scholarly journals Smoothness at Null Infinity and the Structure of Initial Data

2004 ◽  
pp. 121-203 ◽  
Author(s):  
Helmut Friedrich
Keyword(s):  
Initial Data ◽  
Null Infinity

scholarly journals SOLUTIONS OF QUASI-LINEAR WAVE EQUATIONS POLYHOMOGENEOUS AT NULL INFINITY IN HIGH DIMENSIONS

2011 ◽  
Vol 08 (02) ◽  
pp. 269-346 ◽  
Author(s):  
PIOTR T. CHRUŚCIEL ◽  
ROGER TAGNE WAFO
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Maxwell Equations ◽  
Wave Equations ◽  
Hyperbolic Systems ◽  
Small Data ◽  
Linear Wave ◽  
High Dimensions ◽  
Null Infinity ◽  
Linear Wave Equations

We prove propagation of weighted Sobolev regularity for solutions of the hyperboloidal Cauchy problem for a class of quasi-linear symmetric hyperbolic systems, under structure conditions compatible with the Einstein–Maxwell equations in space-time dimensions n + 1 ≥ 7. Similarly we prove propagation of polyhomogeneity in dimensions n + 1 ≥ 9. As a byproduct we obtain, in those last dimensions, polyhomogeneity at null infinity of small data solutions of vacuum Einstein, or Einstein–Maxwell equations evolving out of initial data which are stationary outside of a ball.

Null infinity is not a good initial‐data surface

1978 ◽  
Vol 19 (6) ◽  
pp. 1300-1303 ◽  
Author(s):  
Robert Geroch
Keyword(s):  
Initial Data ◽  
Null Infinity ◽  
Data Surface

scholarly journals The Maxwell field on the Schwarzschild space–time: behaviour near spatial infinity

2007 ◽  
Vol 463 (2086) ◽  
pp. 2609-2630 ◽  
Author(s):  
Juan Antonio Valiente Kroon
Keyword(s):  
Initial Data ◽  
Maxwell Equations ◽  
Geodesic Distance ◽  
Transport Equations ◽  
Maxwell Field ◽  
Spatial Infinity ◽  
Schwarzschild Solution ◽  
Null Infinity ◽  
Completely Regular ◽  
Critical Sets

The behaviour of the Maxwell field near one of the spatial infinities of the Schwarzschild solution is analysed by means of the transport equations implied by the Maxwell equations on the cylinder at spatial infinity. Initial data for the Maxwell equations will be assumed to be expandable in terms of powers of a coordinate ρ measuring the geodesic distance to spatial infinity (in the conformal picture) and such that the highest possible spherical harmonics at order p are 2 p -polar ones. It is shown that if the 2 p -polar harmonics at order p in the initial data satisfy a certain regularity condition, then the solutions to the transport equations at orders p and p +1 are completely regular at the critical sets where null infinity touches spatial infinity. On the other hand, the solutions to the transport equations of order p +2 contain, in general, logarithmic singularities at the critical sets.

On the null-timelike boundary for Maxwell and spin-2 fields in asymptotically flat spaces

2021 ◽  
Vol 18 (02) ◽  
pp. 343-395
Author(s):  
Qing Han ◽  
Lin Zhang
Keyword(s):  
Field Equation ◽  
Initial Data ◽  
Asymptotic Expansions ◽  
Mixed Boundary ◽  
Initial Value ◽  
Null Infinity ◽  
Asymptotically Flat ◽  
Time Estimates ◽  
Lorentzian Metric ◽  
Timelike Hypersurface

We study the Maxwell equation and the spin-2 field equation in Bondi–Sachs coordinates associated with an asymptotically flat Lorentzian metric. We consider the mixed boundary/initial value problem, where the initial data are imposed on a null hypersurface and a boundary value is prescribed on a timelike hypersurface. We establish Sobolev [Formula: see text] space-time estimates for solutions and their asymptotic expansions at the null infinity.

Remarks on nonlinear elastic waves with null conditions

2020 ◽  
Vol 26 ◽  
pp. 121
Author(s):  
Dongbing Zha ◽  
Weimin Peng
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Elastic Waves ◽  
Wave Equations ◽  
Alternative Proof ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Hyperelastic Materials ◽  
Variational Structure ◽  
Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

ON THE EXPERT’S RIGHT TO BE PRESENT AT LEGAL PROCEEDINGS

2016 ◽  
Vol 15 ◽  
pp. 163-171
Author(s):  
M. G. Shcherbakovskiy
Keyword(s):  
Initial Data ◽  
Legal Proceedings ◽  
Reliable Assessment ◽  
Data Acquiring ◽  
Selection Of

The article discusses the reasonsfor an expert to participate in legal proceedings. The gnoseological reason for that consists of the bad quality of materials subject to examination that renders the examination either completely impossible or compromises objective, reasoned and reliable assessment of the findings. The procedural reason consists ofa proscription for an expert to collect evidence himself or herself. The author investigates into the ways of how an expert can participate in legal proceedings. If the defense invites an expert to participate in the proceedings, then it is recommended that his or her involvement should be in the presence of attesting witnesses and recorded in the protocol. In the course of the legal proceedings an expert has the following tasks: adding initial data, acquiring new initial data, understanding the situation of the incident, acquiring new objects to be studied, including samples for examination. An expert’s participation in legal proceedings differs from the participation of a specialist or an examination on the scene of the incident. The author describes the tasks that an expert solves in the course of legal proceedings, the peculiarities ofan investigation experiment practices, the selection of samples for an examination, inspection, interrogation.

UPAYA MENINGKATKAN KEMAMPUAN MEMBACA MELALUI METODE SCRAMBLE PADA PESERTA DIDIK KELAS I SD NEGERI 002 BENTENG KECAMATAN SUNGAI BATANG

2017 ◽  
Vol 6 (2) ◽  
pp. 409
Author(s):  
Reni Marlina
Keyword(s):  
Cooperative Learning ◽  
Initial Data ◽  
Learning Outcomes ◽  
Reading Ability ◽  
First Grade ◽  
Student Learning Outcomes ◽  
Female Students ◽  
Male Students ◽  
Average Analysis ◽  
Before And After

This study aims to improve students' reading ability through the first grade scramble students of SD Negeri 002 Benteng, Kecamatan Sungai Batang, which are 28 students with 11 male students and 17 female students with heterogeneous ability. This study is based on the low ability of students' learning outcomes and lack of awareness of teachers to implement an effective, innovative, and cooperative learning. The study was conducted from September 3, 2016 to October 8, 2016. This study is a classroom action research (PTK) consisting of two cycles. Minimum completeness criteria (KKM) and average analysis are used to determine whether or not improvement of student learning outcomes before and after using the scramble learning model. The results of this study indicate that the number of students who reach KKM in the initial data is only 10 people (36%), cycle I is 16 people (57%), and the second cycle is 25 people (89%). The average student score at baseline was 68.4; cycle I increased to 75,9; in the second cycle increased again to 83,6. Based on the results of this study it can be concluded that the model of learning scramble can improve reading ability in Indonesian language students class I of SD Negeri 002 Benteng, Kecamatan Sungai Batang.

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