In a measure space $(X,{\mathcal {A}},\mu )$, we consider two measurable functions $f,g:E\rightarrow {\mathbb {R}}$, for some $E\in {\mathcal {A}}$. We prove that the property of having equal p-norms when p varies in some infinite set $P\subseteq [1,+\infty )$ is equivalent to the following condition: $\begin{aligned} \mu (\{x\in E:|f(x)|>\alpha \})=\mu (\{x\in E:|g(x)|>\alpha \})\quad \text { for all } \alpha \ge 0. \end{aligned}$

%B Annali di Matematica Pura ed Applicata (1923 -) %V 199 %P 955-968 %G eng %U https://doi.org/10.1007/s10231-019-00907-z %R 10.1007/s10231-019-00907-z %0 Journal Article %J NONLINEAR ANALYSIS %D 2020 %T Periodic solutions of nearly integrable Hamiltonian systems bifurcating from infinite-dimensional tori %A Alessandro Fonda %A Giuliano Klun %A Andrea Sfecci %XWe prove the existence of periodic solutions of some infinite-dimensional nearly integrable Hamiltonian systems, bifurcating from infinite-dimensional tori, by the use of a generalization of the Poincaré–Birkhoff Theorem.

%B NONLINEAR ANALYSIS %G eng %U https://doi.org/10.1016/j.na.2019.111720 %R 10.1016/j.na.2019.111720 %0 Conference Paper %B IUTAM Symposium on Model Order Reduction of Coupled Systems, Stuttgart, Germany, May 22–25, 2018 %D 2020 %T A Reduced Order Approach for the Embedded Shifted Boundary FEM and a Heat Exchange System on Parametrized Geometries %A Efthymios N. Karatzas %A Giovanni Stabile %A Nabib Atallah %A Guglielmo Scovazzi %A Gianluigi Rozza %E Fehr, Jörg %E Haasdonk, Bernard %XA model order reduction technique is combined with an embedded boundary finite element method with a POD-Galerkin strategy. The proposed methodology is applied to parametrized heat transfer problems and we rely on a sufficiently refined shape-regular background mesh to account for parametrized geometries. In particular, the employed embedded boundary element method is the Shifted Boundary Method (SBM) recently proposed. This approach is based on the idea of shifting the location of true boundary conditions to a surrogate boundary, with the goal of avoiding cut cells near the boundary of the computational domain. This combination of methodologies has multiple advantages. In the first place, since the Shifted Boundary Method always relies on the same background mesh, there is no need to update the discretized parametric domain. Secondly, we avoid the treatment of cut cell elements, which usually need particular attention. Thirdly, since the whole background mesh is considered in the reduced basis construction, the SBM allows for a smooth transition of the reduced modes across the immersed domain boundary. The performances of the method are verified in two dimensional heat transfer numerical examples.

%B IUTAM Symposium on Model Order Reduction of Coupled Systems, Stuttgart, Germany, May 22–25, 2018 %I Springer International Publishing %G eng %U https://arxiv.org/abs/1807.07753 %R 10.1007/978-3-030-21013-7_8 %0 Journal Article %J Journal of Numerical Mathematics %D 2019 %T The deal.II Library, Version 9.1 %A Arndt, Daniel %A Bangerth, Wolfgang %A Clevenger, Thomas C. %A Davydov, Denis %A Fehling, Marc %A Garcia-Sanchez, Daniel %A Harper, Graham %A Heister, Timo %A Heltai, Luca %A Kronbichler, Martin %A Maguire Kynch, Ross %A Maier, Matthias %A Pelteret, Jean Paul %A Turcksin, Bruno %A Wells, David %X This paper provides an overview of the new features of the finite element library deal.II, version 9.1. %B Journal of Numerical Mathematics %G eng %R 10.1515/jnma-2019-0064 %0 Journal Article %J Discrete & Computational Geometry %D 2019 %T On the Number of Flats Tangent to Convex Hypersurfaces in Random Position %A Khazhgali Kozhasov %A Antonio Lerario %B Discrete & Computational Geometry %8 Mar %G eng %U https://doi.org/10.1007/s00454-019-00067-0 %R 10.1007/s00454-019-00067-0 %0 Journal Article %J Foundations of Computational Mathematics %D 2019 %T The Real Polynomial Eigenvalue Problem is Well Conditioned on the Average %A Carlos Beltrán %A Khazhgali Kozhasov %XWe study the average condition number for polynomial eigenvalues of collections of matrices drawn from some random matrix ensembles. In particular, we prove that polynomial eigenvalue problems defined by matrices with random Gaussian entries are very well conditioned on the average.

%B Foundations of Computational Mathematics %8 May %G eng %U https://doi.org/10.1007/s10208-019-09414-2 %R 10.1007/s10208-019-09414-2 %0 Journal Article %J Computer Methods in Applied Mechanics and Engineering %D 2019 %T A reduced basis approach for PDEs on parametrized geometries based on the shifted boundary finite element method and application to a Stokes flow %A Karatzas, Efthymios N %A Stabile, Giovanni %A Nouveau, Leo %A Scovazzi, Guglielmo %A Rozza, Gianluigi %XWe propose a model order reduction technique integrating the Shifted Boundary Method (SBM) with a POD-Galerkin strategy. This approach allows to treat more complex parametrized domains in an efficient and straightforward way. The impact of the proposed approach is threefold. First, problems involving parametrizations of complex geometrical shapes and/or large domain deformations can be efficiently solved at full-order by means of the SBM, an unfitted boundary method that avoids remeshing and the tedious handling of cut cells by introducing an approximate surrogate boundary. Second, the computational effort is further reduced by the development of a reduced order model (ROM) technique based on a POD-Galerkin approach. Third, the SBM provides a smooth mapping from the true to the surrogate domain, and for this reason, the stability and performance of the reduced order basis are enhanced. This feature is the net result of the combination of the proposed ROM approach and the SBM. Similarly, the combination of the SBM with a projection-based ROM gives the great advantage of an easy and fast to implement algorithm considering geometrical parametrization with large deformations. The transformation of each geometry to a reference geometry (morphing) is in fact not required. These combined advantages will allow the solution of PDE problems more efficiently. We illustrate the performance of this approach on a number of two-dimensional Stokes flow problems.

