Lie group invariance properties of radiation hydrodynamics equations and their associated similarity solutions

1986 ◽  
Vol 29 (8) ◽  
pp. 2398 ◽  
Author(s):  
Stephen V. Coggeshall ◽  
Roy A. Axford
Keyword(s):  
Lie Group ◽  
Similarity Solutions ◽  
Radiation Hydrodynamics ◽  
Invariance Properties ◽  
Group Invariance ◽  
Hydrodynamics Equations

The general Lie group and similarity solutions for the one-dimensional Vlasov–Maxwell equations

1985 ◽  
Vol 33 (2) ◽  
pp. 219-236 ◽  
Author(s):  
Dana Roberts
Keyword(s):  
Lie Group ◽  
Maxwell Equations ◽  
Transformation Group ◽  
Spatial Dimension ◽  
Similarity Solutions ◽  
One Dimensional ◽  
Special Cases ◽  
General Similarity ◽  
The One ◽  
The Individual

The general Lie point transformation group and the associated reduced differential equations and similarity forms for the solutions are derived here for the coupled (nonlinear) Vlasov–Maxwell equations in one spatial dimension. The case of one species in a background is shown to admit a larger group than the multi-species case. Previous exact solutions are shown to be special cases of the above solutions, and many of the new solutions are found to constrain the form of the distribution function much more than, for example, the BGK solutions do. The individual generators of the Lie group are used to find the possible subgroups. Finally, a simple physical argument is given to show that the asymptotic solution (t→∞) for a one-species, one-dimensional plasma is one of the general similarity solutions.

A Novel Blind Source Separation Method Based on the Extended Natural Gradient

2012 ◽  
Vol 433-440 ◽  
pp. 3570-3576
Author(s):  
Yu Feng Xue ◽  
Yu Jia Wang ◽  
Qiu Dong Sun
Keyword(s):  
Iterative Algorithm ◽  
Blind Source Separation ◽  
Lie Group ◽  
Mathematical Proof ◽  
Source Separation ◽  
Separation Method ◽  
New Method ◽  
Invariance Property ◽  
Natural Gradient ◽  
Group Invariance

In this paper, a new method is introduced to derive the extended natural gradient, which was proposed by Lewicki and Sejnowski in [1]. However, they made their derivation under many approximations, and the proof is also very complicated. To give a more rigors mathematical proof for this gradient, the Lie group invariance property is introduced which makes the proof much easier and straightforward. In addition, an iterative algorithm through Newton's method is also given to estimate the sources efficiently. The results of the experiments confirm the efficiency of the proposed method.

scholarly journals Lie group analysis and similarity solutions for hydro-magnetic Maxwell fluid through a porous medium

2012 ◽  
Vol 2012 (1) ◽  
Author(s):  
Khaled Saad Mekheimer ◽  
Mostafa Fatouh El-Sabbagh ◽  
Rabea Elshennawy Abo-Elkhair
Keyword(s):  
Porous Medium ◽  
Lie Group ◽  
Group Analysis ◽  
Maxwell Fluid ◽  
Similarity Solutions ◽  
Lie Group Analysis
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