduce a new weak maximal dissipation condition and show that it is compatible with cracks constrained to grow smoothly on a smooth curve. In particular, we show existence of dynamic fracture evolutions satisfying this maximal dissipation condition, subject to the above smoothness constraints, and exhibit explicit examples to show that this maximal dissipation principle can indeed rule out stationary cracks as solutions.

Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems - Guglielmo Feltrin; Fabio Zanolin (Adv. Differential Equations 20 (2015), 937–982., 2015)
We prove the existence of positive periodic solutions for the second order nonlinear equation u'' + a(x) g(u) = 0, where g(u) has superlinear growth at zero and at infinity. The weight function a(x) is allowed to change its sign. Necessary and sufficient conditions for the existence of nontrivial solutions are obtained. The proof is based on Mawhin's coincidence degree and applies also to Neumann boundary conditions. Applications are given to the search of positive solutions for a nonlinear PDE in annular domains and for a periodic problem associated to a non-Hamiltonian equation.
AMS Subject Classification: 34B18, 34B15, 34C25, 47H11.

Existence of positive solutions of a superlinear boundary value problem with indefinite weight - Guglielmo Feltrin (Conference Publications, 2015)
We deal with the existence of positive solutions for a two-point boundary value problem associated with the nonlinear second order equation $u''+a(x)g(u)=0$. The weight $a(x)$ is allowed to change sign. We assume that the function $g\colon\mathopen[0,+∞\mathclose[\to\mathbb{R}$ is continuous, $g(0)=0$ and satisfies suitable growth conditions, including the superlinear case $g(s)=s^p$, with $p>1$. In particular we suppose that $g(s)/s$ is large near infinity, but we do not require that $g(s)$ is non-negative in a neighborhood of zero. Using a topological approach based on the Leray-Schauder degree we obtain a result of existence of at least a positive solution that improves previous existence theorems.
Keywords: boundary value problem; indefi nite weight; Positive solution; existence result.; superlinear equation

Experience on vectorizing lattice Boltzmann kernels for multi-and many-core architectures - Calore, Enrico; Nicola Demo; Schifano, Sebastiano Fabio; Tripiccione, Raffaele (International Conference on Parallel Processing and Applied Mathematics, 2015)

Explicit formulas for relaxed disarrangement densities arising from structured deformations - Ana Cristina Barroso; Jose Matias; Marco Morandotti; David R. Owen (SISSA, 2015)
Structured deformations provide a multiscale geometry that captures the contributions at the macrolevel of both smooth geometrical changes and non-smooth geometrical changes (disarrangements) at submacroscopic levels. For each (first-order) structured deformation (g,G) of a continuous body, the tensor field G is known to be a measure of deformations without disarrangements, and M:=∇g−G is known to be a measure of deformations due to disarrangements. The tensor fields G and M together deliver not only standard notions of plastic deformation, but M and its curl deliver the Burgers vector field associated with closed curves in the body and the dislocation density field used in describing geometrical changes in bodies with defects. Recently, Owen and Paroni [13] evaluated explicitly some relaxed energy densities arising in Choksi and Fonseca’s energetics of structured deformations [4] and thereby showed: (1) (trM)+ , the positive part of trM, is a volume density of disarrangements due to submacroscopic separations, (2) (trM)−, the negative part of trM, is a volume density of disarrangements due to submacroscopic switches and interpenetrations, and (3) trM, the absolute value of trM, is a volume density of all three of these non-tangential disarrangements: separations, switches, and interpenetrations. The main contribution of the present research is to show that a different approach to the energetics of structured deformations, that due to Baía, Matias, and Santos [1], confirms the roles of (trM)+, (trM)−, and trM established by Owen and Paroni. In doing so, we give an alternative, shorter proof of Owen and Paroni’s results, and we establish additional explicit formulas for other measures of disarrangements.

Extended affine Weyl groups of BCD type, Frobenius manifolds and their Landau-Ginzburg superpotentials - Boris Dubrovin; Ian A.B. Strachan; Youjin Zhang; Dafeng Zuo (SISSA, 2015)
For the root systems of type Bl, Cl and Dl, we generalize the result of [7] by showing the existence of Frobenius manifold structures on the orbit spaces of the extended affine Weyl groups that correspond to any vertex of the Dynkin diagram instead of a particular choice made in [7]. It also depends on certain additional data. We also construct LG superpotentials for these Frobenius manifold structures.

Fast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a POD-Galerkin method and a vascular shape parametrization - F. Ballarin; Elena Faggiano; Sonia Ippolito; Andrea Manzoni; Alfio Quarteroni; Gianluigi Rozza; Roberto Scrofani (2015)
In this work a reduced-order computational framework for the study of haemodynamics in three-dimensional patient-specific configurations of coronary artery bypass grafts dealing with a wide range of scenarios is proposed. We combine several efficient algorithms to face at the same time both the geometrical complexity involved in the description of the vascular network and the huge computational cost entailed by time dependent patient-specific flow simulations. Medical imaging procedures allow to reconstruct patient-specific configurations from clinical data. A centerlines-based parametrization is proposed to efficiently handle geometrical variations. POD–Galerkin reduced-order models are employed to cut down large computational costs. This computational framework allows to characterize blood flows for different physical and geometrical variations relevant in the clinical practice, such as stenosis factors and anastomosis variations, in a rapid and reliable way. Several numerical results are discussed, highlighting the computational performance of the proposed framework, as well as its capability to perform sensitivity analysis studies, so far out of reach.