%B Computer Methods in Applied Mechanics and Engineering %V 347 %P 568–587 %G eng %U https://arxiv.org/abs/1807.07790 %R 10.1016/j.cma.2018.12.040 %0 Book Section %B Numerical Mathematics and Advanced Applications - ENUMATH 2017 %D 2019 %T A Spectral Element Reduced Basis Method in Parametric CFD %A Hess, Martin W. %A Rozza, Gianluigi %E Radu, Florin Adrian %E Kumar, Kundan %E Berre, Inga %E Nordbotten, Jan Martin %E Pop, Iuliu Sorin %XWe consider the Navier-Stokes equations in a channel with varying Reynolds numbers. The model is discretized with high-order spectral element ansatz functions, resulting in 14 259 degrees of freedom. The steady-state snapshot solu- tions define a reduced order space, which allows to accurately evaluate the steady- state solutions for varying Reynolds number with a reduced order model within a fixed-point iteration. In particular, we compare different aspects of implementing the reduced order model with respect to the use of a spectral element discretization. It is shown, how a multilevel static condensation in the pressure and velocity boundary degrees of freedom can be combined with a reduced order modelling approach to enhance computational times in parametric many-query scenarios.

%B Numerical Mathematics and Advanced Applications - ENUMATH 2017 %I Springer International Publishing %V 126 %G eng %U https://arxiv.org/abs/1712.06432 %& A Spectral Element Reduced Basis Method in Parametric CFD %R 10.1007/978-3-319-96415-7_64 pages = 693–701 %0 Journal Article %J TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS %D 2019 %T On the topological degree of planar maps avoiding normal cones %A Alessandro Fonda %A Giuliano Klun %XThe classical Poincaré-Bohl theorem provides the existence of a zero for a function avoiding external rays. When the domain is convex, the same holds true when avoiding normal cones.

We consider here the possibility of dealing with nonconvex sets having inward corners or cusps, in which cases the normal cone vanishes. This allows us to deal with situations where the topological degree may be strictly greater than $1$.

We construct generic real symmetric tensors with only real eigenvectors or, equivalently, real homogeneous polynomials with the maximum possible finite number of critical points on the sphere.

%B SIAM Journal on Applied Algebra and Geometry %I SIAM %V 2 %P 339–347 %G eng %U https://epubs.siam.org/doi/pdf/10.1137/17M1145902 %R 10.1137/17M1145902 %0 Journal Article %J Journal de Mathématiques Pures et Appliquées %D 2018 %T Minimizing movements for mean curvature flow of droplets with prescribed contact angle %A Giovanni Bellettini %A Matteo Novaga %A Shokhrukh Kholmatov %K Capillary functional %K Mean curvature flow with prescribed contact angle %K Minimizing movements %K Sets of finite perimeter %XWe 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.

%B Journal de Mathématiques Pures et Appliquées %V 117 %P 1 - 58 %G eng %U http://www.sciencedirect.com/science/article/pii/S0021782418300825 %R https://doi.org/10.1016/j.matpur.2018.06.003 %0 Journal Article %J SIAM Journal on Mathematical Analysis %D 2018 %T Minimizing Movements for Mean Curvature Flow of Partitions %A Giovanni Bellettini %A Shokhrukh Kholmatov %XWe 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.

%B SIAM Journal on Mathematical Analysis %V 50 %P 4117-4148 %G eng %U https://doi.org/10.1137/17M1159294 %R 10.1137/17M1159294 %0 Journal Article %J Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences %D 2018 %T Numerical study of the Kadomtsev-Petviashvili equation and dispersive shock waves %A Tamara Grava %A Christian Klein %A Giuseppe Pitton %XA detailed numerical study of the long time behaviour of dispersive shock waves in solutions to the Kadomtsev–Petviashvili (KP) I equation is presented. It is shown that modulated lump solutions emerge from the dispersive shock waves. For the description of dispersive shock waves, Whitham modulation equations for KP are obtained. It is shown that the modulation equations near the soliton line are hyperbolic for the KPII equation while they are elliptic for the KPI equation leading to a focusing effect and the formation of lumps. Such a behaviour is similar to the appearance of breathers for the focusing nonlinear Schrödinger equation in the semiclassical limit.

%B Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences %V 474 %P 20170458 %G eng %U https://royalsocietypublishing.org/doi/abs/10.1098/rspa.2017.0458 %R 10.1098/rspa.2017.0458 %0 Generic %D 2018 %T A Reduced Basis approach for PDEs on parametrized geometries based on the Shifted Boundary Finite Element Method and application to fluid dynamics %A Efthymios N. Karatzas %A Giovanni Stabile %A Leo Nouveau %A Guglielmo Scovazzi %A Gianluigi Rozza %G eng %0 Generic %D 2018 %T A Reduced Order Approach for the Embedded Shifted Boundary FEM and a Heat Exchange System on Parametrized Geometries %A Efthymios N. Karatzas %A Giovanni Stabile %A N. Atallah %A Guglielmo Scovazzi %A Gianluigi Rozza %G eng %0 Journal Article %J Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences %D 2018 %T Symplectic invariants for parabolic orbits and cusp singularities of integrable systems %A Alexey Bolsinov %A Lorenzo Guglielmi %A Elena Kudryavtseva %XWe discuss normal forms and symplectic invariants of parabolic orbits and cuspidal tori in integrable Hamiltonian systems with two degrees of freedom. Such singularities appear in many integrable systems in geometry and mathematical physics and can be considered as the simplest example of degenerate singularities. We also suggest some new techniques which apparently can be used for studying symplectic invariants of degenerate singularities of more general type. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

%B Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences %V 376 %P 20170424 %G eng %U https://royalsocietypublishing.org/doi/abs/10.1098/rsta.2017.0424 %R 10.1098/rsta.2017.0424 %0 Conference Paper %B Theory, Numerics and Applications of Hyperbolic Problems I %D 2018 %T On Uniqueness of Weak Solutions to Transport Equation with Non-smooth Velocity Field %A Paolo Bonicatto %E Klingenberg, Christian %E Westdickenberg, Michael %B Theory, Numerics and Applications of Hyperbolic Problems I %I Springer International Publishing %C Cham %P 191–203 %@ 978-3-319-91545-6 %G eng %U https://link.springer.com/chapter/10.1007/978-3-319-91545-6_15 %R 10.1007/978-3-319-91545-6_15 %0 Journal Article %J JOURNAL OF NUMERICAL MATHEMATICS %D 2017 %T The deal.II Library, Version 8.5 %A Arndt, Daniel %A W. Bangerth %A Denis Davydov %A Timo Heister %A Luca Heltai %A Martin Kronbichler %A Matthias Maier %A Pelteret, Jean-Paul %A B. Turcksin %A David Wells %B JOURNAL OF NUMERICAL MATHEMATICS %V 25 %P 137–145 %G eng %U https://www.dealii.org/deal85-preprint.pdf %R 10.1515/jnma-2017-0058 %0 Journal Article %J Communications on Pure & Applied Analysis %D 2017 %T Minimizers of anisotropic perimeters with cylindrical norms %A Giovanni Bellettini %A Matteo Novaga %A Shokhrukh Kholmatov %K anisotropic Bernstein problem; %K minimal cones %K Non parametric minimal surfaces %K Sets of finite perimeter %XWe 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.

