Mathematical modeling and quantitative study of biological motility (in particular, of motility at microscopic scales) is producing new biophysical insight and is offering opportunities for new discoveries at the level of both fundamental science and technology. These range from the explanation of how complex behavior at the level of a single organism emerges from body architecture, to the understanding of collective phenomena in groups of organisms and tissues, and of how these forms of swarm intelligence can be controlled and harnessed in engineering applications, to the elucidation of processes of fundamental biological relevance at the cellular and sub-cellular level. In this paper, some of the most exciting new developments in the fields of locomotion of unicellular organisms, of soft adhesive locomotion across scales, of the study of pore translocation properties of knotted DNA, of the development of synthetic active solid sheets, of the mechanics of the unjamming transition in dense cell collectives, of the mechanics of cell sheet folding in volvocalean algae, and of the self-propulsion of topological defects in active matter are discussed. For each of these topics, we provide a brief state of the art, an example of recent achievements, and some directions for future research.

10aactive matter10aadhesive locomotion10acell motility10acell sheet folding10aknotted DNA10atopological defects10aunicellular swimmers10aunjamming transition1 aAgostinelli, Daniele1 aCerbino, Roberto1 aDel Alamo, Juan, C1 aDeSimone, Antonio1 aHöhn, Stephanie1 aMicheletti, Cristian1 aNoselli, Giovanni1 aSharon, Eran1 aYeomans, Julia uhttp://dx.doi.org/10.3934/mine.202001100488nas a2200121 4500008004100000245009200041210006900133300000700202490000700209100002100216700001600237856011300253 2020 eng d00aMinimizers of the prescribed mean curvature functional in a Jordan domain with no necks0 aMinimizers of the prescribed mean curvature functional in a Jord a760 v261 aLeonardi, G., P.1 aSaracco, G. uhttps://www.math.sissa.it/publication/minimizers-prescribed-mean-curvature-functional-jordan-domain-no-necks00372nas a2200097 4500008004100000245008800041210006900129100002200198700001700220856003700237 2020 eng d00aMultiscale modeling of fiber reinforced materials via non-matching immersed methods0 aMultiscale modeling of fiber reinforced materials via nonmatchin1 aAlzetta, Giovanni1 aHeltai, Luca uhttps://arxiv.org/abs/1906.0388100404nas a2200097 4500008004100000245006600041210006600107100002500173700002000198856008800218 2019 eng d00aMinimality of the ball for a model of charged liquid droplets0 aMinimality of the ball for a model of charged liquid droplets1 aMukoseeva, Ekaterina1 aVescovo, Giulia uhttps://www.math.sissa.it/publication/minimality-ball-model-charged-liquid-droplets00406nas a2200121 4500008004100000245008200041210006900123300001000192490000700202100001700209700002100226856003700247 2019 eng d00aMultiscale modeling of vascularized tissues via non-matching immersed methods0 aMultiscale modeling of vascularized tissues via nonmatching imme ae32640 v351 aHeltai, Luca1 aCaiazzo, Alfonso uhttps://doi.org/10.1002/cnm.326400694nas a2200121 4500008004100000245007500041210006900116260001000185520028900195100002100484700001900505856004800524 2018 en d00aA minimization approach to the wave equation on time-dependent domains0 aminimization approach to the wave equation on timedependent doma bSISSA3 aWe prove the existence of weak solutions to the homogeneous wave equation on a suitable class of time-dependent domains. Using the approach suggested by De Giorgi and developed by Serra and Tilli, such solutions are approximated by minimizers of suitable functionals in space-time.1 aDal Maso, Gianni1 aDe Luca, Lucia uhttp://preprints.sissa.it/handle/1963/3531801671nas a2200205 4500008004100000022001400041245009100055210006900146300001100215490000800226520095800234653002501192653005401217653002501271653002901296100002501325700001901350700002501369856007101394 2018 eng d a0021-782400aMinimizing movements for mean curvature flow of droplets with prescribed contact angle0 aMinimizing movements for mean curvature flow of droplets with pr a1 - 580 v1173 aWe study the mean curvature motion of a droplet flowing by mean curvature on a horizontal hyperplane with a possibly nonconstant prescribed contact angle. Using the solutions constructed as a limit of an approximation algorithm of Almgren–Taylor–Wang and Luckhaus–Sturzenhecker, we show the existence of a weak evolution, and its compatibility with a distributional solution. We also prove various comparison results. Résumé Nous étudions le mouvement par courbure moyenne d'une goutte qui glisse par courbure moyenne sur un hyperplan horizontal avec un angle de contact prescrit éventuellement non constant. En utilisant les solutions construites comme limites d'un algorithme d'approximation dû à Almgren, Taylor et Wang et Luckhaus et Sturzenhecker, nous montrons l'existence d'une évolution faible, et sa compatibilité avec une solution au sens des distributions. Nous démontrons également plusieurs résultats de comparaison.

10aCapillary functional10aMean curvature flow with prescribed contact angle10aMinimizing movements10aSets of finite perimeter1 aBellettini, Giovanni1 aNovaga, Matteo1 aKholmatov, Shokhrukh uhttp://www.sciencedirect.com/science/article/pii/S002178241830082501065nas a2200133 4500008004100000245006300041210006300104300001400167490000700181520065400188100002500842700002500867856003900892 2018 eng d00aMinimizing Movements for Mean Curvature Flow of Partitions0 aMinimizing Movements for Mean Curvature Flow of Partitions a4117-41480 v503 aWe prove the existence of a weak global in time mean curvature flow of a bounded partition of space using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We also prove some consistency results for a minimizing movement solution with smooth and viscosity solutions when the evolution starts from a partition made by a union of bounded sets at a positive distance. In addition, the motion starting from the union of convex sets at a positive distance agrees with the classical mean curvature flow and is stable with respect to the Hausdorff convergence of the initial partitions.

1 aBellettini, Giovanni1 aKholmatov, Shokhrukh uhttps://doi.org/10.1137/17M115929401777nas a2200157 4500008004100000245013300041210006900174260003000243520120300273100001901476700001701495700002101512700001701533700002101550856004801571 2018 eng d00aModel Order Reduction by means of Active Subspaces and Dynamic Mode Decomposition for Parametric Hull Shape Design Hydrodynamics0 aModel Order Reduction by means of Active Subspaces and Dynamic M aTrieste, ItalybIOS Press3 aWe present the results of the application of a parameter space reduction methodology based on active subspaces (AS) to the hull hydrodynamic design problem. Several parametric deformations of an initial hull shape are considered to assess the influence of the shape parameters on the hull wave resistance. Such problem is relevant at the preliminary stages of the ship design, when several flow simulations are carried out by the engineers to establish a certain sensibility with respect to the parameters, which might result in a high number of time consuming hydrodynamic simulations. The main idea of this work is to employ the AS to identify possible lower dimensional structures in the parameter space. The complete pipeline involves the use of free form deformation to parametrize and deform the hull shape, the full order solver based on unsteady potential flow theory with fully nonlinear free surface treatment directly interfaced with CAD, the use of dynamic mode decomposition to reconstruct the final steady state given only few snapshots of the simulation, and the reduction of the parameter space by AS, and shared subspace. Response surface method is used to minimize the total drag.1 aTezzele, Marco1 aDemo, Nicola1 aGadalla, Mahmoud1 aMola, Andrea1 aRozza, Gianluigi uhttp://ebooks.iospress.nl/publication/4927000512nas a2200145 4500008004100000245011100041210006900152300001600221490000700237100002200244700002400266700001600290700002100306856003900327 2018 eng d00aModel Reduction for Parametrized Optimal Control Problems in Environmental Marine Sciences and Engineering0 aModel Reduction for Parametrized Optimal Control Problems in Env aB1055-B10790 v401 aStrazzullo, Maria1 aBallarin, Francesco1 aMosetti, R.1 aRozza, Gianluigi uhttps://doi.org/10.1137/17M115059100454nas a2200133 4500008004100000245005600041210005500097260001600152300001000168490000800178100002300186700002400209856008700233 2018 eng d00aMorpho-elastic model of the tortuous tumour vessels0 aMorphoelastic model of the tortuous tumour vessels bElsevier BV a1–90 v1071 aRiccobelli, Davide1 aCiarletta, Pasquale uhttps://www.math.sissa.it/publication/morpho-elastic-model-tortuous-tumour-vessels00364nas a2200121 4500008004100000022001400041245004900055210004500104300002200149490000700171100001900178856004500197 2017 eng d a1815-065900aThe Malgrange form and Fredholm determinants0 aMalgrange form and Fredholm determinants aPaper No. 046, 120 v131 aBertola, Marco uhttp://dx.doi.org/10.3842/SIGMA.2017.04600465nas a2200133 4500008004100000022001400041245009600055210006900151300001400220490000800234100001900242700002200261856004800283 2017 eng d a0010-361600aMaximal amplitudes of finite-gap solutions for the focusing Nonlinear Schrödinger Equation0 aMaximal amplitudes of finitegap solutions for the focusing Nonli a525–5470 v3541 aBertola, Marco1 aTovbis, Alexander uhttp://dx.doi.org/10.1007/s00220-017-2895-900961nas a2200157 4500008004100000022001400041245007700055210007100132260000800203300001400211490000600225520047300231100002900704700002400733856004600757 2017 eng d a1664-235X00aMean-field quantum dynamics for a mixture of Bose–Einstein condensates0 aMeanfield quantum dynamics for a mixture of Bose–Einstein conden cDec a377–4160 v73 aWe study the effective time evolution of a large quantum system consisting of a mixture of different species of identical bosons in interaction. If the system is initially prepared so as to exhibit condensation in each component, we prove that condensation persists at later times and we show quantitatively that the many-body Schrödinger dynamics is effectively described by a system of coupled cubic non-linear Schrödinger equations, one for each component.

