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: $\begin{aligned} \mu (\{x\in E:|f(x)|\>\alpha \})=\mu (\{x\in E:|g(x)|\>\alpha \})\quad \text { for all } \alpha \ge 0. \end{aligned}$

}, doi = {10.1007/s10231-019-00907-z}, url = {https://doi.org/10.1007/s10231-019-00907-z}, author = {Giuliano Klun} } @article {2020, title = {Periodic solutions of nearly integrable Hamiltonian systems bifurcating from infinite-dimensional tori}, journal = {NONLINEAR ANALYSIS}, year = {2020}, abstract = {We 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 Poincar{\'e}{\textendash}Birkhoff Theorem.

}, issn = {0362-546X}, doi = {10.1016/j.na.2019.111720}, url = {https://doi.org/10.1016/j.na.2019.111720}, author = {Alessandro Fonda and Giuliano Klun and Andrea Sfecci} } @conference {2020, title = {A Reduced Order Approach for the Embedded Shifted Boundary FEM and a Heat Exchange System on Parametrized Geometries}, booktitle = {IUTAM Symposium on Model Order Reduction of Coupled Systems, Stuttgart, Germany, May 22{\textendash}25, 2018}, year = {2020}, publisher = {Springer International Publishing}, organization = {Springer International Publishing}, abstract = {A 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.

}, doi = {10.1007/978-3-030-21013-7_8}, url = {https://arxiv.org/abs/1807.07753}, author = {Efthymios N. Karatzas and Giovanni Stabile and Nabib Atallah and Guglielmo Scovazzi and Gianluigi Rozza}, editor = {Fehr, J{\"o}rg and Haasdonk, Bernard} } @article {ArndtBangerthClevenger-2019-a, title = {The deal.II Library, Version 9.1}, journal = {Journal of Numerical Mathematics}, year = {2019}, abstract = {This paper provides an overview of the new features of the finite element library deal.II, version 9.1.}, issn = {15702820}, doi = {10.1515/jnma-2019-0064}, author = {Arndt, Daniel and Bangerth, Wolfgang and Clevenger, Thomas C. and Davydov, Denis and Fehling, Marc and Garcia-Sanchez, Daniel and Harper, Graham and Heister, Timo and Heltai, Luca and Kronbichler, Martin and Maguire Kynch, Ross and Maier, Matthias and Pelteret, Jean Paul and Turcksin, Bruno and Wells, David} } @article {Kozhasov2019, title = {On the Number of Flats Tangent to Convex Hypersurfaces in Random Position}, journal = {Discrete \& Computational Geometry}, year = {2019}, month = {Mar}, issn = {1432-0444}, doi = {10.1007/s00454-019-00067-0}, url = {https://doi.org/10.1007/s00454-019-00067-0}, author = {Khazhgali Kozhasov and Antonio Lerario} } @article {Beltr{\'a}n2019, title = {The Real Polynomial Eigenvalue Problem is Well Conditioned on the Average}, journal = {Foundations of Computational Mathematics}, year = {2019}, month = {May}, abstract = {We 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.

}, issn = {1615-3383}, doi = {10.1007/s10208-019-09414-2}, url = {https://doi.org/10.1007/s10208-019-09414-2}, author = {Carlos Beltr{\'a}n and Khazhgali Kozhasov} } @article {2019, title = {A reduced basis approach for PDEs on parametrized geometries based on the shifted boundary finite element method and application to a Stokes flow}, journal = {Computer Methods in Applied Mechanics and Engineering}, volume = {347}, year = {2019}, pages = {568{\textendash}587}, abstract = {We 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.

}, doi = {10.1016/j.cma.2018.12.040}, url = {https://arxiv.org/abs/1807.07790}, author = {Karatzas, Efthymios N and Stabile, Giovanni and Nouveau, Leo and Scovazzi, Guglielmo and Rozza, Gianluigi} } @inbook {2019, title = {A Spectral Element Reduced Basis Method in Parametric CFD}, booktitle = {Numerical Mathematics and Advanced Applications - ENUMATH 2017}, volume = {126}, year = {2019}, publisher = {Springer International Publishing}, organization = {Springer International Publishing}, chapter = {A Spectral Element Reduced Basis Method in Parametric CFD}, abstract = {We 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.

