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Cosmology

Yvonne Choquet-Bruhat

in General Relativity and the Einstein Equations

Published in print:
2008
Published Online:
May 2009
ISBN:
9780199230723
eISBN:
9780191710872
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199230723.003.0005
Subject:
Mathematics, Applied Mathematics

This chapter begins with a discussion of the cosmological principle. It then covers isotropic and homogeneous Riemannian manifolds, Robertson–Walker spacetimes, Friedmann–Lemaître models, homogeneous ... More


THE NEUMANN PROBLEM AND PROBLEMS ON MANIFOLDS

Juan Luis Vázquez

in The Porous Medium Equation: Mathematical Theory

Published in print:
2006
Published Online:
September 2007
ISBN:
9780198569039
eISBN:
9780191717468
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198569039.003.0011
Subject:
Mathematics, Mathematical Physics

This chapter completes the investigation of previous chapters on the Dirichlet and Cauchy problems by applying the techniques to other important problems. It selects two directions, the Neumann ... More


Classical And Quantum Gravity. Riemannian Manifolds And Tensors

JEAN ZINN-JUSTIN

in Quantum Field Theory and Critical Phenomena

Published in print:
2002
Published Online:
January 2010
ISBN:
9780198509233
eISBN:
9780191708732
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198509233.003.0022
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter has two purposes; to present the few elements of differential geometry which are required in different places in this volume and to provide, for completeness, a short introduction to the ... More


Stochastic Differential Equations: Langevin, Fokker–Planck Equations

JEAN ZINN-JUSTIN

in Quantum Field Theory and Critical Phenomena

Published in print:
2002
Published Online:
January 2010
ISBN:
9780198509233
eISBN:
9780191708732
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198509233.003.0004
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter discusses Langevin equations, that is, stochastic differential equations related to diffusion processes, brownian motion, or random walk. From the Langevin equation, the Fokker–Planck ... More


Riemannian manifolds

Nathalie Deruelle and Jean-Philippe Uzan

in Relativity in Modern Physics

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198786399
eISBN:
9780191828669
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198786399.003.0064
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter is about Riemannian manifolds. It first discusses the metric manifold and the Levi-Civita connection, determining if the metric is Riemannian or Lorentzian. Next, the chapter turns to ... More


Modifying Distances

Orlitsky Alon

in Semi-Supervised Learning

Published in print:
2006
Published Online:
August 2013
ISBN:
9780262033589
eISBN:
9780262255899
Item type:
chapter
Publisher:
The MIT Press
DOI:
10.7551/mitpress/9780262033589.003.0017
Subject:
Computer Science, Machine Learning

This chapter discusses density-based metrics induced by Riemannian manifold structures. It presents asymptotically consistent methods to estimate and compute these metrics and present upper and lower ... More


Dehn Functions

Timothy Riley

in Office Hours with a Geometric Group Theorist

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691158662
eISBN:
9781400885398
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691158662.003.0008
Subject:
Mathematics, Geometry / Topology

This chapter is concerned with Dehn functions. It begins by presenting jigsaw puzzles that are somewhat different from the conventional kind and explains how to solve them. It then considers a ... More


Riemannian manifolds

Nathalie Deruelle and Jean-Philippe Uzan

in Relativity in Modern Physics

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198786399
eISBN:
9780191828669
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198786399.003.0042
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter introduces the Riemann tensor characterizing curved spacetimes, and then the metric tensor, which allows lengths and durations to be defined. As shown in the preceding chapter, ... More


On Eigenfunction Restriction Estimates and L4-Bounds for Compact Surfaces with Nonpositive Curvature

Christopher D. Sogge and Steve Zelditch

Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, and Stephen Wainger (eds)

in Advances in Analysis: The Legacy of Elias M. Stein

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691159416
eISBN:
9781400848935
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691159416.003.0018
Subject:
Mathematics, Numerical Analysis

This chapter discusses a “restriction theorem,” which is related to certain Littlewood–Paley estimates for eigenfunctions. The main step in proving this theorem is to see that an estimate involving a ... More


Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188)

Christopher D. Sogge

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691160757
eISBN:
9781400850549
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691160757.001.0001
Subject:
Mathematics, Numerical Analysis

Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. The book gives a proof of the sharp Weyl ... More


The Cartan structure equations

Nathalie Deruelle and Jean-Philippe Uzan

in Relativity in Modern Physics

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198786399
eISBN:
9780191828669
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198786399.003.0065
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter focuses on Cartan structure equations. It first introduces a 1-form and its exterior derivative, before turning to a study of the connection and torsion forms, thereby expressing the ... More


A review: The Laplacian and the d’Alembertian

Christopher D. Sogge

in Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188)

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691160757
eISBN:
9781400850549
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691160757.003.0001
Subject:
Mathematics, Numerical Analysis

This chapter reviews the Laplacian and the d'Alembertian. It begins with a brief discussion on the solution of wave equation both in Euclidean space and on manifolds and how this knowledge can be ... More


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