Topic
Area theorem
About: Area theorem is a research topic. Over the lifetime, 146 publications have been published within this topic receiving 3293 citations.
Papers published on a yearly basis
Papers
Book•
1 Jan 1973
TL;DR: In this article, the authors defined the boundary values of Riemann maps and defined the corresponding boundary values for bounded analytic functions in the Bergman space of the Dirichlet problem.
Abstract: of Volume II- 13 Return to Basics- 1 Regions and Curves- 2 Derivatives and Other Recollections- 3 Harmonic Conjugates and Primitives- 4 Analytic Arcs and the Reflection Principle- 5 Boundary Values for Bounded Analytic Functions- 14 Conformal Equivalence for Simply Connected Regions- 1 Elementary Properties and Examples- 2 Crosscuts- 3 Prime Ends- 4 Impressions of a Prime End- 5 Boundary Values of Riemann Maps- 6 The Area Theorem- 7 Disk Mappings: The Class S- 15 Conformal Equivalence for Finitely Connected Regions- 1 Analysis on a Finitely Connected Region- 2 Conformal Equivalence with an Analytic Jordan Region- 3 Boundary Values for a Conformal Equivalence Between Finitely Connected Jordan Regions- 4 Convergence of Univalent Functions- 5 Conformal Equivalence with a Circularly Slit Annulus- 6 Conformal Equivalence with a Circularly Slit Disk- 7 Conformal Equivalence with a Circular Region- 16 Analytic Covering Maps- 1 Results for Abstract Covering Spaces- 2 Analytic Covering Spaces- 3 The Modular Function- 4 Applications of the Modular Function- 5 The Existence of the Universal Analytic Covering Map- 17 De Branges's Proof of the Bieberbach Conjecture- 1 Subordination- 2 Loewner Chains- 3 Loewner's Differential Equation- 4 The Milin Conjecture- 5 Some Special Functions- 6 The Proof of de Branges's Theorem- 18 Some Fundamental Concepts from Analysis- 1 Bergman Spaces of Analytic and Harmonic Functions- 2 Partitions of Unity- 3 Convolution in Euclidean Space- 4 Distributions- 5 The Cauchy Transform- 6 An Application: Rational Approximation- 7 Fourier Series and Cesaro Sums- 19 Harmonic Functions Redux- 1 Harmonic Functions on the Disk- 2 Fatou's Theorem- 3 Semicontinuous Functions- 4 Subharmonic Functions- 5 The Logarithmic Potential- 6 An Application: Approximation by Harmonic Functions- 7 The Dirichlet Problem- 8 Harmonic Majorants- 9 The Green Function- 10 Regular Points for the Dirichlet Problem- 11 The Dirichlet Principle and Sobolev Spaces- 20 Hardy Spaces on the Disk- 1 Definitions and Elementary Properties- 2 The Nevanlinna Class- 3 Factorization of Functions in the Nevanlinna Class- 4 The Disk Algebra- 5 The Invariant Subspaces of Hp- 6 Szego's Theorem- 21 Potential Theory in the Plane- 1 Harmonic Measure- 2 The Sweep of a Measure- 3 The Robin Constant- 4 The Green Potential- 5 Polar Sets- 6 More on Regular Points- 7 Logarithmic Capacity: Part 1- 8 Some Applications and Examples of Logarithmic Capacity- 9 Removable Singularities for Functions in the Bergman Space- 10 Logarithmic Capacity: Part 2- 11 The Transfinite Diameter and Logarithmic Capacity- 12 The Refinement of a Subharmonic Function- 13 The Fine Topology- 14 Wiener's criterion for Regular Points- References- List of Symbols
1,178 citations
TL;DR: In this paper, a boundary derivative expansion is proposed to determine the location of the event horizon in the bulk as a local function of the fluid dynamical variables, and a natural map from the boundary to the horizon using ingoing null geodesics is defined.