}, doi = {10.1007/978-3-319-96415-7_64 pages = 693{\textendash}701}, url = {https://arxiv.org/abs/1712.06432}, author = {Hess, Martin W. and Rozza, Gianluigi}, editor = {Radu, Florin Adrian and Kumar, Kundan and Berre, Inga and Nordbotten, Jan Martin and Pop, Iuliu Sorin} } @article {2019, title = {On the topological degree of planar maps avoiding normal cones}, journal = {TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS}, volume = {53}, number = {SISSA;04/2019/MATE}, year = {2019}, pages = {825-845}, publisher = {SISSA}, abstract = {The classical Poincar{\'e}-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.

}, doi = {10.1137/17M1145902}, url = {https://epubs.siam.org/doi/pdf/10.1137/17M1145902}, author = {Khazhgali Kozhasov} } @article {BELLETTINI20181, title = {Minimizing movements for mean curvature flow of droplets with prescribed contact angle}, journal = {Journal de Math{\'e}matiques Pures et Appliqu{\'e}es}, volume = {117}, year = {2018}, pages = {1 - 58}, abstract = {We 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{\textendash}Taylor{\textendash}Wang and Luckhaus{\textendash}Sturzenhecker, we show the existence of a weak evolution, and its compatibility with a distributional solution. We also prove various comparison results. R{\'e}sum{\'e} Nous {\'e}tudions le mouvement par courbure moyenne d{\textquoteright}une goutte qui glisse par courbure moyenne sur un hyperplan horizontal avec un angle de contact prescrit {\'e}ventuellement non constant. En utilisant les solutions construites comme limites d{\textquoteright}un algorithme d{\textquoteright}approximation d{\^u} {\`a} Almgren, Taylor et Wang et Luckhaus et Sturzenhecker, nous montrons l{\textquoteright}existence d{\textquoteright}une {\'e}volution faible, et sa compatibilit{\'e} avec une solution au sens des distributions. Nous d{\'e}montrons {\'e}galement plusieurs r{\'e}sultats de comparaison.

}, keywords = {Capillary functional, Mean curvature flow with prescribed contact angle, Minimizing movements, Sets of finite perimeter}, issn = {0021-7824}, doi = {https://doi.org/10.1016/j.matpur.2018.06.003}, url = {http://www.sciencedirect.com/science/article/pii/S0021782418300825}, author = {Giovanni Bellettini and Matteo Novaga and Shokhrukh Kholmatov} } @article {doi:10.1137/17M1159294, title = {Minimizing Movements for Mean Curvature Flow of Partitions}, journal = {SIAM Journal on Mathematical Analysis}, volume = {50}, number = {4}, year = {2018}, pages = {4117-4148}, abstract = {We 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.

}, doi = {10.1137/17M1159294}, url = {https://doi.org/10.1137/17M1159294}, author = {Giovanni Bellettini and Shokhrukh Kholmatov} } @article {doi:10.1098/rspa.2017.0458, title = {Numerical study of the Kadomtsev-Petviashvili equation and dispersive shock waves}, journal = {Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences}, volume = {474}, number = {2210}, year = {2018}, pages = {20170458}, abstract = {A detailed numerical study of the long time behaviour of dispersive shock waves in solutions to the Kadomtsev{\textendash}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 Schr{\"o}dinger equation in the semiclassical limit.