%B Communications on Pure & Applied Analysis %V 16 %P 1427 %G eng %U http://aimsciences.org//article/id/47054f15-00c7-40b7-9da1-4c0b1d0a103d %R 10.3934/cpaa.2017068 %0 Journal Article %J COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING %D 2017 %T A natural framework for isogeometric fluid-structure interaction based on BEM-shell coupling %A Luca Heltai %A Kiendl, J. %A Antonio DeSimone %A Alessandro Reali %B COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING %V 316 %P 522–546 %G eng %U http://cdsads.u-strasbg.fr/abs/2017CMAME.316..522H %R 10.1016/j.cma.2016.08.008 %0 Report %D 2017 %T Random spectrahedra %A Paul Breiding %A Khazhgali Kozhasov %A Antonio Lerario %G eng %0 Journal Article %J Inventiones Mathematicae %D 2017 %T Symplectic geometry of the moduli space of projective structures in homological coordinates %A Marco Bertola %A Dmitry Korotkin %A Chaya Norton %B Inventiones Mathematicae %P 1–56 %8 06 %G eng %U https://arxiv.org/abs/1506.07918 %0 Journal Article %J ARCHIVE OF NUMERICAL SOFTWARE %D 2016 %T The deal.II Library, Version 8.3 %A W. Bangerth %A Timo Heister %A Luca Heltai %A G. Kanschat %A Martin Kronbichler %A Matthias Maier %A B. Turcksin %B ARCHIVE OF NUMERICAL SOFTWARE %V 4 %P 1–11 %G eng %U http://nbn-resolving.de/urn:nbn:de:bsz:16-ans-231226 %R 10.11588/ans.2016.100.23122 %0 Journal Article %J JOURNAL OF NUMERICAL MATHEMATICS %D 2016 %T The deal.II library, Version 8.4 %A W. Bangerth %A Denis Davydov %A Timo Heister %A Luca Heltai %A G. Kanschat %A Martin Kronbichler %A Matthias Maier %A B. Turcksin %A David Wells %B JOURNAL OF NUMERICAL MATHEMATICS %V 24 %P 135–141 %G eng %U https://www.math.clemson.edu/ heister/preprints/deal84-preprint.pdf %R 10.1515/jnma-2016-1045 %0 Journal Article %J JOURNAL OF SCIENTIFIC COMPUTING %D 2016 %T Error Estimates of B-spline based finite-element method for the wind-driven ocean circulation %A Rotundo, N. %A Kim, T. -Y. %A Jiang, W. %A Luca Heltai %A Fried, E. %B JOURNAL OF SCIENTIFIC COMPUTING %V 69 %P 430–459 %G eng %R 10.1007/s10915-016-0201-1 %0 Report %D 2016 %T Large KAM tori for perturbations of the dNLS equation %A Massimiliano Berti %A Thomas Kappeler %A Riccardo Montalto %X We prove that small, semi-linear Hamiltonian perturbations of the defocusing nonlinear Schr\"odinger (dNLS) equation on the circle have an abundance of invariant tori of any size and (finite) dimension which support quasi-periodic solutions. When compared with previous results the novelty consists in considering perturbations which do not satisfy any symmetry condition (they may depend on x in an arbitrary way) and need not be analytic. The main difficulty is posed by pairs of almost resonant dNLS frequencies. The proof is based on the integrability of the dNLS equation, in particular the fact that the nonlinear part of the Birkhoff coordinates is one smoothing. We implement a Newton-Nash-Moser iteration scheme to construct the invariant tori. The key point is the reduction of linearized operators, coming up in the iteration scheme, to 2×2 block diagonal ones with constant coefficients together with sharp asymptotic estimates of their eigenvalues. %G en %U http://preprints.sissa.it/handle/1963/35284 %1 35589 %2 Mathematics %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-05-30T09:09:35Z No. of bitstreams: 1 1603.09252.pdf: 1306610 bytes, checksum: 1e34137dcb21eb2e2a8e93d0c2d009a3 (MD5) %0 Journal Article %J Journal of Noncommutative Geometry %D 2016 %T Pimsner algebras and Gysin sequences from principal circle actions %A Francesca Arici %A Jens Kaad %A Giovanni Landi %B Journal of Noncommutative Geometry %V 10 %P 29–64 %G eng %U http://hdl.handle.net/2066/162951 %R 10.4171/jncg/228 %0 Journal Article %J Communications in Number Theory and Physics %D 2016 %T Refined node polynomials via long edge graphs %A Lothar Göttsche %A Benjamin Kipkirui Kikwai %XThe generating functions of the Severi degrees for sufficiently ample line bundles on algebraic surfaces are multiplicative in the topological invariants of the surface and the line bundle. Recently new proofs of this fact were given for toric surfaces by Block, Colley, Kennedy and Liu, Osserman, using tropical geometry and in particular the combinatorial tool of long-edged graphs. In the first part of this paper these results are for $\mathbb{P}^2$ and rational ruled surfaces generalised to refined Severi degrees. In the second part of the paper we give a number of mostly conjectural generalisations of this result to singular surfaces, and curves with prescribed multiple points. The formulas involve modular forms and theta functions.

