We investigate the integrability of two-dimensional invariant distributions (tangent sub-bundles) which arise naturally in the context of dynamical systems on 3-manifolds. In particular, we prove unique integrability of dynamically dominated and volume-dominated Lipschitz continuous invariant decompositions as well as distributions with some other regularity conditions.

This chapter presents an overview of model order reduction – a new paradigm in the field of simulation-based engineering sciences, and one that can tackle the challenges and leverage the opportunities of modern ICT technologies. Despite the impressive progress attained by simulation capabilities and techniques, a number of challenging problems remain intractable. These problems are of different nature, but are common to many branches of science and engineering. Among them are those related to high-dimensional problems, problems involving very different time scales, models defined in degenerate domains with at least one of the characteristic dimensions much smaller than the others, model requiring real-time simulation, and parametric models. All these problems represent a challenge for standard mesh-based discretization techniques; yet the ability to solve these problems efficiently would open unexplored routes for real-time simulation, inverse analysis, uncertainty quantification and propagation, real-time optimization, and simulation-based control – critical needs in many branches of science and engineering. Model order reduction offers new simulation alternatives by circumventing, or at least alleviating, otherwise intractable computational challenges. In the present chapter, we revisit three of these model reduction techniques: proper orthogonal decomposition, proper generalized decomposition, and reduced basis methodologies.} preprint = {http://preprints.sissa.it/xmlui/bitstream/handle/1963/35194/ECM_MOR.pdf?sequence=1&isAllowed=y

%B Encyclopedia of Computational Mechanics Second Edition %I John Wiley & Sons %P 1-36 %G eng %& Model Reduction Methods %R 10.1002/9781119176817.ecm2110 %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 International Journal of Mathematics %D 2016 %T A Frobenius theorem for corank-1 continuous distributions in dimensions two and three %A Stefano Luzzatto %A Sina Türeli %A Khadim Mbacke War %XWe formulate a notion of (uniform) asymptotic involutivity and show that it implies (unique) integrability of corank-1 continuous distributions in dimensions three or less. This generalizes and extends a classical Frobenius theorem, which says that an involutive C1 distribution is uniquely integrable.

%B International Journal of Mathematics %V 27 %P 1650061 %G eng %U https://doi.org/10.1142/S0129167X16500610 %R 10.1142/S0129167X16500610 %0 Journal Article %J Dynamical Systems %D 2016 %T Integrability of C1 invariant splittings %A Stefano Luzzatto %A Sina Türeli %A Khadim Mbacke War %XWe derive some new conditions for integrability of dynamically defined C1 invariant splittings, formulated in terms of the singular values of the iterates of the derivative of the diffeomorphism which defines the splitting.