1 aMichelangeli, Alessandro1 aOlgiati, Alessandro uhttps://doi.org/10.1007/s13324-016-0147-301160nas a2200205 4500008004100000022001400041245006400055210006400119300000900183490000700192520049200199653003500691653001800726653003600744653002900780100002500809700001900834700002500853856007600878 2017 eng d a1534-039200aMinimizers of anisotropic perimeters with cylindrical norms0 aMinimizers of anisotropic perimeters with cylindrical norms a14270 v163 aWe study various regularity properties of minimizers of the Φ–perimeter, where Φ is a norm. Under suitable assumptions on Φ and on the dimension of the ambient space, we prove that the boundary of a cartesian minimizer is locally a Lipschitz graph out of a closed singular set of small Hausdorff dimension. Moreover, we show the following anisotropic Bernstein-type result: any entire cartesian minimizer is the subgraph of a monotone function depending only on one variable.

10aanisotropic Bernstein problem;10aminimal cones10aNon parametric minimal surfaces10aSets of finite perimeter1 aBellettini, Giovanni1 aNovaga, Matteo1 aKholmatov, Shokhrukh uhttp://aimsciences.org//article/id/47054f15-00c7-40b7-9da1-4c0b1d0a103d01991nas a2200157 4500008004100000245002800041210002800069260002200097300000900119520158000128100002401708700002001732700002101752700001901773856004101792 2017 eng d00aModel Reduction Methods0 aModel Reduction Methods bJohn Wiley & Sons a1-363 aThis chapter presents an overview of model order reduction – a new paradigm in the field of simulation-based engineering sciences, and one that can tackle the challenges and leverage the opportunities of modern ICT technologies. Despite the impressive progress attained by simulation capabilities and techniques, a number of challenging problems remain intractable. These problems are of different nature, but are common to many branches of science and engineering. Among them are those related to high-dimensional problems, problems involving very different time scales, models defined in degenerate domains with at least one of the characteristic dimensions much smaller than the others, model requiring real-time simulation, and parametric models. All these problems represent a challenge for standard mesh-based discretization techniques; yet the ability to solve these problems efficiently would open unexplored routes for real-time simulation, inverse analysis, uncertainty quantification and propagation, real-time optimization, and simulation-based control – critical needs in many branches of science and engineering. Model order reduction offers new simulation alternatives by circumventing, or at least alleviating, otherwise intractable computational challenges. In the present chapter, we revisit three of these model reduction techniques: proper orthogonal decomposition, proper generalized decomposition, and reduced basis methodologies.} preprint = {http://preprints.sissa.it/xmlui/bitstream/handle/1963/35194/ECM_MOR.pdf?sequence=1&isAllowed=y

1 aChinesta, Francisco1 aHuerta, Antonio1 aRozza, Gianluigi1 aWillcox, Karen uhttps://www.math.sissa.it/node/1294904754nas a2200097 4500008004100000245005000041210005000091520445700141100002104598856003704619 2017 eng d00aModuli of semistable sheaves as quiver moduli0 aModuli of semistable sheaves as quiver moduli3 aIn the 1980s Drézet and Le Potier realized moduli spaces of Gieseker-semistable sheaves on P2 as what are now called quiver moduli spaces. We discuss how this construction can be understood using t-structures and exceptional collections on derived categories, and how it can be extended to a similar result on P1×P1.

1 aMaiorana, Andrea uhttps://arxiv.org/abs/1709.0555501736nas a2200181 4500008004100000022001400041245010800055210006900163300000900232490000700241520106900248653003901317653002301356653004001379653003601419100002301455856007601478 2017 eng d a1534-039200aMultiple positive solutions of a sturm-liouville boundary value problem with conflicting nonlinearities0 aMultiple positive solutions of a sturmliouville boundary value p a10830 v163 aWe study the second order nonlinear differential equation

\begindocument $ u'' + \sum\limits_i = 1^m α_ia_i(x)g_i(u) - \sum\limits_j = 1^m + 1 β_jb_j(x)k_j(u) = 0,\rm $ \enddocument

where $\alpha_i, \beta_j>0$, $a_i(x), b_j(x)$ are non-negative Lebesgue integrable functions defined in $\mathopen[0, L\mathclose]$, and the nonlinearities $g_i(s), k_j(s)$ are continuous, positive and satisfy suitable growth conditions, as to cover the classical superlinear equation $u"+a(x)u.p = 0$, with $p>1$.When the positive parameters $\beta_j$ are sufficiently large, we prove the existence of at least $2.m-1$positive solutions for the Sturm-Liouville boundary value problems associated with the equation.The proof is based on the Leray-Schauder topological degree for locally compact operators on open and possibly unbounded sets.Finally, we deal with radially symmetric positive solutions for the Dirichlet problems associated with elliptic PDEs.

10aLeray-Schauder topological degree;10apositive solutions10aSturm-Liouville boundary conditions10aSuperlinear indefinite problems1 aFeltrin, Guglielmo uhttp://aimsciences.org//article/id/1163b042-0c64-4597-b25c-3494b268e5a101598nas a2200217 4500008004100000022001400041245010600055210006900161300001600230490000800246520083500254653002301089653002501112653003601137653003201173653002601205653003601231100002301267700001901290856007101309 2017 eng d a0022-039600aMultiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree0 aMultiplicity of positive periodic solutions in the superlinear i a4255 - 42910 v2623 aWe study the periodic boundary value problem associated with the second order nonlinear differential equationu″+cu′+(a+(t)−μa−(t))g(u)=0, where g(u) has superlinear growth at zero and at infinity, a(t) is a periodic sign-changing weight, c∈R and μ>0 is a real parameter. Our model includes (for c=0) the so-called nonlinear Hill's equation. We prove the existence of 2m−1 positive solutions when a(t) has m positive humps separated by m negative ones (in a periodicity interval) and μ is sufficiently large, thus giving a complete solution to a problem raised by G.J. Butler in 1976. The proof is based on Mawhin's coincidence degree defined in open (possibly unbounded) sets and applies also to Neumann boundary conditions. Our method also provides a topological approach to detect subharmonic solutions.

10aCoincidence degree10aMultiplicity results10aNeumann boundary value problems10aPositive periodic solutions10asubharmonic solutions10aSuperlinear indefinite problems1 aFeltrin, Guglielmo1 aZanolin, Fabio uhttp://www.sciencedirect.com/science/article/pii/S002203961730021900690nas a2200109 4500008004100000245007500041210006900116520030100185100002100486700002200507856005100529 2016 en d00aA model for the quasistatic growth of cracks with fractional dimension0 amodel for the quasistatic growth of cracks with fractional dimen3 aWe study a variational model for the quasistatic growth of cracks with fractional dimension in brittle materials. We give a minimal set of properties of the collection of admissible cracks which ensure the existence of a quasistatic evolution. Both the antiplane and the planar cases are treated.1 aDal Maso, Gianni1 aMorandotti, Marco uhttp://urania.sissa.it/xmlui/handle/1963/3517500391nas a2200133 4500008004100000245003600041210003500077260001000112100002400122700002000146700002100166700001900187856005100206 2016 en d00aModel Order Reduction: a survey0 aModel Order Reduction a survey bWiley1 aChinesta, Francisco1 aHuerta, Antonio1 aRozza, Gianluigi1 aWillcox, Karen uhttp://urania.sissa.it/xmlui/handle/1963/3519401034nas a2200145 4500008004100000022001400041245006600055210006600121260000800187300001600195490000800211520060300219100002000822856004600842 2016 eng d a1432-182300aMoser–Trudinger inequalities for singular Liouville systems0 aMoser–Trudinger inequalities for singular Liouville systems cApr a1169–11900 v2823 aIn this paper we prove a Moser–Trudinger inequality for the Euler–Lagrange functional of general singular Liouville systems on a compact surface. We characterize the values of the parameters which yield coercivity for the functional, hence the existence of energy-minimizing solutions for the system, and we give necessary conditions for boundedness from below. We also provide a sharp inequality under assuming the coefficients of the system to be non-positive outside the diagonal. The proofs use a concentration-compactness alternative, Pohožaev-type identities and blow-up analysis.

