scholarly journals Transformation of the integral using Hylleraas coordinates in N-dimensions

2006 ◽  
Vol 84 (9) ◽  
pp. 857-859
Author(s):  
Soma Mukhopadhyay ◽  
Ashok Chatterjee
Keyword(s):  
Three Dimensional ◽  
Simple Derivation ◽  
Hylleraas Coordinates ◽  
Dimensional Integral

The integral ∫ F(r, r′, |r–r′|) dr dr′ where r and r′ are N-dimensional position vectors can be transformed into a simple three-dimensional integral using Hylleraas coordinates. A simple derivation of this result is presented. PACS Nos.: 31.15.–p; 31.15.Ja; 03.65.–w

scholarly journals The on-axis magnetic well and Mercier's criterion for arbitrary stellarator geometries

2021 ◽  
Vol 87 (2) ◽  
Author(s):  
P. Kim ◽  
R. Jorge ◽  
W. Dorland
Keyword(s):  
Magnetic Field ◽  
Original Work ◽  
Three Dimensional ◽  
Radial Distance ◽  
Analytical Form ◽  
One Dimensional ◽  
Modern Notation ◽  
Magnetic Well ◽  
First Time ◽  
Dimensional Integral

A simplified analytical form of the on-axis magnetic well and Mercier's criterion for interchange instabilities for arbitrary three-dimensional magnetic field geometries is derived. For this purpose, a near-axis expansion based on a direct coordinate approach is used by expressing the toroidal magnetic flux in terms of powers of the radial distance to the magnetic axis. For the first time, the magnetic well and Mercier's criterion are then written as a one-dimensional integral with respect to the axis arclength. When compared with the original work of Mercier, the derivation here is presented using modern notation and in a more streamlined manner that highlights essential steps. Finally, these expressions are verified numerically using several quasisymmetric and non-quasisymmetric stellarator configurations including Wendelstein 7-X.

Advances in three-dimensional integral imaging: sensing, display, and applications [Invited]

2013 ◽  
Vol 52 (4) ◽  
pp. 546 ◽  
Author(s):  
Xiao Xiao ◽  
Bahram Javidi ◽  
Manuel Martinez-Corral ◽  
Adrian Stern
Keyword(s):  
Three Dimensional ◽  
Integral Imaging ◽  
Dimensional Integral

scholarly journals LOOPS, SURFACES AND GRASSMANN REPRESENTATION IN TWO- AND THREE-DIMENSIONAL ISING MODELS

1999 ◽  
Vol 14 (29) ◽  
pp. 4549-4574 ◽  
Author(s):  
C. R. GATTRINGER ◽  
S. JAIMUNGAL ◽  
G. W. SEMENOFF
Keyword(s):  
Three Dimensional ◽  
Ising Models ◽  
Algebraic Representation ◽  
Simple Derivation ◽  
Counting Problems ◽  
Intrinsic Geometry ◽  
Geometry Factor ◽  
Loop Expansion ◽  
Loop Surface

We construct an algebraic representation of the geometrical objects (loop and surface variables) dual to the spins in 2 and 3D Ising models. This algebraic calculus is simpler than dealing with the geometrical objects, in particular when analyzing geometry factors and counting problems. For the 2D case we give the corrected loop expansion of the free energy and the radius of convergence for this series. For the 3D case we give a simple derivation of the geometry factor which prevents overcounting of surfaces in the intrinsic geometry representation of the partition function, and find a classification of the surfaces to be summed over. For 2 and 3D we derive a compact formula for 2n-point functions in loop (surface) representation.

Optical acquiring technique of three-dimensional integral imaging based on optimal pick-up distance

2011 ◽  
Vol 19 (11) ◽  
pp. 2805-2811 ◽  
Author(s):  
焦小雪 JIAO Xiao-xue ◽  
赵星 ZHAO Xing ◽  
杨勇 YANG Yong ◽  
方志良 FANG Zhi-liang ◽  
袁小聪 YUAN Xiao-cong
Keyword(s):  
Three Dimensional ◽  
Integral Imaging ◽  
Dimensional Integral
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