%B Communications in Number Theory and Physics %I International Press of Boston %V 10 %P 193–234 %G eng %U http://dx.doi.org/10.4310/CNTP.2016.v10.n2.a2 %R 10.4310/CNTP.2016.v10.n2.a2 %0 Journal Article %J Journal of Mathematical Analysis and Applications %D 2016 %T On Sobolev instability of the interior problem of tomography %A Marco Bertola %A Alexander Katsevich %A Alexander Tovbis %B Journal of Mathematical Analysis and Applications %G eng %0 Journal Article %J Archive of Numerical Software, vol. 3, n. 100, (2015), pages : 1-8 %D 2015 %T The deal.II Library, Version 8.2 %A W. Bangerth %A Timo Heister %A Luca Heltai %A G. Kanschat %A Martin Kronbichler %A Matthias Maier %A B. Turcksin %A T. D. Young %X This paper provides an overview of the new features of the finite element library deal.II version 8.2 %B Archive of Numerical Software, vol. 3, n. 100, (2015), pages : 1-8 %G en %U http://urania.sissa.it/xmlui/handle/1963/34464 %1 34637 %2 Mathematics %4 1 %# MAT/08 %$ Submitted by Luca Heltai (heltai@sissa.it) on 2015-04-20T15:40:46Z No. of bitstreams: 1 18031-44567-1-PB (1).pdf: 137014 bytes, checksum: 9a5fddeccfa389c32ca89a466e653074 (MD5) %R 10.11588/ans.2015.100.18031 %0 Journal Article %D 2015 %T Gli abachi: antichi strumenti precursori delle moderne macchine da calcolo %A Giuliano Klun %G eng %U http://hdl.handle.net/10077/10884 %0 Journal Article %J Analysis & PDE %D 2015 %T A topological join construction and the Toda system on compact surfaces of arbitrary genus %A Aleks Jevnikar %A Kallel, Sadok %A Andrea Malchiodi %B Analysis & PDE %I Mathematical Sciences Publishers %V 8 %P 1963–2027 %G eng %R 10.2140/apde.2015.8.1963 %0 Report %D 2015 %T Translation and adaptation of Birman's paper "On the theory of self-adjoint extensions of positive definite operators" (1956) %A Mikhail Khotyakov %A Alessandro Michelangeli %X 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. %I SISSA %G en %U http://urania.sissa.it/xmlui/handle/1963/34443 %1 34570 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2015-02-13T10:07:37Z No. of bitstreams: 1 SISSA_preprint_08-2015-MATE(1).pdf: 2044497 bytes, checksum: 092e3561b5722c604c896ccc97db7a76 (MD5) %0 Journal Article %D 2014 %T Adler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras %A Alberto De Sole %A Victor G. Kac %A Daniele Valeri %X 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. %I SISSA %G eng %U http://hdl.handle.net/1963/7242 %0 Journal Article %J Communications in Mathematical Physics 331, nr. 2 (2014) 623-676 %D 2014 %T Classical W-algebras and generalized Drinfeld-Sokolov hierarchies for minimal and short nilpotents %A Alberto De Sole %A Victor G. Kac %A Daniele Valeri %X 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. %B Communications in Mathematical Physics 331, nr. 2 (2014) 623-676 %I SISSA %G en %U http://hdl.handle.net/1963/6979 %1 6967 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Daniele Valeri (dvaleri@sissa.it) on 2013-07-13T15:27:20Z No. of bitstreams: 1 1306.1684v1.pdf: 580565 bytes, checksum: de8a5fc99d3eb9f1c4f85b4a2a6592e9 (MD5) %R 10.1007/s00220-014-2049-2 %0 Journal Article %J Communications in Mathematical Physics 331, nr. 3 (2014) 1155-1190 %D 2014 %T Dirac reduction for Poisson vertex algebras %A Alberto De Sole %A Victor G. Kac %A Daniele Valeri %X 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. %B Communications in Mathematical Physics 331, nr. 3 (2014) 1155-1190 %I SISSA %G en %U http://hdl.handle.net/1963/6980 %1 6968 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Daniele Valeri (dvaleri@sissa.it) on 2013-07-13T15:28:44Z No. of bitstreams: 1 1306.6589v1.pdf: 417586 bytes, checksum: 261351372e0d72d5e2d46b6664f19fd4 (MD5) %R 10.1007/s00220-014-2103-0 %0 Journal Article %J Mathematische Annalen %D 2014 %T Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds %A Kuwert, Ernst %A Andrea Mondino %A Johannes Schygulla %XWe 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 $f:{\mathbb{S}}^2 \rightarrow M$ minimizing the $L^2$ integral of the second fundamental form. Assuming instead that $K^M \leq 2 $ and that there is some point $\bar{x}\in M$ with scalar curvature $R^M(\bar{x})>6$, we obtain a smooth minimizer $f:{\mathbb{S}}^2 \rightarrow M$ for the functional $\int \frac{1}{4}|H|^2+1$, where $H$ is the mean curvature.

%B Mathematische Annalen %V 359 %P 379–425 %8 Jun %G eng %U https://doi.org/10.1007/s00208-013-1005-3 %R 10.1007/s00208-013-1005-3 %0 Report %D 2014 %T Integrability of Dirac reduced bi-Hamiltonian equations %A Alberto De Sole %A Victor G. Kac %A Daniele Valeri %X 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. %I SISSA %G en %U http://hdl.handle.net/1963/7247 %1 7286 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Daniele Valeri (dvaleri@sissa.it) on 2014-01-24T09:47:51Z No. of bitstreams: 1 1401.6006v1.pdf: 249123 bytes, checksum: 3b83fabf790160e10ecb05aa89251970 (MD5) %0 Journal Article %D 2014 %T On an isomonodromy deformation equation without the Painlevé property %A Boris Dubrovin %A Andrey Kapaev %X 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 $2\times2$ 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. %I Maik Nauka-Interperiodica Publishing %G en %U http://hdl.handle.net/1963/6466 %1 6410 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-08T11:29:56Z No. of bitstreams: 1 1301.7211v2.pdf: 415681 bytes, checksum: f606a3b5df4b201ae37a9d6fa1b79016 (MD5) %R 10.1134/S1061920814010026 %0 Journal Article %D 2014 %T Local behavior of fractional p-minimizers %A Agnese Di Castro %A Tuomo Kuusi %A Giampiero Palatucci %K fractional Sobolev spaces %XWe extend the De Giorgi-Nash Moser theory to nonlocal, possibly degerate integro-differential operators

