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

1 aChinesta, Francisco1 aHuerta, Antonio1 aRozza, Gianluigi1 aWillcox, Karen uhttps://www.math.sissa.it/node/1294900573nas a2200205 4500008004100000245003700041210003000078300001400108490000700122100001700129700001900146700001800165700001700183700001700200700002400217700002000241700001700261700001700278856007200295 2016 eng d00aThe deal.II library, Version 8.40 adealII library Version 84 a135–1410 v241 aBangerth, W.1 aDavydov, Denis1 aHeister, Timo1 aHeltai, Luca1 aKanschat, G.1 aKronbichler, Martin1 aMaier, Matthias1 aTurcksin, B.1 aWells, David uhttps://www.math.clemson.edu/ heister/preprints/deal84-preprint.pdf00789nas a2200145 4500008004100000245009000041210006900131300001200200490000700212520031400219100002200533700001800555700002400573856004600597 2016 eng d00aA Frobenius theorem for corank-1 continuous distributions in dimensions two and three0 aFrobenius theorem for corank1 continuous distributions in dimens a16500610 v273 aWe 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.

1 aLuzzatto, Stefano1 aTüreli, Sina1 aWar, Khadim, Mbacke uhttps://doi.org/10.1142/S0129167X1650061000676nas a2200157 4500008004100000245004500041210004500086260002100131300001000152490000700162520023500169100002200404700001800426700002400444856005000468 2016 eng d00aIntegrability of C1 invariant splittings0 aIntegrability of C1 invariant splittings bTaylor & Francis a79-880 v313 aWe 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.

1 aLuzzatto, Stefano1 aTüreli, Sina1 aWar, Khadim, Mbacke uhttps://doi.org/10.1080/14689367.2015.105748000539nas a2200109 4500008004100000245007800041210006900119260001000188520009700198100002400295856011000319 2016 en d00aIntegrability of continuous bundles and applications to dynamical systems0 aIntegrability of continuous bundles and applications to dynamica bSISSA3 aIn this dissertation we study the problem of integrability of bundles with low regularities.1 aWar, Khadim, Mbacke uhttps://www.math.sissa.it/publication/integrability-continuous-bundles-and-applications-dynamical-systems00391nas a2200133 4500008004100000245003600041210003500077260001000112100002400122700002000146700002100166700001900187856005100206 2016 en d00aModel Order Reduction: a survey0 aModel Order Reduction a survey bWiley1 aChinesta, Francisco1 aHuerta, Antonio1 aRozza, Gianluigi1 aWillcox, Karen uhttp://urania.sissa.it/xmlui/handle/1963/3519400921nas a2200121 4500008004100000245008300041210006900124260001300193520050800206100001800714700001600732856005100748 2014 en d00aInfinite-dimensional Frobenius manifolds underlying the Toda lattice hierarchy0 aInfinitedimensional Frobenius manifolds underlying the Toda latt bElsevier3 aFollowing 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.1 aWu, Chaozhong1 aZuo, Dafeng uhttp://urania.sissa.it/xmlui/handle/1963/3502600809nas a2200157 4500008004100000245003700041210003700078260002300115520037700138653001900515100002000534700002000554700002200574700001900596856003600615 2013 en d00aExpanded degenerations and pairs0 aExpanded degenerations and pairs bTaylor and Francis3 aSince 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.10aExpanded pairs1 aAbramovich, Dan1 aCadman, Charles1 aFantechi, Barbara1 aWise, Jonathan uhttp://hdl.handle.net/1963/738301229nas a2200169 4500008004100000245006500041210006200106260001000168520073800178100001600916700003100932700001500963700001200978700001400990700001901004856003601023 2011 en d00aD-branes, surface operators, and ADHM quiver representations0 aDbranes surface operators and ADHM quiver representations bSISSA3 aA 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.1 aBruzzo, Ugo1 aDiaconescu, Duiliu-Emanuel1 aYardim, M.1 aPan, G.1 aZhang, Yi1 aWu-yen, Chuang uhttp://hdl.handle.net/1963/413301064nas a2200193 4500008004100000020002200041245004100063210003700104260002800141300001400169520049000183100002300673700002200696700002100718700002400739700001900763700001600782856007200798 2011 eng d a978-1-4419-9554-400aThe Monge Problem in Geodesic Spaces0 aMonge Problem in Geodesic Spaces aBoston, MAbSpringer US a217–2333 aWe address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish non branching geodesic space. We show that we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce an assumption on the transport problem π which implies that the conditional probabilities of the first marginal on each geodesic are continuous. It is known that this regularity is sufficient for the construction of an optimal transport map.