}, doi = {10.1098/rspa.2017.0458}, url = {https://royalsocietypublishing.org/doi/abs/10.1098/rspa.2017.0458}, author = {Tamara Grava and Christian Klein and Giuseppe Pitton} } @booklet {1807.07790, title = {A Reduced Basis approach for PDEs on parametrized geometries based on the Shifted Boundary Finite Element Method and application to fluid dynamics}, year = {2018}, author = {Efthymios N. Karatzas and Giovanni Stabile and Leo Nouveau and Guglielmo Scovazzi and Gianluigi Rozza} } @booklet {1807.07753, title = {A Reduced Order Approach for the Embedded Shifted Boundary FEM and a Heat Exchange System on Parametrized Geometries}, year = {2018}, author = {Efthymios N. Karatzas and Giovanni Stabile and N. Atallah and Guglielmo Scovazzi and Gianluigi Rozza} } @article {doi:10.1098/rsta.2017.0424, title = {Symplectic invariants for parabolic orbits and cusp singularities of integrable systems}, journal = {Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences}, volume = {376}, number = {2131}, year = {2018}, pages = {20170424}, abstract = {We 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 {\textquoteleft}Finite dimensional integrable systems: new trends and methods{\textquoteright}.

}, doi = {10.1098/rsta.2017.0424}, url = {https://royalsocietypublishing.org/doi/abs/10.1098/rsta.2017.0424}, author = {Alexey Bolsinov and Lorenzo Guglielmi and Elena Kudryavtseva} } @conference {10.1007/978-3-319-91545-6_15, title = {On Uniqueness of Weak Solutions to Transport Equation with Non-smooth Velocity Field}, booktitle = {Theory, Numerics and Applications of Hyperbolic Problems I}, year = {2018}, pages = {191{\textendash}203}, publisher = {Springer International Publishing}, organization = {Springer International Publishing}, address = {Cham}, isbn = {978-3-319-91545-6}, doi = {10.1007/978-3-319-91545-6_15}, url = {https://link.springer.com/chapter/10.1007/978-3-319-91545-6_15}, author = {Paolo Bonicatto}, editor = {Klingenberg, Christian and Westdickenberg, Michael} } @article {20.500.11767_47950, title = {The deal.II Library, Version 8.5}, journal = {JOURNAL OF NUMERICAL MATHEMATICS}, volume = {25}, year = {2017}, pages = {137{\textendash}145}, doi = {10.1515/jnma-2017-0058}, url = {https://www.dealii.org/deal85-preprint.pdf}, author = {Arndt, Daniel and W. Bangerth and Denis Davydov and Timo Heister and Luca Heltai and Martin Kronbichler and Matthias Maier and Pelteret, Jean-Paul and B. Turcksin and David Wells} } @article {1534-0392_2017_4_1427, title = {Minimizers of anisotropic perimeters with cylindrical norms}, journal = {Communications on Pure \& Applied Analysis}, volume = {16}, number = {1534-0392_2017_4_142}, year = {2017}, pages = {1427}, abstract = {We study various regularity properties of minimizers of the Φ{\textendash}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.

