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Strictly positive definite kernels on the 2-sphere: beyond radial symmetry [article]

Janin Jäger
2020 arXiv   pre-print
The paper introduces a new characterisation of strictly positive definiteness for kernels on the 2-sphere without assuming the kernel to be radially (isotropic) or axially symmetric.  ...  These generalise the result derived by Chen et al. 2003 for radial kernels to non-radial kernels.  ...  Acknowledgement The work of J. Jäger was supported by the Justus Liebig University's postdoctoral fellowship Just'Us  ... 
arXiv:2005.02798v1 fatcat:w2bifxqqprbohjc3l2huu4lygu

Pulling Apart 2-spheres in 4-manifolds [article]

Rob Schneiderman, Peter Teichner
2015 arXiv   pre-print
The order 1 intersection invariant is shown to be the complete obstruction to pulling apart 2-spheres in any 4-manifold after taking connected sums with finitely many copies of S^2\times S^2; and the order  ...  The accompanying higher-order intersection invariants provide a geometric generalization of Milnor's link-homotopy invariants, and can give the complete obstruction to pulling apart 2-spheres in certain  ...  Definition 34 For a quadruple of 2-spheres A = A 1 , A 2 , A 3 , A 4 X with X simply connected and A in the radical of λ 0 on π 2 X, define the order 2 intersection relations INT W 2 (A) < Λ 2 (4) to be  ... 
arXiv:1210.5534v2 fatcat:c3iagvigl5cklj6xnbhcvblmgi

How isotropic kernels perform on simple invariants [article]

Jonas Paccolat, Stefano Spigler, Matthieu Wyart
2020 arXiv   pre-print
We investigate how the training curve of isotropic kernel methods depends on the symmetry of the task to be learned, in several settings.  ...  We argue and confirm numerically that for large bandwidth, β = d-1+ξ/3d-3+ξ, where ξ∈ (0,2) is the exponent characterizing the singularity of the kernel at the origin.  ...  This work was partially supported by the grant from the Simons Foundation (#454953 Matthieu Wyart). M.W. thanks the Swiss National Science Foundation for support under Grant No. 200021-165509.  ... 
arXiv:2006.09754v5 fatcat:zh6n4tjzkbdprohbrojpoi2btu

Matter from Space [chapter]

Domenico Giulini
2018 Einstein Studies  
I review this picture of 'Geometrodynamics' and comment on various developments after Einstein.  ...  General Relativity offers the possibility to model attributes of matter, like mass, momentum, angular momentum, spin, chirality etc. from pure space, endowed only with a single field that represents its  ...  Acknowledgements I sincerely thank the organisers of the Beyond-Einstein conference at Mainz University for inviting me to this most stimulating and pleasant meeting.  ... 
doi:10.1007/978-1-4939-7708-6_12 fatcat:ouxdee6m2zenbpmuewnc2chsva

Matter from Space [article]

Domenico Giulini
2009 arXiv   pre-print
I review this picture of 'Geometrodynamics' and comment on various developments after Einstein.  ...  General Relativity offers the possibility to model attributes of matter, like mass, momentum, angular momentum, spin, chirality etc. from pure space, endowed only with a single field that represents its  ...  Acknowledgements I sincerely thank the organisers of the Beyond-Einstein conference at Mainz University for inviting me to this most stimulating and pleasant meeting.  ... 
arXiv:0910.2574v2 fatcat:5irmhfjzenfgvkzbm23fbcoepy

Mapping-class groups of 3-manifolds in canonical quantum gravity [article]

Domenico Giulini
2006 arXiv   pre-print
We also attempt to give a flavor of the mathematical ideas involved.  ...  Mapping-class groups of 3-manifolds feature as symmetry groups in canonical quantum gravity.  ...  Note that this is not obvious from the definition of irreducibility, since non-zero elements of π 2 need not be representable as embedded 2-spheres.  ... 
arXiv:math-ph/0606066v1 fatcat:k4vrbbzqhfdwrhrdijrk7esy5y

Conformal tightness of holographic scaling in black hole thermodynamics

Horacio E Camblong, Carlos R Ordóñez
2013 Classical and quantum gravity  
The near-horizon conformal symmetry of nonextremal black holes is shown to be a mandatory ingredient for the holographic scaling of the scalar-field contribution to the black hole entropy.  ...  Specifically, the conformal SO(2,1) invariance of the radial degree of freedom conspires with the area proportionality of the angular momentum sums to yield a robust holographic outcome.  ...  One of us (C.R.O.) would like to thank the Kavli Institute for Theoretical Physics for generous support and for providing a wonderful environment during the initial stages of this work.  ... 
doi:10.1088/0264-9381/30/17/175007 fatcat:fu3owltvf5ewvhfuohpowuq5py

Random matrix theory and symmetric spaces

M. Caselle, U. Magnea
2004 Physics reports  
In Part I the theory of symmetric spaces is reviewed with particular emphasis on the ideas relevant for appreciating the correspondence with random matrix theories.  ...  In particular the Cartan classification of triplets of symmetric spaces with positive, zero and negative curvature gives rise to a new classification of random matrix ensembles.  ...  The coset SO(3)/SO(2) can be identified with the unit 2-sphere S 2 , a positive curvature space (cf. subsection 2.3).  ... 
doi:10.1016/j.physrep.2003.12.004 fatcat:4eywchyfofcfroti47xgcmygke