1 aBianchini, Stefano1 aCavalletti, Fabio1 aBressan, Alberto1 aChen, Gui-Qiang, G.1 aLewicka, Marta1 aWang, Dehua uhttps://www.math.sissa.it/publication/monge-problem-geodesic-spaces02590nas a2200265 4500008004100000245013200041210006900173260001000242520175000252100001702002700002402019700002002043700001902063700002102082700001802103700003002121700001802151700001702169700001702186700002002203700002202223700002402245700001902269856003602288 2010 en d00aGene expression analysis of the emergence of epileptiform activity after focal injection of kainic acid into mouse hippocampus.0 aGene expression analysis of the emergence of epileptiform activi bWiley3 aWe 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.

1 aMotti, Dario1 aLe Duigou, Caroline1 aChemaly, Nicole1 aWittner, Lucia1 aLazarevic, Dejan1 aKrmac, Helena1 aMarstrand, Troels, Torben1 aValen, Eivind1 aSanges, Remo1 aStupka, Elia1 aSandelin, Albin1 aCherubini, Enrico1 aGustincich, Stefano1 aMiles, Richard uhttp://hdl.handle.net/1963/448000848nas a2200133 4500008004100000245008600041210006900127300001400196490000700210520038600217100001900603700001900622856007300641 2010 eng d00aA global compactness result for the p-Laplacian involving critical nonlinearities0 aglobal compactness result for the pLaplacian involving critical a469–4930 v283 aWe prove a representation theorem for Palais-Smale sequences involving the p-Laplacian and critical nonlinearities. Applications are given to a critical problem.

1 aMercuri, Carlo1 aWillem, Michel uhttp://www.aimsciences.org/journals/displayArticles.jsp?paperID=509701148nas a2200121 4500008004300000245006300043210006200106520076200168100002000930700002200950700001800972856003600990 2008 en_Ud 00aTransition layer for the heterogeneous Allen-Cahn equation0 aTransition layer for the heterogeneous AllenCahn equation3 aWe 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 $j \\\\to + \\\\infty$.1 aMahmoudi, Fethi1 aMalchiodi, Andrea1 aWei, Juncheng uhttp://hdl.handle.net/1963/265600318nas a2200097 4500008004300000245005100043210005000094100002200144700001800166856003600184 2007 en_Ud 00aBoundary interface for the Allen-Cahn equation0 aBoundary interface for the AllenCahn equation1 aMalchiodi, Andrea1 aWei, Juncheng uhttp://hdl.handle.net/1963/202700420nas a2200121 4500008004100000245006400041210006300105260003700168100002200205700001700227700001800244856003600262 2007 en d00aBoundary-clustered interfaces for the Allen–Cahn equation0 aBoundaryclustered interfaces for the Allen–Cahn equation bMathematical Sciences Publishers1 aMalchiodi, Andrea1 aNi, Wei-Ming1 aWei, Juncheng uhttp://hdl.handle.net/1963/508900855nas a2200133 4500008004300000245005000043210005000093520045800143100002400601700002200625700002000647700001800667856003600685 2007 en_Ud 00aDirac operators on all Podles quantum spheres0 aDirac operators on all Podles quantum spheres3 aWe 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.1 aD'Andrea, Francesco1 aDabrowski, Ludwik1 aLandi, Giovanni1 aWagner, Elmar uhttp://hdl.handle.net/1963/217700421nas a2200121 4500008004300000245008100043210006900124260001300193100002200206700001700228700001800245856003600263 2005 en_Ud 00aMultiple clustered layer solutions for semilinear Neumann problems on a ball0 aMultiple clustered layer solutions for semilinear Neumann proble bElsevier1 aMalchiodi, Andrea1 aNi, Wei-Ming1 aWei, Juncheng uhttp://hdl.handle.net/1963/3532