1 aBattaglia, Luca uhttps://doi.org/10.1007/s00209-015-1584-701781nas a2200157 4500008004100000022001400041245006600055210006600121260000800187300000700195490000700202520131900209100002601528700002201554856004701576 2016 eng d a1292-895X00aMotion planning and motility maps for flagellar microswimmers0 aMotion planning and motility maps for flagellar microswimmers cJul a720 v393 aWe study two microswimmers consisting of a spherical rigid head and a passive elastic tail. In the first one the tail is clamped to the head, and the system oscillates under the action of an external torque. In the second one, head and tail are connected by a joint allowing the angle between them to vary periodically, as a result of an oscillating internal torque. Previous studies on these models were restricted to sinusoidal actuations, showing that the swimmers can propel while moving on average along a straight line, in the direction given by the symmetry axis around which beating takes place. We extend these results to motions produced by generic (non-sinusoidal) periodic actuations within the regime of small compliance of the tail. We find that modulation in the velocity of actuation can provide a mechanism to select different directions of motion. With velocity-modulated inputs, the externally actuated swimmer can translate laterally with respect to the symmetry axis of beating, while the internally actuated one is able to move along curved trajectories. The governing equations are analysed with an asymptotic perturbation scheme, providing explicit formulas, whose results are expressed through motility maps. Asymptotic approximations are further validated by numerical simulations.

1 aCicconofri, Giancarlo1 aDeSimone, Antonio uhttps://doi.org/10.1140/epje/i2016-16072-y01951nas a2200169 4500008004100000245009300041210006900134260001300203300000800216490000700224520142100231100002101652700001901673700001701692700002101709856005101730 2016 en d00aA multi-physics reduced order model for the analysis of Lead Fast Reactor single channel0 amultiphysics reduced order model for the analysis of Lead Fast R bElsevier a2080 v873 aIn this work, a Reduced Basis method, with basis functions sampled by a Proper Orthogonal Decomposition technique, has been employed to develop a reduced order model of a multi-physics parametrized Lead-cooled Fast Reactor single-channel. Being the first time that a reduced order model is developed in this context, the work focused on a methodological approach and the coupling between the neutronics and the heat transfer, where the thermal feedbacks on neutronics are explicitly taken into account, in time-invariant settings. In order to address the potential of such approach, two different kinds of varying parameters have been considered, namely one related to a geometric quantity (i.e., the inner radius of the fuel pellet) and one related to a physical quantity (i.e., the inlet lead velocity). The capabilities of the presented reduced order model (ROM) have been tested and compared with a high-fidelity finite element model (upon which the ROM has been constructed) on different aspects. In particular, the comparison focused on the system reactivity prediction (with and without thermal feedbacks on neutronics), the neutron flux and temperature field reconstruction, and on the computational time. The outcomes provided by the reduced order model are in good agreement with the high-fidelity finite element ones, and a computational speed-up of at least three orders of magnitude is achieved as well.1 aSartori, Alberto1 aCammi, Antonio1 aLuzzi, Lelio1 aRozza, Gianluigi uhttp://urania.sissa.it/xmlui/handle/1963/3519101319nas a2200109 4500008004100000245011100041210006900152520088700221100002901108700002101137856005101158 2016 en d00aMultiplicity of self-adjoint realisations of the (2+1)-fermionic model of Ter-Martirosyan--Skornyakov type0 aMultiplicity of selfadjoint realisations of the 21fermionic mode3 aWe reconstruct the whole family of self-adjoint Hamiltonians of Ter-Martirosyan- Skornyakov type for a system of two identical fermions coupled with a third particle of different nature through an interaction of zero range. We proceed through an operator-theoretic approach based on the self-adjoint extension theory of Kreĭn, Višiik, and Birman. We identify the explicit `Kreĭn-Višik-Birman extension param- eter' as an operator on the `space of charges' for this model (the `Kreĭn space') and we come to formulate a sharp conjecture on the dimensionality of its kernel. Based on our conjecture, for which we also discuss an amount of evidence, we explain the emergence of a multiplicity of extensions in a suitable regime of masses and we re- produce for the first time the previous partial constructions obtained by means of an alternative quadratic form approach.1 aMichelangeli, Alessandro1 aOttolini, Andrea uhttp://urania.sissa.it/xmlui/handle/1963/3526700496nas a2200109 4500008004100000245010000041210006900141260001000210653001300220100002600233856012700259 2015 en d00aMathematical Models of Locomotion: Legged Crawling, Snake-like Motility, and Flagellar Swimming0 aMathematical Models of Locomotion Legged Crawling Snakelike Moti bSISSA10aMotility1 aCicconofri, Giancarlo uhttps://www.math.sissa.it/publication/mathematical-models-locomotion-legged-crawling-snake-motility-and-flagellar-swimming00454nas a2200133 4500008004100000022001400041245009000055210006900145300001100214490000600225100001900231700002200250856004800272 2015 eng d a1664-236800aMeromorphic differentials with imaginary periods on degenerating hyperelliptic curves0 aMeromorphic differentials with imaginary periods on degenerating a1–220 v51 aBertola, Marco1 aTovbis, Alexander uhttp://dx.doi.org/10.1007/s13324-014-0088-700678nas a2200169 4500008004100000245013400041210006900175300001400244490000700258100001800265700002100283700002000304700002100324700001900345700001900364856012500383 2015 eng d00aModel order reduction of parameterized systems ({MoRePaS}): Preface to the special issue of advances in computational mathematics0 aModel order reduction of parameterized systems MoRePaS Preface t a955–9600 v411 aBenner, Peter1 aOhlberger, Mario1 aPatera, Anthony1 aRozza, Gianluigi1 aSorensen, D.C.1 aUrban, Karsten uhttps://www.math.sissa.it/publication/model-order-reduction-parameterized-systems-morepas-preface-special-issue-advances01569nas a2200181 4500008004100000022001400041245006000055210005800115300001400173490000700187520100500194653001901199653002201218653002801240100002601268700002201294856007101316 2015 eng d a0020-746200aMotility of a model bristle-bot: A theoretical analysis0 aMotility of a model bristlebot A theoretical analysis a233 - 2390 v763 aBristle-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.

10aBristle-robots10aCrawling motility10aFrictional interactions1 aCicconofri, Giancarlo1 aDeSimone, Antonio uhttp://www.sciencedirect.com/science/article/pii/S002074621500002501808nas a2200121 4500008004100000245011400041210006900155260001000224520121600234653008901450100001901539856012801558 2015 en d00aMultidimensional Poisson Vertex Algebras and Poisson cohomology of Hamiltonian operators of hydrodynamic type0 aMultidimensional Poisson Vertex Algebras and Poisson cohomology bSISSA3 aThe 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.10aPoisson Vertex Algebras, Poisson brackets, Hamiltonian operators, Integrable Systems1 aCasati, Matteo uhttps://www.math.sissa.it/publication/multidimensional-poisson-vertex-algebras-and-poisson-cohomology-hamiltonian-operators01516nas a2200133 4500008004100000245012100041210006900162260001300231520102900244100002101273700001501294700002201309856005101331 2015 en d00aMultilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations0 aMultilevel and weighted reduced basis method for stochastic opti bSpringer3 aIn 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.1 aRozza, Gianluigi1 aChen, Peng1 aQuarteroni, Alfio uhttp://urania.sissa.it/xmlui/handle/1963/3449101194nas a2200121 4500008004100000245008200041210006900123260001300192520077400205100002300979700001901002856005101021 2015 en d00aMultiple positive solutions for a superlinear problem: a topological approach0 aMultiple positive solutions for a superlinear problem a topologi bElsevier3 aWe 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.

1 aFeltrin, Guglielmo1 aZanolin, Fabio uhttp://urania.sissa.it/xmlui/handle/1963/3514700683nas a2200109 4500008004100000245005300041210005300094260001300147520034200160100002000502856005100522 2014 en d00aMaximal generalized solution of eikonal equation0 aMaximal generalized solution of eikonal equation bElsevier3 aWe 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.1 aZagatti, Sandro uhttp://urania.sissa.it/xmlui/handle/1963/3464201625nas a2200133 4500008004100000245007900041210006900120260001300189520117000202100002301372700002001395700002501415856005101440 2014 en d00aMinimal Liouville gravity correlation numbers from Douglas string equation0 aMinimal Liouville gravity correlation numbers from Douglas strin bSpringer3 aWe 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 $\tau_{m,n}$ 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 $\tau_{m,n}$ are related in a non-linear fashion to the natural coupling constants $\lambda_{m,n}$ 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}.1 aBelavin, Alexander1 aDubrovin, Boris1 aMukhametzhanov, Baur uhttp://urania.sissa.it/xmlui/handle/1963/3458801687nas a2200145 4500008004100000245005600041210005400097260001000151300001100161490000700172520119700179653007701376100001801453856007001471 2014 en d00aA model for crack growth with branching and kinking0 amodel for crack growth with branching and kinking bSISSA a63-1100 v893 aWe 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.

10aquasistatic crack evolution, branching, kinking, Griffith\\\'s criterion1 aRacca, Simone uhttps://content.iospress.com/articles/asymptotic-analysis/asy123301650nas a2200145 4500008004100000245007300041210006900114260001300183520112000196100001801316700002001334700002201354700002101376856010701397 2014 en d00aModel Order Reduction in Fluid Dynamics: Challenges and Perspectives0 aModel Order Reduction in Fluid Dynamics Challenges and Perspecti bSpringer3 aThis 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.1 aLassila, Toni1 aManzoni, Andrea1 aQuarteroni, Alfio1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/model-order-reduction-fluid-dynamics-challenges-and-perspectives01391nas a2200109 4500008004100000245005300041210005000094260001300144520105400157100001901211856005101230 2014 en d00aA modular spectral triple for κ-Minkowski space0 amodular spectral triple for κMinkowski space bElsevier3 aWe 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.1 aMatassa, Marco uhttp://urania.sissa.it/xmlui/handle/1963/3489500433nas a2200121 4500008004100000245006200041210005900103300001100162490000600173100002000179700002200199856009000221 2014 eng d00aA Moser-Trudinger inequality for the singular Toda system0 aMoserTrudinger inequality for the singular Toda system a1–230 v91 aBattaglia, Luca1 aMalchiodi, Andrea uhttps://www.math.sissa.it/publication/moser-trudinger-inequality-singular-toda-system01318nas a2200157 4500008004100000022001400041245005900055210005800114260000800172300000700180490000900187520087300196100002501069700002201094856004401116 2014 eng d a1029-847900aM-theory interpretation of the real topological string0 aMtheory interpretation of the real topological string cAug a540 v20143 aWe 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.