Abstract: Spacetime geometries dual to arbitrary fluid flows in strongly coupled N=4 super Yang Mills theory have recently been constructed perturbatively in the long wavelength limit. We demonstrate that these geometries all have regular event horizons, and determine the location of the horizon order by order in a boundary derivative expansion. Intriguingly, the derivative expansion allows us to determine the location of the event horizon in the bulk as a local function of the fluid dynamical variables. We define a natural map from the boundary to the horizon using ingoing null geodesics. The area-form on spatial sections of the horizon can then be pulled back to the boundary to define a local entropy current for the dual field theory in the hydrodynamic limit. The area theorem of general relativity guarantees the positivity of the divergence of the entropy current thus constructed.
297 citations
TL;DR: In this article, a boundary derivative expansion is proposed to determine the location of the event horizon in the bulk as a local function of the fluid dynamical variables, and a natural map from the boundary to the horizon using ingoing null geodesics is defined.
Abstract: Spacetime geometries dual to arbitrary fluid flows in strongly coupled = 4 super Yang Mills theory have recently been constructed perturbatively in the long wavelength limit. We demonstrate that these geometries all have regular event horizons, and determine the location of the horizon order by order in a boundary derivative expansion. Intriguingly, the derivative expansion allows us to determine the location of the event horizon in the bulk as a local function of the fluid dynamical variables. We define a natural map from the boundary to the horizon using ingoing null geodesics. The area-form on spatial sections of the horizon can then be pulled back to the boundary to define a local entropy current for the dual field theory in the hydrodynamic limit. The area theorem of general relativity guarantees the positivity of the divergence of the entropy current thus constructed.
227 citations
TL;DR: It is demonstrated that in order for the design rate of an ensemble to approach the capacity under BP decoding the component codes have to be perfectly matched, a statement which is well known for the special case of transmission over the binary erasure channel.
Abstract: There is a fundamental relationship between belief propagation (BP) and maximum a posteriori decoding. The case of transmission over the binary erasure channel was investigated in detail in a companion paper (C. MEacuteasson, A. Montanari, and R. Urbanke, "Maxwell's construction: The hidden bridge between iterative and maximum a posteriori decoding," IEEE Transactions on Information Theory, submitted for publication). This paper investigates the extension to general memoryless channels (paying special attention to the binary case). An area theorem for transmission over general memoryless channels is introduced and some of its many consequences are discussed. We show that this area theorem gives rise to an upper bound on the maximum a posteriori threshold for sparse graph codes. In situations where this bound is tight, the extrinsic soft bit estimates delivered by the BP decoder coincide with the correct a posteriori probabilities above the maximum a posteriori threshold. More generally, it is conjectured that the fundamental relationship between the maximum a posteriori probability (MAP) and the BP decoder which was observed for transmission over the binary erasure channel carries over to the general case. We finally demonstrate that in order for the design rate of an ensemble to approach the capacity under BP decoding the component codes have to be perfectly matched, a statement which is well known for the special case of transmission over the binary erasure channel.
164 citations
TL;DR: In this paper, an analytical solution to the cubic-quintic Ginzbug-Landau equation is derived for dissipative optical solitons, where the energy does not scale inversely with the pulse duration, and in addition there is an upper limit to the energy.
Abstract: Soliton area theorems express the pulse energy as a function of the pulse shape and the system parameters. From an analytical solution to the cubic-quintic Ginzbug-Landau equation, we derive an area theorem for dissipative optical solitons. In contrast to area theorems for conservative optical solitons, the energy does not scale inversely with the pulse duration, and in addition there is an upper limit to the energy. Energy quantization explains the existence of, and conditions for, multiple-pulse solutions. The theoretical predictions are confirmed with numerical simulations and experiments in the context of dissipative soliton fiber lasers.
118 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 7 |
| 2020 | 3 |
| 2019 | 6 |
| 2018 | 13 |
| 2017 | 9 |
| 2016 | 10 |