Spatiotemporal Pattern Extraction by Spectral Analysis of Vector-Valued Observables

Dimitrios Giannakis, Abbas Ourmazd, Joanna Slawinska, Zhizhen Zhao
2019 Journal of nonlinear science  
Moreover, the kernel construction naturally quotients out dynamical symmetries in the data and exhibits an asymptotic commutativity property with the Koopman evolution operator of the system, enabling  ...  Our approach, called vector-valued spectral analysis (VSA), is based on an eigendecomposition of a kernel integral operator acting on a Hilbert space of vector-valued observables of the system, taking  ...  First, Gaussian kernels are strictly positive and bounded below on compact sets.  ... 
doi:10.1007/s00332-019-09548-1 fatcat:ilmercswgra6xik7fo5bvlxtta

Constraints on the topology of higher dimensional black holes [article]

Gregory J. Galloway
2011 arXiv   pre-print
Here we review some of the known constraints on horizon topology for higher dimensional black holes.  ...  Then cross sections of the event horizon are topologically 2-spheres.  ...  Hence, Σ must be a 2-sphere. Actually, the torus T 2 (g = 1) arises as a borderline case in the proof.  ... 
arXiv:1111.5356v1 fatcat:ten3zgzjlrapfgn3micq6rwxsu

Aspects of Positive Scalar Curvature and Topology I

M. Walsh
2017 Irish Mathematical Society Bulletin  
Whether or not a smooth manifold admits a Riemannian metric whose scalar curvature function is strictly positive is a problem which has been extensively studied by geometers and topologists alike.  ...  We provide a brief survey, aimed at the non-expert, which is intended to provide a gentle introduction to some of the work done on these deep questions. 2010 Mathematics Subject Classification. 46A99,  ...  I would like to thank my former colleagues there for their friendship and support over the years.  ... 
doi:10.33232/bims.0080.45.68 fatcat:nnl5ehnydrftrii7ndf4xino5e

Numerical initial data deformation exploiting a gluing construction: I. Exterior asymptotic Schwarzschild [article]

Boris Daszuta, Jörg Frauendiener
2019 arXiv   pre-print
The generality of the method is illustrated in a comparison of the ADM mass of EAG initial data sets featuring internal BBHs as modelled by Brill-Lindquist and Misner data.In contrast to the recent work  ...  of Doulis and Rinne (2016), and Pook-Kolb and Giulini (2018) we do not make use of the York-Lichnerowicz conformal framework to reformulate the constraints.  ...  ACKNOWLEDGEMENTS The authors are grateful for a University of Otago PhD scholarship to BD and a University of Otago Research Grant to JF. APPENDIX  ... 
arXiv:1902.05670v1 fatcat:vnby72o4qndg7eguwxf3xca5eu

Anisotropic Einstein data with isotropic non negative prescribed scalar curvature

Bernold Fiedler, Juliette Hell, Brian Smith
2015 Annales de l'Institut Henri Poincare. Analyse non linéar  
Our results are based on center manifold theory for quasilinear parabolic equations and on equivariant bifurcation theory for not necessarily self-similar solutions of a self-similarly rescaled equation  ...  The metric itself is regular up to and including the surface at the blow up radius, which is a minimal surface.  ...  Acknowledgement This work was supported by the Deutsche Forschungsgemeinschaft, SFB 647 "Space-Time-Matter".  ... 
doi:10.1016/j.anihpc.2014.01.002 fatcat:5mv26qb7xjgejlxklw4wf6vvk4

Counting of states in Higgs theories

Mark P. Hertzberg, Mudit Jain
2019 Physical Review D  
Since gauge fields fall off exponentially from sources there are no conserved charges or symmetries in the Higgs phase; so the Higgs mechanism is the removal of symmetry from the theory.  ...  global symmetry, as is well known.  ...  A family of conserved charges arises from taking α(x) to be independent of radius, but proportional to a spherical harmonic α(x) ∝ Y lm (ϕ,φ), (73) where (ϕ,φ) are polar angles on the 2-sphere at spatial  ... 
doi:10.1103/physrevd.99.065015 fatcat:wl3pxq3ybbd2zerbteg2gix3z4

Two species of vortices in massive gauged non-linear sigma models

A. Alonso-Izquierdo, W. García Fuertes, J. Mateos Guilarte
2015 Journal of High Energy Physics  
In the simplest n=1 case, where the target manifold is the S^2 sphere, we describe the scalar fields by means of stereographic maps.  ...  Two different situations are envisaged: either the vacuum orbit lies on a parallel in the sphere, or one pole and the same parallel form the vacuum orbit.  ...  Acknowledgments After the first version of this work had been completed and sent to the arXives, we learned that vorticial solutions on S 2 like those reported here had been previously found in [17]  ... 
doi:10.1007/jhep02(2015)139 fatcat:umv3u36ldffd5d3ksvslgwpehy
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