%I SISSA %G en %1 7301 %2 Mathematics %4 1 %# MAT/05 ANALISI MATEMATICA %$ Submitted by Giampiero Palatucci (palatucc@sissa.it) on 2014-03-03T08:54:40Z No. of bitstreams: 1 DKP.pdf: 459581 bytes, checksum: 58cf9e0f905a3932505ab467eefd39e9 (MD5) %0 Journal Article %J Comm. Pure Appl. Math. %D 2014 %T Singular Value Decomposition of a Finite Hilbert Transform Defined on Several Intervals and the Interior Problem of Tomography: The Riemann-Hilbert Problem Approach %A Marco Bertola %A Alexander Katsevich %A Alexander Tovbis %B Comm. Pure Appl. Math. %G eng %0 Report %D 2014 %T Structure of classical (finite and affine) W-algebras %A Alberto De Sole %A Victor G. Kac %A Daniele Valeri %X First, we derive an explicit formula for the Poisson bracket of the classical finite W-algebra W^{fin}(g,f), the algebra of polynomial functions on the Slodowy slice associated to a simple Lie algebra g and its nilpotent element f. On the other hand, we produce an explicit set of generators and we derive an explicit formula for the Poisson vertex algebra structure of the classical affine W-algebra W(g,f). As an immediate consequence, we obtain a Poisson algebra isomorphism between W^{fin}(g,f) and the Zhu algebra of W(g,f). We also study the generalized Miura map for classical W-algebras. %I SISSA %G en %U http://hdl.handle.net/1963/7314 %1 7359 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Daniele Valeri (dvaleri@sissa.it) on 2014-04-04T09:52:02Z No. of bitstreams: 1 1404.0715v1.pdf: 386481 bytes, checksum: bafb532b11dfa1100008853a1f4cd543 (MD5) %0 Journal Article %D 2014 %T Weighted quantile correlation test for the logistic family %A Ferenc Balogh %A Éva Krauczi %X We summarize the results of investigating the asymptotic behavior of the weighted quantile correlation tests for the location-scale family associated to the logistic distribution. Explicit representations of the limiting distribution are given in terms of integrals of weighted Brownian bridges or alternatively as infinite series of independent Gaussian random variables. The power of this test and the test for the location logistic family against some alternatives are demonstrated by numerical simulations. %I University of Szeged %G en %U http://urania.sissa.it/xmlui/handle/1963/35025 %1 35261 %2 Mathematics %4 1 %$ Submitted by gfeltrin@sissa.it (gfeltrin@sissa.it) on 2015-11-17T09:50:18Z No. of bitstreams: 1 preprint2014.pdf: 364753 bytes, checksum: 77f35846fb1cee29a66ca18ff89d5331 (MD5) %R 10.14232/actasm-013-809-8 %0 Journal Article %J Communications in Mathematical Physics 323, nr. 2 (2013) 663-711 %D 2013 %T Classical W-algebras and generalized Drinfeld-Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras %A Alberto De Sole %A Victor G. Kac %A Daniele Valeri %X We provide a description of the Drinfeld-Sokolov Hamiltonian reduction for the construction of classical W-algebras within the framework of Poisson vertex algebras. In this context, the gauge group action on the phase space is translated in terms of (the exponential of) a Lie conformal algebra action on the space of functions. Following the ideas of Drinfeld and Sokolov, we then establish under certain sufficient conditions the applicability of the Lenard-Magri scheme of integrability and the existence of the corresponding integrable hierarchy of bi-Hamiltonian equations. %B Communications in Mathematical Physics 323, nr. 2 (2013) 663-711 %I Springer %G en %U http://hdl.handle.net/1963/6978 %1 6966 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Daniele Valeri (dvaleri@sissa.it) on 2013-07-13T15:24:57Z No. of bitstreams: 1 1207.6286v3.pdf: 588052 bytes, checksum: a5630eb1399cba992d56be98f25cdc9c (MD5) %R 10.1007/s00220-013-1785-z %0 Report %D 2013 %T On critical behaviour in systems of Hamiltonian partial differential equations %A Boris Dubrovin %A Tamara Grava %A Christian Klein %A Antonio Moro %XWe study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlev\'e-I (P$_I$) equation or its fourth order analogue P$_I^2$. As concrete examples we discuss nonlinear Schr\"odinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture.

%I SISSA %G en %1 7280 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Tamara Grava (grava@sissa.it) on 2014-01-14T18:10:19Z No. of bitstreams: 1 EHfinal_3.pdf: 7760169 bytes, checksum: 1e98e693fbceb1268a5acd269dd9b03e (MD5) %0 Report %D 2013 %T The deal.II Library, Version 8.1 %A W. Bangerth %A Timo Heister %A Luca Heltai %A G. Kanschat %A Martin Kronbichler %A Matthias Maier %A B. Turcksin %A T. D. Young %X This paper provides an overview of the new features of the finite element library deal.II version 8.0. %I SISSA %G en %U http://hdl.handle.net/1963/7236 %1 7272 %2 Mathematics %4 1 %# MAT/08 ANALISI NUMERICA %$ Submitted by Luca Heltai (heltai@sissa.it) on 2013-12-10T10:43:08Z No. of bitstreams: 1 1312.2266v1.pdf: 245445 bytes, checksum: e00fcd6d7d9ac6e2076208958b07eef3 (MD5) %0 Journal Article %J Communications in Applied and Industrial Mathematics %D 2013 %T Free Form Deformation Techniques Applied to 3D Shape Optimization Problems %A Anwar Koshakji %A Alfio Quarteroni %A Gianluigi Rozza %X The purpose of this work is to analyse and study an efficient parametrization technique for a 3D shape optimization problem. After a brief review of the techniques and approaches already available in literature, we recall the Free Form Deformation parametrization, a technique which proved to be efficient and at the same time versatile, allowing to manage complex shapes even with few parameters. We tested and studied the FFD technique by establishing a path, from the geometry definition, to the method implementation, and finally to the simulation and to the optimization of the shape. In particular, we have studied a bulb and a rudder of a race sailing boat as model applications, where we have tested a complete procedure from Computer-Aided-Design to build the geometrical model to discretization and mesh generation. %B Communications in Applied and Industrial Mathematics %G eng %R 10.1685/journal.caim.452 %0 Journal Article %J Proc. Amer. Math. Soc. %D 2013 %T Inversion formulae for the $\romancosh$-weighted Hilbert transform %A Marco Bertola %A Katsevich, A. %A Alexander Tovbis %B Proc. Amer. Math. Soc. %V 141 %P 2703–2718 %G eng %U http://dx.doi.org/10.1090/S0002-9939-2013-11642-4 %R 10.1090/S0002-9939-2013-11642-4 %0 Report %D 2013 %T On the tritronquée solutions of P$_I^2$ %A Tamara Grava %A Andrey Kapaev %A Christian Klein %XFor equation P$_I^2$, the second member in the P$_I$ hierarchy, we prove existence of various degenerate solutions depending on the complex parameter $t$ and evaluate the asymptotics in the complex $x$ plane for $|x|\to\infty$ and $t=o(x^{2/3})$. Using this result, we identify the most degenerate solutions $u^{(m)}(x,t)$, $\hat u^{(m)}(x,t)$, $m=0,\dots,6$, called {\em tritronqu\'ee}, describe the quasi-linear Stokes phenomenon and find the large $n$ asymptotics of the coefficients in a formal expansion of these solutions. We supplement our findings by a numerical study of the tritronqu\'ee solutions.