1 aPiazzalunga, Nicolò1 aUranga, Angel, M. uhttps://doi.org/10.1007/JHEP08(2014)05400625nas a2200157 4500008004100000245011600041210006900157260001700226300001400243490000700257100001500264700002300279700002200302700001800324856012500342 2013 eng d00aMacroscopic contact angle and liquid drops on rough solid surfaces via homogenization and numerical simulations0 aMacroscopic contact angle and liquid drops on rough solid surfac bEDP Sciences a837–8580 v471 aCacace, S.1 aChambolle, Antonin1 aDeSimone, Antonio1 aFedeli, Livio uhttps://www.math.sissa.it/publication/macroscopic-contact-angle-and-liquid-drops-rough-solid-surfaces-homogenization-and01676nas a2200145 4500008004100000245009400041210006900135260001000204520112400214653008201338100002001420700002501440700002901465856003601494 2013 en d00aMinimal partitions and image classification using a gradient-free perimeter approximation0 aMinimal partitions and image classification using a gradientfree bSISSA3 aIn this paper a new mathematically-founded method for the optimal partitioning of domains, with applications to the classification of greyscale and color images, is proposed. Since optimal partition problems are in general ill-posed, some regularization strategy is required. Here we regularize by a non-standard approximation of the total interface length, which does not involve the gradient of approximate characteristic functions, in contrast to the classical Modica-Mortola approximation. Instead, it involves a system of uncoupled linear partial differential equations and nevertheless shows $\Gamma$-convergence properties in appropriate function spaces. This approach leads to an alternating algorithm that ensures a decrease of the objective function at each iteration, and which always provides a partition, even during the iterations. The efficiency of this algorithm is illustrated by various numerical examples. Among them we consider binary and multilabel minimal partition problems including supervised or automatic image classification, inpainting, texture pattern identification and deblurring.10aImage classification, deblurring, optimal partitions, perimeter approximation1 aAmstutz, Samuel1 aVan Goethem, Nicolas1 aNovotny, Antonio, André uhttp://hdl.handle.net/1963/697600516nas a2200109 4500008004100000245010700041210006900148260001000217653003200227100002000259856012700279 2013 en d00aMinimality and stability results for a class of free-discontinuity and nonlocal isoperimetric problems0 aMinimality and stability results for a class of freediscontinuit bSISSA10afree-discontinuity problems1 aBonacini, Marco uhttps://www.math.sissa.it/publication/minimality-and-stability-results-class-free-discontinuity-and-nonlocal-isoperimetric00733nas a2200133 4500008004100000245005300041210005300094520025800147653004800405100002200453700001600475700002400491856008400515 2013 en d00aMonads for framed sheaves on Hirzebruch surfaces0 aMonads for framed sheaves on Hirzebruch surfaces3 aWe define monads for framed torsion-free sheaves on Hirzebruch surfaces and use them to construct moduli spaces for these objects. These moduli spaces are smooth algebraic varieties, and we show that they are fine by constructing a universal monad.10aMonads, framed sheaves, Hirzebruch surfaces1 aBartocci, Claudio1 aBruzzo, Ugo1 aRava, Claudio, L.S. uhttps://www.math.sissa.it/publication/monads-framed-sheaves-hirzebruch-surfaces01330nas a2200157 4500008004100000022001400041245005900055210005500114260000800169300001400177490000800191520088200199100002301081700002201104856004601126 2013 eng d a1432-091600aThe Monge Problem for Distance Cost in Geodesic Spaces0 aMonge Problem for Distance Cost in Geodesic Spaces cMar a615–6730 v3183 aWe address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dLis a geodesic Borel distance which makes (X, dL) a non branching geodesic space. We show that under the assumption that geodesics are d-continuous and locally compact, we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce two assumptions on the transport problem π which imply that the conditional probabilities of the first marginal on each geodesic are continuous or absolutely continuous w.r.t. the 1-dimensional Hausdorff distance induced by dL. It is known that this regularity is sufficient for the construction of a transport map. We study also the dynamics of transport along the geodesic, the stability of our conditions and show that in this setting dL-cyclical monotonicity is not sufficient for optimality.

1 aBianchini, Stefano1 aCavalletti, Fabio uhttps://doi.org/10.1007/s00220-013-1663-800430nas a2200121 4500008004100000245009500041210006900136260001300205653001400218100001800232700002200250856003600272 2013 en d00aMultiplicity result for a nonhomogeneous Yamabe type equation involving the Kohn Laplacian0 aMultiplicity result for a nonhomogeneous Yamabe type equation in bElsevier10aCR-Yamabe1 aMaalaoui, Ali1 aMartino, Vittorio uhttp://hdl.handle.net/1963/737401914nas a2200145 4500008004100000020001800041245010100059210006900160260003100229520140500260653002201665100002301687700002201710856003601732 2012 en d a978160511380700aMathematical and numerical modeling of liquid crystal elastomer phase transition and deformation0 aMathematical and numerical modeling of liquid crystal elastomer bCambridge University Press3 aLiquid crystal (in particular, nematic) elastomers consist of cross-linked flexible polymer chains with embedded stiff rod molecules that allow them to behave as a rubber and a liquid crystal. Nematic elastomers are characterized by a phase transition from isotropic to nematic past a temperature threshold. They behave as rubber at high temperature and show nematic behavior below the temperature threshold. Such transition is reversible. While in the nematic phase, the rod molecules are aligned along the direction of the "nematic director". This molecular rearrangement induces a stretch in the polymer chains and hence macroscopic spontaneous deformations. The coupling between nematic order parameter and deformation gives rise to interesting phenomena with a potential for new interesting applications. In the biological field, the ability to considerably change their length makes them very promising as artificial muscles actuators. Their tunable optical properties make them suitable, for example, as lenses for new imaging systems. We present a mathematical model able to describe the behavior of nematic elastomers and numerical simulations reproducing such peculiar behavior. We use a geometrically linear version of the Warner and Terentjev model [1] and consider cooling experiments and stretching experiments in the direction perpendicular to the one of the director at cross-linking.10aArtificial muscle1 aDe Luca, Mariarita1 aDeSimone, Antonio uhttp://hdl.handle.net/1963/702001486nas a2200121 4500008004100000245006100041210006000102260005400162520106700216100002201283700002301305856003601328 2012 en d00aModeling and control of quantum systems: An introduction0 aModeling and control of quantum systems An introduction bInstitute of Electrical and Electronics Engineers3 aThe scope of this work is to provide a self-contained introduction to a selection of basic theoretical aspects in the modeling and control of quantum mechanical systems, as well as a brief survey on the main approaches to control synthesis. While part of the existing theory, especially in the open-loop setting, stems directly from classical control theory (most notably geometric control and optimal control), a number of tools specifically tailored for quantum systems have been developed since the 1980s, in order to take into account their distinctive features: the probabilistic nature of atomic-scale physical systems, the effect of dissipation and the irreversible character of the measurements have all proved to be critical in feedback-design problems. The relevant dynamical models for both closed and open quantum systems are presented, along with the main results on their controllability and stability. A brief review of several currently available control design methods is meant to provide the interested reader with a roadmap for further studies1 aAltafini, Claudio1 aTicozzi, Francesco uhttp://hdl.handle.net/1963/650500979nas a2200133 4500008004100000245010200041210006900143260001000212520051900222100001600741700002600757700002600783856003600809 2012 en d00aModuli of symplectic instanton vector bundles of higher rank on projective space $\\mathbb{P}^3$0 aModuli of symplectic instanton vector bundles of higher rank on bSISSA3 aSymplectic instanton vector bundles on the projective space $\\mathbb{P}^3$ constitute a natural generalization of mathematical instantons of rank 2. We study the moduli space $I_{n,r}$ of rank-$2r$ symplectic instanton vector bundles on $\\mathbb{P}^3$ with $r\\ge2$ and second Chern class $n\\ge r,\\ n\\equiv r({\\rm mod}2)$. We give an explicit construction of an irreducible component $I^*_{n,r}$ of this space for each such value of $n$ and show that $I^*_{n,r}$ has the expected dimension $4n(r+1)-r(2r+1)$.1 aBruzzo, Ugo1 aMarkushevich, Dimitri1 aTikhomirov, Alexander uhttp://hdl.handle.net/1963/465601072nas a2200121 4500008004300000245008700043210006900130260002800199520065000227100001700877700002000894856003600914 2012 en_Ud 00aModuli spaces of noncommutative instantons: gauging away noncommutative parameters0 aModuli spaces of noncommutative instantons gauging away noncommu bOxford University Press3 aUsing the theory of noncommutative geometry in a braided monoidal category, we improve upon a previous construction of noncommutative families of instantons of arbitrary charge on the deformed sphere S^4_\\\\theta. We formulate a notion of noncommutative parameter spaces for families of instantons and we explore what it means for such families to be gauge equivalent, as well as showing how to remove gauge parameters using a noncommutative quotient construction. Although the parameter spaces are a priori noncommutative, we show that one may always recover a classical parameter space by making an appropriate choice of gauge transformation.1 aBrain, Simon1 aLandi, Giovanni uhttp://hdl.handle.net/1963/377700591nas a2200145 4500008004100000022001400041245003800055210003400093260000800127300001400135490000700149520022100156100002200377856004600399 2012 eng d a1432-083500aThe Monge problem in Wiener space0 aMonge problem in Wiener space cSep a101–1240 v453 aWe address the Monge problem in the abstract Wiener space and we give an existence result provided both marginal measures are absolutely continuous with respect to the infinite dimensional Gaussian measure γ.

