Author’s Closure: Discussion on “Exact expansions of arbitrary tensor functions F(A) and their derivatives”

2005 ◽  
Vol 42 (15) ◽  
pp. 4516-4517
Author(s):  
Jia Lu
Keyword(s):  
Arbitrary Tensor

scholarly journals The Formula of Grangeat for Tensor Fields of Arbitrary Order innDimensions

2007 ◽  
Vol 2007 ◽  
pp. 1-4 ◽  
Author(s):  
T. Schuster
Keyword(s):  
Radon Transform ◽  
Tensor Field ◽  
Arbitrary Order ◽  
Cone Beam ◽  
Inversion Method ◽  
Tensor Fields ◽  
Arbitrary Tensor

The cone beam transform of a tensor field of orderminn≥2dimensions is considered. We prove that the image of a tensor field under this transform is related to a derivative of then-dimensional Radon transform applied to a projection of the tensor field. Actually the relation we show reduces form=0andn=3to the well-known formula of Grangeat. In that sense, the paper contains a generalization of Grangeat's formula to arbitrary tensor fields in any dimension. We further briefly explain the importance of that formula for the problem of tensor field tomography. Unfortunately, form>0, an inversion method cannot be derived immediately. Thus, we point out the possibility to calculate reconstruction kernels for the cone beam transform using Grangeat's formula.

scholarly journals A general method for rotational averages

2021 ◽  
Vol 2090 (1) ◽  
pp. 012041
Author(s):  
Reed Nessler ◽  
Tuguldur Kh. Begzjav
Keyword(s):  
Closed Form ◽  
Direction Cosine ◽  
Nonlinear Spectroscopy ◽  
Tensor Rank ◽  
Closed Form Expression ◽  
Form Expression ◽  
Oriented Molecules ◽  
Rotationally Invariant ◽  
Arbitrary Tensor ◽  
General Method

Abstract The theory of nonlinear spectroscopy on randomly oriented molecules leads to the problem of averaging molecular quantities over random rotation. We solve this problem for arbitrary tensor rank by deriving a closed-form expression for the rotationally invariant tensor of averaged direction cosine products. From it, we obtain some useful new facts about this tensor. Our results serve to speed the inherently lengthy calculations of nonlinear optics.

ON DIFFRACTION BY A HALF-PLANE IN AN ARBITRARY ANISOTROPIC MEDIUM

1967 ◽  
Vol 45 (8) ◽  
pp. 2561-2579 ◽  
Author(s):  
R. A. Hurd
Keyword(s):  
Plane Wave ◽  
Half Plane ◽  
Kernel Matrix ◽  
Polynomial Matrices ◽  
Correct Form ◽  
Complete Listing ◽  
Arbitrary Plane ◽  
Solvable Problems ◽  
Arbitrary Tensor ◽  
Tensor Permittivity

A method of uncoupling simultaneous Wiener–Hopf equations is developed. If the kernel-matrix G of the equations has the form G = Γ1(A + Γ1B), where Γ1, Γ2 are scalars and A and B are polynomial matrices, then the method works if the elements of A and B satisfy a certain equation called the "criterion".The method is applied to the diffraction of an arbitrary plane wave by a conducting half-plane in a medium with arbitrary tensor permittivity. It is found that in certain circumstances, G has the correct form Γ1(A + Γ2B). Application of the criterion then yields in principle a catalogue of solvable problems. In practice a complete listing has not been obtained because of the amount of algebra involved. However, a partial catalogue has been prepared. It includes most of the previously solved problems plus one or two which have not yet been considered. An example is briefly considered.

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