}, keywords = {anisotropic Bernstein problem;, minimal cones, Non parametric minimal surfaces, Sets of finite perimeter}, issn = {1534-0392}, doi = {10.3934/cpaa.2017068}, url = {http://aimsciences.org//article/id/47054f15-00c7-40b7-9da1-4c0b1d0a103d}, author = {Giovanni Bellettini and Matteo Novaga and Shokhrukh Kholmatov} } @article {20.500.11767_11953, title = {A natural framework for isogeometric fluid-structure interaction based on BEM-shell coupling}, journal = {COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING}, volume = {316}, year = {2017}, pages = {522{\textendash}546}, doi = {10.1016/j.cma.2016.08.008}, url = {http://cdsads.u-strasbg.fr/abs/2017CMAME.316..522H}, author = {Luca Heltai and Kiendl, J. and Antonio DeSimone and Alessandro Reali} } @article {1711.08253, title = {Random spectrahedra}, year = {2017}, author = {Paul Breiding and Khazhgali Kozhasov and Antonio Lerario} } @article {Bertola:2015tg, title = {Symplectic geometry of the moduli space of projective structures in homological coordinates}, journal = {Inventiones Mathematicae}, year = {2017}, month = {06}, pages = {1{\textendash}56}, url = {https://arxiv.org/abs/1506.07918}, author = {Marco Bertola and Dmitry Korotkin and Chaya Norton} } @article {20.500.11767_14497, title = {The deal.II Library, Version 8.3}, journal = {ARCHIVE OF NUMERICAL SOFTWARE}, volume = {4}, year = {2016}, pages = {1{\textendash}11}, doi = {10.11588/ans.2016.100.23122}, url = {http://nbn-resolving.de/urn:nbn:de:bsz:16-ans-231226}, author = {W. Bangerth and Timo Heister and Luca Heltai and G. Kanschat and Martin Kronbichler and Matthias Maier and B. Turcksin} } @article {20.500.11767_14498, title = {The deal.II library, Version 8.4}, journal = {JOURNAL OF NUMERICAL MATHEMATICS}, volume = {24}, year = {2016}, pages = {135{\textendash}141}, doi = {10.1515/jnma-2016-1045}, url = {https://www.math.clemson.edu/ heister/preprints/deal84-preprint.pdf}, author = {W. Bangerth and Denis Davydov and Timo Heister and Luca Heltai and G. Kanschat and Martin Kronbichler and Matthias Maier and B. Turcksin and David Wells} } @article {20.500.11767_11949, title = {Error Estimates of B-spline based finite-element method for the wind-driven ocean circulation}, journal = {JOURNAL OF SCIENTIFIC COMPUTING}, volume = {69}, year = {2016}, pages = {430{\textendash}459}, doi = {10.1007/s10915-016-0201-1}, author = {Rotundo, N. and Kim, T. -Y. and Jiang, W. and Luca Heltai and Fried, E.} } @article {2016, title = {Large KAM tori for perturbations of the dNLS equation}, number = {arXiv;1603.09252}, year = {2016}, abstract = {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{\texttimes}2 block diagonal ones with constant coefficients together with sharp asymptotic estimates of their eigenvalues.}, url = {http://preprints.sissa.it/handle/1963/35284}, author = {Massimiliano Berti and Thomas Kappeler and Riccardo Montalto} } @article {arici2016pimsner, title = {Pimsner algebras and Gysin sequences from principal circle actions}, journal = {Journal of Noncommutative Geometry}, volume = {10}, year = {2016}, pages = {29{\textendash}64}, issn = {1661-6952}, doi = {10.4171/jncg/228}, url = {http://hdl.handle.net/2066/162951}, author = {Francesca Arici and Jens Kaad and Giovanni Landi} } @article {gottsche2016refined, title = {Refined node polynomials via long edge graphs}, journal = {Communications in Number Theory and Physics}, volume = {10}, number = {2}, year = {2016}, pages = {193{\textendash}234}, publisher = {International Press of Boston}, abstract = {The 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.