%I SISSA %G en %1 7282 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Tamara Grava (grava@sissa.it) on 2014-01-14T18:35:53Z No. of bitstreams: 1 tritronquee_coeff.pdf: 753719 bytes, checksum: 812d268b2abe25ccbcc69eb40ff75f1f (MD5) %0 Journal Article %J Nucleic Acids Research, Volume 40, Issue 15, August 2012, Pages 7132-7149 %D 2012 %T Detection of transcriptional triggers in the dynamics of microbial growth: application to the respiratory-versatile bacterium Shewanella oneidensis %A Q Beg %A Mattia Zampieri %A N Klitgord %A S Collins %A M Serres %A Daniel Segrè %A Claudio Altafini %X The capacity of microorganisms to respond to variable external conditions requires a coordination of environment-sensing mechanisms and decisionmaking regulatory circuits. Here, we seek to understand the interplay between these two processes by combining high-throughput measurement of time-dependent mRNA profiles with a novel computational approach that searches for key genetic triggers of transcriptional changes. Our approach helped us understand the regulatory strategies of a respiratorily versatile bacterium with promising bioenergy and bioremediation applications, Shewanella oneidensis, in minimal and rich media. By comparing expression profiles across these two conditions, we unveiled components of the transcriptional program that depend mainly on the growth phase. Conversely, by integrating our time-dependent data with a previously available large compendium of static perturbation responses, we identified transcriptional changes that cannot be explained solely by internal network dynamics, but are rather triggered by specific genes acting as key mediators of an environment-dependent response. These transcriptional triggers include known and novel regulators that respond to carbon, nitrogen and oxygen limitation. Our analysis suggests a sequence of physiological responses, including a coupling between nitrogen depletion and glycogen storage, partially recapitulated through dynamic flux balance analysis, and experimentally confirmed by metabolite measurements. Our approach is broadly applicable to other systems %B Nucleic Acids Research, Volume 40, Issue 15, August 2012, Pages 7132-7149 %I SISSA %G en %U http://hdl.handle.net/1963/6506 %1 6452 %2 Mathematics %4 1 %$ Approved for entry into archive by Lucio Lubiana (lubiana@sissa.it) on 2013-02-27T15:39:07Z (GMT) No. of bitstreams: 0 %R 10.1093/nar/gks467 %0 Journal Article %J Physica D 241, nr. 23-24 (2012): 2246-2264 %D 2012 %T Numerical study of the small dispersion limit of the Korteweg-de Vries equation and asymptotic solutions %A Tamara Grava %A Christian Klein %K Korteweg-de Vries equation %X We study numerically the small dispersion limit for the Korteweg-de Vries (KdV) equation $u_t+6uu_x+\epsilon^{2}u_{xxx}=0$ for $\epsilon\ll1$ and give a quantitative comparison of the numerical solution with various asymptotic formulae for small $\epsilon$ in the whole $(x,t)$-plane. The matching of the asymptotic solutions is studied numerically. %B Physica D 241, nr. 23-24 (2012): 2246-2264 %I Elsevier %G en %1 7069 %2 Physics %4 -1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2013-09-16T14:49:23Z No. of bitstreams: 1 1202.0962v2.pdf: 2652650 bytes, checksum: d8678338138745b35d8515af39f85d18 (MD5) %R 10.1016/j.physd.2012.04.001 %0 Journal Article %J SIAM J. Appl. Math. 71 (2011) 983-1008 %D 2011 %T Numerical Study of breakup in generalized Korteweg-de Vries and Kawahara equations %A Boris Dubrovin %A Tamara Grava %A Christian Klein %X This article is concerned with a conjecture in [B. Dubrovin, Comm. Math. Phys., 267 (2006), pp. 117–139] on the formation of dispersive shocks in a class of Hamiltonian dispersive regularizations of the quasi-linear transport equation. The regularizations are characterized by two arbitrary functions of one variable, where the condition of integrability implies that one of these functions must not vanish. It is shown numerically for a large class of equations that the local behavior of their solution near the point of gradient catastrophe for the transport equation is described by a special solution of a Painlevé-type equation. This local description holds also for solutions to equations where blowup can occur in finite time. Furthermore, it is shown that a solution of the dispersive equations away from the point of gradient catastrophe is approximated by a solution of the transport equation with the same initial data, modulo terms of order $\\\\epsilon^2$, where $\\\\epsilon^2$ is the small dispersion parameter. Corrections up to order $\\\\epsilon^4$ are obtained and tested numerically. %B SIAM J. Appl. Math. 71 (2011) 983-1008 %I SIAM %G en %U http://hdl.handle.net/1963/4951 %1 4732 %2 Mathematics %3 Mathematical Physics %4 -1 %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2011-10-27T12:18:45Z\\nNo. of bitstreams: 1\\n1101.0268v1.pdf: 522533 bytes, checksum: d9e2df220724f918ec3b888cef3593d4 (MD5) %R 10.1137/100819783 %0 Journal Article %J The European journal of neuroscience. 2010 Oct; 32(8):1364-79 %D 2010 %T Gene expression analysis of the emergence of epileptiform activity after focal injection of kainic acid into mouse hippocampus. %A Dario Motti %A Caroline Le Duigou %A Nicole Chemaly %A Lucia Wittner %A Dejan Lazarevic %A Helena Krmac %A Troels Torben Marstrand %A Eivind Valen %A Remo Sanges %A Elia Stupka %A Albin Sandelin %A Enrico Cherubini %A Stefano Gustincich %A Richard Miles %XWe report gene profiling data on genomic processes underlying the progression towards recurrent seizures after injection of kainic acid (KA) into the mouse hippocampus. Focal injection enabled us to separate the effects of proepileptic stimuli initiated by KA injection. Both the injected and contralateral hippocampus participated in the status epilepticus. However, neuronal death induced by KA treatment was restricted to the injected hippocampus, although there was some contralateral axonal degeneration. We profiled gene expression changes in dorsal and ventral regions of both the injected and contralateral hippocampus. Changes were detected in the expression of 1526 transcripts in samples from three time-points: (i) during the KA-induced status epilepticus, (ii) at 2 weeks, before recurrent seizures emerged, and (iii) at 6 months after seizures emerged. Grouping genes with similar spatio-temporal changes revealed an early transcriptional response, strong immune, cell death and growth responses at 2 weeks and an activation of immune and extracellular matrix genes persisting at 6 months. Immunostaining for proteins coded by genes identified from array studies provided evidence for gliogenesis and suggested that the proteoglycan biglycan is synthesized by astrocytes and contributes to a glial scar. Gene changes at 6 months after KA injection were largely restricted to tissue from the injection site. This suggests that either recurrent seizures might depend on maintained processes including immune responses and changes in extracellular matrix proteins near the injection site or alternatively might result from processes, such as growth, distant from the injection site and terminated while seizures are maintained.

