On Numerically Implementable Explicit Formulas for the Solutions to the 2D and 3D Equations $$\operatorname{div}(\alpha(w)\nabla w)=0$$ and $$\operatorname{div}(\beta\nabla w)=0$$ with Cauchy Data on an Analytic Boundary

2021 ◽  
Vol 55 (1) ◽  
pp. 52-58
Author(s):  
A. S. Demidov
Author(s):  
Fahimeh Mokhtari ◽  
Jan A. Sanders

To compute the unique formal normal form of families of vector fields with nilpotent linear part, we choose a basis of the Lie algebra consisting of orbits under the action of the nilpotent linear part. This creates a new problem: to find explicit formulas for the structure constants in this new basis. These are well known in the 2D case, and recently expressions were found for the 3D case by ad hoc methods. The goal of the this paper is to formulate a systematic approach to this calculation. We propose to do this using a rational method for the inversion of the Clebsch–Gordan coefficients. We illustrate the method on a family of 3D vector fields and compute the unique formal normal form for the Euler family both in the 2D and 3D cases, followed by an application to the computation of the unique normal form of the Rössler equation.


Author(s):  
P.M. Rice ◽  
MJ. Kim ◽  
R.W. Carpenter

Extrinsic gettering of Cu on near-surface dislocations in Si has been the topic of recent investigation. It was shown that the Cu precipitated hetergeneously on dislocations as Cu silicide along with voids, and also with a secondary planar precipitate of unknown composition. Here we report the results of investigations of the sense of the strain fields about the large (~100 nm) silicide precipitates, and further analysis of the small (~10-20 nm) planar precipitates.Numerous dark field images were analyzed in accordance with Ashby and Brown's criteria for determining the sense of the strain fields about precipitates. While the situation is complicated by the presence of dislocations and secondary precipitates, micrographs like those shown in Fig. 1(a) and 1(b) tend to show anomalously wide strain fields with the dark side on the side of negative g, indicating the strain fields about the silicide precipitates are vacancy in nature. This is in conflict with information reported on the η'' phase (the Cu silicide phase presumed to precipitate within the bulk) whose interstitial strain field is considered responsible for the interstitial Si atoms which cause the bounding dislocation to expand during star colony growth.


2021 ◽  
Author(s):  
Ruoyang Liu ◽  
Ke Tian Tan ◽  
Yifan Gong ◽  
Yongzhi Chen ◽  
Zhuoer Li ◽  
...  

Covalent organic frameworks offer a molecular platform for integrating organic units into periodically ordered yet extended 2D and 3D polymers to create topologically well-defined polygonal lattices and built-in discrete micropores and/or mesopores.


2012 ◽  
Author(s):  
Michael Sackllah ◽  
Denny Yu ◽  
Charles Woolley ◽  
Steven Kasten ◽  
Thomas J. Armstrong

Author(s):  
Denny Yu ◽  
Michael Sackllah ◽  
Charles Woolley ◽  
Steven Kasten ◽  
Thomas J. Armstrong
Keyword(s):  

2014 ◽  
Vol 75 (S 02) ◽  
Author(s):  
Gerlig Widmann ◽  
P. Schullian ◽  
R. Hoermann ◽  
E. Gassner ◽  
H. Riechelmann ◽  
...  

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