}, doi = {10.4310/CNTP.2016.v10.n2.a2}, url = {http://dx.doi.org/10.4310/CNTP.2016.v10.n2.a2}, author = {Lothar G{\"o}ttsche and Benjamin Kipkirui Kikwai} } @article {BKT-sobolev, title = {On Sobolev instability of the interior problem of tomography}, journal = {Journal of Mathematical Analysis and Applications}, year = {2016}, author = {Marco Bertola and Alexander Katsevich and Alexander Tovbis} } @article {2015, title = {The deal.II Library, Version 8.2}, journal = {Archive of Numerical Software, vol. 3, n. 100, (2015), pages : 1-8}, year = {2015}, abstract = {This paper provides an overview of the new features of the finite element library deal.II version 8.2}, doi = {10.11588/ans.2015.100.18031}, url = {http://urania.sissa.it/xmlui/handle/1963/34464}, author = {W. Bangerth and Timo Heister and Luca Heltai and G. Kanschat and Martin Kronbichler and Matthias Maier and B. Turcksin and T. D. Young} } @article {2015, title = {Gli abachi: antichi strumenti precursori delle moderne macchine da calcolo}, year = {2015}, url = {http://hdl.handle.net/10077/10884}, author = {Giuliano Klun} } @article {jevnikar2015topological, title = {A topological join construction and the Toda system on compact surfaces of arbitrary genus}, journal = {Analysis \& PDE}, volume = {8}, number = {8}, year = {2015}, pages = {1963{\textendash}2027}, publisher = {Mathematical Sciences Publishers}, doi = {10.2140/apde.2015.8.1963}, author = {Aleks Jevnikar and Kallel, Sadok and Andrea Malchiodi} } @article {2015, title = {Translation and adaptation of Birman{\textquoteright}s paper "On the theory of self-adjoint extensions of positive definite operators" (1956)}, number = {SISSA;08/2015/MATE}, year = {2015}, institution = {SISSA}, abstract = {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.}, url = {http://urania.sissa.it/xmlui/handle/1963/34443}, author = {Mikhail Khotyakov and Alessandro Michelangeli} } @article {2014, title = {Adler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras}, number = {arXiv:1401.2082}, year = {2014}, note = {45 pages}, publisher = {SISSA}, abstract = {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.}, url = {http://hdl.handle.net/1963/7242}, author = {Alberto De Sole and Victor G. Kac and Daniele Valeri} } @article {2013, title = {Classical W-algebras and generalized Drinfeld-Sokolov hierarchies for minimal and short nilpotents}, journal = {Communications in Mathematical Physics 331, nr. 2 (2014) 623-676}, number = {arXiv:1306.1684;}, year = {2014}, note = {46 pages}, publisher = {SISSA}, abstract = {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{\textquoteright}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.}, doi = {10.1007/s00220-014-2049-2}, url = {http://hdl.handle.net/1963/6979}, author = {Alberto De Sole and Victor G. Kac and Daniele Valeri} } @article {2013, title = {Dirac reduction for Poisson vertex algebras}, journal = {Communications in Mathematical Physics 331, nr. 3 (2014) 1155-1190}, number = {arXiv:1306.6589;}, year = {2014}, note = {31 pages}, publisher = {SISSA}, abstract = {We construct an analogue of Dirac{\textquoteright}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{\textquoteright}s reduction of an arbitrary non-local Poisson structure. We apply this construction to an example of a generalized Drinfeld-Sokolov hierarchy.}, doi = {10.1007/s00220-014-2103-0}, url = {http://hdl.handle.net/1963/6980}, author = {Alberto De Sole and Victor G. Kac and Daniele Valeri} } @article {Kuwert2014, title = {Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds}, journal = {Mathematische Annalen}, volume = {359}, number = {1}, year = {2014}, month = {Jun}, pages = {379{\textendash}425}, abstract = {We study curvature functionals for immersed 2-spheres in a compact, three-dimensional Riemannian manifold $M$. Under the assumption that the sectional curvature $K^M$ is strictly positive, we prove the existence of a smooth immersion $f:{\mathbb{S}}^2 \rightarrow M$ 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.

}, issn = {1432-1807}, doi = {10.1007/s00208-013-1005-3}, url = {https://doi.org/10.1007/s00208-013-1005-3}, author = {Kuwert, Ernst and Andrea Mondino and Johannes Schygulla} } @article {2014, title = {Integrability of Dirac reduced bi-Hamiltonian equations}, number = {arXiv:1401.6006;}, year = {2014}, note = {15 pages}, institution = {SISSA}, abstract = {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{\textquoteright}s, obtained by Dirac reduction from some generalized Drinfeld-Sokolov hierarchies.}, url = {http://hdl.handle.net/1963/7247}, author = {Alberto De Sole and Victor G. Kac and Daniele Valeri} } @article {2013, title = {On an isomonodromy deformation equation without the Painlev{\'e} property}, number = {Russian Journal of Mathematical Physics}, year = {2014}, note = {34 pages, 8 figures, references added}, publisher = {Maik Nauka-Interperiodica Publishing}, abstract = {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.}, doi = {10.1134/S1061920814010026}, url = {http://hdl.handle.net/1963/6466}, author = {Boris Dubrovin and Andrey Kapaev} } @article {11062, title = {Local behavior of fractional p-minimizers}, number = {SISSA preprint;10/2014/MATE}, year = {2014}, publisher = {SISSA}, abstract = {We extend the De Giorgi-Nash Moser theory to nonlocal, possibly degerate integro-differential operators