1 aCavalletti, Fabio uhttps://doi.org/10.1007/s00526-011-0452-501120nas a2200133 4500008004300000245010500043210006900148260001300217520065000230100001900880700002500899700002600924856003600950 2011 en_Ud 00aThe matching property of infinitesimal isometries on elliptic surfaces and elasticity on thin shells0 amatching property of infinitesimal isometries on elliptic surfac bSpringer3 aUsing the notion of Γ-convergence, we discuss the limiting behavior of the three-dimensional nonlinear elastic energy for thin elliptic shells, as their thickness h converges to zero, under the assumption that the elastic energy of deformations scales like h β with 2 < β < 4. We establish that, for the given scaling regime, the limiting theory reduces to linear pure bending. Two major ingredients of the proofs are the density of smooth infinitesimal isometries in the space of W 2,2 first order infinitesimal isometries, and a result on matching smooth infinitesimal isometries with exact isometric immersions on smooth elliptic surfaces.1 aLewicka, Marta1 aMora, Maria Giovanna1 aPakzad, Mohammad Reza uhttp://hdl.handle.net/1963/339201329nas a2200169 4500008004100000022001400041245008700055210006900142260000800211300001400219490000700233520081100240100001801051700002201069700002201091856004601113 2011 eng d a1432-095900aMetastable equilibria of capillary drops on solid surfaces: a phase field approach0 aMetastable equilibria of capillary drops on solid surfaces a pha cSep a453–4710 v233 aWe discuss a phase field model for the numerical simulation of metastable equilibria of capillary drops resting on rough solid surfaces and for the description of contact angle hysteresis phenomena in wetting. The model is able to reproduce observed transitions of drops on micropillars from Cassie–Baxter to Wenzel states. When supplemented with a dissipation potential which describes energy losses due to frictional forces resisting the motion of the contact line, the model can describe metastable states such as drops in equilibrium on vertical glass plates. The reliability of the model is assessed by a detailed comparison of its predictions with experimental data on the maximal size of water drops that can stick on vertical glass plates which have undergone different surface treatments.

1 aFedeli, Livio1 aTurco, Alessandro1 aDeSimone, Antonio uhttps://doi.org/10.1007/s00161-011-0189-601532nas a2200277 4500008004100000022001600041245006500057210006300122260009400185300001600279490000900295520056700304653002100871653002200892653002200914653002400936653003200960653002500992653002101017653002901038653002401067653002401091100002401115700001901139856009601158 2011 eng d a{0218-2025}00aA MODEL FOR CRACK PROPAGATION BASED ON VISCOUS APPROXIMATION0 aMODEL FOR CRACK PROPAGATION BASED ON VISCOUS APPROXIMATION a{5 TOH TUCK LINK, SINGAPORE 596224, SINGAPORE}b{WORLD SCIENTIFIC PUBL CO PTE LTD}c{OCT} a{2019-2047}0 v{21}3 a{In the setting of antiplane linearized elasticity, we show the existence of quasistatic evolutions of cracks in brittle materials by using a vanishing viscosity approach, thus taking into account local minimization. The main feature of our model is that the path followed by the crack need not be prescribed a priori: indeed, it is found as the limit (in the sense of Hausdorff convergence) of curves obtained by an incremental procedure. The result is based on a continuity property for the energy release rate in a suitable class of admissible cracks.}

10aBrittle fracture10aCrack propagation10aenergy derivative10aenergy release rate10afree-discontinuity problems10aGriffith's criterion10alocal minimizers10astress intensity factor}10avanishing viscosity10a{Variational models1 aLazzaroni, Giuliano1 aToader, Rodica uhttps://www.math.sissa.it/publication/model-crack-propagation-based-viscous-approximation-000841nas a2200121 4500008004100000245005200041210005200093260002600145520047000171100001600641700002600657856003600683 2011 en d00aModuli of framed sheaves on projective surfaces0 aModuli of framed sheaves on projective surfaces bDocumenta Mathematica3 aWe show that there exists a fine moduli space for torsion-free sheaves on a\\r\\nprojective surface, which have a \\\"good framing\\\" on a big and nef divisor. This\\r\\nmoduli space is a quasi-projective scheme. This is accomplished by showing that such framed sheaves may be considered as stable pairs in the sense of\\r\\nHuybrechts and Lehn. We characterize the obstruction to the smoothness of the moduli space, and discuss some examples on rational surfaces.1 aBruzzo, Ugo1 aMarkushevich, Dimitri uhttp://hdl.handle.net/1963/512601064nas a2200193 4500008004100000020002200041245004100063210003700104260002800141300001400169520049000183100002300673700002200696700002100718700002400739700001900763700001600782856007200798 2011 eng d a978-1-4419-9554-400aThe Monge Problem in Geodesic Spaces0 aMonge Problem in Geodesic Spaces aBoston, MAbSpringer US a217–2333 aWe address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish non branching geodesic space. We show that we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce an assumption on the transport problem π which implies that the conditional probabilities of the first marginal on each geodesic are continuous. It is known that this regularity is sufficient for the construction of an optimal transport map.

1 aBianchini, Stefano1 aCavalletti, Fabio1 aBressan, Alberto1 aChen, Gui-Qiang, G.1 aLewicka, Marta1 aWang, Dehua uhttps://www.math.sissa.it/publication/monge-problem-geodesic-spaces00490nas a2200145 4500008004100000245008500041210006900126260002500195300001200220490000700232100001700239700001900256700002300275856004600298 2011 eng d00aMulti-physics modelling and sensitivity analysis of olympic rowing boat dynamics0 aMultiphysics modelling and sensitivity analysis of olympic rowin bSpringer Naturecnov a85–940 v141 aMola, Andrea1 aGhommem, Mehdi1 aHajj, Muhammad, R. uhttps://doi.org/10.1007/s12283-011-0075-200786nas a2200121 4500008004100000245007600041210006900117300001400186490000600200520033000206100002600536856010200562 2011 eng d00aMultiplicity of solutions for a mean field equation on compact surfaces0 aMultiplicity of solutions for a mean field equation on compact s a245–2570 v43 aWe consider a scalar field equation on compact surfaces which has variational structure. When the surface is a torus and a physical parameter ρ belongs to $(8\pi, 4\pi^2 )$ we show under some extra assumptions that, as conjectured in [9], the functional admits at least three saddle points other than a local minimum.

