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Book Review: Optimal transport: old and new

Alessio Figalli
2010 Bulletin of the American Mathematical Society  
However, since the book was completed in 2008, it misses some of very recent developments linking the MTW tensor with the geometry of the manifold (see for instance [4] for a recent account on these  ...  This fact shows that the MTW tensor is a nonlocal version of the sectional curvature and the MTW condition implies nonnegative sectional curvature.  ...  Alessio Figalli The University of Texas at Austin E-mail address: figalli@math.utexas.edu  ... 
doi:10.1090/s0273-0979-10-01285-1 fatcat:ptvff7qmhvcpjlbpj4fzc5nzd4

When optimal transport meets information geometry

Gabriel Khan, Jun Zhang
2022 Information Geometry  
Our piece also serves as an introduction to the Special Issue on Optimal Transport of the journal Information Geometry.  ...  During the recent years, there has been a surge of research endeavors that cut across these two areas and explore their links and interactions.  ...  Acknowledgement The authors would like to thank Leonard Wong for his helpful comments on the manuscript. G.K. is partially supported by a  ... 
doi:10.1007/s41884-022-00066-w fatcat:vikp7dg7qvcirlpddp57y6glka

On pseudoparticle solutions in Yang's theory of gravity

Eckehard W. Mielke
1981 General Relativity and Gravitation  
the Rainich conditions of Wheeler's geometrodynamics.  ...  methods reveal that the gravitational analogue of the "instanton" or pseudoparticle solution of Yang-Mills theory was already known to Riemann.  ...  Later on, this point of view has been taken up and developed further by e.g.  ... 
doi:10.1007/bf00756856 fatcat:gepwlplqdvhnpd75cbkl7n7t4u

Pseudo-Riemannian geometry encodes information geometry in optimal transport

Ting-Kam Leonard Wong, Jiaowen Yang
2021 Information Geometry  
As a by-product, we obtain a new information-geometric interpretation of the MTW tensor on the graph of the transport map. This relation sheds light on old and new aspects of information geometry.  ...  Namely, the pseudo-Riemannian framework of Kim and McCann, which provides a geometric perspective on the fundamental Ma–Trudinger–Wang (MTW) condition in the regularity theory of optimal transport maps  ...  Most of the work was done when he was a student at the University of Southern California. We also thank the anonymous reviewers for their careful reading and comments. Declarations  ... 
doi:10.1007/s41884-021-00053-7 pmid:35874116 pmcid:PMC9296067 fatcat:xeiffkarrzhxtde6nv7zhkbase

Transformations for Piola-mapped elements [article]

Francis Aznaran, Robert Kirby, Patrick Farrell
2021 arXiv   pre-print
In this work we apply the novel transformation theory recently developed by Kirby [SMAI-JCM, 4:197-224, 2018] to devise the correct map for transforming the basis on a reference cell to a generic physical  ...  The Arnold-Winther element successfully discretizes the Hellinger-Reissner variational formulation of linear elasticity; its development was one of the key early breakthroughs of the finite element exterior  ...  Acknowledgements The authors are grateful to Endre Süli for comments on Section 6 and to Kaibo Hu for comments on subsection 7.1.  ... 
arXiv:2110.13224v1 fatcat:7j6deumdozgqzjqvphx4oflcxu

Pseudo-Riemannian geometry embeds information geometry in optimal transport [article]

Ting-Kam Leonard Wong, Jiaowen Yang
2021 arXiv   pre-print
As a by-product, we obtain a new information-geometric interpretation of the MTW tensor. This connection sheds light on old and new aspects of information geometry.  ...  Namely, the pseudo-Riemannian framework of Kim and McCann, a geometric perspective on the fundamental Ma-Trudinger-Wang (MTW) condition in the regularity theory of optimal transport maps, encodes the dualistic  ...  Most of the work was done when he was a student at the University of Southern California. We also thank the anonymous reviewers for their careful reading and comments.  ... 
arXiv:1906.00030v4 fatcat:ltvube7efvghplkpaiuhp7eebu

New derivation of the Lagrangian of a perfect fluid with a barotropic equation of state

Olivier Minazzoli, Tiberiu Harko
2012 Physical Review D  
The only assumptions used in the derivation are: 1) the matter part of the Lagrangian does not depend on the derivatives of the metric, and 2) the particle number of the fluid is conserved (∇_σ (ρ u^σ)  ...  the pressure.  ...  Acknowledgments This research was supported by an appointment to the NASA Postdoctoral Program at the Jet Propulsion Laboratory, California Institute of Technology, administered by Oak Ridge Associated  ... 
doi:10.1103/physrevd.86.087502 fatcat:qdftzd3r5vav3mvf6xi2v7fqwy

Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds

Alessio Figalli, Ludovic Rifford, Cédric Villani
2011 Tohoku mathematical journal  
In this paper we continue the investigation of the regularity of optimal transport maps on Riemannian manifolds, in relation with the geometric conditions of Ma-Trudinger-Wang and the geometry of the cut  ...  We derive some sufficient and some necessary conditions to ensure that the optimal transport map is always continuous.  ...  In [17] we showed that the same condition implies the semiconvexity of injectivity domains, without assumption on the Ma-Trudinger-Wang tensor.  ... 
doi:10.2748/tmj/1325886291 fatcat:crdamwlqyfdaxitwthuhjjnb3m