%B Dynamical Systems %I Taylor & Francis %V 31 %P 79-88 %G eng %U https://doi.org/10.1080/14689367.2015.1057480 %R 10.1080/14689367.2015.1057480 %0 Thesis %D 2016 %T Integrability of continuous bundles and applications to dynamical systems %A Khadim Mbacke War %X In this dissertation we study the problem of integrability of bundles with low regularities. %I SISSA %G en %1 35529 %2 Mathematics %4 1 %# MAT/05 %$ Submitted by kmwar@sissa.it (kmwar@sissa.it) on 2016-09-28T12:10:43Z No. of bitstreams: 1 Thesis-Khadim.pdf: 510947 bytes, checksum: 66daf50c9c84de98301953af3433621e (MD5) %0 Book Section %B Wiley Encyclopedia of Computational Mechanics, 2016 %D 2016 %T Model Order Reduction: a survey %A Francisco Chinesta %A Antonio Huerta %A Gianluigi Rozza %A Karen Willcox %B Wiley Encyclopedia of Computational Mechanics, 2016 %I Wiley %G en %U http://urania.sissa.it/xmlui/handle/1963/35194 %1 35470 %2 Mathematics %4 1 %# MAT/08 %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2016-05-27T00:41:12Z No. of bitstreams: 1 ECM_MOR.pdf: 746542 bytes, checksum: 93d6252fe8a175c4378b96bd4192712c (MD5) %0 Journal Article %D 2014 %T Infinite-dimensional Frobenius manifolds underlying the Toda lattice hierarchy %A Chaozhong Wu %A Dafeng Zuo %X Following the approach of Carlet et al. (2011) [9], we construct a class of infinite-dimensional Frobenius manifolds underlying the Toda lattice hierarchy, which are defined on the space of pairs of meromorphic functions with possibly higher-order poles at the origin and at infinity. We also show a connection between these infinite-dimensional Frobenius manifolds and the finite-dimensional Frobenius manifolds on the orbit space of extended affine Weyl groups of type A defined by Dubrovin and Zhang. %I Elsevier %G en %U http://urania.sissa.it/xmlui/handle/1963/35026 %1 35264 %2 Mathematics %4 1 %$ Submitted by gfeltrin@sissa.it (gfeltrin@sissa.it) on 2015-11-17T10:32:31Z No. of bitstreams: 1 preprint2014.pdf: 358025 bytes, checksum: ef718fc5b61dfb8c0e0bd874196cfd13 (MD5) %R 10.1016/j.aim.2014.01.013 %0 Journal Article %J Communications in Algebra. Volume 41, Issue 6, May 2013, Pages 2346-2386 %D 2013 %T Expanded degenerations and pairs %A Dan Abramovich %A Charles Cadman %A Barbara Fantechi %A Jonathan Wise %K Expanded pairs %X Since Jun Li's original definition, several other definitions of expanded pairs and expanded degenerations have appeared in the literature. We explain how these definitions are related and introduce several new variants and perspectives. Among these are the twisted expansions used by Abramovich and Fantechi as a basis for orbifold techniques in degeneation formulas. %B Communications in Algebra. Volume 41, Issue 6, May 2013, Pages 2346-2386 %I Taylor and Francis %G en %U http://hdl.handle.net/1963/7383 %1 7431 %2 Mathematics %4 1 %# MAT/05 ANALISI MATEMATICA %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2014-06-18T09:55:33Z No. of bitstreams: 1 1110.2976v1.pdf: 418919 bytes, checksum: 61308c478c1e11ea1c389bd98e227cd2 (MD5) %R 10.1080/00927872.2012.658589 %0 Report %D 2011 %T D-branes, surface operators, and ADHM quiver representations %A Ugo Bruzzo %A Duiliu-Emanuel Diaconescu %A M. Yardim %A G. Pan %A Yi Zhang %A Chuang Wu-yen %X A supersymmetric quantum mechanical model is constructed for BPS states bound to surface operators in five dimensional SU(r) gauge theories using D-brane engineering. This model represents the effective action of a certain D2-brane configuration, and is naturally obtained by dimensional reduction of a quiver $(0,2)$ gauged linear sigma model. In a special stability chamber, the resulting moduli space of quiver representations is shown to be smooth and isomorphic to a moduli space of framed quotients on the projective plane. A precise conjecture relating a K-theoretic partition function of this moduli space to refined open string invariants of toric lagrangian branes is formulated for conifold and local P^1 x P^1 geometries. %I SISSA %G en %U http://hdl.handle.net/1963/4133 %1 3873 %2 Mathematics %3 Mathematical Physics %4 -1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2011-09-15T08:27:03Z\\r\\nNo. of bitstreams: 1\\r\\n1012.1826v2.pdf: 536824 bytes, checksum: 38cd256a97e7244dc3eaac8073827cc2 (MD5) %0 Conference Paper %B Nonlinear Conservation Laws and Applications %D 2011 %T The Monge Problem in Geodesic Spaces %A Stefano Bianchini %A Fabio Cavalletti %E Alberto Bressan %E Chen, Gui-Qiang G. %E Marta Lewicka %E Wang, Dehua %XWe address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish non branching geodesic space. We show that we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce an assumption on the transport problem π which implies that the conditional probabilities of the first marginal on each geodesic are continuous. It is known that this regularity is sufficient for the construction of an optimal transport map.

%B Nonlinear Conservation Laws and Applications %I Springer US %C Boston, MA %P 217–233 %@ 978-1-4419-9554-4 %G eng %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 Journal Article %J Discrete & Continuous Dynamical Systems-A %D 2010 %T A global compactness result for the p-Laplacian involving critical nonlinearities %A Mercuri, Carlo %A Willem, Michel %XWe prove a representation theorem for Palais-Smale sequences involving the p-Laplacian and critical nonlinearities. Applications are given to a critical problem.