1 aDe Marchis, Francesca uhttps://www.math.sissa.it/publication/multiplicity-solutions-mean-field-equation-compact-surfaces02106nas a2200121 4500008004300000245013400043210006900177260001900246520164000265100002101905700002201926856003601948 2010 en_Ud 00aMonotonicity, frustration, and ordered response: an analysis of the energy landscape of perturbed large-scale biological networks0 aMonotonicity frustration and ordered response an analysis of the bBioMed Central3 aBackground. \\nFor large-scale biological networks represented as signed graphs, the index of frustration measures how far a network is from a monotone system, i.e., how incoherently the system responds to perturbations.\\nResults. \\nIn this paper we find that the frustration is systematically lower in transcriptional networks (modeled at functional level) than in signaling and metabolic networks (modeled at stoichiometric level). A possible interpretation of this result is in terms of energetic cost of an interaction: an erroneous or contradictory transcriptional action costs much more than a signaling/metabolic error, and therefore must be avoided as much as possible. Averaging over all possible perturbations, however, we also find that unlike for transcriptional networks, in the signaling/metabolic networks the probability of finding the system in its least frustrated configuration tends to be high also in correspondence of a moderate energetic regime, meaning that, in spite of the higher frustration, these networks can achieve a globally ordered response to perturbations even for moderate values of the strength of the interactions. Furthermore, an analysis of the energy landscape shows that signaling and metabolic networks lack energetic barriers around their global optima, a property also favouring global order.\\nConclusion. \\nIn conclusion, transcriptional and signaling/metabolic networks appear to have systematic differences in both the index of frustration and the transition to global order. These differences are interpretable in terms of the different functions of the various classes of networks.1 aIacono, Giovanni1 aAltafini, Claudio uhttp://hdl.handle.net/1963/405500928nas a2200109 4500008004300000245008500043210006900128520053800197100002300735700002400758856003600782 2010 en_Ud 00aMoore-Read Fractional Quantum Hall wavefunctions and SU(2) quiver gauge theories0 aMooreRead Fractional Quantum Hall wavefunctions and SU2 quiver g3 aWe identify Moore-Read wavefunctions, describing non-abelian statistics in fractional quantum Hall systems, with the instanton partition of N=2 superconformal quiver gauge theories at suitable values of masses and \\\\Omega-background parameters. This is obtained by extending to rational conformal field theories the SU(2) gauge quiver/Liouville field theory duality recently found by Alday-Gaiotto-Tachikawa. A direct link between the Moore-Read Hall $n$-body wavefunctions and Z_n-equivariant Donaldson polynomials is pointed out.1 aSantachiara, Raoul1 aTanzini, Alessandro uhttp://hdl.handle.net/1963/385200443nas a2200145 4500008004100000022001400041245004700055210004700102300001500149490000700164100001900171700001600190700001500206856007600221 2009 eng d a1751-811300aMesoscopic colonization in a spectral band0 aMesoscopic colonization in a spectral band a415204, 170 v421 aBertola, Marco1 aLee, S., Y.1 aMo, M., Y. uhttp://0-dx.doi.org.mercury.concordia.ca/10.1088/1751-8113/42/41/41520400903nas a2200145 4500008004100000245006400041210006300105260002900168300001600197490000700213520043600220100003000656700001900686856005200705 2009 eng d00aMinimal disc-type surfaces embedded in a perturbed cylinder0 aMinimal disctype surfaces embedded in a perturbed cylinder bKhayyam Publishing, Inc. a1115–11240 v223 aIn the present note we deal with small perturbations of an infinite cylinder in the 3D euclidian space. We find minimal disc-type surfaces embedded in the cylinder and intersecting its boundary perpendicularly. The existence and localization of those minimal discs is a consequence of a non-degeneracy condition for the critical points of a functional related to the oscillations of the cylinder from the flat configuration.