FEM SUPG stabilisation of mixed isoparametric BEMs: application to linearised free surface flows - Nicola Giuliani; Andrea Mola; Luca Heltai; L. Formaggia (Engineering Analysis with Boundary Elements 59 (2015), pp. 8-22, 2015)
In finite element formulations, transport dominated problems are often stabilised through the Streamline-Upwind-Petrov–Galerkin (SUPG) method. Its application is straightforward when the problem at hand is solved using Galerkin methods. Applications of boundary integral formulations often resort to collocation techniques which are computationally more tractable. In this framework, the Galerkin method and the stabilisation may still be used to successfully apply boundary conditions and resolve instabilities that are frequently observed in transport dominated problems. We apply this technique to an adaptive collocation boundary element method for the solution of stationary potential flows, where we solve a mixed Poisson problem in boundary integral form, with the addition of linearised free surface boundary conditions. We use a mixed boundary element formulation to allow for different finite dimensional spaces describing the flow potential and its normal derivative, and we validate our method simulating the flow around both a submerged body and a surface piercing body. The coupling of mixed surface finite elements and strongly consistent stabilisation techniques with boundary elements opens up the possibility to use non conformal unstructured grids with local refinement, without introducing the inconsistencies of other stabilisation techniques based on up-winding and finite difference schemes.

A general existence result for the Toda system on compact surfaces - Luca Battaglia; Aleks Jevnikar; Andrea Malchiodi; David Ruiz (Advances in Mathematics, 2015)
In this paper we consider the following Toda system of equations on a compact surface:−Δu1=2ρ1(h1eu1∫Σh1eu1dVg−1)−ρ2(h2eu2∫Σh2eu2dVg−1)−Δu1=−4π∑j=1mα1,j(δpj−1),−Δu2=2ρ2(h2eu2∫Σh2eu2dVg−1)−ρ1(h1eu1∫Σh1eu1dVg−1)−Δu2=−4π∑j=1mα2,j(δpj−1), which is motivated by the study of models in non-abelian Chern–Simons theory. Here h1,h2 are smooth positive functions, ρ1,ρ2 two positive parameters, pi points of the surface and α1,i,α2,j non-negative numbers. We prove a general existence result using variational methods. The same analysis applies to the following mean field equation−Δu=ρ1(heu∫ΣheudVg−1)−ρ2(he−u∫Σhe−udVg−1), which arises in fluid dynamics."
Keywords: Geometric PDEs; Min–max schemes; Variational methods

Geodesics and horizontal-path spaces in Carnot groups - Andrei A. Agrachev; Alessandro Gentile; Antonio Lerario (Geometry & Topology, 2015)
We study properties of the space of horizontal paths joining the origin with a vertical point on a generic two-step Carnot group. The energy is a Morse-Bott functional on paths and its critical points (sub-Riemannian geodesics) appear in families (compact critical manifolds) with controlled topology. We study the asymptotic of the number of critical manifolds as the energy grows. The topology of the horizontal-path space is also investigated, and we find asymptotic results for the total Betti number of the sublevels of the energy as it goes to infinity. We interpret these results as local invariants of the sub-Riemannian structure.

Geometric phases in graphene and topological insulators - Domenico Monaco (SISSA, 2015)
This thesis collects three of the publications that the candidate produced during his Ph.D. studies. They all focus on geometric phases in solid state physics. We first study topological phases of 2-dimensional periodic quantum systems, in absence of a spectral gap, like e.g. (multilayer) graphene. A topological invariant n_v in Z, baptized eigenspace vorticity, is attached to any intersection of the energy bands, and characterizes the local topology of the eigenprojectors around that intersection. With the help of explicit models, each associated to a value of n_v in Z, we are able to extract the decay at infinity of the single-band Wannier function w in mono- and bilayer graphene, obtaining |w(x)| <= const |x|^{-2} as |x| tends to infinity. Next, we investigate gapped periodic quantum systems, in presence of time-reversal symmetry. When the time-reversal operator Theta is of bosonic type, i.e. it satisfies Theta^2 = 1, we provide an explicit algorithm to construct a frame of smooth, periodic and time-reversal symmetric (quasi-)Bloch functions, or equivalently a frame of almost-exponentially localized, real-valued (composite) Wannier functions, in dimension d <= 3. In the case instead of a fermionic time-reversal operator, satisfying Theta^2 = -1, we show that the existence of such a Bloch frame is in general topologically obstructed in dimension d=2 and d=3. This obstruction is encoded in Z_2-valued topological invariants, which agree with the ones proposed in the solid state literature by Fu, Kane and Mele.
Keywords: Geometric phases, graphene, topological insulators, Wannier functions, Bloch frames

Gibbs-Markov-Young Structures and Decay of Correlations - Marks Ruziboev (SISSA, 2015)
In this work we study mixing properties of discrete dynamical systems and related to them geometric structure. In the first chapter we show that the direct product of maps with Young towers admits a Young tower whose return times decay at a rate which is bounded above by the slowest of the rates of decay of the return times of the component maps. An application of this result, together with other results in the literature, yields various statistical properties for the direct product of various classes of systems, including Lorenz-like maps, multimodal maps, piecewise $C^2$ interval maps with critical points and singularities, H\'enon maps and partially hyperbolic systems. The second chapter is dedicated to the problem of decay of correlations for continuous observables. First we show that if the underlying system admits Young tower then the rate of decay of correlations for continuous observables can be estimated in terms of modulus of continuity and the decay rate of tail of Young tower. In the rest of the second chapter we study the relations between the rates of decay of correlations for smooth observables and continuous observables. We show that if the rates of decay of correlations is known for $C^r,$ observables ($r\ge 1$) then it is possible to obtain decay of correlations for continuous observables in terms of modulus of continuity.
Keywords: Decay of Correlations, GMY-towers

Gli abachi: antichi strumenti precursori delle moderne macchine da calcolo - Giuliano Klun (2015)

Global well-posedness of the magnetic Hartree equation with non-Strichartz external fields - Alessandro Michelangeli (SISSA, 2015)
We study the magnetic Hartree equation with external fields to which magnetic Strichartz estimates are not necessarily applicable. We characterise the appropriate notion of energy space and in such a space we prove the global well-posedness of the associated initial value problem by means of energy methods only.