}, keywords = {fractional Sobolev spaces}, author = {Agnese Di Castro and Tuomo Kuusi and Giampiero Palatucci} } @article {berto-Tovbis-Katsevich-CPAM, title = {Singular Value Decomposition of a Finite Hilbert Transform Defined on Several Intervals and the Interior Problem of Tomography: The Riemann-Hilbert Problem Approach}, journal = {Comm. Pure Appl. Math.}, year = {2014}, author = {Marco Bertola and Alexander Katsevich and Alexander Tovbis} } @article {2014, title = {Structure of classical (finite and affine) W-algebras}, number = {arXiv:1404.0715;}, year = {2014}, note = {40 pages}, institution = {SISSA}, abstract = {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.}, url = {http://hdl.handle.net/1963/7314}, author = {Alberto De Sole and Victor G. Kac and Daniele Valeri} } @article {2014, title = {Weighted quantile correlation test for the logistic family}, number = {Acta Scientiarum Mathematicarum;volume 80; issue 1-2; pages 307-326;}, year = {2014}, publisher = {University of Szeged}, abstract = {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.}, doi = {10.14232/actasm-013-809-8}, url = {http://urania.sissa.it/xmlui/handle/1963/35025}, author = {Ferenc Balogh and {\'E}va Krauczi} } @article {2012, title = {Classical W-algebras and generalized Drinfeld-Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras}, journal = {Communications in Mathematical Physics 323, nr. 2 (2013) 663-711}, number = {arXiv:1207.6286;}, year = {2013}, note = {43 pages. Second version with minor editing and corrections}, publisher = {Springer}, abstract = {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.}, doi = {10.1007/s00220-013-1785-z}, url = {http://hdl.handle.net/1963/6978}, author = {Alberto De Sole and Victor G. Kac and Daniele Valeri} } @article {10978, title = {On critical behaviour in systems of Hamiltonian partial differential equations}, year = {2013}, institution = {SISSA}, abstract = {We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlev\'e-I (P$_I$) equation or its fourth order analogue P$_I^2$. As concrete examples we discuss nonlinear Schr\"odinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture.

}, author = {Boris Dubrovin and Tamara Grava and Christian Klein and Antonio Moro} } @article {2013, title = {The deal.II Library, Version 8.1}, number = {arXiv:1312.2266;}, year = {2013}, note = {5 pages}, institution = {SISSA}, abstract = {This paper provides an overview of the new features of the finite element library deal.II version 8.0.}, url = {http://hdl.handle.net/1963/7236}, author = {W. Bangerth and Timo Heister and Luca Heltai and G. Kanschat and Martin Kronbichler and Matthias Maier and B. Turcksin and T. D. Young} } @article {KoshakjiQuarteroniRozza2013, title = {Free Form Deformation Techniques Applied to 3D Shape Optimization Problems}, journal = {Communications in Applied and Industrial Mathematics}, year = {2013}, abstract = {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.}, doi = {10.1685/journal.caim.452}, author = {Anwar Koshakji and Alfio Quarteroni and Gianluigi Rozza} } @article {BertolaKatsevichTovbis1, title = {Inversion formulae for the $\romancosh$-weighted Hilbert transform}, journal = {Proc. Amer. Math. Soc.}, volume = {141}, number = {8}, year = {2013}, pages = {2703{\textendash}2718}, issn = {0002-9939}, doi = {10.1090/S0002-9939-2013-11642-4}, url = {http://dx.doi.org/10.1090/S0002-9939-2013-11642-4}, author = {Marco Bertola and Katsevich, A. and Alexander Tovbis} } @article {10979, title = {On the tritronqu{\'e}e solutions of P$_I^2$}, year = {2013}, institution = {SISSA}, abstract = {For equation P$_I^2$, the second member in the P$_I$ hierarchy, we prove existence of various degenerate solutions depending on the complex parameter $t$ and evaluate the asymptotics in the complex $x$ plane for $|x|\to\infty$ 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.