%B The European journal of neuroscience. 2010 Oct; 32(8):1364-79 %I Wiley %G en %U http://hdl.handle.net/1963/4480 %1 4244 %2 Neuroscience %3 Neurobiology %4 -1 %$ Approved for entry into archive by Andrea Wehrenfennig (andreaw@sissa.it) on 2011-10-05T07:58:51Z (GMT) No. of bitstreams: 0 %R 10.1111/j.1460-9568.2010.07403.x %0 Report %D 2010 %T Numerical Solution of the Small Dispersion Limit of the Camassa-Holm and Whitham Equations and Multiscale Expansions %A Simonetta Abenda %A Tamara Grava %A Christian Klein %X The small dispersion limit of solutions to the Camassa-Holm (CH) equation is characterized by the appearance of a zone of rapid modulated oscillations. An asymptotic description of these oscillations is given, for short times, by the one-phase solution to the CH equation, where the branch points of the corresponding elliptic curve depend on the physical coordinates via the Whitham equations. We present a conjecture for the phase of the asymptotic solution. A numerical study of this limit for smooth hump-like initial data provides strong evidence for the validity of this conjecture.... %G en_US %U http://hdl.handle.net/1963/3840 %1 487 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2010-02-05T10:20:56Z\\nNo. of bitstreams: 1\\n0909.1020v1.pdf: 613403 bytes, checksum: be892250a6d664faff51d74b323fea67 (MD5) %0 Journal Article %J J. Nonlinear Sci. 19 (2009) 57-94 %D 2009 %T On universality of critical behaviour in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the \\\\it tritronquée solution to the Painlevé-I equation %A Boris Dubrovin %A Tamara Grava %A Christian Klein %X We argue that the critical behaviour near the point of ``gradient catastrophe\\\" of the solution to the Cauchy problem for the focusing nonlinear Schr\\\\\\\"odinger equation $ i\\\\epsilon \\\\psi_t +\\\\frac{\\\\epsilon^2}2\\\\psi_{xx}+ |\\\\psi|^2 \\\\psi =0$ with analytic initial data of the form $\\\\psi(x,0;\\\\epsilon) =A(x) e^{\\\\frac{i}{\\\\epsilon} S(x)}$ is approximately described by a particular solution to the Painlev\\\\\\\'e-I equation. %B J. Nonlinear Sci. 19 (2009) 57-94 %G en_US %U http://hdl.handle.net/1963/2525 %1 1593 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-12-11T13:11:26Z\\nNo. of bitstreams: 1\\n0704.0501v3.pdf: 568190 bytes, checksum: cf8471fc01eea53ce252339eda81b3cd (MD5) %R 10.1007/s00332-008-9025-y %0 Journal Article %J Proc. R. Soc. A 464 (2008) 733-757 %D 2008 %T Numerical study of a multiscale expansion of the Korteweg-de Vries equation and Painlevé-II equation %A Tamara Grava %A Christian Klein %X The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order $\\\\e^2$, $\\\\e\\\\ll 1$, is characterized by the appearance of a zone of rapid modulated oscillations. These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wave-number and frequency are not constant but evolve according to the Whitham equations. Whereas the difference between the KdV and the asymptotic solution decreases as $\\\\epsilon$ in the interior of the Whitham oscillatory zone, it is known to be only of order $\\\\epsilon^{1/3}$ near the leading edge of this zone. To obtain a more accurate description near the leading edge of the oscillatory zone we present a multiscale expansion of the solution of KdV in terms of the Hastings-McLeod solution of the Painlev\\\\\\\'e-II equation. We show numerically that the resulting multiscale solution approximates the KdV solution, in the small dispersion limit, to the order $\\\\epsilon^{2/3}$. %B Proc. R. Soc. A 464 (2008) 733-757 %G en_US %U http://hdl.handle.net/1963/2592 %1 1530 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-02-25T15:08:44Z\\nNo. of bitstreams: 1\\n0708.0638v3.pdf: 453744 bytes, checksum: 05291095860df236125f0d9f8c676fbb (MD5) %R 10.1098/rspa.2007.0249 %0 Journal Article %J Ann. Inst. Henri Poincare-Prob. Stat. 43 (2007) 399-415 %D 2007 %T On finite-dimensional projections of distributions for solutions of randomly forced PDE\\\'s %A Andrei A. Agrachev %A Sergei Kuksin %A Andrey Sarychev %A Armen Shirikyan %X The paper is devoted to studying the image of probability measures on a Hilbert space under finite-dimensional analytic maps. We establish sufficient conditions under which the image of a measure has a density with respect to the Lebesgue measure and continuously depends on the map. The results obtained are applied to the 2D Navier-Stokes equations perturbed by various random forces of low dimension. %B Ann. Inst. Henri Poincare-Prob. Stat. 43 (2007) 399-415 %G en_US %U http://hdl.handle.net/1963/2012 %1 2184 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-08-31T09:21:55Z\\nNo. of bitstreams: 1\\nmathAP0603295v1.pdf: 316127 bytes, checksum: 6686ad78dcb2dbc5efb64a959b30750c (MD5) %R 10.1016/j.anihpb.2006.06.001 %0 Report %D 2007 %T Numerical solution of the small dispersion limit of Korteweg de Vries and Whitham equations %A Tamara Grava %A Christian Klein %X The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order $\\\\epsilon^2$, is characterized by the appearance of a zone of rapid modulated oscillations of wave-length of order $\\\\epsilon$. These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wave-number and frequency are not constant but evolve according to the Whitham equations. In this manuscript we give a quantitative analysis of the discrepancy between the