Hilbert schemes of points of OP1(-n) as quiver varieties - Ugo Bruzzo (arXiv:1504.02987 [math.AG], 2015)
Relying on a representation of framed torsion-free sheaves on Hirzebruch surfaces in terms of monads, we construct ADHM data for the Hilbert scheme of points of the total space of the line bundle $\mathcal O(-n)$ on $\mathbb P^1$. This ADHM description is then used to realize these Hilbert schemes as quiver varieties.

Homogenization problems in the Calculus of Variations: an overview - Jose Matias; Marco Morandotti (SISSA, 2015)
In this note we present a brief overview of variational methods to solve homogenization problems. The purpose is to give a first insight on the subject by presenting some fundamental theoretical tools, both classical and modern. We conclude by mentioning some open problems.
DEDICATED TO PROF. ORLANDO LOPES

Hourglass stabilization and the virtual element method - Andrea Cangiani; Manzini, G.; Russo, A.; Sukumar, N. (Internat. J. Numer. Methods Engrg., 2015)

Hourglass stabilization and the virtual element method - Andrea Cangiani; Manzini, G.; Russo, A.; Sukumar, N. (International Journal for Numerical Methods in Engineering, 2015)
SummaryIn this paper, we establish the connections between the virtual element method (VEM) and the hourglass control techniques that have been developed since the early 1980s to stabilize underintegrated C0 Lagrange finite element methods. In the VEM, the bilinear form is decomposed into two parts: a consistent term that reproduces a given polynomial space and a correction term that provides stability. The essential ingredients of -continuous VEMs on polygonal and polyhedral meshes are described, which reveals that the variational approach adopted in the VEM affords a generalized and robust means to stabilize underintegrated finite elements. We focus on the heat conduction (Poisson) equation and present a virtual element approach for the isoparametric four-node quadrilateral and eight-node hexahedral elements. In addition, we show quantitative comparisons of the consistency and stabilization matrices in the VEM with those in the hourglass control method of Belytschko and coworkers. Numerical examples in two and three dimensions are presented for different stabilization parameters, which reveals that the method satisfies the patch test and delivers optimal rates of convergence in the L2 norm and the H1 seminorm for Poisson problems on quadrilateral, hexahedral, and arbitrary polygonal meshes. Copyright © 2015 John Wiley & Sons, Ltd.
Keywords: consistency matrix; hourglass control; polygonal and polyhedral finite elements; stabilization matrix; underintegration; virtual element method

Integrability of Continuous Tangent Sub-bundles - Sina Türeli (SISSA, 2015)
In this thesis, the main aim is to study the integrability properties of continuous tangent sub-bundles, especially those that arise in the study of dynamical systems. After the introduction and examples part we start by studying integrability of such sub-bundles under different regularity and dynamical assumptions. Then we formulate a continuous version of the classical Frobenius theorem and state some applications to such bundles, to ODE and PDE. Finally we close of by stating some ongoing work related to interactions between integrability, sub-Riemannian geometry and contact geometry.
Keywords: Dynamical Systems, Global Analysis, Frobenius Theorem, Integrability

Interaction functionals, Glimm approximations and Lagrangian structure of BV solutions for Hyperbolic Systems of Conservations Laws - Stefano Modena (SISSA, 2015)
This thesis is a contribution to the mathematical theory of Hyperbolic Conservation Laws. Three are the main results which we collect in this work. The first and the second result (denoted in the thesis by Theorem A and Theorem B respectively) deal with the following problem. The most comprehensive result about existence, uniqueness and stability of the solution to the Cauchy problem \begin{equation}\tag{$\mathcal C$} \label{E:abstract} \begin{cases} u_t + F(u)_x = 0, \\u(0, x) = \bar u(x), \end{cases} \end{equation} where $F: \R^N \to \R^N$ is strictly hyperbolic, $u = u(t,x) \in \R^N$, $t \geq 0$, $x \in \R$, $\TV(\bar u) \ll 1$, can be found in [Bianchini, Bressan 2005], where the well-posedness of \eqref{E:abstract} is proved by means of vanishing viscosity approximations. After the paper [Bianchini, Bressan 2005], however, it seemed worthwhile to develop a \emph{purely hyperbolic} theory (based, as in the genuinely nonlinear case, on Glimm or wavefront tracking approximations, and not on vanishing viscosity parabolic approximations) to prove existence, uniqueness and stability results. The reason of this interest can be mainly found in the fact that hyperbolic approximate solutions are much easier to study and to visualize than parabolic ones. Theorems A and B in this thesis are a contribution to this line of research. In particular, Theorem A proves an estimate on the change of the speed of the wavefronts present in a Glimm approximate solution when two of them interact; Theorem B proves the convergence of the Glimm approximate solutions to the weak admissible solution of \eqref{E:abstract} and provides also an estimate on the rate of convergence. Both theorems are proved in the most general setting when no assumption on $F$ is made except the strict hyperbolicity. The third result of the thesis, denoted by Theorem C, deals with the Lagrangian structure of the solution to \eqref{E:abstract}. The notion of Lagrangian flow is a well-established concept in the theory of the transport equation and in the study of some particular system of conservation laws, like the Euler equation. However, as far as we know, the general system of conservations laws \eqref{E:abstract} has never been studied from a Lagrangian point of view. This is exactly the subject of Theorem C, where a Lagrangian representation for the solution to the system \eqref{E:abstract} is explicitly constructed. The main reasons which led us to look for a Lagrangian representation of the solution of \eqref{E:abstract} are two: on one side, this Lagrangian representation provides the continuous counterpart in the exact solution of \eqref{E:abstract} to the well established theory of wavefront approximations; on the other side, it can lead to a deeper understanding of the behavior of the solutions in the general setting, when the characteristic field are not genuinely nonlinear or linearly degenerate.
Keywords: Hyperbolic conservation laws