1 aFall, Mouhamed, Moustapha1 aMercuri, Carlo uhttps://projecteuclid.org/euclid.die/135601940700433nas a2200157 4500008004100000245004500041210004300086260001500129300001400144490000700158100001800165700001700183700001700200700002100217856003700238 2009 eng d00aA model for the dynamics of rowing boats0 amodel for the dynamics of rowing boats bWileycsep a119–1430 v611 aFormaggia, L.1 aMiglio, Edie1 aMola, Andrea1 aMontano, Antonio uhttps://doi.org/10.1002/fld.194000606nas a2200133 4500008004300000245006900043210006700112260002300179520017500202100002200377700001800399700001900417856003600436 2009 en_Ud 00aA model for the orbifold Chow ring of weighted projective spaces0 amodel for the orbifold Chow ring of weighted projective spaces bTaylor and Francis3 aWe construct an isomorphism of graded Frobenius algebras between the orbifold Chow ring of weighted projective spaces and graded algebras of groups of roots of the unity.1 aBoissiere, Samuel1 aMann, Etienne1 aPerroni, Fabio uhttp://hdl.handle.net/1963/358900389nas a2200109 4500008004100000245005500041210005500096300001200151490000700163100001900170856009000189 2009 eng d00aMoment determinants as isomonodromic tau functions0 aMoment determinants as isomonodromic tau functions a29–500 v221 aBertola, Marco uhttps://www.math.sissa.it/publication/moment-determinants-isomonodromic-tau-functions01344nas a2200145 4500008004300000245009700043210006900140260001900209520085100228100002001079700002101099700002001120700002201140856003601162 2009 en_Ud 00amRNA stability and the unfolding of gene expression in the long-period yeast metabolic cycle0 amRNA stability and the unfolding of gene expression in the longp bBioMed Central3 aBackground: In yeast, genome-wide periodic patterns associated with energy-metabolic oscillations have been shown recently for both short (approx. 40 min) and long (approx. 300 min) periods.\\nResults: The dynamical regulation due to mRNA stability is found to be an important aspect of the genome-wide coordination of the long-period yeast metabolic cycle. It is shown that for periodic genes, arranged in classes according either to expression profile or to function, the pulses of mRNA abundance have phase and width which are directly proportional to the corresponding turnover rates.\\nConclusion: The cascade of events occurring during the yeast metabolic cycle (and their correlation with mRNA turnover) reflects to a large extent the gene expression program observable in other dynamical contexts such as the response to stresses/stimuli.1 aSoranzo, Nicola1 aZampieri, Mattia1 aFarina, Lorenzo1 aAltafini, Claudio uhttp://hdl.handle.net/1963/363000334nas a2200085 4500008004300000245008000043210006900123100002000192856003600212 2008 en_Ud 00aMinimization of non quasiconvex functionals by integro-extremization method0 aMinimization of non quasiconvex functionals by integroextremizat1 aZagatti, Sandro uhttp://hdl.handle.net/1963/276100355nas a2200085 4500008004300000245010100043210006900144100002000213856003600233 2008 en_Ud 00aMinimizers of non convex scalar functionals and viscosity solutions of Hamilton-Jacobi equations0 aMinimizers of non convex scalar functionals and viscosity soluti1 aZagatti, Sandro uhttp://hdl.handle.net/1963/276000352nas a2200097 4500008004300000245006500043210006500108260002300173100002200196856003600218 2008 en_Ud 00aMorse theory and a scalar field equation on compact surfaces0 aMorse theory and a scalar field equation on compact surfaces bKhayyam Publishing1 aMalchiodi, Andrea uhttp://hdl.handle.net/1963/353100342nas a2200097 4500008004300000245006300043210006200106100002400168700001600192856003600208 2008 en_Ud 00aMultiple bound states for the Schroedinger-Poisson problem0 aMultiple bound states for the SchroedingerPoisson problem1 aAmbrosetti, Antonio1 aRuiz, David uhttp://hdl.handle.net/1963/267900522nas a2200145 4500008004100000245006600041210006500107260002300172300001200195490000800207100001900215700002400234700001900258856009900277 2007 eng d00aMassless scalar field in a two-dimensional de Sitter universe0 aMassless scalar field in a twodimensional de Sitter universe aBaselbBirkhäuser a27–380 v2511 aBertola, Marco1 aCorbetta, Francesco1 aMoschella, Ugo uhttps://www.math.sissa.it/publication/massless-scalar-field-two-dimensional-de-sitter-universe00410nas a2200097 4500008004300000245012400043210006900167100002000236700002000256856003600276 2007 en_Ud 00aOn the Maz\\\'ya inequalities: existence and multiplicity results for an elliptic problem involving cylindrical weights0 aMazya inequalities existence and multiplicity results for an ell1 aGazzini, Marita1 aMusina, Roberta uhttp://hdl.handle.net/1963/252201055nas a2200109 4500008004300000245006600043210006600109520069300175100001600868700002500884856003600909 2007 en_Ud 00aMetrics on semistable and numerically effective Higgs bundles0 aMetrics on semistable and numerically effective Higgs bundles3 aWe consider fibre metrics on Higgs vector bundles on compact K\\\\\\\"ahler manifolds, providing notions of numerical effectiveness and numerical flatness in terms of such metrics. We prove several properties of bundles satisfying such conditions and in particular we show that numerically flat Higgs bundles have vanishing Chern classes, and that they admit filtrations whose quotients are stable flat Higgs bundles. We compare these definitions with those previously given in the case of projective varieties. Finally we study the relations between numerically effectiveness and semistability, providing semistability criteria for Higgs bundles on projective manifolds of any dimension.1 aBruzzo, Ugo1 aGrana-Otero, Beatriz uhttp://hdl.handle.net/1963/184000408nas a2200109 4500008004300000245008800043210006900131100002400200700002200224700001600246856003600262 2007 en_Ud 00aMulti-bump solitons to linearly coupled systems of nonlinear Schrödinger equations0 aMultibump solitons to linearly coupled systems of nonlinear Schr1 aAmbrosetti, Antonio1 aColorado, Eduardo1 aRuiz, David uhttp://hdl.handle.net/1963/183500691nas a2200109 4500008004100000245006700041210006500108260001000173520034100183100002100524856003600545 2006 en d00aMatching Procedure for the Sixth Painlevé Equation (May 2006)0 aMatching Procedure for the Sixth Painlevé Equation May 2006 bSISSA3 aWe present a constructive procedure to obtain the critical behavior of\r\nPainleve\' VI transcendents and solve the connection problem. This procedure\r\nyields two and one parameter families of solutions, including trigonometric and\r\nlogarithmic behaviors, and three classes of solutions with Taylor expansion at\r\na critical point.1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/652401933nas a2200145 4500008004100000245004900041210004900090260002900139520150600168100002001674700002001694700002201714700001501736856003601751 2005 en d00aMinimal surfaces in pseudohermitian geometry0 aMinimal surfaces in pseudohermitian geometry bScuola Normale Superiore3 aWe consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group. We interpret the p-mean curvature not only as the tangential sublaplacian of a defining function, but also as the curvature of a characteristic curve, and as a quantity in terms of calibration geometry. As a differential equation, the p-minimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set, we formulate some {\em extension} theorems, which describe how the characteristic curves meet the singular set. This allows us to classify the entire solutions to this equation and to solve a Bernstein-type problem (for graphs over the $xy$-plane) in the Heisenberg group $H_1$. In $H_{1}$, identified with the Euclidean space $R^{3}$, the p-minimal surfaces are classical ruled surfaces with the rulings generated by Legendrian lines. We also prove a uniqueness theorem for the Dirichlet problem under a condition on the size of the singular set in two dimensions, and generalize to higher dimensions without any size control condition. We also show that there are no closed, connected, $C^{2}$ smoothly immersed constant p-mean curvature or p-minimal surfaces of genus greater than one in the standard $S^{3}.$ This fact continues to hold when $S^{3}$ is replaced by a general spherical pseudohermitian 3-manifold.1 aCheng, Jih-Hsin1 aHwang, JennFang1 aMalchiodi, Andrea1 aYang, Paul uhttp://hdl.handle.net/1963/457900521nas a2200097 4500008004300000245006000043210005300103520021100156100002000367856003600387 2005 en_Ud 00aOn the Minimum Problem for Nonconvex Scalar Functionals0 aMinimum Problem for Nonconvex Scalar Functionals3 aWe study the minimum problem for scalar nonconvex functionals defined on Sobolev maps satisfying a Dirichlet boundary condition and refine well-known existence results under standard regularity assumptions.1 aZagatti, Sandro uhttp://hdl.handle.net/1963/276401078nas a2200109 4500008004300000245007500043210006900118520070500187100002200892700001800914856003600932 2005 en_Ud 00aModulation of the Camassa-Holm equation and reciprocal transformations0 aModulation of the CamassaHolm equation and reciprocal transforma3 aWe derive the modulation equations or Whitham equations for the Camassa-Holm (CH) equation. We show that the modulation equations are hyperbolic and admit bi-Hamiltonian structure. Furthermore they are connected by a reciprocal transformation to the modulation equations of the first negative flow of the Korteweg de Vries (KdV) equation. The reciprocal transformation is generated by the Casimir of the second Poisson bracket of the KdV averaged flow. We show that the geometry of the bi-Hamiltonian structure of the KdV and CH modulation equations is quite different: indeed the KdV averaged bi-Hamiltonian structure can always be related to a semisimple Frobenius manifold while the CH one cannot.1 aAbenda, Simonetta1 aGrava, Tamara uhttp://hdl.handle.net/1963/230500421nas a2200121 4500008004300000245008100043210006900124260001300193100002200206700001700228700001800245856003600263 2005 en_Ud 00aMultiple clustered layer solutions for semilinear Neumann problems on a ball0 aMultiple clustered layer solutions for semilinear Neumann proble bElsevier1 aMalchiodi, Andrea1 aNi, Wei-Ming1 aWei, Juncheng uhttp://hdl.handle.net/1963/353200956nas a2200133 4500008004100000245008400041210006900125260000900194520052900203100001700732700001700749700002000766856003600786 2004 en d00aOn the minimal degree of a common Lyapunov function for planar switched systems0 aminimal degree of a common Lyapunov function for planar switched bIEEE3 aIn this paper, we consider linear switched systems x(t) = Au(t)x(t), x ε Rn, u ε U, and the problem of asymptotic stability for arbitrary switching functions, uniform with respect to switching (UAS for short). We first prove that, given a UAS system, it is always possible to build a polynomial common Lyapunov function. Then our main result is that the degree of that the common polynomial Lyapunov function is not uniformly bounded over all the UAS systems. This result answers a question raised by Dayawansa and Martin.1 aMason, Paolo1 aBoscain, Ugo1 aChitour, Yacine uhttp://hdl.handle.net/1963/483400410nas a2200109 4500008004300000245008000043210006900123260002600192100002200218700002400240856003600264 2004 en_Ud 00aMultidimensional boundary layers for a singularly perturbed Neumann problem0 aMultidimensional boundary layers for a singularly perturbed Neum bDuke University Press1 aMalchiodi, Andrea1 aMontenegro, Marcelo uhttp://hdl.handle.net/1963/296000380nas a2200109 4500008004300000245006700043210006700110260001300177100002400190700002000214856003600234 2004 en_Ud 00aMultiplicity of periodic solutions of nonlinear wave equations0 aMultiplicity of periodic solutions of nonlinear wave equations bElsevier1 aBerti, Massimiliano1 aBolle, Philippe uhttp://hdl.handle.net/1963/297400444nas a2200133 4500008004100000022001400041245005600055210005500111300001600166490000700182100001900189700001500208856008700223 2003 eng d a0305-447000aMixed correlation functions of the two-matrix model0 aMixed correlation functions of the twomatrix model a7733–77500 v361 aBertola, Marco1 aEynard, B. uhttps://www.math.sissa.it/publication/mixed-correlation-functions-two-matrix-model01197nas a2200121 4500008004300000245012700043210006900170260001300239520074500252100002200997700002001019856003601039 2003 en_Ud 00aMotion on submanifolds of noninvariant holonomic constraints for a kinematic control system evolving on a matrix Lie group0 aMotion on submanifolds of noninvariant holonomic constraints for bElsevier3 aFor a control system on a matrix Lie group with one or more configuration constraints that are not left/right invariant, finding the combinations of (kinematic) control inputs satisfying the motion constraints is not a trivial problem. Two methods, one coordinate-dependent and the other coordinate-free are suggested. The first is based on the Wei-Norman formula; the second on the calculation of the annihilator of the coadjoint action of the constraint one-form at each point of the group manifold. The results are applied to a control system on SE(3) with a holonomic inertial constraint involving the noncommutative part in a nontrivial way. The difference in terms of compactness of the result between the two methods is considerable.1 aAltafini, Claudio1 aFrezza, Ruggero uhttp://hdl.handle.net/1963/301800423nas a2200133 4500008004100000245005600041210005500097260001800152100001600170700002100186700002200207700002400229856003600253 2003 en d00aMulti-instanton calculus and equivariant cohomology0 aMultiinstanton calculus and equivariant cohomology bSISSA Library1 aBruzzo, Ugo1 aMorales, Jose F.1 aFucito, Francesco1 aTanzini, Alessandro uhttp://hdl.handle.net/1963/164501609nas a2200121 4500008004100000245009900041210006900140260001800209520118400227100002101411700001901432856003601451 2002 en d00aA model for the quasi-static growth of a brittle fracture: existence and approximation results0 amodel for the quasistatic growth of a brittle fracture existence bSISSA Library3 aWe study a variant of the variational model for the quasi-static growth of brittle fractures proposed by Francfort and Marigo.9 The main feature of our model is that, in the discrete-time formulation, in each step we do not consider absolute minimizers of the energy, but, in a sense, we look for local minimizers which are sufficiently close to the approximate solution obtained in the previous step. This is done by introducing in the variational problem an additional term which penalizes the L2-distance between the approximate solutions at two consecutive times. We study the continuous-time version of this model, obtained by passing to the limit as the time step tends to zero, and show that it satisfies (for almost every time) some minimality conditions which are slightly different from those considered in Refs. 9 and 8, but are still enough to prove (under suitable regularity assumptions on the crack path) that the classical Griffith\\\'s criterion holds at the crack tips. We also prove that, if no initial crack is present and if the data of the problem are sufficiently smooth, no crack will develop in this model, provided the penalization term is large enough.1 aDal Maso, Gianni1 aToader, Rodica uhttp://hdl.handle.net/1963/157101599nas a2200121 4500008004100000245008900041210006900130260001800199520118400217100002101401700001901422856003601441 2002 en d00aA model for the quasi-static growth of brittle fractures based on local minimization0 amodel for the quasistatic growth of brittle fractures based on l bSISSA Library3 aWe study a variant of the variational model for the quasi-static growth of brittle fractures proposed by Francfort and Marigo.9 The main feature of our model is that, in the discrete-time formulation, in each step we do not consider absolute minimizers of the energy, but, in a sense, we look for local minimizers which are sufficiently close to the approximate solution obtained in the previous step. This is done by introducing in the variational problem an additional term which penalizes the L2-distance between the approximate solutions at two consecutive times. We study the continuous-time version of this model, obtained by passing to the limit as the time step tends to zero, and show that it satisfies (for almost every time) some minimality conditions which are slightly different from those considered in Refs. 9 and 8, but are still enough to prove (under suitable regularity assumptions on the crack path) that the classical Griffith\\\'s criterion holds at the crack tips. We also prove that, if no initial crack is present and if the data of the problem are sufficiently smooth, no crack will develop in this model, provided the penalization term is large enough.1 aDal Maso, Gianni1 aToader, Rodica uhttp://hdl.handle.net/1963/162101237nas a2200121 4500008004300000245009800043210006900141260001300210520081600223100002101039700001901060856003601079 2002 en_Ud 00aA Model for the Quasi-Static Growth of Brittle Fractures: Existence and Approximation Results0 aModel for the QuasiStatic Growth of Brittle Fractures Existence bSpringer3 aWe give a precise mathematical formulation of a variational model for the irreversible quasi-static evolution of brittle fractures proposed by G.A. Francfort and J.-J. Marigo, and based on Griffith\\\'s theory of crack growth. In the two-dimensional case we prove an existence result for the quasi-static evolution and show that the total energy is an absolutely continuous function of time, although we can not exclude that the bulk energy and the surface energy may present some jump discontinuities. This existence result is proved by a time discretization process, where at each step a global energy minimization is performed, with the constraint that the new crack contains all cracks formed at the previous time steps. This procedure provides an effective way to approximate the continuous time evolution.1 aDal Maso, Gianni1 aToader, Rodica uhttp://hdl.handle.net/1963/305600546nas a2200109 4500008004300000245008800043210006900131260001000200520016200210100002800372856003600400 2002 en_Ud 00aA multiplicity result for the Schrodinger-Maxwell equations with negative potential0 amultiplicity result for the SchrodingerMaxwell equations with ne bIMPAN3 aWe prove the existence of a sequence of radial solutions with negative energy of the Schrödinger-Maxwell equations under the action of a negative potential.1 aCoclite, Giuseppe Maria uhttp://hdl.handle.net/1963/305300451nas a2200109 4500008004100000245005400041210005400095260003300149520009900182100002400281856003600305 2002 en d00aMultiplicity results for the Yamabe problem on Sn0 aMultiplicity results for the Yamabe problem on Sn bNational Academy of Sciences3 aWe discuss some results related to the existence of multiple solutions for the Yamabe problem.1 aAmbrosetti, Antonio uhttp://hdl.handle.net/1963/588501054nas a2200121 4500008004100000245008200041210006900123260001800192520064300210100002100853700002200874856003600896 2001 en d00aA monotonicity approach to nonlinear Dirichlet problems in perforated domains0 amonotonicity approach to nonlinear Dirichlet problems in perfora bSISSA Library3 aWe study the asymptotic behaviour of solutions to Dirichlet problems in perforated domains for nonlinear elliptic equations associated with monotone operators. The main difference with respect to the previous papers on this subject is that no uniformity is assumed in the monotonicity condition. Under a very general hypothesis on the holes of the domains, we construct a limit equation, which is satisfied by the weak limits of the solutions. The additional term in the limit problem depends only on the local behaviour of the holes, which can be expressed in terms of suitable nonlinear capacities associated with the monotone operator.1 aDal Maso, Gianni1 aSkrypnik, Igor V. uhttp://hdl.handle.net/1963/155500380nas a2200109 4500008004100000245006800041210006700109260001800176100001700194700002300211856003600234 2001 en d00aMorse properties for the minimum time function on 2-D manifolds0 aMorse properties for the minimum time function on 2D manifolds bSISSA Library1 aBoscain, Ugo1 aPiccoli, Benedetto uhttp://hdl.handle.net/1963/154100437nas a2200133 4500008004100000245006500041210005600106260001800162100001600180700002200196700002400218700002500242856003600267 2001 en d00aOn the Multi-Instanton Measure for Super Yang-Mills Theories0 aMultiInstanton Measure for Super YangMills Theories bSISSA Library1 aBruzzo, Ugo1 aFucito, Francesco1 aTanzini, Alessandro1 aTravaglini, Gabriele uhttp://hdl.handle.net/1963/153100362nas a2200097 4500008004300000245008100043210006900124260001300193100002200206856003600228 2001 en_Ud 00aMultiple positive solutions of some elliptic equations in \\\\bold R\\\\sp N0 aMultiple positive solutions of some elliptic equations in bold R bElsevier1 aMalchiodi, Andrea uhttp://hdl.handle.net/1963/309400433nas a2200121 4500008004100000245008200041210006900123260001800192100002400210700002200234700001900256856003600275 2001 en d00aMultiplicity results for some nonlinear Schrodinger equations with potentials0 aMultiplicity results for some nonlinear Schrodinger equations wi bSISSA Library1 aAmbrosetti, Antonio1 aMalchiodi, Andrea1 aSecchi, Simone uhttp://hdl.handle.net/1963/156400916nas a2200109 4500008004300000245006700043210006600110260000900176520056500185100002000750856003600770 2000 en_Ud 00aMinimization of functionals of the gradient by Baire's theorem0 aMinimization of functionals of the gradient by Baires theorem bSIAM3 aWe give sufficient conditions for the existence of solutions of the minimum problem $$ {\mathcal{P}}_{u_0}: \qquad \hbox{Minimize}\quad \int_\Omega g(Du(x))dx, \quad u\in u_0 + W_0^{1,p}(\Omega,{\mathbb{R}}), $$ based on the structure of the epigraph of the lower convex envelope of g, which is assumed be lower semicontinuous and to grow at infinity faster than the power p with p larger than the dimension of the space. No convexity conditions are required on g, and no assumptions are made on the boundary datum $u_0\in W_0^{1,p}(\Omega,\mathbb{R})$.

