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

JF - Encyclopedia of Computational Mechanics Second Edition PB - John Wiley & Sons ER - TY - JOUR T1 - The deal.II library, Version 8.4 JF - JOURNAL OF NUMERICAL MATHEMATICS Y1 - 2016 A1 - W. Bangerth A1 - Denis Davydov A1 - Timo Heister A1 - Luca Heltai A1 - G. Kanschat A1 - Martin Kronbichler A1 - Matthias Maier A1 - B. Turcksin A1 - David Wells VL - 24 UR - https://www.math.clemson.edu/ heister/preprints/deal84-preprint.pdf ER - TY - JOUR T1 - A Frobenius theorem for corank-1 continuous distributions in dimensions two and three JF - International Journal of Mathematics Y1 - 2016 A1 - Stefano Luzzatto A1 - Sina Türeli A1 - Khadim Mbacke War AB -We 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.

VL - 27 UR - https://doi.org/10.1142/S0129167X16500610 ER - TY - JOUR T1 - Integrability of C1 invariant splittings JF - Dynamical Systems Y1 - 2016 A1 - Stefano Luzzatto A1 - Sina Türeli A1 - Khadim Mbacke War AB -We 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.

PB - Taylor & Francis VL - 31 UR - https://doi.org/10.1080/14689367.2015.1057480 ER - TY - THES T1 - Integrability of continuous bundles and applications to dynamical systems Y1 - 2016 A1 - Khadim Mbacke War AB - In this dissertation we study the problem of integrability of bundles with low regularities. PB - SISSA U1 - 35529 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - CHAP T1 - Model Order Reduction: a survey T2 - Wiley Encyclopedia of Computational Mechanics, 2016 Y1 - 2016 A1 - Francisco Chinesta A1 - Antonio Huerta A1 - Gianluigi Rozza A1 - Karen Willcox JF - Wiley Encyclopedia of Computational Mechanics, 2016 PB - Wiley UR - http://urania.sissa.it/xmlui/handle/1963/35194 U1 - 35470 U2 - Mathematics U4 - 1 U5 - MAT/08 ER - TY - JOUR T1 - Infinite-dimensional Frobenius manifolds underlying the Toda lattice hierarchy Y1 - 2014 A1 - Chaozhong Wu A1 - Dafeng Zuo AB - 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. PB - Elsevier UR - http://urania.sissa.it/xmlui/handle/1963/35026 U1 - 35264 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Expanded degenerations and pairs JF - Communications in Algebra. Volume 41, Issue 6, May 2013, Pages 2346-2386 Y1 - 2013 A1 - Dan Abramovich A1 - Charles Cadman A1 - Barbara Fantechi A1 - Jonathan Wise KW - Expanded pairs AB - 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. PB - Taylor and Francis UR - http://hdl.handle.net/1963/7383 N1 - This article is composed of 41 pages and is recorded in PDF format U1 - 7431 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - RPRT T1 - D-branes, surface operators, and ADHM quiver representations Y1 - 2011 A1 - Ugo Bruzzo A1 - Duiliu-Emanuel Diaconescu A1 - M. Yardim A1 - G. Pan A1 - Yi Zhang A1 - Chuang Wu-yen AB - 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. PB - SISSA UR - http://hdl.handle.net/1963/4133 N1 - 45 pages, v2: minor corrections U1 - 3873 U2 - Mathematics U3 - Mathematical Physics U4 - -1 ER - TY - CONF T1 - The Monge Problem in Geodesic Spaces T2 - Nonlinear Conservation Laws and Applications Y1 - 2011 A1 - Stefano Bianchini A1 - Fabio Cavalletti ED - Alberto Bressan ED - Chen, Gui-Qiang G. ED - Marta Lewicka ED - Wang, Dehua AB -We 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.

JF - Nonlinear Conservation Laws and Applications PB - Springer US CY - Boston, MA SN - 978-1-4419-9554-4 ER - TY - JOUR T1 - Gene expression analysis of the emergence of epileptiform activity after focal injection of kainic acid into mouse hippocampus. JF - The European journal of neuroscience. 2010 Oct; 32(8):1364-79 Y1 - 2010 A1 - Dario Motti A1 - Caroline Le Duigou A1 - Nicole Chemaly A1 - Lucia Wittner A1 - Dejan Lazarevic A1 - Helena Krmac A1 - Troels Torben Marstrand A1 - Eivind Valen A1 - Remo Sanges A1 - Elia Stupka A1 - Albin Sandelin A1 - Enrico Cherubini A1 - Stefano Gustincich A1 - Richard Miles AB -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.

PB - Wiley UR - http://hdl.handle.net/1963/4480 U1 - 4244 U2 - Neuroscience U3 - Neurobiology U4 - -1 ER - TY - JOUR T1 - A global compactness result for the p-Laplacian involving critical nonlinearities JF - Discrete & Continuous Dynamical Systems-A Y1 - 2010 A1 - Mercuri, Carlo A1 - Willem, Michel AB -We prove a representation theorem for Palais-Smale sequences involving the p-Laplacian and critical nonlinearities. Applications are given to a critical problem.

VL - 28 UR - http://www.aimsciences.org/journals/displayArticles.jsp?paperID=5097 ER - TY - JOUR T1 - Transition layer for the heterogeneous Allen-Cahn equation JF - Ann. Inst. H. Poincare Anal. Non Lineaire 25 (2008) 609-631 Y1 - 2008 A1 - Fethi Mahmoudi A1 - Andrea Malchiodi A1 - Juncheng Wei AB - 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 $j \\\\to + \\\\infty$. UR - http://hdl.handle.net/1963/2656 U1 - 1467 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Boundary interface for the Allen-Cahn equation JF - J. Fixed Point Theory Appl. 1 (2007) 305-336 Y1 - 2007 A1 - Andrea Malchiodi A1 - Juncheng Wei UR - http://hdl.handle.net/1963/2027 U1 - 2169 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Boundary-clustered interfaces for the Allen–Cahn equation JF - Pacific Journal of Mathematics 229 (2007), No. 2, 447–468 Y1 - 2007 A1 - Andrea Malchiodi A1 - Wei-Ming Ni A1 - Juncheng Wei PB - Mathematical Sciences Publishers UR - http://hdl.handle.net/1963/5089 U1 - 4905 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Dirac operators on all Podles quantum spheres JF - J. Noncomm. Geom. 1 (2007) 213-239 Y1 - 2007 A1 - Francesco D'Andrea A1 - Ludwik Dabrowski A1 - Giovanni Landi A1 - Elmar Wagner AB - 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. UR - http://hdl.handle.net/1963/2177 U1 - 2067 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Multiple clustered layer solutions for semilinear Neumann problems on a ball JF - Ann. Inst. H. Poincare Anal. Non Lineaire 22 (2005) 143-163 Y1 - 2005 A1 - Andrea Malchiodi A1 - Wei-Ming Ni A1 - Juncheng Wei PB - Elsevier UR - http://hdl.handle.net/1963/3532 U1 - 732 U2 - Mathematics U3 - Functional Analysis and Applications ER -