Liquid crystal elastomer strips as soft crawlers - Antonio DeSimone; Paolo Gidoni; Giovanni Noselli (Journal of the Mechanics and Physics of Solids, 2015)
In this paper, we speculate on a possible application of Liquid Crystal Elastomers to the field of soft robotics. In particular, we study a concept for limbless locomotion that is amenable to miniaturisation. For this purpose, we formulate and solve the evolution equations for a strip of nematic elastomer, subject to directional frictional interactions with a flat solid substrate, and cyclically actuated by a spatially uniform, time-periodic stimulus (e.g., temperature change). The presence of frictional forces that are sensitive to the direction of sliding transforms reciprocal, ‘breathing-like’ deformations into directed forward motion. We derive formulas quantifying this motion in the case of distributed friction, by solving a differential inclusion for the displacement field. The simpler case of concentrated frictional interactions at the two ends of the strip is also solved, in order to provide a benchmark to compare the continuously distributed case with a finite-dimensional benchmark. We also provide explicit formulas for the axial force along the crawler body.
Keywords: Crawling motility; Directional surfaces; Frictional interactions; Liquid crystal elastomers; Soft biomimetic robots

Mathematical Models of Locomotion: Legged Crawling, Snake-like Motility, and Flagellar Swimming - Giancarlo Cicconofri (SISSA, 2015)
Keywords: Motility

Meromorphic differentials with imaginary periods on degenerating hyperelliptic curves - Marco Bertola; Alexander Tovbis (Anal. Math. Phys., 2015)

Model order reduction of parameterized systems (MoRePaS): Preface to the special issue of advances in computational mathematics - Peter Benner; Mario Ohlberger; Anthony Patera; Gianluigi Rozza; Sorensen, D.C.; Karsten Urban (Advances in Computational Mathematics, 2015)

Motility of a model bristle-bot: A theoretical analysis - Giancarlo Cicconofri; Antonio DeSimone (International Journal of Non-Linear Mechanics, 2015)
Bristle-bots are legged robots that can be easily made out of a toothbrush head and a small vibrating engine. Despite their simple appearance, the mechanism enabling them to propel themselves by exploiting friction with the substrate is far from trivial. Numerical experiments on a model bristle-bot have been able to reproduce such a mechanism revealing, in addition, the ability to switch direction of motion by varying the vibration frequency. This paper provides a detailed account of these phenomena through a fully analytical treatment of the model. The equations of motion are solved through an expansion in terms of a properly chosen small parameter. The convergence of the expansion is rigorously proven. In addition, the analysis delivers formulas for the average velocity of the robot and for the frequency at which the direction switch takes place. A quantitative description of the mechanism for the friction modulation underlying the motility of the bristle-bot is also provided.
Keywords: Bristle-robots; Crawling motility; Frictional interactions

Multidimensional Poisson Vertex Algebras and Poisson cohomology of Hamiltonian operators of hydrodynamic type - Matteo Casati (SISSA, 2015)
The Poisson brackets of hydrodynamic type, also called Dubrovin-Novikov brackets, constitute the Hamiltonian structure of a broad class of evolutionary PDEs, that are ubiquitous in the theory of Integrable Systems, ranging from Hopf equation to the principal hierarchy of a Frobenius manifold. They can be regarded as an analogue of the classical Poisson brackets, defined on an infinite dimensional space of maps Σ → M between two manifolds. Our main problem is the study of Poisson-Lichnerowicz cohomology of such space when dim Σ > 1. We introduce the notion of multidimensional Poisson Vertex Algebras, generalizing and adapting the theory by A. Barakat, A. De Sole, and V. Kac [Poisson Vertex Algebras in the theory of Hamiltonian equations, 2009]; within this framework we explicitly compute the first nontrivial cohomology groups for an arbitrary Poisson bracket of hydrodynamic type, in the case dim Σ = dim M = 2. For the case of the so-called scalar brackets, namely the ones for which dim M = 1, we give a complete description on their Poisson–Lichnerowicz cohomology. From this computations it follows, already in the particular case dim Σ = 2, that the cohomology is infinite dimensional.
161 pages
Keywords: Poisson Vertex Algebras, Poisson brackets, Hamiltonian operators, Integrable Systems

Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations - Gianluigi Rozza; Peng Chen; Alfio Quarteroni (Numerische Mathematik, (2015), 36 p. Article in Press, 2015)
In this paper we develop and analyze a multilevel weighted reduced basis method for solving stochastic optimal control problems constrained by Stokes equations. We prove the analytic regularity of the optimal solution in the probability space under certain assumptions on the random input data. The finite element method and the stochastic collocation method are employed for the numerical approximation of the problem in the deterministic space and the probability space, respectively, resulting in many large-scale optimality systems to solve. In order to reduce the unaffordable computational effort, we propose a reduced basis method using a multilevel greedy algorithm in combination with isotropic and anisotropic sparse-grid techniques. A weighted a posteriori error bound highlights the contribution stemming from each method. Numerical tests on stochastic dimensions ranging from 10 to 100 demonstrate that our method is very efficient, especially for solving high-dimensional and large-scale optimization problems.