General relativistic symmetry of electron spin torque

Akitomo Tachibana
2011 Journal of Mathematical Chemistry  
On the Minkowski space, the relativistic quantum mechanical motion of electron is described by 4-spinor.  ...  In our recent paper, the electron spin torque is found to be counter-balanced by the chiral electron density.  ...  For massive particles, helicity is not conserved since the direction of linear momentum depends on the inertial frame of observer.  ... 
doi:10.1007/s10910-011-9943-z fatcat:icjl6ckeffas7j4lwstyitmreu

Tissue- and column-specific measurements from multi-parameter mapping of the human cervical spinal cord at 3 T

R. S. Samson, O. Ciccarelli, C. Kachramanoglou, L. Brightman, A. Lutti, D. L. Thomas, N. Weiskopf, C. A. M. Wheeler-Kingshott
2013 NMR in Biomedicine  
In summary, an optimised cervical cord multi-parameter mapping protocol was developed, enabling tissue-and column-specific measurements to be made.  ...  This technique has the potential to provide insight into the pathological processes occurring in the cervical cord affected by neurological disorders.  ...  Scanning was performed at the Wellcome Trust Centre for Neuroimaging, which is funded by the Wellcome Trust. The authors would also like to thank all the participants of this study.  ... 
doi:10.1002/nbm.3022 pmid:24105923 pmcid:PMC4034603 fatcat:jdoutrcwy5eidaydfp4igunr5y

Kähler-Ricci Flow preserves negative anti-bisectional curvature [article]

Gabriel Khan, Fangyang Zheng
2021 arXiv   pre-print
In particular, we showed that the MTW tensor for a cost function c(x,y)=Ψ(x-y) can be understood as the anti-bisectional curvature of an associated Kähler metric defined on a tube domain.  ...  The correspondence between the anti-bisectional curvature and the MTW tensor provides a meaningful sense in which the anti-bisectional curvature can have a sign (i.e., be positive or negative).  ...  As such, if we consider a two dimensional Ψ-cost which is KR weakly regular but has points where the MTW tensor is strongly positive, after a short period of time we can expect the MTW tensor to be strictly  ... 
arXiv:2011.07181v2 fatcat:ggz6vgkzujbkhb5wqw3s7lv5ne

The Monge-Ampère equation and its link to optimal transportation [article]

Guido De Philippis, Alessio Figalli
2013 arXiv   pre-print
We survey the (old and new) regularity theory for the Monge-Amp\'ere equation, show its connection to optimal transportation, and describe the regularity properties of a general class of Monge-Amp\'ere  ...  Acknowledgments: We wish to thank Neil Trudinger for several useful comments on a preliminary version of this paper. The second author is partially supported by NSF Grant DMS-1262411.  ...  Both authors acknowledge the support of the ERC ADG Grant GeMeThNES.  ... 
arXiv:1310.6167v1 fatcat:fhqoid6i2zbrjmkysxk4xh4yg4

The Ma–Trudinger–Wang curvature for natural mechanical actions

Paul W. Y. Lee, Robert J. McCann
2010 Calculus of Variations and Partial Differential Equations  
We go on to study gentle perturbations of the free action by a potential, and deduce conditions on the potential which guarantee either that the corresponding cost satisfies the more restrictive condition  ...  In particular, the quartic potential of the anharmonic oscillator satisfies (A3s) in the perturbative regime.  ...  (A3w): the cost satisfies the weak MTW condition on if ≥ 0 on , (A3s): the cost satisfies the strong MTW condition on if it satisfies the weak MTW condition on and ( , , 1 ) = 0 only if = 0 or 1 = 0, (  ... 
doi:10.1007/s00526-010-0362-y fatcat:4nynafmz3rgetcg3jfq7oiowwm

HARM: A Numerical Scheme for General Relativistic Magnetohydrodynamics

Charles F. Gammie, Jonathan C. McKinney, Gabor Toth
2003 Astrophysical Journal  
Only the covariant form of the metric in a coordinate basis is required to specify the geometry. We describe code performance on a full suite of test problems in both special and general relativity.  ...  On smooth flows we show that it converges at second order. We conclude by showing some results from the evolution of a magnetized torus near a rotating black hole.  ...  Serguei Komissarov and the referee, Luca Del Zanna, made valuable comments on the manuscript. This work was supported by NSF ITR grant PHY 02-05155, by NSF PECASE grant AST 00-93091, by a NASA  ... 
doi:10.1086/374594 fatcat:xvfxgrnd7reutdsuvoudgwzvwa

The Monge–Ampère equation and its link to optimal transportation

Guido De Philippis, Alessio Figalli
2014 Bulletin of the American Mathematical Society  
By the regularity theory developed by Ivochkina [68], Krylov [78], and Caffarelli, Nirenberg, and Spruck [32], using the continuity method (see Section 2.2 for a description of this method), one obtains  ...  We now give a brief overview on the existence and regularity theory for the Monge-Ampère equation.  ...  Acknowledgments The authors wish to thank Neil Trudinger for several useful comments on a preliminary version of this paper. The second author is partially supported by NSF Grant DMS-1262411.  ... 
doi:10.1090/s0273-0979-2014-01459-4 fatcat:e7j4tpt62zbmrmvbpdw6vpqtxa
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