%B Discrete & Continuous Dynamical Systems-A %V 28 %P 469–493 %G eng %U http://www.aimsciences.org/journals/displayArticles.jsp?paperID=5097 %R 10.3934/dcds.2010.28.469 %0 Journal Article %J Ann. Inst. H. Poincare Anal. Non Lineaire 25 (2008) 609-631 %D 2008 %T Transition layer for the heterogeneous Allen-Cahn equation %A Fethi Mahmoudi %A Andrea Malchiodi %A Juncheng Wei %X We consider the equation $\\\\e^{2}\\\\Delta u=(u-a(x))(u^2-1)$ in $\\\\Omega$, $\\\\frac{\\\\partial u}{\\\\partial \\\\nu} =0$ on $\\\\partial \\\\Omega$, where $\\\\Omega$ is a smooth and bounded domain in $\\\\R^n$, $\\\\nu$ the outer unit normal to $\\\\pa\\\\Omega$, and $a$ a smooth function satisfying $-10} and {a<0}. Assuming $\\\\nabla a \\\\neq 0$ on $K$ and $a\\\\ne 0$ on $\\\\partial \\\\Omega$, we show that there exists a sequence $\\\\e_j \\\\to 0$ such that the above equation has a solution $u_{\\\\e_j}$ which converges uniformly to $\\\\pm 1$ on the compact sets of $\\\\O_{\\\\pm}$ as $j \\\\to + \\\\infty$. %B Ann. Inst. H. Poincare Anal. Non Lineaire 25 (2008) 609-631 %G en_US %U http://hdl.handle.net/1963/2656 %1 1467 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-05-12T13:00:33Z\\nNo. of bitstreams: 1\\n0702878v1.pdf: 345060 bytes, checksum: a1e9182e6448c835b1c66b4f226b0b8d (MD5) %R 10.1016/j.anihpc.2007.03.008 %0 Journal Article %J J. Fixed Point Theory Appl. 1 (2007) 305-336 %D 2007 %T Boundary interface for the Allen-Cahn equation %A Andrea Malchiodi %A Juncheng Wei %B J. Fixed Point Theory Appl. 1 (2007) 305-336 %G en_US %U http://hdl.handle.net/1963/2027 %1 2169 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-09-03T11:10:26Z\\nNo. of bitstreams: 1\\nMalchiodiWei07.pdf: 379788 bytes, checksum: 4c87f76e7db9858a9acfeca0a8ec75c1 (MD5) %R 10.1007/s11784-007-0016-7 %0 Journal Article %J Pacific Journal of Mathematics 229 (2007), No. 2, 447–468 %D 2007 %T Boundary-clustered interfaces for the Allen–Cahn equation %A Andrea Malchiodi %A Wei-Ming Ni %A Juncheng Wei %B Pacific Journal of Mathematics 229 (2007), No. 2, 447–468 %I Mathematical Sciences Publishers %G en %U http://hdl.handle.net/1963/5089 %1 4905 %2 Mathematics %3 Functional Analysis and Applications %4 -1 %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2011-11-17T13:19:29Z\\nNo. of bitstreams: 1\\nmnw-pjm-final.pdf: 287316 bytes, checksum: 2a18e3af66be7b9b793a528138cf493a (MD5) %0 Journal Article %J J. Noncomm. Geom. 1 (2007) 213-239 %D 2007 %T Dirac operators on all Podles quantum spheres %A Francesco D'Andrea %A Ludwik Dabrowski %A Giovanni Landi %A Elmar Wagner %X We construct spectral triples on all Podles quantum spheres. These noncommutative geometries are equivariant for a left action of $U_q(su(2))$ and are regular, even and of metric dimension 2. They are all isospectral to the undeformed round geometry of the 2-sphere. There is also an equivariant real structure for which both the commutant property and the first order condition for the Dirac operators are valid up to infinitesimals of arbitrary order. %B J. Noncomm. Geom. 1 (2007) 213-239 %G en_US %U http://hdl.handle.net/1963/2177 %1 2067 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-04T09:28:47Z\\nNo. of bitstreams: 1\\n0606480v2.pdf: 249691 bytes, checksum: b7ae4969eee716046a815f5e66a249fb (MD5) %R 10.4171/JNCG/5 %0 Journal Article %J Ann. Inst. H. Poincare Anal. Non Lineaire 22 (2005) 143-163 %D 2005 %T Multiple clustered layer solutions for semilinear Neumann problems on a ball %A Andrea Malchiodi %A Wei-Ming Ni %A Juncheng Wei %B Ann. Inst. H. Poincare Anal. Non Lineaire 22 (2005) 143-163 %I Elsevier %G en_US %U http://hdl.handle.net/1963/3532 %1 732 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2009-02-20T10:15:49Z\\nNo. of bitstreams: 1\\nMalchiodiNiWei04.pdf: 340366 bytes, checksum: a0bb6df9ca59c557fdacb5e7142ecf3d (MD5) %R 10.1016/j.anihpc.2004.05.003