}, author = {Tamara Grava and Andrey Kapaev and Christian Klein} } @article {2012, title = {Detection of transcriptional triggers in the dynamics of microbial growth: application to the respiratory-versatile bacterium Shewanella oneidensis}, journal = {Nucleic Acids Research, Volume 40, Issue 15, August 2012, Pages 7132-7149}, year = {2012}, publisher = {SISSA}, abstract = {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}, doi = {10.1093/nar/gks467}, url = {http://hdl.handle.net/1963/6506}, author = {Q Beg and Mattia Zampieri and N Klitgord and S Collins and M Serres and Daniel Segr{\`e} and Claudio Altafini} } @article {2012, title = {Numerical study of the small dispersion limit of the Korteweg-de Vries equation and asymptotic solutions}, journal = {Physica D 241, nr. 23-24 (2012): 2246-2264}, number = {arXiv:1202.0962;}, year = {2012}, publisher = {Elsevier}, abstract = {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.}, keywords = {Korteweg-de Vries equation}, doi = {10.1016/j.physd.2012.04.001}, author = {Tamara Grava and Christian Klein} } @article {2011, title = {Numerical Study of breakup in generalized Korteweg-de Vries and Kawahara equations}, journal = {SIAM J. Appl. Math. 71 (2011) 983-1008}, number = {arXiv:1101.0268;}, year = {2011}, publisher = {SIAM}, abstract = {This article is concerned with a conjecture in [B. Dubrovin, Comm. Math. Phys., 267 (2006), pp. 117{\textendash}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 Painlev{\'e}-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.}, doi = {10.1137/100819783}, url = {http://hdl.handle.net/1963/4951}, author = {Boris Dubrovin and Tamara Grava and Christian Klein} } @article {2010, title = {Gene expression analysis of the emergence of epileptiform activity after focal injection of kainic acid into mouse hippocampus.}, journal = {The European journal of neuroscience. 2010 Oct; 32(8):1364-79}, number = {PMID:20950280;}, year = {2010}, publisher = {Wiley}, abstract = {We 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.