Multiple positive solutions for a superlinear problem: a topological approach - Guglielmo Feltrin; Fabio Zanolin (J. Differential Equations 259 (2015), 925–963., 2015)
We study the multiplicity of positive solutions for a two-point boundary value problem associated to the nonlinear second order equation u''+f(x,u)=0. We allow x ↦ f(x,s) to change its sign in order to cover the case of scalar equations with indefinite weight. Roughly speaking, our main assumptions require that f(x,s)/s is below λ_1 as s→0^+ and above λ_1 as s→+∞. In particular, we can deal with the situation in which f(x,s) has a superlinear growth at zero and at infinity. We propose a new approach based on the topological degree which provides the multiplicity of solutions. Applications are given for u'' + a(x) g(u) = 0, where we prove the existence of 2^n-1 positive solutions when a(x) has n positive humps and a^-(x) is sufficiently large.
Work presented at the "Special Session 21" of the "10th AIMS Conference on Dynamical Systems, Differential Equations and Applications" (Madrid, July 7-11, 2014).

N=2 supersymmetric gauge theories on S^2xS^2 and Liouville Gravity - Aditya Bawane; Giulio Bonelli; Massimiliano Ronzani; Alessandro Tanzini (Journal of High Energy Physics, 2015)
We consider $\mathcal{N}=2$ supersymmetric gauge theories on four manifolds admitting an isometry. Generalized Killing spinor equations are derived from the consistency of supersymmetry algebrae and solved in the case of four manifolds admitting a $U(1)$ isometry. This is used to explicitly compute the supersymmetric path integral on $S^2 × S^2$ via equivariant localization. The building blocks of the resulting partition function are shown to contain the three point functions and the conformal blocks of Liouville Gravity.

A Nash-Moser approach to KAM theory - Massimiliano Berti; Bolle, Philippe (Hamiltonian partial differential equations and applications, 2015)

Normal matrix models and orthogonal polynomials for a class of potentials with discrete rotational symmetries - Dario Merzi (SISSA, 2015)
In this thesis we are going to study normal random matrix models which generalize naturally the polynomially perturbed Ginibre ensamble, focusing in particular on their eigenvalue distribution and on the asymptotics of the associated orthogonal polynomials. \\ The main result we are going to present are the following: \begin{itemize} \item we describe the explicit derivation of the equilibrium measure for a class of potentials with discrete rotational symmetries, namely of the form \[V(z)=|z|^{2n}-t(z^{d}+\bar{z}^{d})\qquad n,d\in\mathbb{N},\ \ d\leq2n\ \ t>0 .\] \item We obtain the strong asymptotics for the orthogonal polynomials associated to the weight \[ e^{-NV(z)},\quad V(z)=|z|^{2s}-t(z^s+\bar{z}^{s}) \qquad z \in \mathbb{C},\;s\in \mathbb{N},\quad t>0,\] and we will show how the density of their zeroes is related to the eigenvalue distribution of the corresponding matrix model; \item We show how the conformal maps used to describe the support of the equilibrium measure for polynomial perturbation of the potential $V(z)=|z|^{2n}$ lead to a natural generalization of the concept of polynomial curves introduced in by Elbau. \end{itemize}
Keywords: Mathematical Physics

A note on compactness properties of the singular Toda system - Luca Battaglia; Gabriele Mancini (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur., 2015)
In this note, we consider blow-up for solutions of the SU(3) Toda system on compact surfaces. In particular, we give a complete proof of a compactness result stated by Jost, Lin and Wang and we extend it to the case of singular systems. This is a necessary tool to find solutions through variational methods.

Onofri-Type Inequalities for Singular Liouville Equations - Gabriele Mancini (Springer US, 2015)
We study the blow-up behavior of minimizing sequences for the singular Moser–Trudinger functional on compact surfaces. Assuming non-existence of minimum points, we give an estimate for the infimum value of the functional. This result can be applied to give sharp Onofri-type inequalities on the sphere in the presence of at most two singularities.

The partition function of the extended $r$-reduced Kadomtsev-Petviashvili hierarchy - Marco Bertola; Di Yang (J. Phys. A, 2015)

A permanence theorem for local dynamical systems - Alessandro Fonda; Paolo Gidoni (Nonlinear Analysis: Theory, Methods & Applications, 2015)
We provide a necessary and sufficient condition for permanence related to a local dynamical system on a suitable topological space. We then present an illustrative application to a Lotka–Volterra predator–prey model with intraspecific competition.
Nonlinear Partial Differential Equations, in honor of Enzo Mitidieri for his 60th birthday
Keywords: Lotka–Volterra; permanence; Predator–prey; Uniform persistence

The phototransduction machinery in the rod outer segment has a strong efficacy gradient - Monica Mazzolini; Giuseppe Facchetti; L. Andolfi; R. Proietti Zaccaria; S. Tuccio; J. Treud; Claudio Altafini; Enzo M. Di Fabrizio; Marco Lazzarino; G. Rapp; Vincent Torre (National Academy of Sciences, 2015)
Open Access article

Poisson cohomology of scalar multidimensional Dubrovin-Novikov brackets - Guido Carlet; Matteo Casati; Sergey Shadrin (2015)
We compute the Poisson cohomology of a scalar Poisson bracket of Dubrovin-Novikov type with D independent variables. We find that the second and third cohomology groups are generically non-vanishing in D>1. Hence, in contrast with the D=1 case, the deformation theory in the multivariable case is non-trivial.