1 aZagatti, Sandro uhttp://hdl.handle.net/1963/351101068nas a2200121 4500008004300000245007400043210007000117260001300187520067000200100002000870700002000890856003600910 2000 en_Ud 00aMonodromy of certain Painlevé-VI transcendents and reflection groups0 aMonodromy of certain PainlevéVI transcendents and reflection gro bSpringer3 aWe study the global analytic properties of the solutions of a particular family of Painleve\\\' VI equations with the parameters $\\\\beta=\\\\gamma=0$, $\\\\delta={1\\\\over2}$ and $\\\\alpha$ arbitrary. We introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space. The analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group. This result is used to classify all the algebraic solutions of our Painleve\\\' VI equation.1 aDubrovin, Boris1 aMazzocco, Marta uhttp://hdl.handle.net/1963/288201398nas a2200133 4500008004100000245006400041210006000105260001300165520099200178100002101170700001801191700001901209856003601228 1999 en d00aThe method of Poisson pairs in the theory of nonlinear PDEs0 amethod of Poisson pairs in the theory of nonlinear PDEs bSpringer3 aThe aim of these lectures is to show that the methods of classical Hamiltonian mechanics can be profitably used to solve certain classes of nonlinear partial differential equations. The prototype of these equations is the well-known Korteweg-de Vries (KdV) equation.\\nIn these lectures we touch the following subjects:\\ni) the birth and the role of the method of Poisson pairs inside the theory of the KdV equation;\\nii) the theoretical basis of the method of Poisson pairs;\\niii) the Gel\\\'fand-Zakharevich theory of integrable systems on bi-Hamiltonian manifolds;\\niv) the Hamiltonian interpretation of the Sato picture of the KdV flows and of its linearization on an infinite-dimensional Grassmannian manifold.\\nv) the reduction technique(s) and its use to construct classes of solutions;\\nvi) the role of the technique of separation of variables in the study of the reduced systems;\\nvii) some relations intertwining the method of Poisson pairs with the method of Lax pairs.1 aFalqui, Gregorio1 aMagri, Franco1 aPedroni, Marco uhttp://hdl.handle.net/1963/135000599nas a2200121 4500008004100000245006400041210005600105260001300161520022100174100002400395700002200419856003600441 1999 en d00aA multiplicity result for the Yamabe problem on $S\\\\sp n$0 amultiplicity result for the Yamabe problem on Ssp n bElsevier3 aWe prove a multiplicity result for the Yamabe problem on the manifold (S, g), where g is a perturbation of the standard metric g0 of Sn. Solutions are found by variational methods via an abstract perturbation result.1 aAmbrosetti, Antonio1 aMalchiodi, Andrea uhttp://hdl.handle.net/1963/126400706nas a2200121 4500008004300000245006300043210006200106260001300168520032800181100001600509700002300525856003600548 1998 en_Ud 00aMirror Symmetry on K3 Surfaces as a Hyper-Kähler Rotation0 aMirror Symmetry on K3 Surfaces as a HyperKähler Rotation bSpringer3 aWe show that under the hypotheses of Strominger, Yau and Zaslow\\\'s paper, a mirror partner of a K3 surface $X$ with a fibration in special Lagrangian tori can be obtained by rotating the complex structure of $X$ within its hyperk\\\\\\\"ahler family of complex structures. The same hypotheses force the B-field to vanish.1 aBruzzo, Ugo1 aSanguinetti, Guido uhttp://hdl.handle.net/1963/288800394nas a2200109 4500008004100000245008600041210006900127260001000196653002100206100002100227856003600248 1990 en d00aModuli Spaces and Geometrical Aspects of Two-Dimensional Conformal Field Theories0 aModuli Spaces and Geometrical Aspects of TwoDimensional Conforma bSISSA10aAlgebraic curves1 aFalqui, Gregorio uhttp://hdl.handle.net/1963/555200412nas a2200097 4500008004100000245012200041210006900163260001800232100002900250856003500279 1988 en d00aMethods of stochastic stability and properties of the Gribov horizon in the stochastic quantization of gauge theories0 aMethods of stochastic stability and properties of the Gribov hor bSISSA Library1 aDell'Antonio, Gianfausto uhttp://hdl.handle.net/1963/81700605nas a2200121 4500008004100000245006100041210006000102260003200162520020700194100002700401700002000428856003500448 1985 en d00aMaximal acceleration and Sakharov's limiting temperature0 aMaximal acceleration and Sakharovs limiting temperature bSocietà Italiana di Fisica3 aIt is shown that Sakharov's maximal temperature, derived by him from astrophysical considerations, is a straightforward consequence of the maximal acceleration introduced by us in previous works.

1 aCaianiello, Eduardo R.1 aLandi, Giovanni uhttp://hdl.handle.net/1963/372