numerical solution of the KdV equation in the small dispersion limit and the corresponding approximate solution for values of $\\\\epsilon$ between $10^{-1}$ and $10^{-3}$. The numerical results are compatible with a difference of order $\\\\epsilon$ within the `interior\\\' of the Whitham oscillatory zone, of order $\\\\epsilon^{1/3}$ at the left boundary outside the Whitham zone and of order $\\\\epsilon^{1/2}$ at the right boundary outside the Whitham zone. %B Comm. Pure Appl. Math. 60 (2007) 1623-1664 %G en_US %U http://hdl.handle.net/1963/1788 %1 2756 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2006-03-30T13:45:26Z\\nNo. of bitstreams: 1\\n91FM-2005.pdf: 905542 bytes, checksum: 8505fe7c8ac2e5f1da7248d62ae542b2 (MD5) %R 10.1002/cpa.20183 %0 Report %D 2007 %T Numerical study of a multiscale expansion of KdV and Camassa-Holm equation %A Tamara Grava %A Christian Klein %X We study numerically solutions to the Korteweg-de Vries and Camassa-Holm equation close to the breakup of the corresponding solution to the dispersionless equation. The solutions are compared with the properly rescaled numerical solution to a fourth order ordinary differential equation, the second member of the Painlev\\\\\\\'e I hierarchy. It is shown that this solution gives a valid asymptotic description of the solutions close to breakup. We present a detailed analysis of the situation and compare the Korteweg-de Vries solution quantitatively with asymptotic solutions obtained via the solution of the Hopf and the Whitham equations. We give a qualitative analysis for the Camassa-Holm equation %G en_US %U http://hdl.handle.net/1963/2527 %1 1591 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-12-12T12:39:23Z\\nNo. of bitstreams: 1\\n0702038v1.pdf: 367188 bytes, checksum: f88c96f6ff42a7c5378de0118866d4bb (MD5) %0 Journal Article %J Calc. Var. Partial Differential Equations 27 (2006) 233-253 %D 2006 %T 2-d stability of the Néel wall %A Antonio DeSimone %A Hans Knuepfer %A Felix Otto %X We are interested in thin-film samples in micromagnetism, where the magnetization m is a 2-d unit-length vector field. More precisely we are interested in transition layers which connect two opposite magnetizations, so called Néel walls. %B Calc. Var. Partial Differential Equations 27 (2006) 233-253 %G en_US %U http://hdl.handle.net/1963/2194 %1 2050 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-10T12:14:45Z\\nNo. of bitstreams: 1\\nstability.pdf: 226255 bytes, checksum: dec90ada941ac865d9451848ca2faa5f (MD5) %R 10.1007/s00526-006-0019-z %0 Book Section %B The science of hysteresis / eds. Giorgio Bertotti, Isaak D. Mayergoyz. - Amsterdam: Elsevier, 2006. Vol.2, 269-381. %D 2006 %T Recent analytical developments in micromagnetics %A Antonio DeSimone %A Robert V. Kohn %A Stefan Müller %A Felix Otto %B The science of hysteresis / eds. Giorgio Bertotti, Isaak D. Mayergoyz. - Amsterdam: Elsevier, 2006. Vol.2, 269-381. %@ 978-0-12-480874-4 %G en_US %U http://hdl.handle.net/1963/2230 %1 2014 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-15T15:16:57Z\\nNo. of bitstreams: 1\\nDKMO.pdf: 1644450 bytes, checksum: 4c1079e4854e372d31af9a285f87f400 (MD5) %0 Journal Article %J Teoret. Mat. Fiz. %D 2003 %T The duality of spectral curves that arises in two-matrix models %A Marco Bertola %A B. Eynard %A Kharnad, Dzh. %B Teoret. Mat. Fiz. %V 134 %P 32–45 %G eng %0 Journal Article %J Mod. Phys. Lett. A 18 (2003) 2371-2379 %D 2003 %T Non-linear sigma-models in noncommutative geometry: fields with values in finite spaces %A Ludwik Dabrowski %A Thomas Krajewski %A Giovanni Landi %X We study sigma-models on noncommutative spaces, notably on noncommutative tori. We construct instanton solutions carrying a nontrivial topological charge q and satisfying a Belavin-Polyakov bound. The moduli space of these instantons is conjectured to consists of an ordinary torus endowed with a complex structure times a projective space $CP^{q-1}$. %B Mod. Phys. Lett. A 18 (2003) 2371-2379 %I World Scientific %G en_US %U http://hdl.handle.net/1963/3215 %1 1086 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-10-30T11:23:21Z\\nNo. of bitstreams: 1\\n0309143v1.pdf: 165627 bytes, checksum: c79b1a62edf34ae51819b5e8d752db8b (MD5) %R 10.1142/S0217732303012593 %0 Journal Article %J Int. J. Mod. Phys. B 14 (2000) 2367-2382 %D 2000 %T Some Properties of Non-linear sigma-Models in Noncommutative Geometry %A Ludwik Dabrowski %A Thomas Krajewski %A Giovanni Landi %X We introduce non-linear $\\\\sigma$-models in the framework of noncommutative geometry with special emphasis on models defined on the noncommutative torus. We choose as target spaces the two point space and the circle and illustrate some characteristic features of the corresponding $\\\\sigma$-models. In particular we construct a $\\\\sigma$-model instanton with topological charge equal to 1. We also define and investigate some properties of a noncommutative analogue of the Wess-Zumino-Witten model. %B Int. J. Mod. Phys. B 14 (2000) 2367-2382 %I SISSA Library %G en %U http://hdl.handle.net/1963/1373 %1 3082 %2 Mathematics %3 Mathematical Physics %$ Made available in DSpace on 2004-09-01T12:57:00Z (GMT). No. of bitstreams: 1\\nhep-th0003099.pdf: 185777 bytes, checksum: 30806664c895808c1cb1afe5e6364f9f (MD5)\\n Previous issue date: 1999 %R 10.1142/S0217979200001898