Principal circle bundles, Pimsner algebras and Gysin sequences - Francesca Arici (SISSA, 2015)
Principal circle bundles and Gysin sequences play a crucial role in mathematical physics, in particular in Chern-Simons theories and T-duality. This works focuses on the noncommutative topology of principal circle bundles: we investigate the connections between noncommutative principal circle bundles, Pimsner algebras and strongly graded algebras. At the C*-algebraic level, we start from a self-Morita equivalence bimodule E for a C*-algebra B which we think of as a non commutative line bundle over the `base space’ algebra B. The corresponding Pimsner algebra O_E, is then the total space algebra of an associated circle bundle. A natural six term exact sequence, an analogue of the Gysin sequence for circle bundles, relates the KK-theories of O_E and of the base space B. We illustrate several results with the examples of quantum weighted projective and lens spaces.

Quadratic Interaction Functional for General Systems of Conservation Laws - Stefano Bianchini; Stefano Modena (Communications in Mathematical Physics, 2015)
For the Glimm scheme approximation to the solution of the system of conservation laws in one space dimension with initial data u 0 with small total variation, we prove a quadratic (w.r.t. Tot. Var. ( u 0)) interaction estimate, which has been used in the literature for stability and convergence results. No assumptions on the structure of the flux f are made (apart from smoothness), and this estimate is the natural extension of the Glimm type interaction estimate for genuinely nonlinear systems. More precisely, we obtain the following results: a new analysis of the interaction estimates of simple waves;

Reduced basis approximation and a-posteriori error estimation for the coupled Stokes-Darcy system - Immanuel Martini; Gianluigi Rozza; Bernard Haasdonk (Advances in Computational Mathematics, 2015)
The coupling of a free flow with a flow through porous media has many potential applications in several fields related with computational science and engineering, such as blood flows, environmental problems or food technologies. We present a reduced basis method for such coupled problems. The reduced basis method is a model order reduction method applied in the context of parametrized systems. Our approach is based on a heterogeneous domain decomposition formulation, namely the Stokes-Darcy problem. Thanks to an offline/online-decomposition, computational times can be drastically reduced. At the same time the induced error can be bounded by fast evaluable a-posteriori error bounds. In the offline-phase the proposed algorithms make use of the decomposed problem structure. Rigorous a-posteriori error bounds are developed, indicating the accuracy of certain lifting operators used in the offline-phase as well as the accuracy of the reduced coupled system. Also, a strategy separately bounding pressure and velocity errors is extended. Numerical experiments dealing with groundwater flow scenarios demonstrate the efficiency of the approach as well as the limitations regarding a-posteriori error estimation.
Keywords: Domain decomposition; Error estimation; Non-coercive problem; Porous medium equation; Reduced basis method; Stokes flow

Reduced basis approximation of parametrized advection-diffusion PDEs with high Péclet number - Pacciarini, P.; Gianluigi Rozza (Lecture Notes in Computational Science and Engineering, 2015)
In this work we show some results about the reduced basis approximation of advection dominated parametrized problems, i.e. advection-diffusion problems with high Péclet number. These problems are of great importance in several engineering applications and it is well known that their numerical approximation can be affected by instability phenomena. In this work we compare two possible stabilization strategies in the framework of the reduced basis method, by showing numerical results obtained for a steady advection-diffusion problem.

Reduced basis approximation of parametrized optimal flow control problems for the Stokes equations - Federico Negri; Andrea Manzoni; Gianluigi Rozza (Computers and Mathematics with Applications, 2015)
This paper extends the reduced basis method for the solution of parametrized optimal control problems presented in Negri et al. (2013) to the case of noncoercive (elliptic) equations, such as the Stokes equations. We discuss both the theoretical properties-with particular emphasis on the stability of the resulting double nested saddle-point problems and on aggregated error estimates-and the computational aspects of the method. Then, we apply it to solve a benchmark vorticity minimization problem for a parametrized bluff body immersed in a two or a three-dimensional flow through boundary control, demonstrating the effectivity of the methodology.

