Results 11 to 20 of about 61,943 (76)
Geodesics in the space of relatively Kähler metrics
Journal of the London Mathematical Society, Volume 108, Issue 3, Page 1036-1081, September 2023., 2023 Abstract We derive the geodesic equation for relatively Kähler metrics on fibrations and prove that any two such metrics with fibrewise constant scalar curvature are joined by a unique smooth geodesic. We then show convexity of the log‐norm functional for this setting along geodesics, which yields simple proofs of Dervan and Sektnan's uniqueness result
Michael Hallam
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Degenerate nonlinear parabolic equations with discontinuous diffusion coefficients
Journal of the London Mathematical Society, Volume 104, Issue 2, Page 688-746, September 2021., 2021 Abstract This paper is devoted to the study of some nonlinear parabolic equations with discontinuous diffusion intensities. Such problems appear naturally in physical and biological models. Our analysis is based on variational techniques and in particular on gradient flows in the space of probability measures equipped with the distance arising in the ...
Dohyun Kwon, Alpár Richárd Mészáros
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Three ways to solve partial differential equations with neural networks — A review
GAMM-Mitteilungen, Volume 44, Issue 2, June 2021., 2021 Abstract Neural networks are increasingly used to construct numerical solution methods for partial differential equations. In this expository review, we introduce and contrast three important recent approaches attractive in their simplicity and their suitability for high‐dimensional problems: physics‐informed neural networks, methods based on the ...
Jan Blechschmidt, Oliver G. Ernst
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Error estimation for second‐order partial differential equations in nonvariational form
Numerical Methods for Partial Differential Equations, Volume 37, Issue 3, Page 2190-2221, May 2021., 2021 Abstract Second‐order partial differential equations (PDEs) in nondivergence form are considered. Equations of this kind typically arise as subproblems for the solution of Hamilton–Jacobi–Bellman equations in the context of stochastic optimal control, or as the linearization of fully nonlinear second‐order PDEs. The nondivergence form in these problems
Jan Blechschmidt +2 more
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The Sinkhorn algorithm, parabolic optimal transport and geometric Monge–Ampère equations [PDF]
Numerische Mathematik, 2017 We show that the discrete Sinkhorn algorithm—as applied in the setting of Optimal Transport on a compact manifold—converges to the solution of a fully non-linear parabolic PDE of Monge–Ampère type, in a large-scale limit.
R. Berman
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Canonical complex extensions of Kähler manifolds
Journal of the London Mathematical Society, Volume 101, Issue 2, Page 786-827, April 2020., 2020 Abstract Given a complex manifold X, any Kähler class defines an affine bundle over X, and any Kähler form in the given class defines a totally real embedding of X into this affine bundle. We formulate conditions under which the affine bundles arising this way are Stein and relate this question to other natural positivity conditions on the tangent ...
Daniel Greb, Michael Lennox Wong
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Simplified Geometric Approach to Freeform Beam Shaper Design
International Journal of Optics, Volume 2020, Issue 1, 2020., 2020 Beam of light shaping process can be considered ultimate, if both irradiance and wavefront spatial distributions are under control and both can be shaped arbitrarily. In order to keep these two quantities determined simultaneously, it is required to apply at least two powered refractive or reflective surfaces.
Jacek Wojtanowski +2 more
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Symmetry of solutions to parabolic Monge-Ampère equations
Boundary Value Problems, 2013 In this paper, we study the parabolic Monge-Ampère equation −utdet(D2u)=f(t,u)in Ω×(0,T]. Using the method of moving planes, we show that any parabolically convex solution is symmetric with respect to some hyperplane.
Limei Dai
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Attractors and Finite‐Dimensional Behaviour in the 2D Navier‐Stokes Equations
International Scholarly Research Notices, Volume 2013, Issue 1, 2013., 2013 The purpose of this review is to give a broad outline of the dynamical systems approach to the two‐dimensional Navier‐Stokes equations. This example has led to much of the theory of infinite‐dimensional dynamical systems, which is now well developed.
James C. Robinson +3 more
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Parabolic Monge-Ampere equations giving rise to a free boundary: The worn stone model
, 2015 This paper deals with several qualitative properties of solutions of some parabolic equations associated to the Monge--Ampere operator arising in suitable formulations of the Gauss curvature flow and the worn stone problems.
G. Díaz, J. I. Díaz
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