}, doi = {10.1111/j.1460-9568.2010.07403.x}, url = {http://hdl.handle.net/1963/4480}, author = {Dario Motti and Caroline Le Duigou and Nicole Chemaly and Lucia Wittner and Dejan Lazarevic and Helena Krmac and Troels Torben Marstrand and Eivind Valen and Remo Sanges and Elia Stupka and Albin Sandelin and Enrico Cherubini and Stefano Gustincich and Richard Miles} } @article {2010, title = {Numerical Solution of the Small Dispersion Limit of the Camassa-Holm and Whitham Equations and Multiscale Expansions}, number = {SISSA;10/2010/FM}, year = {2010}, abstract = {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....}, url = {http://hdl.handle.net/1963/3840}, author = {Simonetta Abenda and Tamara Grava and Christian Klein} } @article {2009, title = {On universality of critical behaviour in the focusing nonlinear Schr{\"o}dinger equation, elliptic umbilic catastrophe and the {\\\\it tritronqu{\'e}e} solution to the Painlev{\'e}-I equation}, journal = {J. Nonlinear Sci. 19 (2009) 57-94}, number = {arXiv.org;0704.0501}, year = {2009}, abstract = {We argue that the critical behaviour near the point of {\textquoteleft}{\textquoteleft}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.}, doi = {10.1007/s00332-008-9025-y}, url = {http://hdl.handle.net/1963/2525}, author = {Boris Dubrovin and Tamara Grava and Christian Klein} } @article {2008, title = {Numerical study of a multiscale expansion of the Korteweg-de Vries equation and Painlev{\'e}-II equation}, journal = {Proc. R. Soc. A 464 (2008) 733-757}, number = {arXiv.org;0708.0638v3}, year = {2008}, abstract = {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 $\\\\epsilon^{2/3}$.}, doi = {10.1098/rspa.2007.0249}, url = {http://hdl.handle.net/1963/2592}, author = {Tamara Grava and Christian Klein} } @article {2007, title = {On finite-dimensional projections of distributions for solutions of randomly forced PDE\\\'s}, journal = {Ann. Inst. Henri Poincare-Prob. Stat. 43 (2007) 399-415}, number = {arXiv.org;math/0603295}, year = {2007}, abstract = {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.}, doi = {10.1016/j.anihpb.2006.06.001}, url = {http://hdl.handle.net/1963/2012}, author = {Andrei A. Agrachev and Sergei Kuksin and Andrey Sarychev and Armen Shirikyan} } @article {2007, title = {Numerical solution of the small dispersion limit of Korteweg de Vries and Whitham equations}, number = {SISSA;91/2005/FM}, year = {2007}, abstract = {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 {\textquoteleft}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.}, doi = {10.1002/cpa.20183}, url = {http://hdl.handle.net/1963/1788}, author = {Tamara Grava and Christian Klein} } @article {2007, title = {Numerical study of a multiscale expansion of KdV and Camassa-Holm equation}, number = {arXiv.org;math-ph/0702038v1}, year = {2007}, abstract = {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}, url = {http://hdl.handle.net/1963/2527}, author = {Tamara Grava and Christian Klein} } @article {2006, title = {2-d stability of the N{\'e}el wall}, journal = {Calc. Var. Partial Differential Equations 27 (2006) 233-253}, year = {2006}, abstract = {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 N{\'e}el walls.}, doi = {10.1007/s00526-006-0019-z}, url = {http://hdl.handle.net/1963/2194}, author = {Antonio DeSimone and Hans Knuepfer and Felix Otto} } @inbook {2006, title = {Recent analytical developments in micromagnetics}, booktitle = {The science of hysteresis / eds. Giorgio Bertotti, Isaak D. Mayergoyz. - Amsterdam: Elsevier, 2006. Vol.2, 269-381.}, number = {SISSA;76/2004/M}, year = {2006}, isbn = {978-0-12-480874-4}, url = {http://hdl.handle.net/1963/2230}, author = {Antonio DeSimone and Robert V. Kohn and Stefan M{\"u}ller and Felix Otto} } @article {Bertola:Duality, title = {The duality of spectral curves that arises in two-matrix models}, journal = {Teoret. Mat. Fiz.}, volume = {134}, number = {1}, year = {2003}, pages = {32{\textendash}45}, issn = {0564-6162}, author = {Marco Bertola and B. Eynard and Kharnad, Dzh.} } @article {2003, title = {Non-linear sigma-models in noncommutative geometry: fields with values in finite spaces}, journal = {Mod. Phys. Lett. A 18 (2003) 2371-2379}, number = {arXiv.org;math/0309143v1}, year = {2003}, publisher = {World Scientific}, abstract = {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 $CP^{q-1}$.}, doi = {10.1142/S0217732303012593}, url = {http://hdl.handle.net/1963/3215}, author = {Ludwik Dabrowski and Thomas Krajewski and Giovanni Landi} } @article {2000, title = {Some Properties of Non-linear sigma-Models in Noncommutative Geometry}, journal = {Int. J. Mod. Phys. B 14 (2000) 2367-2382}, number = {SISSA;158/99/FM}, year = {2000}, publisher = {SISSA Library}, abstract = {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.}, doi = {10.1142/S0217979200001898}, url = {http://hdl.handle.net/1963/1373}, author = {Ludwik Dabrowski and Thomas Krajewski and Giovanni Landi} }