Reduced Basis Isogeometric Methods (RB-IGA) for the real-time simulation of potential flows about parametrized NACA airfoils - Andrea Manzoni; Filippo Salmoiraghi; Luca Heltai (Comput Methods Appl Mech Eng. 2015;284:1147–1180, 2015)
We present a Reduced Basis (RB) method based on Isogeometric Analysis (IGA) for the rapid and reliable evaluation of PDE systems characterized by complex geometrical features. At the current state of the art, this is the first case of coupling between RB and IGA methods. The construction of the RB method relies on an Isogeometric Boundary Element Method (IGA-BEM) as the high-fidelity technique, allowing a direct interface with Computer Aided Design (CAD) tools. A suitable Empirical Interpolation Method (EIM) ensures an efficient offline/online decomposition between the construction and the evaluation of the RB method. We consider the real-time simulation of potential flows past airfoils, parametrized with respect to the angle of attack and the NACA number identifying their shape, and we provide a validation of our methodology with respect to experimental data and reference numerical codes, showing in both cases a very good agreement.We present a Reduced Basis (RB) method based on Isogeometric Analysis (IGA) for the rapid and reliable evaluation of PDE systems characterized by complex geometrical features. At the current state of the art, this is the first case of coupling between RB and IGA methods. The construction of the RB method relies on an Isogeometric Boundary Element Method (IGA-BEM) as the high-fidelity technique, allowing a direct interface with Computer Aided Design (CAD) tools. A suitable Empirical Interpolation Method (EIM) ensures an efficient offline/online decomposition between the construction and the evaluation of the RB method. We consider the real-time simulation of potential flows past airfoils, parametrized with respect to the angle of attack and the NACA number identifying their shape, and we provide a validation of our methodology with respect to experimental data and reference numerical codes, showing in both cases a very good agreement.

A Reduced Order Model for the Simulation of Mooring Cable Dynamics - Giovanni Stabile; Hermann G. Matthies; Claudio Borri (Computational Methods in Marine Engineering VI – MARINE2015, 2015)

The relaxed area of maps from the plane to the plane with a line discontinuity, and the role of semicartesian surfaces. - Lucia Tealdi (SISSA, 2015)
In this thesis we study the relaxation of the area functional w.r.t. the L^1 topology of a map from a bounded planar domain with values in the plane and jumping on a segment. We estimate from above the singular contribution of this functional due to the presence of the jump in terms of the infimum of the area among a suitable family of surfaces that we call semicartesian surfaces. In our analysis, we also introduce a different notion of area, namely the relaxation of the area w.r.t. a convergence stronger than the L^1 convergence, whose singular contribution is completely characterized in terms of suitable semicartesian area minimizing problems. We propose also some examples of maps for which the two notions of relaxation are different: these examples underline the highly non-local behaviour of the L^1-relaxation, and justify the introduction of the other functional. Some result about the existence of a semicartesian area-minimizing surface is also provided.
Keywords: Area functional

Results on the minimization of the Dirichlet functional among semicartesian parametrizations - Lucia Tealdi; Giovanni Bellettini; Maurizio Paolini (2015)
We start to investigate the existence of conformal minimizers for the Dirichlet functional in the setting of the so-called semicartesian parametrizations, adapting to this context some techniques used in solving the classical Plateau's problem. The final goal is to find area minimizing semicartesian parametrizations spanning a Jordan curve obtained as union of two graphs; this problem appeared in the study of the relaxed area functional for maps from the plane to the plane jumping on a line.
The article is compsed of 18 pages and is recorded in PDF format

Rigidity of three-dimensional lattices and dimension reduction in heterogeneous nanowires - Giuliano Lazzaroni; Mariapia Palombaro; Anja Schlomerkemper (SISSA, 2015)
In the context of nanowire heterostructures we perform a discrete to continuum limit of the corresponding free energy by means of Γ-convergence techniques. Nearest neighbours are identified by employing the notions of Voronoi diagrams and Delaunay triangulations. The scaling of the nanowire is done in such a way that we perform not only a continuum limit but a dimension reduction simultaneously. The main part of the proof is a discrete geometric rigidity result that we announced in an earlier work and show here in detail for a variety of three-dimensional lattices. We perform the passage from discrete to continuum twice: once for a system that compensates a lattice mismatch between two parts of the heterogeneous nanowire without defects and once for a system that creates dislocations. It turns out that we can verify the experimentally observed fact that the nanowires show dislocations when the radius of the specimen is large

Schödinger operators on half-line with shrinking potentials at the origin - Gianfausto Dell'Antonio; Alessandro Michelangeli (SISSA, 2015)
We discuss the general model of a Schrödinger quantum particle constrained on a straight half-line with given self-adjoint boundary condition at the origin and an interaction potential supported around the origin. We study the limit when the range of the potential scales to zero and its magnitude blows up. We show that in the limit the dynamics is generated by a self-adjoint negative Laplacian on the half-line, with a possible preservation or modification of the boundary condition at the origin, depending on the magnitude of the scaling and of the strength of the potential.

Semicartesian surfaces and the relaxed area of maps from the plane to the plane with a line discontinuity - Lucia Tealdi; Giovanni Bellettini; Maurizio Paolini (2015)
We address the problem of estimating the area of the graph of a map u, defined on a bounded planar domain O and taking values in the plane, jumping on a segment J, either compactly contained in O or having both the end points on the boundary of O. We define the relaxation of the area functional w.r.t. a sort of uniform convergence, and we characterize it in terms of the infimum of the area among those surfaces in the space spanning the graphs of the traces of u on the two side of J and having what we have called a semicartesian structure. We exhibit examples showing that the relaxed area functional w.r.t the L^1 convergence may depend also on the values of u far from J, and on the relative position of J w.r.t. the boundary of O; these examples confirm the non-local behaviour of the L^1 relaxed area functional, and justify the interest in studying the relaxation w.r.t. a stronger convergence. We prove also that the L^1 relaxed area functional in non-subadditive for a rather class of maps.
The preprint is compsed of 37 pages and is recorded in PDF format

Sharp Inequalities and Blow-up Analysis for Singular Moser-Trudinger Embeddings. - Gabriele Mancini (SISSA, 2015)
We investigate existence of solutions for a singular Liouville equation on S^2 and prove sharp Onofri-type inequalities for a Moser-Trudinger functional in the presence of singular potentials. As a consequence we obtain existence of extremal functions for the Moser-Trudinger embedding on compact surfaces with conical singularities. Finally we study the blow-up behavior for sequences of solutions Liouville-type systems and prove a compactness condition which plays an important role in the variational analysis of Toda systems.
Keywords: Moser-Trudinger

Singular Liouville Equations on S^2: Sharp Inequalities and Existence Results - Gabriele Mancini (2015)
We prove a sharp Onofri-type inequality and non-existence of extremals for a Moser-Tudinger functional on S^2 in the presence of potentials having positive order singularities. We also investigate the existence of critical points and give some sufficient conditions under symmetry or nondegeneracy assumptions.

