scholarly journals A note on the solvability of a tensor equation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Shenglong Hu
Keyword(s):  
Tensor Equation

<p style='text-indent:20px;'>In this note, we generalize a solvability result for a tensor equation with a nonsingular leading tensor to the case with possibly a singular leading tensor.</p>

A Tensor Equation of Elliptic Type

1953 ◽  
Vol 5 ◽  
pp. 524-535 ◽  
Author(s):  
G. F. D. Duff
Keyword(s):  
Riemannian Manifold ◽  
Differential Equations ◽  
Tensor Field ◽  
Differential Operators ◽  
Elliptic Type ◽  
Tensor Equation ◽  
Tensor Theory ◽  
Elliptic Differential Equations ◽  
Invariant Differential ◽  
Image Position

The theory of the systems of partial differential equations which arise in connection with the invariant differential operators on a Riemannian manifold may be developed by methods based on those of potential theory. It is therefore natural to consider in the same context the theory of elliptic differential equations, in particular those which are self-adjoint. Some results for a tensor equation in which appears, in addition to the operator Δ of tensor theory, a matrix or double tensor field defined on the manifold, are here presented. The equation may be writtenin a notation explained below.

scholarly journals Iterative solution of quadratic tensor equations for mutual polarisation

2002 ◽  
Vol 32 (5) ◽  
pp. 301-312 ◽  
Author(s):  
Wynand S. Verwoerd
Keyword(s):  
Dipole Moment ◽  
Linear Approximation ◽  
Iterative Solution ◽  
Second Order ◽  
Local Fields ◽  
Bulk Materials ◽  
Tensor Equation ◽  
Order Tensor ◽  
Intrinsic Values ◽  
Induced Dipole

To describe mutual polarisation in bulk materials containing high polarisability molecules, local fields beyond the linear approximation need to be included. A second order tensor equation is formulated, and it describes this in the case of crystalline or at least locally ordered materials such as an idealised polymer. It is shown that this equation is solved by a set of recursion equations that relate the induced dipole moment, linear polarisability, and first hyperpolarisability in the material to the intrinsic values of the same properties of isolated molecules. From these, macroscopic susceptibility tensors up to second order can be calculated for the material.

Modified Newton integration neural algorithm for solving the multi-linear M-tensor equation

2020 ◽  
Vol 96 ◽  
pp. 106674
Author(s):  
Haoen Huang ◽  
Dongyang Fu ◽  
Jiazheng Zhang ◽  
Xiuchun Xiao ◽  
Guancheng Wang ◽  
...  
Keyword(s):  
Tensor Equation

The linear bi-spatial tensor equation ?i j AiXBj= C

1996 ◽  
Vol 17 (10) ◽  
pp. 979-986
Author(s):  
Chen Yuming ◽  
Xiao Heng ◽  
Li Jianbo
Keyword(s):  
Tensor Equation

Intrinsic solution to the n-dimensional tensor equation ∑r = 1mUm − rXUr − 1 = C

1994 ◽  
Vol 115 (3-4) ◽  
pp. 359-364 ◽  
Author(s):  
Guo Zhong-heng ◽  
Li Jian-bo ◽  
Xiao Heng ◽  
Chen Yu-ming
Keyword(s):  
Tensor Equation
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