Some results on anisotropic mean curvature and other phase-transition problems - Stefano Amato (SISSA, 2015)
The present thesis is divided into three parts. In the first part, we analyze a suitable regularization — which we call nonlinear multidomain model — of the motion of a hypersurface under smooth anisotropic mean curvature flow. The second part of the thesis deals with crystalline mean curvature of facets of a solid set of R^3 . Finally, in the third part we study a phase-transition model for Plateau’s type problems based on the theory of coverings and of BV functions.
Keywords: Anisotropic mean curvature

Stability of closed gaps for the alternating Kronig-Penney Hamiltonian - Alessandro Michelangeli; Domenico Monaco (SISSA, 2015)
We consider the Kronig-Penney model for a quantum crystal with equispaced periodic delta-interactions of alternating strength. For this model all spectral gaps at the centre of the Brillouin zone are known to vanish, although so far this noticeable property has only been proved through a very delicate analysis of the discriminant of the corresponding ODE and the associated monodromy matrix. We provide a new, alternative proof by showing that this model can be approximated, in the norm resolvent sense, by a model of regular periodic interactions with finite range for which all gaps at the centre of the Brillouin zone are still vanishing. In particular this shows that the vanishing gap property is stable in the sense that it is present also for the "physical" approximants and is not only a feature of the idealised model of zero-range interactions.

Stability of the (2+2)-fermionic system with zero-range interaction - Alessandro Michelangeli; Paul Pfeiffer (2015)
We introduce a 3D model, and we study its stability, consisting of two distinct pairs of identical fermions coupled with a two-body interaction between fermions of different species, whose effective range is essentially zero (a so called (2+2)-fermionic system with zero-range interaction). The interaction is modelled by implementing the the celebrated (and ubiquitous, in the literature of this field) Bethe-Peierls contact condition with given two-body scattering length within the Krein-Visik-Birman theory of extensions of semi-bounded symmetric operators, in order to make the Hamiltonian a well-defined (self-adjoint) physical observable. After deriving the expression for the associated energy quadratic form, we show analytically and numerically that the energy of the model is bounded below, thus describing a stable system.
This SISSA preprint has 17 pages and recorded in PDF format

Stable regular critical points of the Mumford-Shah functional are local minimizers - Marco Bonacini; Massimiliano Morini (Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 2015)
<span style="color: #2e2e2e; font-family: NexusSerif, Georgia, 'Times New Roman', Times, STIXGeneral, 'Cambria Math', 'Lucida Sans Unicode', 'Microsoft Sans Serif', 'Segoe UI Symbol', 'Arial Unicode MS', serif; font-size: 18px;">In this paper it is shown that any regular critical point of the Mumford–Shah functional, with positive definite second variation, is an isolated local minimizer with respect to competitors which are sufficiently close in the $L^1$ <span style="box-sizing: border-box; margin: 0px; padding: 0px; color: #2e2e2e; font-family: NexusSerif, Georgia, 'Times New Roman', Times, STIXGeneral, 'Cambria Math', 'Lucida Sans Unicode', 'Microsoft Sans Serif', 'Segoe UI Symbol', 'Arial Unicode MS', serif; font-size: 18px;">-topology. A global <a style="box-sizing: border-box; margin: 0px; padding: 0px; background-color: transparent; text-decoration-line: none; color: #0c7dbb; word-break: break-word;" title="Learn more about minimality from ScienceDirect's AI-generated Topic Pages" href="https://www.sciencedirect.com/topics/mathematics/minimality">minimality result in small tubular neighborhoods of the <a style="box-sizing: border-box; margin: 0px; padding: 0px; background-color: transparent; text-decoration-line: none; color: #0c7dbb; word-break: break-word;" title="Learn more about Discontinuities from ScienceDirect's AI-generated Topic Pages" href="https://www.sciencedirect.com/topics/mathematics/discontinuities">discontinuity set is also established.
Keywords: Mumford-Shah functional

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.
The article is composed of 23 pages and recorded in PDF format

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)
The PHD thesis is composed of 112 pages and is recorded in PDF format
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.

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.
45 pages

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.
46 pages

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 $\vec{H}$.
14 pages, 1 figure
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.
16 pages

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.
The PhD thesis is composed of 211 pages and is recorded in PDF format
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.
This is an open access article under the CC BY-NC-ND license

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.
31 pages

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.
This paper resumes the main part of the Bachelor thesis of the second author, discussed in 2013 at the University of Trieste.

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.
15 pages

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.
34 pages, 8 figures, references added

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.
cited By (since 1996)0; Article in Press

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.
Book title: Optimal transportation

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 (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.
2014 22nd International Conference on Nuclear Engineering, ICONE 2014; Prague; Czech Republic; 7 July 2014 through 11 July 2014; Code 109131;

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.
This article is composed if 33 pages and recorded in PDF format
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.